42 terms

AP Stats Chapter 9

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When is a confidence interval used?
When your goal is to estimate a population parameter
Significance test
A formal procedure for comparing observed data with a claim (also called hypothesis) whose truth we want to assess
Basic idea of significance test
An outcome that would rarely happen if were true is good evidence that the claim is not true
Null hypothesis (Ho)
The claim tested by a statistical test. The test is designed to assess the strength of evidence against the null hypothesis. Often the null hypothesis is a statement of "no difference"
Alternative Hypothesis (Ha)
A claim about the population for which we are trying to find evidence
What is the form of the null hypothesis
Ho: parameter=value
What are the possible forms of the alternative hypotheses
Ha: parameter<value
Ha: parameter>value
Ha: parameter≠value
When is the alternative hypothesis one-sided?
if it states that the parameter is larger than or smaller than the null hypothesis value
When is the alternative hypothesis two-sided?
If it states that the parameter is different from the null hypothesis value
Do hypotheses always refer to populations or samples?
Populations (p, μ, σ)
When is it correct to write a hypothesis about a sample statistic?
NEVER- do not use p-hat or x-bar
P-value
The probability the statistic would take a value as extreme or more extreme than the one actually observed. The smaller the p-value, the stronger the evidence against Ho provided by the data. The p-value is calculated assuming Ho is true
What are you assuming when calculating the p-value
That the null hypothesis is true (right until proven wrong)
What are small p-values evidence for
Evidence AGAINST Ho because they say that the observed result is unlikely to occur by chance when Ho is true
What do large p-values tell us?
They fail to give convincing evidence against Ho because they say that the observed result is likely to occur by chance when Ho is true
What are the 2 conclusions you can make?
Reject Ho OR fail to reject Ho
When do you reject Ho?
If our sample result is too unlikely to have happened by chance assuming Ho is true. (Reject when p-value is small). When you reject Ho you conclude Ha
When do you fail to reject Ho
When the p-value is large (in context). Any time when you aren't rejecting Ho, you fail to reject it, and therefore you cannot conclude Ha
Significance level
A fixed level that we regard as decisive. (Often a=0.05)
What do we do when our p-value is less than the chosen significance level (a)?
We say that the result is statistically significant - and we reject the null hypothesis Ho to conclude Ha
What do we do when our p-value is greater than the chosen significance level (a)?
We fail to reject Ho and cannot Ha - the result is not statistically significant
How do you find the probability of type I error?
This is the same as the significance level a
How do you find the probability of type II error?
The probability of type two error is ß, and can be found if you are given an alternate mean or proportion to prove
What is type I error?
When we conclude to reject Ho when Ho is true
What is Type II error?
When we conclude to fail to reject Ho when it is false.
What is power of a test?
The power of a test against a specific alternative is the probability that the test will reject Ho at a chosen significance level a when the specified alternative value of the parameter is true. ( probability of detecting a difference of the size you hope to find) you want higher power usually
How do you calculate power of a test?
Power=1-ß
Or 1-p(type II error)
How do you increase power?
Increasing significance level α, or invreasing sample size both increase power by reducing the probability of type II error
What are the conditions for inferencing proportions or means?
RANDOM- sample has to be random or experiment has to be randomly assigned
NORMAL- np>10, n(1-p)>10 or n>30 or population distribution is normal
INDEPENDENT- if sampling w/o replacement, n<1/10N or in an experiment we can assume that results of one subject do not affect another
*test statistic
Measures how far a sample statistic diverges from what we would expect if the null hypothesis Ho were true?
Test statistic for proportions
Z= p-hat - p/ sq rt( p(1-p)/n)
Z Test statistic for means
Z= x-bar-μ/sq rt (n)
What is a two sided significance test similar to?
A confidence interval- for example, a significance test at the 0.05 level is similar to a 95% confidence interval
Which test is used for proportions?
ALWAYS A Z-TEST: Population proportions and are based on z values from the standard normal distribution
Which test is used for means?
A z- test if the population standard deviation is given (usually not the case). If not, you use the sample standard deviation and use a t-distribution and test.
How do you find degrees of freedom and when do you use it?
Df=n-1, used for t distributions
Formula for a one sample t test
T= x-bar-μ/(Sx/sq rt(n))
When the alternative hypothesis has a [not equal to sign], what do you have to do with the probability?
Double it- as you need the probability for both tails
To perform a one sample t test, which conditions must be met?
NORMAL: Population distribution is normal or n>30
INDEPENDENT: population is at least 10 times as large as the sample
Will you get an exact value when using table B for t tests?
No- there will be a range
When we have paired data, which parameter are we looking at?
μd or mean difference- you must state how you subtract the two observations.
How do we test paired data?
If the conditions for inference are met, we can use one-sample t procedures to perform inference about the mean difference