Exam 13: Inference About Comparing Two Populations

In testing the hypothesis $H _ { 0 } : \mu _ { D } = 5$ vs. $H _ { 1 } : \mu _ { D } > 5$ , two random samples from two normal populations produced the following statistics: $n _ { D } = 20 , \bar { x } _ { D } = 9 \text {, and } s _ { D } = 7.5$ . What conclusion can we draw at the 1% significance level?

NARRBEGIN: Engine Wear Engine Wear To compare the wearing of two types of automobile engines, 1 and 2, an experimenter chose to "pair" the measurements, comparing the wear for the two types of engines on each of 7 automobiles, as shown below. Autamobile 1 2 3 4 5 6 7 Engine 1 15 7 9 10 13 11 Engine 2 12 18 8 9 12 11 10 NARREND -(Engine Wear Narrative} Determine whether these data are sufficient to infer at the 10% significance level that the two types of engines wear differently.

The test for the equality of two population variances assumes that each of the two populations is normally distributed.

The F-distribution is the sampling distribution of the ratio of:

NARRBEGIN: Profit Margin Profit Margin An investor is considering two types of investment. She is quite satisfied that the expected profit margin on Investment 1 is higher than the expected profit margin on Investment 2. However, she is quite concerned that the risk associated with Investment 1 is higher than that of Investment 2. To help make her decision, she randomly selects seven monthly profit margins on investment 1 and ten monthly profit margins on investment 2. She finds that the sample variances of Investments 1 and 2 are 225 and 118, respectively.NARREND -{Profit Margin Narrative} Estimate with 95% confidence the ratio of the two population variances.

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} What is the p-value of the test?

NARRBEGIN: Mass Production Line Mass Production Line A quality control examiner keeps a tally sheet of the number of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. Froducts Mass Production Line Acceptable Unacceptable 1 152 48 2 136 54 NARREND -{Mass Production Line Narrative} What is the p-value of the test? Explain how to use it for testing the hypotheses.

In testing the difference between two population means using two independent samples, we use the pooled variance in estimating the standard error of the sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ if:

NARRBEGIN: Mass Production Line Mass Production Line A quality control examiner keeps a tally sheet of the number of acceptable and unacceptable products that come off two different production lines. The completed sheet is shown below. Froducts Mass Production Line Acceptable Unacceptable 1 152 48 2 136 54 NARREND -{Mass Production Line Narrative} Can the inspector infer at the 5% significance level that production line 1 is doing a better job than production line 2?

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} Can the consultant conclude at the 5% significance level that the husband's estimate is higher than the wife's estimate?

If there are 10 pairs of data in a matched pairs experiment, the degrees of freedom for the corresponding t-test is 18.

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 -A behaviorist has performed the following experiment. For each of 10 sets of identical twins who were born 30 years ago, he recorded their annual incomes, according to which twin was born first. The results (in $1,000s) are shown below. Can he infer at 5% significance level that there is a difference in income between the twins? Twin Set First Born Second Born 1 32 44 2 36 43 3 21 28 4 30 39 5 49 51 6 27 25 7 39 32 8 38 42 9 56 64 10 44 44

The unequal-variances test statistic of $\mu _ { 1 } - \mu _ { 2 }$ has an approximate ____________________ distribution with n₁ + n₂ - 2 degrees of freedom.

Unless we can conclude that the population variances are equal, we cannot use the pooled variance estimate.

In testing for the equality of two population variances, when the populations are normally distributed, the 10% level of significance has been used. To determine the rejection region, it will be necessary to refer to the F table corresponding to an upper-tail area of:

The best estimator of the difference between two population means $\mu _ { 1 } - \mu _ { 2 }$ is the difference between two sample means $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ .

When testing for the difference between two population variances with sample sizes of n₁ = 8 and n₂ = 10, the degrees of freedom are:

A political analyst in Iowa surveys a random sample of registered Republicans and compares the results with those obtained from a random sample of registered Democrats . This would be an example of two independent samples.

A test is being conducted to test the difference between two population means using data that are gathered from a matched pairs experiment. If the paired differences are normal, then the distribution used for testing is the:

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} Explain how to use the interval estimate to test the hypotheses.