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Stat Ch 9
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Gravity
Terms in this set (38)
significance test
A formal procedure for using observed data to decide between two competing claims (called hypothesis).
Ask if sample data give convincing evidence against the H0 and in favor of the Ha.
Answers the question "How likely is it to get a result like this just by chance when null hypothesis is true?"
hypothesis
Statements about a parameter, like the population proportion p or population μ.
Should express the hopes or suspicions we have BEFORE we see the data.
Always refers to a population, not sample, so HAVE to state H0 and Ha in terms of population parameters (cannot use p-hat or x-bar)
null hypothesis (H0)
The claim that we weigh evidence against in a stat. test. a statement of "no difference"
Form H0: parameter = value
alternative hypothesis (Ha)
The claim about the population that we are trying to find evidence for
Ha : parameter < value, Ha: parameter > value, or Ha: parameter ≠ value.
Ha one-sided
States that a parameter is LARGER THAN the null hypothesis value or if it states that a parameter is
SMALLER THAN the null value
Significance test w/one-sided Ha = one-sided test
Ha two-sided
States that a parameter is DIFFERENT FROM the null hypothesis (could be smaller or larger)
Significance test w/two-sided Ha = two-sided test
test of significance
another name for significance test
hypothesis test
another name for significance test
test of hypotheses
another name for significance test
the reasoning of sig. test
An outcome that would rarely happen if the H0 were true is good evidence that the H0 is not true.
p-value
The probability that measures the strength of the evidence against H0 and in favor of Ha.
The probability, computed assuming H0 is true, that the statistic (p-hat or x-bar) would take a value as extreme as or more extreme than the one actually observed, in the direction specified by Ha.
small p-value
evidence against H0 bc they say that the observed result is unlikely to occur when H0 is true.
P-value small → reject H0 → convincing evidence for Ha (in context)
Large p-value
Fail to give convincing evidence against H0 and in favor of Ha bc they say that the observed result is likely to occur by chance alone when H0 is true.
P-value large → fail to reject H0 → not convincing evidence for Ha (in context)
reject H0
If our sample result is too unlikely to have happened by chance assuming H0 is true, we reject H0 and say there is convincing evidence for Ha.
fail to reject H0
Vice versa. Not convincing evidence for Ha.
statistically significant at level α
It's when our P-value is less than the chosen α in a sig. test.
In that case, we reject the H0 and conclude that there is convincing evidence for Ha.
P-value < α → reject H0 → convincing evidence for Ha (in context)
P-value ≥ α → fail to reject H0 → not convincing evidence for Ha (in context)
most commonly used sig. level
α = 0.05.
3 component of sig. test
1) an explicit comparison of the P-value to a stated significance level.
2) a decision about the null hypothesis: reject or fail to reject H0.
3) a statement in the context of the problem about whether or not there is convincing evidence for Ha.
Type I Error
we reject H0 when we shouldn't have. (H0 is true)
Type II Error
we fail to reject H0 when Ha is true
Significance and Type I Error
The significance level of α of any fixed-level test is the probability of Type I Error. That is, α is the probability that the test will reject the H0 when Ha is actually true.
Conditions for Performing a Sig. Test About a Proportion
* Random
* 10%
* Large counts (np0 and n(1-p0) are at least 10)
test statistic
Measures how far a sample statistics diverges from what we would expect if the null hypothesis H0 were true, in standard units.
(statistic - parameter)/ standard deviation of statistic
Significance Test: A four-step process
State: What hypothesis do you want to test, and at what sig. level? Define any parameters you use.
Plan: Choose the approp. inference method. Check conditions.
Do: If the conditions are met, perform calculations. Compute the test statistic, find the p-value.
Conclude: Make a decision about the hypothesis in context of the problem.
One-sample z test for a Proportion
Find the p-value by calculating the probability of getting a z statistic this large or larger in the direction specified by the Ha.
P-value in a one-sided test
the area in one tail of a standard Normal distribution- the tail specified by Ha.
use normal Cdf
P-value in a two-sided test
the area in both tails of a standard Normal distribution.
use 1-Prop- ZTest
Confidence Interval in sig. test
Gives more information; gives an approx. set of P0's that would not be rejected by a two-sided test at the ___ significance level.
Use when it's a two-sided test to find the actual proportion p or population μ because you can't find the actual value using the two-sided test.
A two-sided test at significance level α and a 100(1 − α)% confidence interval (a 95% confidence interval if α = 0.05) give similar information about the population parameter.
Power
"The power of a test against a specific alternative" is the probability that the test will CORRECTLY REJECT H0 at a chosen sig. level α when the specified alternative value of the parameter is true.
Power and Type II Error
The power of a test against any alternative is 1 minus the probability of a Type II error for that alternative; that is, power = 1 − β.
How to increase the power of a significance test to detect when H0 is false and Ha is true.
* Increase the sample size.
* Increase the significance level α.
* Increase the difference between the null and alternative parameter values that is important to detect
Conditions for Performing a Significance Test about a Mean
* Random
* 10%
* 1. population has a normal distribution. 2. n ≥ 30. 3. when data is sketched, no strong outliers or skewness.
One-Sample T Test for a Mean
Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha in the t distribution w/df = n-1.
paired data
Study designs that involve making two observations on the same individual or one observation on each of two similar individuals result in paired data.
paired t procedures
When paired data result from measuring the same quantitative variable twice, we can MAKE COMPARISONS BY ANALYZING THE DIFFERENCE IN EACH PAIR.
If the conditions for inferences are met, we can use "One-Sample TProcedures" to perform inference about the mean difference μd. These methods are sometimes called paired t procedures.
We (don't) have convincing evidence that the true mean difference ...(context)... is positive for subjects like these.
If you are not sampling...
don't need to check the 10% condition.
To determine a sample size
* Significance level (How much risk of a Type I Error - rejecting H0 when H0 is actually true- are we willing to accept? If a Type I error has serious consequences, we might opt for α = 0.01. Otherwise, we should choose α = 0.05 or α = 0.10. Recall that using a higher significance level would decrease the Type II error probability and increase the power.)
* Effect Size (How large a difference between the null parameter value and the actual parameter value is important for us to detect?)
* Power (What chance do we want our study to have to detect a difference of the size we think is important?)
a summary of influences on "How large a sample do I need?"
If you insist on a smaller significance level (such as 1% rather than 5%), you have to take a larger sample. A smaller significance level requires stronger evidence to reject the null hypothesis.
If you insist on higher power (such as 0.99 rather than 0.90), you will need a larger sample. Higher power gives a better chance of detecting a difference when it really exists.
At any significance level and desired power, detecting a small difference between the null and alternative parameter values requires a larger sample than detecting a large difference.
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