Exam 9: Inferences From Two Samples

Provide an appropriate response. -What is the effect on the P-value when a test is changed from two-tailed hypothesis with = and ? to one-tailed hypothesis such as $\geq$ and

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 -Suppose you wish to test the claim that $\mu _ { \mathrm { d } }$ , the mean value of the differences $\mathrm { d }$ for a population of paired data, is greater than 0 . Given a sample of $\mathrm { n } = 15$ and a significance level of $\alpha = 0.01$ , what criterion would be used for rejecting the null hypothesis?

Use the traditional method to test the given hypothesis. 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. At the 0.05 significance level, test the claim that the recognition rates are the same in both states.

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. -Consider the set of differences between two dependent sets: 84, 85, 83, 63, 61, 100, 98. Round to the nearest tenth. A)13.1 B)16.2 C)15.3 D)15.7

Compute the test statistic used to test the null hypothesis that p₁ = p₂ -A report on the nightly news broadcast stated that 13 out of 111 households with pet dogs were burglarized and 23 out of 190 without pet dogs were burglarized.

Suppose that you wish to perform a traditional hypothesis test to test a claim regarding two means. Give an example of a situation in which the matched pairs test would be appropriate and give an example of a situation in which it would be appropriate to perform a test for large and independent samples.

Use the traditional method to test the given hypothesis. Assume that the samples are independent and that they have been randomly selected -7 of 8,500 people vaccinated against a certain disease later developed the disease. 18 of 10,000 people vaccinated with a placebo later developed the disease. Test the claim that the vaccine is effective in lowering the incidence of the disease. Use a significance level of 0.02.

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. -x1=65,n1=90 and x2=79,n2=116,construct a 98% confidence interival for the difference berween population proportions $\mathrm { p } _ { 1 } - \mathrm { p } _ { 2 }$ .

Use the traditional method of hypothesis testing to test the given claim about the means of two populations. Assume that two dependent samples have been randomly selected from normally distributed populations. -Five students took a math test before and after tutoring. Their scores were as follows. Subject A B C D E Before 77 69 66 72 69 After 81 78 64 75 81 Using a 0.01 level of significance, test the claim that the tutoring has an effect on the math scores.

Solve the problem. -A confidence interval estimate of the ratio $\sigma \frac { 2 } { 1 } / \sigma \frac { 2 } { 2 }$ can be found using the following expression: $\left(\frac{\mathrm{s}_{1}^{2}}{\mathrm{~s}_{2}^{2}} \cdot \frac{1}{\mathrm{~F}_{\mathrm{R}}}\right)<\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}<\left(\frac{\mathrm{s}_{1}^{2}}{\mathrm{~s}_{2}^{2}} \cdot \frac{1}{\mathrm{~F}_{\mathrm{L}}}\right)$ where $\mathrm { F } _ { \mathrm { R } }$ is found in the standard way and $\mathrm { F } _ { \mathrm { L } }$ is found as follows: interchange the degrees of freedom, and then take the reciprocal of the resulting F value found in table A-5. A manager at a bank is interested in the standard deviation of the waiting times when a single waiting line is us when individual lines are used. Obtain a $95 \%$ confidence interval for $\sigma \frac { 2 } { 1 } / \sigma \frac { 2 } { 2 }$ given the following sample data: Sample 1: multiple waiting lines: $\mathrm { n } _ { 1 } = 13 , \mathrm {~s} _ { 1 } = 2.1$ minutes Sample 2: single waiting line: $\mathrm { n } _ { 2 } = 16 , \mathrm {~s} _ { 2 } = 1.2$ minutes

Solve the problem. -The sample size needed to estimate the difference between two population proportions to within a margin of error $\mathrm { E }$ with a confidence level of $1 - \alpha$ can be found as follows: in the expression $E = z _ { \alpha / 2} \sqrt { \frac { p _ { 1 } q _ { 1 } } { \mathrm { n } _ { 1 } } + \frac { \mathrm { p } _ { 2 } \mathrm { q } _ { 2 } } { \mathrm { n } _ { 2 } } }$ replace $\mathrm { n } _ { 1 }$ and $\mathrm { n } _ { 2 }$ by $\mathrm { n }$ (assuming both samples have the same size) and replace each of $\mathrm { p } _ { 1 } , \mathrm { q } _ { 1 } , \mathrm { p } _ { 2 }$ , and $\mathrm { q } _ { 2 }$ by $0.5$ (because their values are not known). Then solve for $n$ . Use this approach to find the size of each sample if you want to estimate the difference between the proportions . and women who plan to vote in the next presidential election. Assume that you want $99 \%$ confidence that your no more than $0.03$ .

A researcher wishes to determine whether listening to music affects students' performance on memory test. He randomly selects 50 students and has each student perform a memory test once while listening to music and once without listening to music. He obtains the mean and standard deviation of the 50 "with music" scores and obtains the mean and standard deviation of the 50 "without music scores". He then performs a hypothesis test for two means assuming large and independent samples. Is this approach appropriate? If not, how would you proceed.

Determine whether the following statement regarding the hypothesis test for two population proportions is true or false: However small the difference between two population proportions, for sufficiently large sample sizes, the null hypothesis of equal population proportions is likely to be rejected.

Construct the indicated confidence interval for the difference between the two population means. Assume that the two samples are independent and that they have been randomly selected. -Two types of flares are tested for their burning times (in minutes)and sample results are given below. Brand X Brand Y n=35 =40 =19.4 =15.1 =1.4 =0.8 Construct a $95 \%$ confidence interval for the differences $\mu _ { X } - \mu _ { Y }$ based on the sample data.

Solve the problem. -A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. A 95% confidence interval for the difference between the population means is (1.4, 8.7). If the hypothesis test is based on the same samples, which of the following do you know for sure: A: The hypothesis $\mu _ { 1 } = \mu _ { 2 }$ would be rejected at the $5 \%$ level of significance. B: The hypothesis $\mu _ { 1 } = \mu _ { 2 }$ would be rejected at the $10 \%$ level of significance. C: The hypothesis $\mu _ { 1 } = \mu _ { 2 }$ would be rejected at the $1 \%$ level of significance.

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 500 people aged $20 - 24,22 \%$ were smokers. In a random sample of 450 people aged $25 - 29,14 \%$ were smokers. Construct a $95 \%$ confidence interval for the difference between the population proportions $\mathrm { p } _ { 1 } - \mathrm { p } 2$ .

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 -1, 5, -4, 2. Round to the nearest tenth.

Provide an appropriate response. -Describe the process for testing hypothesis about two means when the samples are dependent. Compare this process to the methods of hypothesis testing for one mean in Chapter 7.

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. -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 5 percent level of significance. $\mathrm { H } _ { \mathrm { O } } : \mu _ { \mathrm { D } } = 0 \text { against } \mathrm { H } _ { 1 } : \mu _ { \mathrm { D } } \neq 0 \text {. }$ What is the value of the appropriate test statistic?

The two data sets are dependent. Find $\bar { d }$ to the nearest tenth. -A hypothesis test is to be performed to test the equality of two population means. The sample sizes are large and the samples are independent. Give an expression for the population standard deviation of the $\left( \bar { x } _ { 1 } - \bar { x } _ { 2 } \right)$ values in terms of $\mathrm { s } _ { 1 } , \mathrm {~s} _ { 2 } , \mathrm { n } _ { 1 }$ , and $\mathrm { n } _ { 2 }$ .