Exam 9: Inferences Based on a Two Samples: Confidence Intervals and Tests of Hypotheses

A marketing study was conducted to compare the mean age of male and female purchasers of a certain product. Random and independent samples were selected for both male and female purchasers of the product. What type of analysis should be used to compare the mean age of male and female purchasers?

Which of the following represents the difference in two population proportions? A) $p _ { 1 } - p _ { 2 }$ B) $\mu _ { 1 } - \mu _ { 2 }$ C) $\frac { \sigma _ { 1 } ^ { 2 } } { \sigma _ { 2 } ^ { 2 } }$ D) $p _ { 1 } + p _ { 2 }$

An inventor has developed a new spray coating that is designed to improve the wear of bicycle tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: 1 1.452 0.785 Bicycle Coated Tire (C) Non-Coated Tire () 1 1.452 0.785 2 1.634 0.844 \downarrow \downarrow \downarrow 50 1.211 0.954 Coated Non-Coated Difference Mean 1.38 0.85 0.53 Std. Dev. 0.12 0.11 0.06 Sample Size 50 50 50 Identify the correct null and alternative hypothesis for testing whether the new spray coating improves the mean wear of the bicycle tires (which would restult in a larger amount of tread left on the tire). A) $H _ { 0 } : \mu _ { \mathrm { d } } = 0$ vs. $\mathrm { H } _ { \mathrm { a } } : \mu _ { \mathrm { d } } < 0$ B) $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 0$ vs. $\mathrm { H } _ { \mathrm { a } } : \mu _ { \mathrm { d } } > 0$ C) $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 0$ vs. $\mathrm { H } _ { \mathrm { a } } : \mu _ { \mathrm { d } } \neq 0$

A paired difference experiment produced the following results. $n _ { d } = 40 , \bar { x } _ { 1 } = 18.4 , \bar { x } _ { 2 } = 19.7 , \bar { d } = - 1.3 , s _ { d } ^ { 2 } = 5$ Perform the appropriate test to determine whether there is sufficient evidence to conclude that $\mu 1 < \mu 2$ using $\alpha = .10$ .

When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Suppose 45 of the females and 60 of the males responded that they were able to give blood. Find the test statistic that would be used if it is desired to test to determine if a difference exists between the proportion of the females and males who responds to the call to donate blood.

An inventor has developed a new spray coating that is designed to improve the wear of bicycle tires. To test the new coating, the inventor randomly selects one of the two tires on each of 50 bicycles to be coated with the new spray. The bicycle is then driven for 100 miles and the amount of the depth of the tread left on the two bicycle tires is measured (in millimeters). It is desired to determine whether the new spray coating improves the wear of the bicycle tires. The data and summary information is shown below: 1 1.452 0.785 Bicycle Coated Tire () Non-Coated Tire (N) 1 1.452 0.785 2 1.634 0.844 \downarrow \downarrow \downarrow 50 1.211 0.954 Coated Non-Coated Difference Mean 1.38 0.85 0.53 Std. Dev. 0.12 0.11 0.06 Sample Size 50 50 50 Use the summary data to construct a $90 \%$ confidence interval for the difference between the means. A) $0.53 \pm 0.01663$ B) $0.53 \pm 0.04512$ C) $0.53 \pm 0.01396$ D) $0.53 \pm 0.03787$

Independent random samples selected from two normal populations produced the following sample means and standard deviations. Sample 1 Sample 2 =14 =11 =7.1 =8.4 =2.3 =2.9 Find and interpret the $95 \%$ confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ .

In order for the results of a paired difference experiment to be unbiased, the experimental units in each pair must be chosen independently of one another.

Independent random samples, each containing 1,000 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 475 and 550 successes, respectively. Test $H _ { 0 }$ : ( $p 1$ $\left. - p _ { 2 } \right) = 0$ against $H _ { \mathrm { a } } : \left( p _ { 1 } - p _ { 2 } \right) \neq 0$ . Use $\alpha = .01$ .

Determine whether the sample sizes are large enough to conclude that the sampling distributions are approximately normal. $n _ { 1 } = 45 , n _ { 2 } = 52 , \hat { p } _ { 1 } = .3 , \hat { p } _ { 2 } = .6$

Find F.05 where $v _ { 1 } = 8 \text { and } v _ { 2 } = 11$

Calculate the degrees of freedom associated with a small-sample test of hypothesis for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ , assuming $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } \text { and } n _ { 1 } = n _ { 2 } = 16$

One indication of how strong the real estate market is performing is the proportion of properties that sell in less than 30 days after being listed. Of the condominiums in a Florida beach community that sold in the first six months of 2006, 75 of the 115 sampled had been on the market less than 30 days. For the first six months of 2007, 25 of the 85 sampled had been on the market less than 30 days. Test the hypothesis that the proportion of condominiums that sold within 30 days decreased from 2006 to 2007. Use α = .01.

Suppose it desired to compare two physical education training programs for preadolescent girls. A total of 82 girls are randomly selected, with 41 assigned to each program. After three 6-week periods on the program, each girl is given a fitness test that yields a score between 0 and 100. The means and variances of the scores for the two groups are shown in the table. n Program 1 41 78.5 201.7 Program 2 41 75.8 259.6 Test to determine if the variances of the two programs differ. Use \alpha=.05 .

Specify the appropriate rejection region for testing $\mathrm { H } _ { 0 : } : \sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ in the given situation. - $\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 } ^ { 2 } < \sigma \underset { 2 } { 2 } ; \alpha = 0.01 , \mathrm { n } _ { 1 } = 16 , \mathrm { n } _ { 2 } = 31$ A) F $> 3.21$ B) $F > 2.70$ C) $F > 2.64$ D) $| \mathrm { F } | > 3.21$

When blood levels are low at an area hospital, a call goes out to local residents to give blood. The blood center is interested in determining which sex - males or females - is more likely to respond. Random, independent samples of 60 females and 100 males were each asked if they would be willing to give blood when called by a local hospital. A success is defined as a person who responds to the call and donates blood. The goal is to compare the percentage of the successes of the male and female responses. Find the rejection region that would be used if it is desired to test to determine if a difference exists between the proportion of the females and males who responds to the call to donate blood. Use α = 0.10. A) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { z } > 1.645$ . B) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { z } < - 1.96$ . C) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { z } < - 1.645$ or $\mathrm { z } > 1.645$ . D) Reject $\mathrm { H } _ { 0 }$ if $\mathrm { z } < - 1.96$ or $\mathrm { z } > 1.96$ .

We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarkets and recording the price of a six-pack of each brand of cola at each supermarket. The data are shown in the following table: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Price }$ Supermarket Brand 1 Brand 2 Difference 1 \ 2.25 \ 2.30 \ -0.05 2 2.47 2.45 0.02 3 2.38 2.44 -0.06 4 2.27 2.29 -0.02 5 2.15 2.25 -0.10 6 2.25 2.25 0.00 7 2.36 2.42 -0.06 8 2.37 2.40 -0.03 =2.3125 =2.3500 =-0.0375 =0.1007 =0.0859 =0.0381 If the problem above represented a paired difference, what assumptions are needed for a confidence interval for the mean difference to be valid?

In order to compare the means of two populations, independent random samples of 225 observations are selected from each population with the following results. Sample 1 Sample 2 =478 =481 =14.2 =11.2 Test the null hypothesis $H _ { 0 } : \left( \mu _ { 1 } - \mu _ { 2 } \right) = 0$ against the alternative hypothesis $H _ { \mathrm { a } } : \left( \mu _ { 1 } - \mu _ { 2 } \right) \neq 0$ using $\alpha = .10$ . Give the significance level, and interpret the result. Give the significance level, and interpret the result. 4 Conduct Pooled Hypothesis Test

A certain manufacturer is interested in evaluating two alternative manufacturing plans consisting of different machine layouts. Because of union rules, hours of operation vary greatly for this particular manufacturer from one day to the next. Twenty-eight random working days were selected and each plan was monitored and the number of items produced each day was recorded. Some of the collected data is shown below: DAY PLAN 1 OUTPUT PLAN 2 OUTPUT 1 1234 units 1311 units 2 1355 units 1366 units 3 1300 units 1289 units What assumptions are necessary for the above test to be valid?

Which supermarket has the lowest prices in town? All claim to be cheaper, but an independent agency recently was asked to investigate this question. The agency randomly selected 100 items common to each of two supermarkets (labeled A and B) and recorded the prices charged by each supermarket. The summary results are provided below: =2.09 =1.99 =.10 =0.22 =0.19 =.03 Assuming a matched pairs design, which of the following assumptions is necessary for a confidence interval for the mean difference to be valid?