Exam 13: Inference About Comparing Two Populations

A matched pairs experiment decreases variability (compared to two independent samples).

If you are testing to see if a weight loss program is working, and you subtract the weights after - before for a group of 10 people, the alternative hypothesis is that the mean difference is ____________________ 0.

NARRBEGIN: Starting Salary Starting Salary In testing the hypotheses $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = 0$ vs. $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } \neq 0$ , two random samples from two populations of college of business graduates majoring in global marketing and international business produced the following statistics regarding their starting salaries (in $1000s): $n _ { 1 } = 50$ , $\bar { x } _ { 1 } = 35$ , $s _ { 1 } = 28$ , $n _ { 2 } = 50$ , $\bar { x } _ { 2 } = 30$ , and $s _ { 2 } = 10$ . (Assume the salaries have normal distributions.) NARREND -The service manager of a car dealer wants to determine if owners of new cars (two years old or less) tune up their cars more frequently than owners of older cars (more than two years old). From his records he takes a random sample of ten new cars and ten older cars and determines the number of times the cars were tuned up in the last 12 months. The data follow. Do these data allow the service station owner to infer at the 10% significance level that new car owners tune up their cars more frequently than older car owners? Frequency of Tunsups in past 12 Month New Car Owmers Old Cars Owners 6 4 3 2 3 1 3 2 4 3 3 2 6 2 5 3 5 2 4 1

The F-distribution can only have non-negative values.

Suppose we randomly selected 250 people, and on the basis of their responses to a survey we assigned them to one of two groups: high-risk group and low-risk group. We then recorded the blood pressure for the members of each group. Such data are called:

The expected value of the difference of two sample means equals the difference of the corresponding population means when:

When comparing two population variances, we use the ratio $\sigma _ { 1 } ^ { 2 } / \sigma _ { 2 } ^ { 2 }$ rather than the difference $\sigma _ { 1 } ^ { 2 } - \sigma _ { 2 } ^ { 2 }$ .

The sampling distribution of the ratio of two sample variances $s _ { 1 } ^ { 2 } / s _ { 2 } ^ { 2 }$ is said to be F-distributed provided that:

The expected value of the difference between two sample proportions is the difference between their corresponding population proportions.

NARRBEGIN: Starting Salary Starting Salary In testing the hypotheses $H _ { 0 } : \mu _ { 1 } - \mu _ { 2 } = 0$ vs. $H _ { 1 } : \mu _ { 1 } - \mu _ { 2 } \neq 0$ , two random samples from two populations of college of business graduates majoring in global marketing and international business produced the following statistics regarding their starting salaries (in $1000s): $n _ { 1 } = 50$ , $\bar { x } _ { 1 } = 35$ , $s _ { 1 } = 28$ , $n _ { 2 } = 50$ , $\bar { x } _ { 2 } = 30$ , and $s _ { 2 } = 10$ . (Assume the salaries have normal distributions.) NARREND -{Starting Salary Narrative} What conclusion can we draw at the 5% significance level?

The test for the mean difference in a matched pairs design requires the differences to have a(n) ____________________ distribution.

Independent samples are those for which the selection process for one is not related to the selection process for the other.

Two independent samples of sizes 20 and 30 are randomly selected from two normally distributed populations. Assume that the population variances are unknown but equal. In order to test the difference between the population means, $\mu _ { 1 } - \mu _ { 2 }$ , the sampling distribution of the sample mean difference, $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ , is:

NARRBEGIN: Undergraduates' Test Scor Undergraduates' Test Scores 35 undergraduate students who completed two years of college were asked to take a basic mathematics test. The mean and standard deviation of their scores were 75.1 and 12.8, respectively. In a random sample of 50 students who only completed high school, the mean and standard deviation of the test scores were 72.1 and 14.6, respectively.NARREND -{Undergraduates' Test Scores Narrative} Can we infer at the 10% significance level that a difference exists between the two groups?

NARRBEGIN: Clothing Expenditures Clothing Expenditures A marketing consultant was in the process of studying the perceptions of married couples concerning their monthly clothing expenditures. He believed that the husband's perception would be higher than the wife's. To judge his belief, he takes a random sample of ten married couples and asks each spouse to estimate the family clothing expenditure (in dollars) during the previous month. The data are shown below. Couple Husband Wife 1 380 270 2 280 300 3 215 185 4 350 320 5 210 180 6 410 390 7 250 250 8 360 320 9 180 170 10 400 330 NARREND -{Clothing Expenditures Narrative} Briefly describe what the interval estimate tells you.

Two measurements from the same individuals is an example of data collected from a matched pairs experiment.

NARRBEGIN: Senatorial Election Senatorial Election A political poll immediately prior to a senatorial election reveals that 145 out of 250 male voters and 105 out of 200 female voters intend to vote for the Democrat candidate.NARREND -{Senatorial Election Narrative} Explain how to use the interval estimate to test the hypotheses.

In comparing the difference in means with a matched pairs experiment, the variable under consideration is $\bar { x } _ { D }$ , where the subscript D refers to the difference.

When two population variances are ____________________ we estimate each population variance with its sample variance. The test statistic of $\mu _ { 1 } - \mu _ { 2 }$ is approximately Student t-distributed with n₁ + n₂ - 2 degrees of freedom.

The statistical distribution used for testing the difference between two population variances is the