188 terms

What Can Go Wrong: Two Types of Errors

In hypothesis testing, the following decisions can occur:

In hypothesis testing, the following decisions can occur:

GOOD

•If the null hypothesis is true and we do not reject it, it is a correct decision.

•If the null hypothesis is false and we reject it, it is a correct decision.

BAD

•If the null hypothesis is true, but we reject it. This is a type I error.

•If the null hypothesis is false, but we fail to reject it. This is a type II error.

*****8Type I and type II errors are not caused by mistakes.***** The data provide evidence for a conclusion that is false. It's no one's fault!

•If the null hypothesis is true and we do not reject it, it is a correct decision.

•If the null hypothesis is false and we reject it, it is a correct decision.

BAD

•If the null hypothesis is true, but we reject it. This is a type I error.

•If the null hypothesis is false, but we fail to reject it. This is a type II error.

***

Tests About μ When σ is Unknown

The t-test for the Population Mean

What can we use to replace σ

the best you can do is find the sample standard deviation, S, and use it instead of σ

four steps of the t-test for the population mean when using standard deviation of a population.

STATE: State the problem.

PLAN: In this step there are no changes:

•State the parameter of interest in context and the symbol used (e.g. μ = mean PA of all BYU students...)

•The null hypothesis has the form:

H o :μ=μ o Ho:μ=μo

•The alternative hypothesis takes one of the following three forms (depending on the context):

1 H a :μ<μ o Ha:μ<μo (one-sided)

2 H a :μ>μ o Ha:μ>μo (one-sided)

3 H a :μ≠μ o Ha:μ≠μo (two-sided)

•Choose level of significance

SOLVE:

•Collect data

•Check the conditions under which the t-test can be safely used and summarizing the data

PLAN: In this step there are no changes:

•State the parameter of interest in context and the symbol used (e.g. μ = mean PA of all BYU students...)

•The null hypothesis has the form:

H o :μ=μ o Ho:μ=μo

•The alternative hypothesis takes one of the following three forms (depending on the context):

1 H a :μ<μ o Ha:μ<μo (one-sided)

2 H a :μ>μ o Ha:μ>μo (one-sided)

3 H a :μ≠μ o Ha:μ≠μo (two-sided)

•Choose level of significance

SOLVE:

•Collect data

•Check the conditions under which the t-test can be safely used and summarizing the data

what is needed to complete a z test when using sample standard devation

The sample is random (or at least can be considered random in context).

normal distribution/ large sample

non normal, but with a large sample

normal distribution/ large sample

non normal, but with a large sample

when you use s instead of the S what is the statistic called?

standardized score of x

standard error of x ¯

standard devation of sample/ square root of N

t distribution

another bell-shaped (unimodal and symmetric) distribution, like the normal distribution; and the center of the t distribution is standardized at zero, like the center of the normal distribution.

different from the normal distribution? (through the spread)

slightly less area near the expected central value

more area in the "tails

appropriate model in certain cases where there is more variability than would be predicted by the normal distribution.

EXAMPLE: values, which have more variability (or "volatility," to use the economic term) than would be predicted by the normal distribution.

The t distributions that are closer to normal are said to have higher "degrees of freedom"

different from the normal distribution? (through the spread)

slightly less area near the expected central value

more area in the "tails

appropriate model in certain cases where there is more variability than would be predicted by the normal distribution.

EXAMPLE: values, which have more variability (or "volatility," to use the economic term) than would be predicted by the normal distribution.

The t distributions that are closer to normal are said to have higher "degrees of freedom"

The t-distribution

shows itself only when s is usted instead of sigma

Why would this single change (using "s" in place of "sigma") result in a sampling distribution that is the t distribution instead of the standard normal (Z) distribution?

more variability.s (the standard deviation of the sample data) varies from sample to sample, and therefore it's another source of variation.

in a normal Z scoring what is the only variable with variability?

X bar

in a T scoring when the s is used instead of Sigma what are the two variables that will change

x bar

s ( sample standard deviation)

s ( sample standard deviation)

score that arises in the context of a test for a mean is a t score with (n - 1) degrees of freedom.

degrees of freedom depend on the sample size in the study.

\higher degrees of freedom indicate that the t distribution is closer to normal ( LARGER SAMPLE MAKES IT MORE NORMAL)

\higher degrees of freedom indicate that the t distribution is closer to normal ( LARGER SAMPLE MAKES IT MORE NORMAL)

the effect of the t distribution is most important for a study with a

relatively small sample size

when Ho is true (i.e., when μ=μ 0 μ=μ0), our test statistic has a t distribution with (n-1) d.f., and this is the distribution under which we find p-values.

...

For a large sample size (n), the null distribution of the test statistic is approximately Z, so whether we use t(n - 1) or Z to calculate the p-values should not make a big difference.

...

Finding the p-value and drawing conclusions from a T TEST

1.solve your normal t equation

a. you will now have your T score

b. negative t scores are equivalent to positive t scores.

2. calculate the degrees of freedom

a. using the formula, df = n - 1.

b.•If the degrees of freedom appropriate for your test is not found on the table shown in Figure 2, always round down to the nearest degrees of freedom.

3.Find your confidence level

a. this is found in the columns of DOF

4. determine if the test is one sided or two

5. find the intervals that the p could be within. ( not as precise as the Z scoring)

6. CONCLUDE

•If the test statistic you have calculated is greater than the one given in the table, assume that your p-value is less than the smallest value given on the table (for example, less than 0.0005 for one-sided tests and less than 0.001 for two-sided tests).

P score will usually be larger than that of the Z score analysis. this is because the tail is fatter in the T scoring.

a. you will now have your T score

b. negative t scores are equivalent to positive t scores.

2. calculate the degrees of freedom

a. using the formula, df = n - 1.

b.•If the degrees of freedom appropriate for your test is not found on the table shown in Figure 2, always round down to the nearest degrees of freedom.

3.Find your confidence level

a. this is found in the columns of DOF

4. determine if the test is one sided or two

5. find the intervals that the p could be within. ( not as precise as the Z scoring)

6. CONCLUDE

•If the test statistic you have calculated is greater than the one given in the table, assume that your p-value is less than the smallest value given on the table (for example, less than 0.0005 for one-sided tests and less than 0.001 for two-sided tests).

P score will usually be larger than that of the Z score analysis. this is because the tail is fatter in the T scoring.

Z test for the population proportion

1. tess the population proportion ( P)

2. variable of interest is categorical

3. population proportion is known

2. variable of interest is categorical

3. population proportion is known

Z test for the population Mean

1. testing the population mean ( miu)

2. variable is quantitative

3. population standard devation is know ( sigma)

2. variable is quantitative

3. population standard devation is know ( sigma)

T test for population mean

1.testing the population mean ( miu)

2. response variable is quantitative

3. population standard deviation is k unknown, so sample standard deviation is used instead

2. response variable is quantitative

3. population standard deviation is k unknown, so sample standard deviation is used instead

The process of hypothesis testing has four steps:

1. stating the null and alternative hypotheses ( Ho and Ha)

a. z test for the population proportion Null:( Ho:P=Po) alternative hypothesis: Ha: P cannot =Po or Ha:p<Po or Ha:P>Po

b . a. z test for the population MeanNull:( Ho:Miu=Miuo) alternative hypothesis: Ha: Miu cannot =Miu(o) or Ha:Miu<Miu(o) or Ha:Miu>Miu(o)

c. t test of population mean MeanNull:( Ho:Miu=Miuo) alternative hypothesis: Ha: Miu cannot =Miu(o) or Ha:Miu<Miu(o) or Ha:Miu>Miu(o)

2.Obtaining a random sample (or at least one that can be considered random) and collecting data. Using the data:

Check that the conditions under which the test can be reliably used are met.

****For the z-test for the Population Proportion, we can reliably use the test is if the following conditions holds: np0≥ 10 and n(1-p0)≥ 10** 10***

*Summarize the data using a test statistic.( t test, Z test or the Zp test)

3. Finding the p-value of the test.

4.Making conclusions.

a. z test for the population proportion Null:( Ho:P=Po) alternative hypothesis: Ha: P cannot =Po or Ha:p<Po or Ha:P>Po

b . a. z test for the population MeanNull:( Ho:Miu=Miuo) alternative hypothesis: Ha: Miu cannot =Miu(o) or Ha:Miu<Miu(o) or Ha:Miu>Miu(o)

c. t test of population mean MeanNull:( Ho:Miu=Miuo) alternative hypothesis: Ha: Miu cannot =Miu(o) or Ha:Miu<Miu(o) or Ha:Miu>Miu(o)

2.Obtaining a random sample (or at least one that can be considered random) and collecting data. Using the data:

Check that the conditions under which the test can be reliably used are met.

**

*Summarize the data using a test statistic.( t test, Z test or the Zp test)

3. Finding the p-value of the test.

4.Making conclusions.

The p-value

****probability of getting data like those observed (or even more extreme) assuming that the null hypothesis is true,

****8p-value is a measure of the evidence against Ho.

******If the p-value is small, the data present enough evidence to reject H0 (and accept Ha).

****If the p-value is not small, the data do not provide enough evidence to reject H0.

****8p-value is a measure of the evidence against Ho.

******If the p-value is small, the data present enough evidence to reject H0 (and accept Ha).

****If the p-value is not small, the data do not provide enough evidence to reject H0.

Additional 'Big Ideas' about hypothesis Testing.

**Results that are based on a larger sample carry more weight, and therefore results that are not significant

**Even a very small and practically unimportant effect becomes statistically significant with a large enough sample size. The distinction between statistical significance and practical importance should therefore always be considered.

**For given data, the p-value of the two-sided test is always twice as large as the p-value of the one-sided test. It is therefore harder to reject H0

**95% confidence intervals can be used in order to carry out two-sided tests (at the .05 significance level).

**If the results are significant, it might be of interest to follow up the tests with a confidence interval in order to get insight into the actual value of the parameter of interest.

**Even a very small and practically unimportant effect becomes statistically significant with a large enough sample size. The distinction between statistical significance and practical importance should therefore always be considered.

**For given data, the p-value of the two-sided test is always twice as large as the p-value of the one-sided test. It is therefore harder to reject H0

**95% confidence intervals can be used in order to carry out two-sided tests (at the .05 significance level).

**If the results are significant, it might be of interest to follow up the tests with a confidence interval in order to get insight into the actual value of the parameter of interest.

Point estimation

estimating an unknown parameter with a single value that is computed from the sample.

Interval estimation

estimating an unknown parameter by an interval of plausible values. To each such interval we attach a level of confidence that indeed the interval captures the value of the unknown parameter and hence the name confidence intervals.

Hypothesis testing

four-step process in which we are assessing evidence provided by the data in favor or against some claim about the population parameter.

WHEN THERE ARE TWO VARIABLES Ho:

There is no relationship between X and Y.

WHEN THERE ARE TWO VARIABLES Ha:

There is a significant relationship between X and Y.

Categorical (explanatory) and (quantitative response)

(quantitative) response Y

(category) of the explanatory X

In general, then, making inferences about the relationship between X and Y in Case C→Q ((((boils down to comparing the means of Y in the sub-populations,))))) which are created by the categories defined in X (say k categories).

K= class ----( how many different Categorical subjects are there) K=2 when its gender (treated different than k>2)

k>2------ whenever there is more than two catergorical subjects. treated different than k=2)

(category) of the explanatory X

In general, then, making inferences about the relationship between X and Y in Case C→Q ((((boils down to comparing the means of Y in the sub-populations,))))) which are created by the categories defined in X (say k categories).

K= class ----( how many different Categorical subjects are there) K=2 when its gender (treated different than k>2)

k>2------ whenever there is more than two catergorical subjects. treated different than k=2)

Differentiating between one-sample and multi-sample

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within each of the five inferential methods we will focus on:

•When the inferential method is appropriate for use.

•Under what conditions the procedure can safely be used.

•The conceptual idea behind the test (as it is usually captured by the test statistic).

•How to use software to carry out the procedure in order to get the p-value of the test.

•Interpreting the results in the context of the problem.

•Under what conditions the procedure can safely be used.

•The conceptual idea behind the test (as it is usually captured by the test statistic).

•How to use software to carry out the procedure in order to get the p-value of the test.

•Interpreting the results in the context of the problem.

Differentiating between one-sample and multi-sample

independent group

matched/paired/linked /dependent samples

matched/paired/linked /dependent samples

independent group design

(sub-population 2) has the other value

n= different for each group

when we have two independent samples, the comparison of the reaction times is a comparison between two groups

n= different for each group

when we have two independent samples, the comparison of the reaction times is a comparison between two groups

matched/paired/linked /dependent design

when the sub populations match up realy well with one another.

n=same for both groups.

In matched pairs, the comparison between the reaction times is done for each individual.

n=same for both groups.

In matched pairs, the comparison between the reaction times is done for each individual.

Paired t-test ( Comparing Two Means—Matched Pairs (Paired t-test)

dependent groups

dependent groups

dependent

One of the most common cases where dependent samples occur is when both samples have the same subjects and they are "paired by subject."

matched pairs design and record each sampled student's SAT score before and after the SAT prep classes are attended:

two values come from a single set of subjects

One of the most common cases where dependent samples occur is when both samples have the same subjects and they are "paired by subject."

matched pairs design and record each sampled student's SAT score before and after the SAT prep classes are attended:

two values come from a single set of subjects

dependent groups

Every observation in one sample is linked with an observation in the other sample

where the paired design is used

"natural pairs," such as siblings, twins, or couples

Sat scores before and after ( same group)

Sat scores before and after ( same group)

Identify a matched pairs

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The Paired t-test

The idea behind the paired t-test is to reduce this two-sample situation, where we are comparing two means, to a single sample situation where we are doing inference on a single mean, and then use a simple t-test that we introduced in the previous module.

1.how do you put it into one simple comparison? ( you take Miu 1- miu2. this will give the difference which then can be used to see the true value.

2. you wil then you use the difference to find the T statistic. there are 4 steps.

1.insead of Mo you will now Use Md ( mean difference)

2.Checking Conditions and Calculating the Test Statistic

a. random samples

b. Normal or larg sample size

3. replace x-bar with Xbar(d) and replace s with S(d) in the t equation.

4. find the P value

p-value tells us that there is very little chance of getting data like those observed (or even more extreme) if the null hypothesis were true.

5. conclusion in context

If the p-value is small, there is a significant difference between what was observed in the sample and what was claimed in Ho, so we reject Ho and conclude that the categorical explanatory variable does affect the quantitative response variable as specified in Ha. If the p-value is not small, we do not have enough statistical evidence to reject Ho. In particular, if a cutoff probability, α (significance level), is specified, we reject Ho if the p-value is less than α. Otherwise, we do not reject Ho.

Typically, as in our example, one of the measurements occurs before a treatment/intervention (2 beers in our case), and the other measurement after the treatment/intervention.

1.how do you put it into one simple comparison? ( you take Miu 1- miu2. this will give the difference which then can be used to see the true value.

2. you wil then you use the difference to find the T statistic. there are 4 steps.

1.insead of Mo you will now Use Md ( mean difference)

2.Checking Conditions and Calculating the Test Statistic

a. random samples

b. Normal or larg sample size

3. replace x-bar with Xbar(d) and replace s with S(d) in the t equation.

4. find the P value

p-value tells us that there is very little chance of getting data like those observed (or even more extreme) if the null hypothesis were true.

5. conclusion in context

If the p-value is small, there is a significant difference between what was observed in the sample and what was claimed in Ho, so we reject Ho and conclude that the categorical explanatory variable does affect the quantitative response variable as specified in Ha. If the p-value is not small, we do not have enough statistical evidence to reject Ho. In particular, if a cutoff probability, α (significance level), is specified, we reject Ho if the p-value is less than α. Otherwise, we do not reject Ho.

Typically, as in our example, one of the measurements occurs before a treatment/intervention (2 beers in our case), and the other measurement after the treatment/intervention.

p-value

dicating that there is a % chance of obtaining data like those observed (or even more extreme) assuming that Ho is true

Paired t confidence interval

Xd +/- (T*(Sd/square root of N))

1. Find your T*

a. do this by (n-1) to find DF

b. after finding DF cross reference this with your Confidence level.

c. this will give you the T*

2. after finding T* calculate out your intervals with the above formula.

As we've seen in previous tests, as well as in the matched pairs case, the 95% confidence interval for μ d μd can be used for testing in the two-sided case (H 0 :μ d =0 H0:μd=0 vs. H a :μ d ≠0 Ha:μd≠0):

If the null value, 0, falls outside the confidence interval, we reject Ho.

If the null value, 0, falls inside the confidence interval, we fail to reject Ho.

1. Find your T*

a. do this by (n-1) to find DF

b. after finding DF cross reference this with your Confidence level.

c. this will give you the T*

2. after finding T* calculate out your intervals with the above formula.

As we've seen in previous tests, as well as in the matched pairs case, the 95% confidence interval for μ d μd can be used for testing in the two-sided case (H 0 :μ d =0 H0:μd=0 vs. H a :μ d ≠0 Ha:μd≠0):

If the null value, 0, falls outside the confidence interval, we reject Ho.

If the null value, 0, falls inside the confidence interval, we fail to reject Ho.

•The paired t-test

is used to compare two population means when the two samples (drawn from the two populations) are dependent in the sense that every observation in one sample can be linked to an observation in the other sample. Such a design is called "matched pairs."

•The idea behind the paired t-test is to reduce the data from two samples to just one sample of the differences, and use these observed differences as data for inference about a single mean — the mean of the differences, μ d

•The paired t-test is therefore simply a one-sample t-test for the mean of the differences μ d μd, where the null value is 0.

•The idea behind the paired t-test is to reduce the data from two samples to just one sample of the differences, and use these observed differences as data for inference about a single mean — the mean of the differences, μ d

•The paired t-test is therefore simply a one-sample t-test for the mean of the differences μ d μd, where the null value is 0.

matched paired design

•The most common case in which the matched pairs design is used is when the same subjects are measured twice, usually before and then after some kind of treatment and/or intervention. Another classic case are studies involving twins.

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Statistical Inference

where we draw conclusions about a population based on the data obtained from a sample chosen from it.

Point Estimation

, we estimate an unknown parameter using a single number that is calculated from the sample data.

examples

1.X-bar for Miu

2.P-hat for P

examples

1.X-bar for Miu

2.P-hat for P

Interval Estimation

we estimate an unknown parameter using an interval of values that is likely to contain the true value of that parameter

Hypothesis Testing

we have some claim about the population, and we check whether or not the data obtained from the sample provide evidence against this claim.

Inference For One Variable

When the variable of interest is categorical, the population parameter that we will infer about is the population proportion (p) associated with that variable

When the variable of interest is quantitative, the population parameter that we infer about is the population mean (μ) associated with that variable

When the variable of interest is quantitative, the population parameter that we infer about is the population mean (μ) associated with that variable

Point estimates

is the form of statistical inference in which, based on the sample data, we estimate the unknown parameter of interest using a single value (hence the name point estimation).

Sampling distribution of a statistic

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Probability theory

an unbiased estimator

Any particular sample mean might turn out to be less than the actual population mean, or it might turn out to be more. But in the long run, such sample means are "on target" in that they will not underestimate any more or less often than they overestimate

Any particular sample mean might turn out to be less than the actual population mean, or it might turn out to be more. But in the long run, such sample means are "on target" in that they will not underestimate any more or less often than they overestimate

p ˆ pˆ, is centered at

population proportion p (as long as the sample is taken at random), thus making p ˆ pˆ an unbiased estimator for p.

X-bar is centered at??

sampling distributions of X ¯ ¯ ¯ X¯ and found that as long as a sample is taken at random, the distribution of sample means is exactly centered at the value of population mean

when is the point estimates unbiased?

Our point estimates are truly unbiased estimates for the population parameter only if the sample is random and the study design is not flawed.

when is the point estimator closer to parameter?

when the population in large.

Point estimation is simple and intuitive, but also a bit problematic. Here is why:

very unlikely that the value of x ¯ x¯ will fall exactly at μ .

interval estimation

to enhance the simple point estimates by supplying information about the size of the error attached.

1. find a confidence %

2. Calculate the interval

- used for z score and t score

3. theory is that Miu or P will be within the interval based off the X- or P-hat used.

1. find a confidence %

2. Calculate the interval

- used for z score and t score

3. theory is that Miu or P will be within the interval based off the X- or P-hat used.

how to phrase the interval confidence

"We are 95% confident that the population mean μ falls within 3 units of x ¯ x¯."

Generalizing confidence intervals

1.STATE the problem:

2.PLAN the solution:

•We will use the formula for a one-sample z-confidence interval

•The parameter of interest is the mean score on the math part of the SAT for all community college students in the researcher's state.

•The confidence level we will use is 95%

3. SOLVE:

a.Before we use the formula, we first have to check whether the conditions are met for using the formula. The conditions for this test are:

•Randomness of data: a random sample of 650 students was chosen.

•Normality of the sampling distribution of x ¯ x¯: the sample size is greater than 30.

•σ known

4. interpret what this means in the context of the problem./ Conclude

"we are % confident that the True mean/Proportion of the ....(state the parameter in which you are studying)..is found within the interval ( what you calculated)

(for the population mean for X-bar and Miu)

2.PLAN the solution:

•We will use the formula for a one-sample z-confidence interval

•The parameter of interest is the mean score on the math part of the SAT for all community college students in the researcher's state.

•The confidence level we will use is 95%

3. SOLVE:

a.Before we use the formula, we first have to check whether the conditions are met for using the formula. The conditions for this test are:

•Randomness of data: a random sample of 650 students was chosen.

•Normality of the sampling distribution of x ¯ x¯: the sample size is greater than 30.

•σ known

4. interpret what this means in the context of the problem./ Conclude

"we are % confident that the True mean/Proportion of the ....(state the parameter in which you are studying)..is found within the interval ( what you calculated)

(for the population mean for X-bar and Miu)

Confidence level

there is a % chance that a normal random variable will take a value within (standard devations, or Z*) standard deviations of its mean.

what does the the confidence level mean

the higher the confidnence lever the less precise it is. the lower the more percice.

There is a trade-off between the level of confidence and the precision with which the parameter is estimated.

The price we have to pay for a higher level of confidence is that the unknown population mean will be estimated with less precision (i.e., with a wider confidence interval). If we would like to estimate μ with more precision (i.e. a narrower confidence interval), we will need to sacrifice and report an interval with a lower level of confidence.

There is a trade-off between the level of confidence and the precision with which the parameter is estimated.

The price we have to pay for a higher level of confidence is that the unknown population mean will be estimated with less precision (i.e., with a wider confidence interval). If we would like to estimate μ with more precision (i.e. a narrower confidence interval), we will need to sacrifice and report an interval with a lower level of confidence.

Is there a way to bypass this trade-off in confidence interval ? In other words, is there a way to increase the precision of the interval (i.e., make it narrower) without compromising on the level of confidence?

yes! by increasing the sample size

this is because N is found in the denominator and the larger it is the smaller the effect of the standard devation.

this is because N is found in the denominator and the larger it is the smaller the effect of the standard devation.

Margin of error ?

Z*e times (standard devation/square root of N )

"in charge" of the width (or precision) of the confidence interval

"in charge" of the width (or precision) of the confidence interval

z*

Confidence multiplier

Sample Size Calculations

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sample size determines

size of the margin of error and thus the width, or precision, of our interval estimation

how to calculate N when you want to a precise Margin of error

Take the Margin of error equation and move it around to make a new equation were you are now solving for N. it looks like this.

N=(Z*(Sigma)/Margin of error. )^

This will give you your N

*** when N is not a full number always round up).

N=(Z*(Sigma)/Margin of error. )^

This will give you your N

*** when N is not a full number always round up).

When to use confidence intervals

X_ distribution= normal ( central limit therum can be applied) ***********this is because we can now use the Z** score*********

the sample must be random.

sample size is large ( if not normal already) n > 30),

***********The only situation when we cannot use the confidence interval, then, is when the sample size is small and the variable of interest is not known to have a normal distribution*************

What if σ is unknown?

---replace sigma with the standard devation fo the sample group.

***********once the sigma has been replaced, we can no longer use Z**, but we must use T*

******Unlike the confidence multipliers we have used so far (z**), which depend only on the level of confidence, the new multipliers (t**) have the added complexity that they depend on both the level of confidence and on the sample size (for example: the t** used in a 95% confidence when n = 10 is different from the t** used when n = 40). Due to this added complexity in determining the appropriate t**, we will rely heavily on software in this case.************

the sample must be random.

sample size is large ( if not normal already) n > 30),

*********

What if σ is unknown?

---replace sigma with the standard devation fo the sample group.

*********

****

standard error

s/n(SQUARE ROOTED)

standard deviation

sigma/n(SQUARE ROOTED)

whenever we replace parameters with their sample counterparts in the standard deviation of a statistic, the resulting quantity is called the standard error of the statistic.

whenever we replace parameters with their sample counterparts in the standard deviation of a statistic, the resulting quantity is called the standard error of the statistic.

Population proportion confidence interval

1.population is categorical

2.parameter we are trying to infer about is the population proportion (p)

3. point estimator for the population proportion p is the sample proportion pˆ.

4. estimate±margin of error (p ˆ ±m)

2.parameter we are trying to infer about is the population proportion (p)

3. point estimator for the population proportion p is the sample proportion pˆ.

4. estimate±margin of error (p ˆ ±m)

margin of error is the product of two components:

Z*=confidence multiplier

⋅SD of the estimator

⋅SD of the estimator

p ˆ pˆ has a normal distribution with mean p

...

standard deviation of P^

square root of (p(1-P)/N

**** but when you do not know what P is you must replace it with your point estimator of P^)

******** it is now known as standard error of p^)

**** but when you do not know what P is you must replace it with your point estimator of P^)

******** it is now known as standard error of p^)

A 95% confidence interval for p is

We can be 95% sure that the proportion of all U.S. adults who were already familiar with Viagra by that time was between .61 and .67 (or 61% and 67%).

Two important results that we discussed at length when we talked about the confidence interval for μ also apply here:

1. There is a trade-off between level of confidence and the width (or precision) of the confidence interval. The more precision you would like the confidence interval for p to have, the more you have to pay by having a lower level of confidence.

2. Since n appears in the denominator of the margin of error of the confidence interval for p, for a fixed level of confidence, the larger the sample, the narrower, or more precise it is. This brings us naturally to our next point.

2. Since n appears in the denominator of the margin of error of the confidence interval for p, for a fixed level of confidence, the larger the sample, the narrower, or more precise it is. This brings us naturally to our next point.

Determining Sample Size for a Given Margin of Error in Estimating Proportions

LOOOK INTO THIS

LOOOK INTO THIS

Taker you standard error for P^ and move it around so that you are now solving for P.

N=(Z*)^(1-P(hat))/M Squared.

N=(Z*)^(1-P(hat))/M Squared.

When to use confidence intervals for P

1.n⋅p ˆ ≥10 and n⋅(1−p ˆ )≥10

******** make sure its normal**********

2.

******

2.

margin of error of a poll is determined

1 /√n.

Hypothesis Testing

Assessing evidence provided by the data in favor of or against some claim about the population

how the process of statistical hypothesis testing works

1.Stating the claims:

claim 1:Ho

claim 2: Ha

2.Choosing a sample and collecting data:

3.Assessment of evidence:

is P lower than the significans alpha?

is P larger

was it random

4.Conclusion:

claim 1:Ho

claim 2: Ha

2.Choosing a sample and collecting data:

3.Assessment of evidence:

is P lower than the significans alpha?

is P larger

was it random

4.Conclusion:

questions to ask yourself when assessing. "pay attention to the wording"

1. how likely is it to get data like the data we observed, in which the difference between the males' average and females' average score is as high as 15 points or higher?

2.If the mean concentration in the whole shipment were really the required 245 ppm (i.e., if claim 1 were true), how surprising would it be to observe a sample of 64 portions where the sample mean concentration

is off by 5 ppm or more (as we did)?

3.how likely is it that in a sample of 250 we will find tat the mean number of hourse per week coporate employees work is as high as 47 if the true mean is 40

4. how likely it is that in a sample of 375 we'll find that as low as 16.5 have used marijuana, when the true trate is actually 21.5

2.If the mean concentration in the whole shipment were really the required 245 ppm (i.e., if claim 1 were true), how surprising would it be to observe a sample of 64 portions where the sample mean concentration

is off by 5 ppm or more (as we did)?

3.how likely is it that in a sample of 250 we will find tat the mean number of hourse per week coporate employees work is as high as 47 if the true mean is 40

4. how likely it is that in a sample of 375 we'll find that as low as 16.5 have used marijuana, when the true trate is actually 21.5

null hypothesis

the claim about the population

**** null hypothesis suggests nothing special is going on; in other words, there is no change from the status quo, no difference from the traditional state of affairs, no relationship

**** null hypothesis suggests nothing special is going on; in other words, there is no change from the status quo, no difference from the traditional state of affairs, no relationship

alternative hypothesis

the counter claim that the population claim is wrong.

***disagrees with this, stating that something is going on, or there is a change from the status quo, or there is a difference from the traditional state of affairs.

***disagrees with this, stating that something is going on, or there is a change from the status quo, or there is a difference from the traditional state of affairs.

how to write out Null and Alternateive

Ha: Miu (<,>,not = )

Ho: Miu = stated claim

make surer you have the Ha and Ho with :

Ho: Miu = stated claim

make surer you have the Ha and Ho with :

Hypothesis testing: Choosing a sample and collecting data

...

test statistic

sample statistics to summarize the data

Hypothesis testing: Assessing the evidence

this is the step where we calculate how likely is it to get data like that observed when Ho true. In a sense, this is the heart of the process, since we draw our conclusions based on this probability

****If this probability is very small (see example 2), then that means that it would be very surprising to get data like that observed if H0

*******if this probability is not very small (see example 3) this means that observing data like that observed is not very surprising if H0 were true

****If this probability is very small (see example 2), then that means that it would be very surprising to get data like that observed if H0

*******if this probability is not very small (see example 3) this means that observing data like that observed is not very surprising if H0 were true

Hypothesis testing: Making conclusions.

significance level of the test- base your conclusion off of this. the standard is .05 and must be decided before you test.

•if the p-value < α (usually .05), then the data we got is considered to be "rare (or surprising) enough" when Ho is true, and we say that the data provide significant evidence against Ho, so we reject Ho and accept Ha.

•if the p-value > α (usually .05), then our data are not considered to be "surprising enough" when Ho is true, and we say that our data do not provide enough evidence to reject Ho (or, equivalently, that the data do not provide enough evidence to accept Ha).

It is important to draw your conclusions in context. It is never enough to say: "p-value = ..., and therefore I have enough evidence to reject Ho at the .05 significance level."You should always add: "... and conclude that ... (what it means in the context of the problem)".

***8Either I reject Ho and accept Ha (when the p-value is smaller than the significance level) or I cannot reject Ho (when the p-value is larger than the significance level).

*** P value is between 1-0

•if the p-value < α (usually .05), then the data we got is considered to be "rare (or surprising) enough" when Ho is true, and we say that the data provide significant evidence against Ho, so we reject Ho and accept Ha.

•if the p-value > α (usually .05), then our data are not considered to be "surprising enough" when Ho is true, and we say that our data do not provide enough evidence to reject Ho (or, equivalently, that the data do not provide enough evidence to accept Ha).

It is important to draw your conclusions in context. It is never enough to say: "p-value = ..., and therefore I have enough evidence to reject Ho at the .05 significance level."You should always add: "... and conclude that ... (what it means in the context of the problem)".

***8Either I reject Ho and accept Ha (when the p-value is smaller than the significance level) or I cannot reject Ho (when the p-value is larger than the significance level).

*** P value is between 1-0

Summary of Hypothises testing(proportions known Sigma)

1. know your Ho and Ha

2. know what direction your Ha is solving for

3. decide on Alpha

4.

2. know what direction your Ha is solving for

3. decide on Alpha

4.

z-test for the population(proportions known Sigma)

1.STATE \

2.PLAN

•Select the procedure or formula to use

z test for the population proportion

z test for the population mean

•State the parameter in context (e.g. let p= the proportion of defective discs)

•State the null and alternative hypotheses

•Choose the significance level

3.SOLVE

•Collect relevant data from a ****dom sample**e***

•Check conditions of the procedure

•If conditions are met, calculate test statistic

•Find the p-value, the probability of observing data like those observed assuming the null hypothesis is true

4.CONCLUDE

Based on the p-value, decide whether we have enough evidence to reject Ho (and accept Ha), and draw conclusions in context.

2.PLAN

•Select the procedure or formula to use

z test for the population proportion

z test for the population mean

•State the parameter in context (e.g. let p= the proportion of defective discs)

•State the null and alternative hypotheses

•Choose the significance level

3.SOLVE

•Collect relevant data from a **

•Check conditions of the procedure

•If conditions are met, calculate test statistic

•Find the p-value, the probability of observing data like those observed assuming the null hypothesis is true

4.CONCLUDE

Based on the p-value, decide whether we have enough evidence to reject Ho (and accept Ha), and draw conclusions in context.

null value(proportions known Sigma)

The value that is specified in the null hypothesis

how to know if its a two sided z test(proportions known Sigma)

Differ

has changed

has changed

test statistic(proportions known Sigma)

a measure of how far the sample proportion p ˆ pˆ is from the null value p 0

Z score is the test statistic

Z score is the test statistic

p value info(proportions known Sigma)

P value is more impressive the lareger the sample you have. no matter what.

the smaller it is, the more unlikely it is to get data like those observed when Ho is true, the more evidence it is against Ho.

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

the smaller it is, the more unlikely it is to get data like those observed when Ho is true, the more evidence it is against Ho.

The p-value is the probability of observing a test statistic as extreme as that observed (or even more extreme) assuming that the null hypothesis is true.

Z score(proportions known Sigma)

It represents the difference between the sample proportion (p ˆ pˆ) and the null value (p 0 p0), measured in standard deviations.

** shows how many standard deviations we are away from the center.

We mentioned earlier that to some degree, the test statistic captures the essence of the test. In this case, the test statistic measures the difference between p ˆ pˆ and p 0 p0 in standard deviations. This is exactly what this test is about

test statistic as a measure of evidence in the data against Ho. The larger the test statistic, the "further the data are from Ho" and therefore the more evidence the data provide against Ho.

*****You can think about this test statistic as a measure of evidence in the data against Ho. The larger the test statistic, the "further the data are from Ho" and therefore the more evidence the data provide against Ho.******

** shows how many standard deviations we are away from the center.

We mentioned earlier that to some degree, the test statistic captures the essence of the test. In this case, the test statistic measures the difference between p ˆ pˆ and p 0 p0 in standard deviations. This is exactly what this test is about

test statistic as a measure of evidence in the data against Ho. The larger the test statistic, the "further the data are from Ho" and therefore the more evidence the data provide against Ho.

***

f we take a different random sample and get a test statistic of zero, what can we conclude?(proportions known Sigma)

p-hat = po

Conditions for use of test statistic ( population proportion)(proportions known Sigma)

random sample of size n

normal distribution

n⋅p o ≥10

n⋅(1−p o )≥10

normal distribution

n⋅p o ≥10

n⋅(1−p o )≥10

Finding the p-values of the test(proportions known Sigma)

-inding p-values involves probability calculations about the value of the test statistic assuming that Ho is true.

-the values of the test statistic follow a standard normal distribution

-the values of the test statistic follow a standard normal distribution

Drawing Conclusions Based on the P-Value(proportions known Sigma)

(i) Based on the p-value, determine whether or not the results are significant (i.e., the data present enough evidence to reject Ho).

(ii) State your conclusions in the context of the problem.

EXAMPLES

Since .023 is small (in particular, .023 < .05), the data provide enough evidence to reject Ho and conclude that as a result of the repair the proportion of defective products has been reduced to below .20. The following figure is the complete story of this example, and includes all the steps we went through, starting from stating the hypotheses and ending with our conclusions:

*******8This last part of the four-step process of hypothesis testing is the same across all statistical tests***********

(ii) State your conclusions in the context of the problem.

EXAMPLES

Since .023 is small (in particular, .023 < .05), the data provide enough evidence to reject Ho and conclude that as a result of the repair the proportion of defective products has been reduced to below .20. The following figure is the complete story of this example, and includes all the steps we went through, starting from stating the hypotheses and ending with our conclusions:

*****

Summary of four steps of hypothesis testing(proportions known Sigma)

Step 1: STATE

State the problem.

Step 2: PLAN

State the null and alternative hypotheses:

***state it: Ho: =

***state it: Ha: Miu>/</ not =

Step 3: SOLVE

Obtain data from a sample and:

1. make sure it is random and normal

2. calculate the sample proportion

3. find the test statistic

4. find p

5. compare against significance level

Step 4: CONCLUDE

If the p-value is small (in particular, smaller than the significance level, which is usually .05), the results are significant (in the sense that there is a significant difference between what was observed in the sample and what was claimed in Ho), and so we reject Ho. If the p-value is not small, we do not have enough statistical evidence to reject Ho, and so we continue to believe that Ho may be true. (Remember: In hypothesis testing we never "accept" Ho).

State the problem.

Step 2: PLAN

State the null and alternative hypotheses:

***state it: Ho: =

***state it: Ha: Miu>/</ not =

Step 3: SOLVE

Obtain data from a sample and:

1. make sure it is random and normal

2. calculate the sample proportion

3. find the test statistic

4. find p

5. compare against significance level

Step 4: CONCLUDE

If the p-value is small (in particular, smaller than the significance level, which is usually .05), the results are significant (in the sense that there is a significant difference between what was observed in the sample and what was claimed in Ho), and so we reject Ho. If the p-value is not small, we do not have enough statistical evidence to reject Ho, and so we continue to believe that Ho may be true. (Remember: In hypothesis testing we never "accept" Ho).

The issues regarding hypothesis testing(proportions known Sigma)

1.e effect of sample size on hypothesis testing.

2. Statistical significance vs. practical importance. (This will be discussed in the activity following number 1.)

3. One-sided alternative vs. two-sided alternative—understanding what is going on.

4. Hypothesis testing and confidence intervals—how are they related?

Let's start.

2. Statistical significance vs. practical importance. (This will be discussed in the activity following number 1.)

3. One-sided alternative vs. two-sided alternative—understanding what is going on.

4. Hypothesis testing and confidence intervals—how are they related?

Let's start.

The Effect of Sample Size on Hypothesis Testing(proportions known Sigma)

1.Larger sample sizes give us more information to pin down the true nature of the population.( miu)

2.for the same level of confidence, we can report a smaller margin of error, and get a narrower confidence interval

3. because there is more in the denominator it makes the Z score larger and in turn makes the P values lower.

2.for the same level of confidence, we can report a smaller margin of error, and get a narrower confidence interval

3. because there is more in the denominator it makes the Z score larger and in turn makes the P values lower.

One-Sided Alternative vs. Two-Sided Alternative (proportions known Sigma)

1. wording is different. one sided = higher or lower

and for the 2 sided= differs.

2.the p-value of the two sided-test (example 2**) is twice the p-value of the one-sided test

and for the 2 sided= differs.

2.the p-value of the two sided-test (example 2**) is twice the p-value of the one-sided test

Hypothesis Testing and Confidence Intervals

1.confidence intervals estimate a parameter, and hypothesis testing assesses the evidence in the data against one claim and in favor of another

2.An alternative way to perform this test is to find a 95% confidence interval for p and check:

If p 0 p0 falls outside the confidence interval, reject Ho.

If p 0 p0 falls inside the confidence interval, do not reject Ho.

In other words, if p 0 p0 is not one of the plausible values for p, we reject Ho.

If p 0 p0 is a plausible value for p, we cannot reject Ho

Confidence intervals can be used in order to carry out two-sided tests .

2.An alternative way to perform this test is to find a 95% confidence interval for p and check:

If p 0 p0 falls outside the confidence interval, reject Ho.

If p 0 p0 falls inside the confidence interval, do not reject Ho.

In other words, if p 0 p0 is not one of the plausible values for p, we reject Ho.

If p 0 p0 is a plausible value for p, we cannot reject Ho

Confidence intervals can be used in order to carry out two-sided tests .

statistical significance

...

practical importance

...

The z-test for the population mean

Tests about μ when sigma (σ) is known

t-test for the population mean μ

The test for the population mean when sigma is unkown.

z-test for the population mean involves:

*. Stating the hypotheses Ho and Ha.

*. Collecting relevant data, checking that the data satisfy the conditions which allow us to use this test, and summarizing the data using a test statistic.

*. Finding the p-value of the test, the probability of obtaining data as extreme as those collected (or even more extreme, in the direction of the alternative hypothesis), assuming that the null hypothesis is true. In other words, how likely is it that the only reason for getting data like those observed is sampling variability (and not because Ho is not true)?

*. Drawing conclusions, assessing the significance of the results based on the p-value, and stating our conclusions in context. (Do we or don't we have evidence to reject Ho and accept Ha?)

*. Collecting relevant data, checking that the data satisfy the conditions which allow us to use this test, and summarizing the data using a test statistic.

*. Finding the p-value of the test, the probability of obtaining data as extreme as those collected (or even more extreme, in the direction of the alternative hypothesis), assuming that the null hypothesis is true. In other words, how likely is it that the only reason for getting data like those observed is sampling variability (and not because Ho is not true)?

*. Drawing conclusions, assessing the significance of the results based on the p-value, and stating our conclusions in context. (Do we or don't we have evidence to reject Ho and accept Ha?)

Stating the Hypotheses( mean, known sigma)

have exactly the same structure as the hypotheses for z-test for the population proportion (p):

Collecting Data and Summarizing Them (( mean, known sigma)

1.find the sample mean

2.Find z score.

2.Find z score.

sample means behave as follows:

1.•Center: The mean of the sample means is µ, the population mean.

2.•Spread: The standard deviation of the sample means is σn √ σn.

3.•Shape: The sample means are normally distributed if the variable is normally distributed in the population or the sample size is large enough to guarantee approximate normality.

2.•Spread: The standard deviation of the sample means is σn √ σn.

3.•Shape: The sample means are normally distributed if the variable is normally distributed in the population or the sample size is large enough to guarantee approximate normality.

test statistic is: z ( mean, known sigma)

Z= Xbar-miu/SD/square root of N

tells us how far x ¯ x¯ is from the null value μ 0 μ0 measured in standard deviations.

The larger the test statistic, the more evidence we have against Ho,

Central Limit Theorem gives us criteria for deciding if the z-test for the population mean can be used

tells us how far x ¯ x¯ is from the null value μ 0 μ0 measured in standard deviations.

The larger the test statistic, the more evidence we have against Ho,

Central Limit Theorem gives us criteria for deciding if the z-test for the population mean can be used

Finding the p-value ( mean, known sigma)

same as for Proportions.

Drawing Conclusions( mean, known sigma)

same as for Proportions.

Relating Hypothesis Tests and Confidence Intervals ( mean, known sigma)

If μ 0 μ0 falls outside the confidence interval, reject Ho.

If μ 0 μ0 falls inside the confidence interval, do not reject Ho.

carry out the two-sided test

If μ 0 μ0 falls inside the confidence interval, do not reject Ho.

carry out the two-sided test

Type I ERRORS In hypothesis testin

•If the null hypothesis is true, but we reject it. This is a type I error.

Type I error: The evidence leads the jury to convict an innocent person. By analogy, we reject a true null hypothesis and accept a false alternative hypothesis.

Type I error: The evidence leads the jury to convict an innocent person. By analogy, we reject a true null hypothesis and accept a false alternative hypothesis.

Type 2 ERRORS In hypothesis testin

•If the null hypothesis is false, but we fail to reject it. This is a type II error.

Type II error: The evidence leads the jury to declare a defendant not guilty, when he is in fact guilty. By analogy, we fail to reject a null hypothesis that is false. In other words, we do not accept an alternative hypothesis when it is really true.

Type II error: The evidence leads the jury to declare a defendant not guilty, when he is in fact guilty. By analogy, we fail to reject a null hypothesis that is false. In other words, we do not accept an alternative hypothesis when it is really true.

Type I and Type II Errors

Type I and type II errors are not caused by mistakes. They are the result of random chance. The data provide evidence for a conclusion that is false. It's no one's fault!

How often will you cause an error (1) ? the percentage that is stated as the significance lever. (njust for Type 1 error)

How often will you cause an error (2)? more complicated to calculate, but it is also related to the significance level, but not direcltly.

HOW TO LOWER THE LIKELYHOOD OF THIS??

*****the chance of a type I error increases the chance of a type II error,***** lowering significance level....which in turn raises the confidence leverl...which in turn raises the N needed.

TYPE 1 = lower the significance leverl

TYPE 2 = larger N or larger significance level

How often will you cause an error (1) ? the percentage that is stated as the significance lever. (njust for Type 1 error)

How often will you cause an error (2)? more complicated to calculate, but it is also related to the significance level, but not direcltly.

HOW TO LOWER THE LIKELYHOOD OF THIS??

***

TYPE 1 = lower the significance leverl

TYPE 2 = larger N or larger significance level

General guidelines for choosing a level of significance:

•If the consequences of a type I error are more serious, choose a small level of significance (α).

•If the consequences of a type II error are more serious, choose a larger level of significance (α). But remember that the level of significance is the probability of committing a type I error.

•****In general, we choose the largest level of significance that we can tolerate as the chance of making a type I error.**.****

•If the consequences of a type II error are more serious, choose a larger level of significance (α). But remember that the level of significance is the probability of committing a type I error.

•**

power of the test or just power

denotes the probability of making a Type II error (failing to reject a false null hypothesis),

Power \equation

power =1- β.

Factors that affect power

•Significance level α

•Sample size n

•Effect size, the difference between the actual value of the parameter and the hypothesized null value

•Increasing α decreases β and increases power

•Conversely, decreasing α increases β and decreases power.

•Sample size n

•Effect size, the difference between the actual value of the parameter and the hypothesized null value

•Increasing α decreases β and increases power

•Conversely, decreasing α increases β and decreases power.

Relationship between power and significance level

•Increasing α decreases β and increases power

•Conversely, decreasing α increases β and decreases power.

•Conversely, decreasing α increases β and decreases power.

Relationship between power and sample size

Power increases when the sample size increases. The larger the sample size, the higher the power of the test. When we have more information from the population (larger sample size), we are less likely to make a wrong decision.

Relationship between power and effect size

If the effect size is small, then the probability of detecting this difference is small because the two values are very close to each other. This leads to a low power and a low probability of making a correct decision ( high aplpha). However, if the effect size is large, then it is easier to detect this difference.

Effect size

difference between the actual value of the parameter (for example, µ) and the hypothesized parameter value

Summary of the relationship between power, α, n and effect size.

•For a fixed n, α and β are inversely related (increasing α increases power).

•For a fixed α, β and n are inversely related (increasing sample size increases power).

•The larger the effect size, the smaller β is and the larger the power.

•For a fixed α, β and n are inversely related (increasing sample size increases power).

•The larger the effect size, the smaller β is and the larger the power.

How large a sample do I need

•A small level of significance requires a larger sample.

•Depending on the effect size, higher power requires a larger sample size.

•Detecting a small effect size requires a larger sample size.

•A two-sided test requires a larger sample size than a one-sided test

•Depending on the effect size, higher power requires a larger sample size.

•Detecting a small effect size requires a larger sample size.

•A two-sided test requires a larger sample size than a one-sided test

Tests About μ When σ is Unknown

The t-test for the Population Mean

what do you use when you don't know the standard devation of the POP

the sample group standard dev.

four steps for the t-test:

STATE: State the problem.

PLAN: In this step there are no changes:

•State the parameter of interest in context and the symbol used (e.g. μ = mean PA of all BYU students...)

•The null hypothesis has the form:

H o :μ=μ o Ho:μ=μo

(where μ o μo is the null value).

•The alternative hypothesis takes one of the following three forms (depending on the context):

H a :μ<μ o Ha:μ<μo (one-sided)

H a :μ>μ o Ha:μ>μo (one-sided)

H a :μ≠μ o Ha:μ≠μo (two-sided)

•Choose level of significance

SOLVE:

•Collect data

•Check the conditions under which the t-test can be safely used and summarizing the data

PLAN: In this step there are no changes:

•State the parameter of interest in context and the symbol used (e.g. μ = mean PA of all BYU students...)

•The null hypothesis has the form:

H o :μ=μ o Ho:μ=μo

(where μ o μo is the null value).

•The alternative hypothesis takes one of the following three forms (depending on the context):

H a :μ<μ o Ha:μ<μo (one-sided)

H a :μ>μ o Ha:μ>μo (one-sided)

H a :μ≠μ o Ha:μ≠μo (two-sided)

•Choose level of significance

SOLVE:

•Collect data

•Check the conditions under which the t-test can be safely used and summarizing the data

standard error of x

s/ square root of n

follows a t distribution

follows a t distribution

t distribution.

1. like the normal distribution; and the center of the t distribution is standardized at zero,

2. t distribution ends up being the appropriate model in certain cases where there is more variability than would be predicted by the normal distribution.

2. t distribution ends up being the appropriate model in certain cases where there is more variability than would be predicted by the normal distribution.

how is the t distribution fundamentally different from the normal distribution?

1. The spread.

a. T distributions have fatter tails.

b. slightly less area near the expected central value

a. T distributions have fatter tails.

b. slightly less area near the expected central value

degrees of Freedom

1.t distributions that are closer to normal are said to have higher "degrees of freedom"

2. lower degrees of freedom are not close to normal distribution and have a fat like tail.

2. lower degrees of freedom are not close to normal distribution and have a fat like tail.

Why would this single change (using "s" in place of "sigma") result in a sampling distribution that is the t distribution instead of the standard normal (Z) distribution?

1. more variability

a. So why is there more variability when s is used in place of the unknown sigma?

*******s (the standard deviation of the sample data) varies from sample to sample, and therefore it's another source of variation********

a. So why is there more variability when s is used in place of the unknown sigma?

*****

degrees of freedom

(n - 1)

always round down to the nearest degrees of freedom

always round down to the nearest degrees of freedom

Using t distributions to find a p-value

1. find you t statistic

2. find degrees of freedom

3. find your value which will between 2 usuallyl

4. state what it could be more and less than.

-when you calculate a negative test statistic, look for the positive value of that test statistic to obtain the p-value.

•If the test statistic you have calculated is greater than the one given in the table, assume that your p-value is less than the smallest value given on the table

2. find degrees of freedom

3. find your value which will between 2 usuallyl

4. state what it could be more and less than.

-when you calculate a negative test statistic, look for the positive value of that test statistic to obtain the p-value.

•If the test statistic you have calculated is greater than the one given in the table, assume that your p-value is less than the smallest value given on the table

hypothesis testing for the population mean (μ μ ), we distinguish between two cases:

z test ( sigma known)

t test( sigma not known)

In both cases, the null hypothesis is: H o :μ=μ o Ho:μ=μo

and the alternative, depending on the context, is one of the following:

H a :μ<μ Ha:μ<μ, or H a :μ>μ o Ha:μ>μo, or H a :μ≠μ o Ha:μ≠μo

t test( sigma not known)

In both cases, the null hypothesis is: H o :μ=μ o Ho:μ=μo

and the alternative, depending on the context, is one of the following:

H a :μ<μ Ha:μ<μ, or H a :μ>μ o Ha:μ>μo, or H a :μ≠μ o Ha:μ≠μo

Point estimation

estimating an unknown parameter with a single value that is computed from the sample.

Interval estimation

estimating an unknown parameter by an interval of plausible values.

Hypothesis testing

a four-step process in which we are assessing evidence provided by the data in favor or against some claim about the population parameter.

relationship exists between the variables X and Y in a population of interest. ( comparing 2 groups)

Ho:There is no relationship between X and Y.

Ha:There is a significant relationship between X and Y.

Ha:There is a significant relationship between X and Y.

Categorical explanatory and quantitative response

(quantitative) response Y for each value (category) of the explanatory X.

EXAMPLE

X : year in college (1 = freshmen, 2 = sophomore, 3 = junior, 4 = senior) and

Y : GPA

inference about the relationship between year and GPA has to be based on some kind of comparison of these four means.

making inferences about the relationship between X and Y in Case C→Q boils down to comparing the means of Y in the sub-populations, which are created by the categories defined in X (say k categories).

EXAMPLE

X : year in college (1 = freshmen, 2 = sophomore, 3 = junior, 4 = senior) and

Y : GPA

inference about the relationship between year and GPA has to be based on some kind of comparison of these four means.

making inferences about the relationship between X and Y in Case C→Q boils down to comparing the means of Y in the sub-populations, which are created by the categories defined in X (say k categories).

Differentiating between one-sample and multi-sample

he sample sizes of the two independent samples need not be the same

matched pairs the sample sizes of the two samples must be the same (and thus we used n for both).

matched pairs the sample sizes of the two samples must be the same (and thus we used n for both).

matched-pairs design

In matched pairs, the comparison between the reaction times is done for each individual.

independent samples

when we have two independent samples, the comparison of the reaction times is a comparison between two groups

when do you worry aout independent and matched pairs samples???

only when there are 2 groups to compare against.

when you are comparing more than two samples what will the sampels be ? matched or independent?

always they will be independent.

Comparing Two Means—Matched Pairs (Paired t-test) ( k=2)

1. C→Q of inference about relationships, where the explanatory variable is categorical and the response variable is quantitative

2.the matched pairs t-test, is used when we are comparing two means, and the samples are paired or matched ( all samples are dependant)

EXAMPLES

1. paired by subject. (SAT) before and after

2.such as siblings, twins, or couples

Example test

1. bears

a. categorical ( yes or no)

b. paired test because it was before and after

2.the matched pairs t-test, is used when we are comparing two means, and the samples are paired or matched ( all samples are dependant)

EXAMPLES

1. paired by subject. (SAT) before and after

2.such as siblings, twins, or couples

Example test

1. bears

a. categorical ( yes or no)

b. paired test because it was before and after

The Paired t-test

reduce this two-sample situation, where we are comparing two means,

paired t test.

1. state your 2 samples that are dependent

2. test before and after

3. subtract before from after

4. take the difference means and then find the mean as if it is one sampel

5. take the new x bar and s and plug it into the t test which was previously discussed. ( make sure to find degrees of freedom.

6. solve with the following equation

T=(Xd -0)/(Sd/square root of N)

Remember that Ho: Miu=0 ( we are claiming there is no difference )

used to compare two population means when the two samples (drawn from the two populations) are dependent in the sense that every observation in one sample can be linked to an observation in the other sample. Such a design is called "matched pairs."

2. seen when:

subjects are measured twice, usually before and then after some kind of treatment and/or intervention.

3. why do we do matched t test:

paired t-test is to reduce the data from two samples to just one sample of the differences, and use these observed differences as data for inference about a single mean

4. when can a intervals be used as supportive evidence.:

•A 95% confidence interval for μ d μd can be very insightful after a test has rejected the null hypothesis, and can also be used for testing in the two-sided case.

2. test before and after

3. subtract before from after

4. take the difference means and then find the mean as if it is one sampel

5. take the new x bar and s and plug it into the t test which was previously discussed. ( make sure to find degrees of freedom.

6. solve with the following equation

T=(Xd -0)/(Sd/square root of N)

Remember that Ho: Miu=0 ( we are claiming there is no difference )

used to compare two population means when the two samples (drawn from the two populations) are dependent in the sense that every observation in one sample can be linked to an observation in the other sample. Such a design is called "matched pairs."

2. seen when:

subjects are measured twice, usually before and then after some kind of treatment and/or intervention.

3. why do we do matched t test:

paired t-test is to reduce the data from two samples to just one sample of the differences, and use these observed differences as data for inference about a single mean

4. when can a intervals be used as supportive evidence.:

•A 95% confidence interval for μ d μd can be very insightful after a test has rejected the null hypothesis, and can also be used for testing in the two-sided case.

checking the conditions for the matched paired t test

i.The sample of differences is random

ii: must be normally distributed. ( 30>0)

ii: must be normally distributed. ( 30>0)

what questions should we ask ourslelve before we start to calculate the T Test?

is it a matched pair or independent.

Paired t confidence interval

can only be used in a matched pair system.

1. equation

a. Xd+/-t*(Sd/square root of N)

2. If the null value, 0, falls outside the confidence interval, we reject Ho.

If the null value, 0, falls inside the confidence interval, we fail to reject Ho.

1. equation

a. Xd+/-t*(Sd/square root of N)

2. If the null value, 0, falls outside the confidence interval, we reject Ho.

If the null value, 0, falls inside the confidence interval, we fail to reject Ho.

what is the measure of uncertainty?

Marginal Error

Alpha

Alpha

issues with point estimation

1. they are always wrong

2. change depending on the sample

2. change depending on the sample

good things about interval estimation

1. provide a range of plausible values the parameter could be

2.were confident the interval estimates is right

3. incorporates sampling variability

2.were confident the interval estimates is right

3. incorporates sampling variability

Confidence interval for Miu ( equation)

Xbar+/-Z*(SD/Square root of N)

****** Sigma is known******

****

Confidence multiplier

Z*

Sample size calc. for Quantitative

N=(Z*SD/largest margin of error you want) all squared

Confidence interval ( Equation) for Proportion

since we wont know what P is we must replace it with P-hat .

P-hat=/-Z*=(P-hat(1-P-hat)/N) all this will then be square rooted.

standard error =(P-hat(1-P-hat)/N) all this will then be square rooted.

P-hat=/-Z*=(P-hat(1-P-hat)/N) all this will then be square rooted.

standard error =(P-hat(1-P-hat)/N) all this will then be square rooted.

Sample Size Calc. for Proportion

N=((Z*/largest margin of error)squared)(p-Hat)(1-P-hat)

*** p-hat can be P if it is known.... but it prob wont be.)

*** p-hat can be P if it is known.... but it prob wont be.)

Test statistic for Miu

Z=Xbar - miuo/(Sigma/square root of N)

Test statistic for proportion

Z= (P-hat)-(Po)/((P0)(1-Po)/n) all square rooted.

llook at chapter 30

...

how to calculate s

s=((Xi-Xbar)^2/n-1) all square rooted

t statistic

T=(Xbar-Miu/(s/Square root of N)

Degrees of freedom

N-1

Hypothesis testing (correct)

when you compare a given statistic and try to prove it wrong.

does the confidence interval always contain the true mean or proportion?

no

what does s/ Square root of n estimate?

The standard deviation of the sampling distribution of x ¯

standard error of P- hat

same as standard devation, but P is unknown.

We say a point estimator is unbiased if:

the sampling distribution is centered exactly at the parameter it estimates (correct)

increasing confidence level means you must increase N if you want to get an accurate stat.

...

True or False: The test statistic is a measure of how close the sample proportion is to the null value

true

True or False: When results are practically significant, they are also statistically significant.

...

True or False: If the p-value is statistically significant, then the difference between the observed statistic and the null value is too large to be due to chance variation alone.

true

beta

probability of type 2 error

power

probability of getting the correct result

Alpha

probability of type 1 error

must know for written

STAT

Plan (choose procedure )

- Ho Ha alpha

-one sample z score for the (pop mea)(pop Prop)

-two sided z score

-matched pairs t-procedure for the population mean

-t-procedure for the population mean

SOLVE

CONCLUDE

Plan (choose procedure )

- Ho Ha alpha

-one sample z score for the (pop mea)(pop Prop)

-two sided z score

-matched pairs t-procedure for the population mean

-t-procedure for the population mean

SOLVE

CONCLUDE

Which of the following will lead to a decrease in power?

A decrease in alpha

A decrease in sample size

A decrease in sample size

what are you calculating when you do a matched paired test?

the mean difference in paramater