HYPOTHESIS TEST AND CONFIDENCE INTERVALS FOR 1 AND …
HYPOTHESIS TEST AND CONFIDENCE INTERVALS FOR 1 AND 2 SAMPLE MEANS AND PROPORTIONS and VARIANCES (standard deviations)
Basic Formula: z(or t) = (non z(or t) of interest - mean )/ SD
Comment: for hypothesis test if given a level of significance first set up the rejection area(s) and find the edge(s) of it by using the z(or t) table
Comment: for hypothesis tests the z(or t) from the data is (column 3 – column 4)/column 5
Comment: the confidence interval for each case below is column 3 +/- column 2 * column 5
Comment: in any of the cases below if the population standard deviation(s) are known then use them and z
Comment: sample size needed for CI for mean is [pic], sample size for CI for proportion is [pic](use [pic]to guarantee sample size is large enough, use [pic]in place of [pic]to get reasonable estimate)
|Situation |z/t |Non z (or t) of interest |Mean |Standard Deviation or Standard Error |
|1 sample mean |t, df = n-1 |sample mean |population mean in Ho |[pic]=[pic] |
|Difference of means from |t, df = n-1 |sample mean of the |population difference |[pic]=[pic] Where s is the s.d. of the |
|2 dependent samples | |differences subtracted in|mean in Ho (often 0) |differences. |
|(a.k.a. matched pairs) | |appropriate order | | |
|Difference of means from |t, df = min of the sample|the difference of the two|difference of the |[pic] |
|2 independent samples |sizes - 1 |sample means subtracted |population means in Ho | |
| | |in appropriate order |(often 0) | |
|Difference of means from |t, df =[pic] |the difference of the two|difference of the |[pic] where [pic]=[pic] |
|2 independent samples | |sample means subtracted |population means in Ho | |
|with the assumption that | |in appropriate order |(often 0) | |
|the population standard | | | | |
|deviations are equal | | | | |
|1 sample proportion |z |sample proportion, |population proportion, p,|HT: [pic] CI: [pic] |
| | |p’=number of successes / |in Ho | |
| | |n | | |
|Difference of proportions|z |the difference of the two|difference in population |HT(0 case): [pic]where [pic][pic] |
|(percentages, or | |sample proportions |proportions in Ho (often |CI & HT(non 0 case): |
|probabilities of success)| |subtracted in appropriate|0) (0 and non 0 |[pic] |
|from 2 samples | |order |differences have | |
| | | |different standard | |
| | | |deviations see ( | |
| | | | | |
Confidence interval for population variance: Use [pic] , get two values from the [pic]table and solve for [pic]twice. (Take square root if you want [pic])
Hypothesis test for population variance: Get critical value(s) from [pic]table, and use [pic]where [pic]is from Ho.
Hypothesis test for ratio of two variances: Use [pic]maker sure the top is bigger than the bottom. Keep track of the two different df’s. Use the F-table for the critical value(s). Note that making F(data)>1 even if you have two critical values as in a two-tail test, only the right hand one matters.
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