DOC PREVIEW
STEVENS MA 331 - Lecture 5 Two Population Tests of Means and Proportions

This preview shows page 1-2-15-16-31-32 out of 32 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 32 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Lecture 5 Two population tests of Means and Proportions Section 7 2 objectives z Comparing two means Two sample z distribution Two independent samples t distribution Two sample t test Two sample t confidence interval Robustness Details of the two sample t procedures Comparing two samples A Population 1 Population 2 Sample 2 Sample 1 Which is it B Population We often compare two treatments used on independent samples Sample 2 Sample 1 Is the difference between both treatments due only to variations from the random sampling B Independent samples Subjects in one samples are completely unrelated to subjects in the other sample or does it reflects a true difference in population means A Two sample z distribution std dev is assumed to be known We have two independent SRSs simple random samples coming maybe from two distinct populations with 1 1 and 2 2 We use and x1 x 2 to estimate the unknown 1 and 2 When both populations are normal the sampling distribution of x 1 x2 12 22 is also normal with standard deviation n1 n2 Then the two sample z statistic has the standard normal N 0 1 sampling distribution z x1 x 2 1 2 12 n1 22 n2 Inference Two Populations Known Population Standard Deviations Alternative Hypothesis P value Ha 1 2 P Z z Ha 1 2 P Z z Ha 1 2 2 P Z z Two independent samples t distribution the population s std dev is not known We have two independent SRSs simple random samples coming maybe from two distinct populations with 1 1 and 2 2 unknown We use x1 s1 and x2 s2 to estimate 1 1 and 2 2 respectively To compare the means both populations should be normally distributed However in practice it is enough that the two distributions have similar shapes and that the sample data contain no strong outliers The two sample t statistic follows approximately the t distribution with a standard error SE spread reflecting variation from both samples SE s12 s22 n1 n 2 Conservatively the degrees of freedom is equal to the df smallest of n1 1 n2 1 s12 s22 n1 n2 1 2 x1 x 2 Two sample t test The null hypothesis is that both population means 1 and 2 are equal thus their difference is equal to zero H0 1 2 1 2 0 with either a one sided or a two sided alternative hypothesis We find how many standard errors SE away from 1 2 is x1 x 2 by standardizing with t Because in a two sample test H0 is 1 2 0 we simply use With df smallest n1 1 n2 1 x1 x2 1 2 t SE t x1 x 2 s s n1 n 2 2 1 2 2 Does smoking damage the lungs of children exposed to parental smoking Forced vital capacity FVC is the volume in milliliters of air that an individual can exhale in 6 seconds FVC was obtained for a sample of children not exposed to parental smoking and a group of children exposed to parental smoking Parental smoking FVC Yes No x s n 75 5 9 3 30 88 2 15 1 30 We want to know whether parental smoking decreases children s lung capacity as measured by the FVC test Is the mean FVC lower in the population of children exposed to parental smoking H0 smoke no smoke no 0 Ha smoke no smoke no 0 one sided The difference in sample averages follows approximately the t distribution t 0 2 2 ssmoke sno n smoke n no df 29 We calculate the t statistic t xsmoke xno 2 2 ssmoke sno nsmoke nno Parental smoking 75 5 88 2 9 32 15 12 30 30 12 7 3 9 t 2 9 7 6 FVC x s n Yes 75 5 9 3 30 No 88 2 15 1 30 In table C for df 29 we find t 3 659 p 0 0005 one sided It s a very significant difference we reject H0 Lung capacity is significantly impaired in children of smoking parents Two sample t confidence interval Because we have two independent samples we use the difference between both sample averages x 1 x2 to estimate 1 2 Practical use of t t z C is the area between t and t z We find t in the line of Table C s12 s22 n1 n 2 SE for df smallest n1 1 n2 1 and the column for confidence level C z C The margin of error m is m s12 s22 m t t SE n1 n2 t m t Common mistake A common mistake is to calculate a one sample confidence interval for 1 and then check whether 2 falls within that confidence interval or vice versa This is WRONG because the variability in the sampling distribution for two independent samples is more complex and must take into account variability coming from both samples Hence the more complex formula for the standard error SE s12 s22 n1 n2 Can directed reading activities in the classroom help improve reading ability A class of 21 third graders participates in these activities for 8 weeks while a control classroom of 23 third graders follows the same curriculum without the activities After 8 weeks all children take a reading test scores in table 95 confidence interval for 1 2 with df 20 conservatively t 2 086 s12 s22 CI x1 x2 m m t 2 086 4 31 8 99 n1 n2 With 95 confidence 1 2 falls within 9 96 8 99 or 1 0 to 18 9 Details of the two sample t procedures The true value of the degrees of freedom for a two sample tdistribution is quite lengthy to calculate That s why we use an approximate value df smallest n1 1 n2 1 which errs on the conservative side often smaller than the exact Computer software though gives the exact degrees of freedom or the rounded value for your sample data s12 s22 2 n1 n2 df 2 2 2 2 1 s2 1 s1 n1 1 n1 n2 1 n2 95 confidence interval for the reading ability study using the more precise degrees of freedom Table C t Test Two Sample Assuming Unequal Variances Treatment group Control group Mean 51 476 41 522 Variance 121 162 294 079 Observations 21 23 Hypothesized Mean Difference df 38 t Stat 2 311 P T t one tail 0 013 t Critical one tail 1 686 P T t two tail 0 026 t Critical two tail 2 024 t s12 s22 m t n1 n2 m 2 024 4 31 8 72 SPSS Independent Samples Test Levene s Test for Equality of Variances F Reading Score Equal variances assumed Equal variances not assumed 2 362 Excel Sig 132 t test for Equality of Means t df Sig 2 tailed Mean Difference Std Error Difference 95 Confidence Interval of the Difference Lower Upper 2 267 42 029 9 95445 4 39189 1 09125 18 81765 2 311 37 855 026 9 95445 4 30763 1 23302 18 67588 …


View Full Document

STEVENS MA 331 - Lecture 5 Two Population Tests of Means and Proportions

Download Lecture 5 Two Population Tests of Means and Proportions
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture 5 Two Population Tests of Means and Proportions and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture 5 Two Population Tests of Means and Proportions 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?