Question

A political analyst was curious if younger adults were becoming more conservative. He decided to see if the mean age of registered Republicans was lower than that of registered Democrats. He selected an SRS of 128 registered Republicans from a list of registered Republicans and determined the mean age to be = 39 years, with a standard deviation s1 = 8 years. He also selected an independent SRS of 200 registered Democrats from a list of registered Democrats and determined the mean age to be = 40 years, with a standard deviation s2 = 10 years. Let μ1 and μ2 represent the mean ages of the populations of all registered Republicans and Democrats, respectively. Suppose that the distributions of age in the populations of registered Republicans and of registered Democrats have the same standard deviation. Assume the pooled two-sample t procedures are safe to use. Reference: Ref 7-9 Suppose the researcher had wished to test the hypotheses H0: μ1 = μ2, Ha: μ1 < μ2. The P-value for the pooled two-sample t test is?

between 0.10 and 0.05? Less than 0.01? Between 0.05 and 0.01? Greater than 0.10? Please show the work thank you!!! I'd like to know how this should work :)

Answer 1

Republicans: Sample 1

Sample size : n₁ = 128

Sample age : = 39

Sample standard deviation : s₁ = 8

Democrats : Sample 2

Sample size : n₂ = 200

Sample age : = 40

Sample standard deviation : s₂ = 10

H0: μ1 = μ2, Ha: μ1 < μ2

Left Tailed test :

p-value for left tailed test :

For 326 degrees of freedom ; p(t<-0.9528) = 0.1707

p-value = 0.1707

The P-value for the pooled two-sample t test is  Greater than 0.10

Using t-table : critical values:

Generally critical values for 0.01, 0.05, 0.10 etc are given;

i.e

Critical value for 0.10 means : P(t > tstat) = 0.10  by symmetry it's equivalent to P(t < -t-stat)=0.10

Also for greater degrees of freedom > 100; we may take the row (1000: degrees of freedom);

from the tables : critical value for 0.10 (for 1000 degrees of freedom) = 1.282 i.e

P(t<-1.282) = 0.10; Therefore we can conclude that; P(t<-0.9528) > P(t<-1.282);

p-value = P(t<-0.9528) >   P(t<-1.282)=0.10

p-value > 0.10

The P-value for the pooled two-sample t test is?

Greater than 0.10