
Comparing two population means (large sample size) 
Comparing two population means  large independent samples Comparing sample means of two independent samples with large sample size is similar to comparing a sample mean against a population mean (Chapter 7); the zscore or zstatistics for the standard normal distribution is used to evaluate tests. The only difference is the values for the parameters used in determining the statistics. The hypothesis testing involving two different means study the distribution of their differences:. 1. Know the basic general test statistics used for comparing two population means. If we have two populations or sample distributions the following basic
statistics can be obtained from each:
Large sample sizes studies use the standard normal zscore statistics and small sample size studies use the student t statistics. If we let
(
and
and
be a combined standard deviation for both sample distributions or data
sets, then
For large sample size the standard deviation and test statistics
are:
2. Know how to use appropriate statistics to test if two sample means are equal to each other or if their difference = 0 (large sample size). 3 Types of tests in comparing two sample means: When comparing the sample means, there are 3 questions to consider: Question 1:: Is ? H_{a} (Twotailed test) Question 2: : Is ? H_{a} (Righttailed test) Question 3: : Is ? H_{a} (Lefttailed test) Question 1:
Is
? H_{a} (Twotailed test)
By Examples: Problem 1. Two types of cars are compared for acceleration rate.
40 test runs are recorded for each car and the results for the mean elapsed
time recorded below:
Construct a 98% CI for the difference in the mean elapsed time for the two types of cars. Using this CI, determine if there is a difference in the mean elapsed times? Given difference , n = 40 (large so can use normal approximation of zscore). Step 1  Hypothesis: The claim that , the null hypothesis. The alternate hypothesis is that H_{0} : H_{a} : Step 2. Select level of significance: This is given as (2% = 100  98) So for twotailed test: Step 3. Test statistics and observed value.
Step 4. Determine the critical region (favors H_{a}) For alpha = 0.02 at both ends of intervals: 0.01 and 0.99, z_{a/2}
= 2.326 and z_{1a/2} = 2.326
The critical region is and (reference table) A 98% Confidence Interval for the difference is or (diff: 0.56 to 1.16) shown in graph above Step 5. Make decision. No not reject the null hypothesis if or The observed z = 0.81, and since 0.81 < 2.326 and is not in the critical region, we have no reason to reject H_{0} in favor of H_{a}. Note also that a difference of 0.75 is between the confidence interval of 0.56 and 1.16 the blue region for the null hypothesis acceptance. There the difference between both means are 0. Question 2:
: Is
? H_{a} (Righttailed test)
By Examples: Problem 2. The personnel officer of a large corporation claimed
that college graduates applying for jobs with their firm in the current
year tended to have higher grade point averages than those applying in
the previous year. Samples from the group of applicants gave the following
results:
Is there sufficient evidence to justify the claim at a 5% level of significance? Given difference , n >= 52 (large so can use normal approximation of zscore). Step 1  Hypothesis: The claim that , the null hypothesis. The alternate hypothesis is that H_{0} : H_{a} : Step 2. Select level of significance: This is given as (5%) Step 3. Test statistics and observed value.
Step 4. Determine the critical region (favors H_{a}) For alpha = 0.05 at the upper end of intervals: 0.95, z_{1a/2}
= 1.65
Step 5. Make decision. No not reject the null hypothesis if or Since 2.08 is > 1.65 and in the critical region (red) we reject the null hypothesis that grades are the same both years. The observed z = 2.08 and since 2.08 > 1.65 and is in the critical region, we reject H_{0} in favor of H_{a}. Therefore we conclude that college graduates from current year have higher grades than previous year. Question 3:
: Is ?
H_{a}
(Lefttailed test)
By Examples: Problem 3. A biologist suspected that males age 20  24 have
a lower mean systolic blood pressure than males in the same age group.
Independent random samples produced the following results for systolic
pressure.
Is there sufficient evidence to justify the claim at a 1% level of significance? Given difference , n >= 31 (large so can use normal approximation of zscore) Step 1  Hypothesis: The claim that , the null hypothesis. The alternate hypothesis is that H_{0} : H_{a} : Step 2. Select level of significance: This is given as (1%) Step 3. Test statistics and observed value.
Step 4. Determine the critical region (favors H_{a}) For alpha = 0.01 at the upper end of intervals: 0.99, z_{a/2}
= 2.326
The critical region is (reference table) Step 5. Make decision. No not reject the null hypothesis if Since 2.56 is < 2.326 and in the critical region (red) we reject the null hypothesis that female and male systolic pressure are the same.. The observed z = 2.56 and since 2.56 < 2.326 and is in the critical region, we reject H_{0} in favor of H_{a}. Therefore we conclude that female systolic pressure are lower than male's same age (2024)
