Exam 9: Inferences From Two Samples

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal - =408 =145 =88 =90

Find the appropriate p-value to test the null hypothesis, H₀: p₁ = p₂, using a significance level of 0.05. -A random sampling of sixty pitchers from the National League and fifty-two pitchers from the American League showed that 16 National and 11 American League pitchers had E.R.A's below 3.5.

A random sample of 16 women resulted in blood pressure levels with a standard deviation of 22.1 mm Hg. A random sample of 17 men resulted in blood pressure levels with a standard deviation of 20.8 mm Hg. Use a 0.025 significance level to test the claim that blood pressure levels for women have a larger variance than those for men.

Find the appropriate p-value to test the null hypothesis, H₀: p₁ = p₂, using a significance level of 0.05. - =50 =75 =20 =15

The two data sets are dependent. Find $\bar { d }$ to the nearest tenth. - 56 59 55 63 51 27 25 21 25 22

Solve the problem. -When performing a hypothesis test for the ratio of two population variances, the upper critical $\mathrm { F }$ value is denoted $\mathrm { F } _ { \mathrm { R } }$ . The lower critical $\mathrm { F }$ value, $\mathrm { F } _ { \mathrm { L } }$ , can be found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting $\mathrm { F }$ value found in table $\mathrm { A } - 5$ . $\mathrm { F } _ { \mathrm { R } }$ can be denoted $\mathrm { F } _ { \alpha / 2 }$ and $\mathrm { F } _ { \mathrm { L } }$ can be denoted $\mathrm { F } _ { 1 - \alpha / 2 }$ Find the critical values $\mathrm { F } _ { \mathrm { L } }$ and $\mathrm { F } _ { \mathrm { R } }$ for a two-tailed hypothesis test based on the following values: $n _ { 1 } = 4 , n _ { 2 } = 8 , \alpha = 0.05$

From the sample statistics, find the value of $\bar{ p}$ used to test the hypothesis that the population proportions are equal -Among 900 people selected randomly from among the residents of one city, 14.33% were found to be living below the official poverty line.

Determine whether the samples are independent or consist of matched pairs. -The effectiveness of a headache medicine is tested by measuring the intensity of a headache in patients before and after drug treatment. The data consist of before and after intensities for each patient.

Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test; i.e., describe the values of the test statistic that would result in rejection of the null hypothesis -A farmer has decided to use a new additive to grow his crops. He divided his farm into 10 plots and kept records of the corn yield (in bushels)before and after using the additive. The results are shown below. Plot: 1 2 3 4 5 6 7 8 9 10 Before 9 9 8 7 6 8 5 9 10 11 After 10 9 9 8 7 10 6 10 10 12 You wish to test the following hypothesis at the 1 percent level of significance. $\mathrm { H } _ { 0 } : \mu _ { \mathrm { D } } = 0$ against $\mathrm { H } _ { 1 } : \mu _ { \mathrm { D } } \neq 0$ What decision rule would you use?

Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected. -A marketing survey involves product recognition in New York and California. Of 558 New Yorkers surveyed, 193 knew the product while 196 out of 614 Californians knew the product. Construct a 99% confidence interval for the difference between the two population proportions.

The two data sets are dependent. Find $\bar { d }$ to the nearest tenth. - 8.3 5.3 7.9 8.5 6.8 5.7 8.1 7.5 9.5 7.7 8.1 9.3

Solve the problem. -To test the null hypothesis that the difference between two population proportions is equal to a nonzero constant c, use the test statistic $\mathrm{z}=\frac{\left(\hat{\mathrm{p}}_{1}-\hat{\mathrm{p}}_{2}\right)-\mathrm{c}}{\sqrt{\hat{p}_{1}\left(1-\hat{p_{1}}\right) / \mathrm{n}_{1}+\hat{p_{2}\left(1-\hat{p_{2}}\right) / \mathrm{n}_{2}}}}$ As long as $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ are both large, the sampling distribution of the test statistic $\mathrm { z }$ will be approximately the standard normal distribution. Given the sample data below, test the claim that the proportion of male voters who plan to vote Republican at the next presidential election is 15 percentage points more than the percentage of female voters who plan to vote Republican. Use the P-value method of hypothesis testing and use a significa level of $0.10$ . Men: $\mathrm { n } _ { 1 } = 250 , \mathrm { x } _ { 1 } = 146$ Women: $\mathrm { n } _ { 2 } = 202 , \mathrm { x } _ { 2 } = 103$

Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected -A researcher finds that of 1,000 people who said that they attend a religious service at least once a week, 31 stopped to help a person with car trouble. Of 1,200 people interviewed who had not attended a religious service at least once a month, 22 stopped to help a person with car trouble. At the 0.05 significance level, test the claim that the two proportions are equal.

Provide an appropriate response. -How does finding the error estimate and confidence intervals for dependent samples compare to the methods for one mean from Chapter 7?

Solve the problem. -When performing a hypothesis test for the ratio of two population variances, the upper critical $\mathrm { F }$ value is denoted $\mathrm { F } _ { \mathrm { R } }$ . The lower critical $\mathrm { F }$ value, $\mathrm { F } _ { \mathrm { L } }$ , can be found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting $\mathrm { F }$ value found in table $\mathrm { A } - 5$ . $\mathrm { F } _ { \mathrm { R } }$ can be denoted $\mathrm { F } _ { \alpha / 2 }$ and $\mathrm { F } _ { \mathrm { L } }$ can be denoted $\mathrm { F } _ { 1 } - \alpha / 2$ . Find the critical values $F _ { L }$ and $F _ { R }$ for a two-tailed hypothesis test based on the following values: $\mathrm { n } _ { 1 } = 10 , \mathrm { n } _ { 2 } = 16 , \alpha = 0.05$

Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected. -In a random sample of 300 women, $50 \%$ favored stricter gun control legislation. In a random sample of 200 men, $26 \%$ favored stricter gun control legislation. Construct a $98 \%$ confidence interval for the difference between the population proportions $\mathrm { p } 1$ - $\mathrm { p } 2$ .

A researcher wishes to compare how students at two different schools perform on a math test. He randomly selects 40 students from each school and obtains their test scores. He pairs the first score from school A with the first school from school B, the second score from school A with the second school from school B and so on. He then performs a hypothesis test for matched pairs. Is this approach valid? Why or why not? If it is not valid, how should the researcher have proceeded?

Assume that you want to test the claim that the paired sample data come from a population for which the mean difference is $\mu _ { \mathrm { d } } = 0$ . Compute the value of the $t$ test statistic. -The differences between two sets of dependent data are 3, 21, 3, 27, 15. Round to the nearest tenth.

Solve the problem. -A manager at a bank is interested in the standard deviation of the waiting times when a single waiting line is used and when individual lines are used. He wishes to test the claim that the population standard deviation for waiting times when multiple lines are used is greater than the population standard deviation for waiting times when a single line is used. Find the P-value for a test of this claim given the following sample data. You won't be able to find the exact P-value, but will be able to give a range of possible values. Sample 1: multiple waiting lines: $\mathrm { n } _ { 1 } = 13 , \mathrm {~s} _ { 1 } = 2.1$ minutes Sample 2: single waiting line: $\mathrm { n } _ { 2 } = 16 , \mathrm {~s} _ { 2 } = 1.1$ minutes

Construct the indicated confidence interval for the difference between population proportions p1 - p2. Assume that the samples are independent and that they have been randomly selected. - $\mathrm { x } _ { 1 } = 27 , \mathrm { n } _ { 1 } = 67$ and $\mathrm { x } _ { 2 } = 27 , \mathrm { n } _ { 2 } = 72 ;$ Construct a $95 \%$ confidence interval for the difference between population proportions $\mathrm { P } 1 - \mathrm { P } 2$ .