Exam 8: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

We sampled 100 men and 100 women and asked: "Do you think the environment is a major concern?" Of those sampled, 67 women and 53 men responded that they believed it is. For the confidence interval procedure to work properly, what additional assumptions must be satisfied?

A confidence interval for $\left( \mu _ { 1 } - \mu _ { 2 } \right)$ is $( 5,8 )$ . Which of the following inferences is correct?

Using paired differences removes sources of variation that tend to inflate $\sigma ^ { 2 }$

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: Bicycle Coated Tire () 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 Use the summary data to calculate the test statistic to determine if the new spray coating improves the mean wear of the bicycle tires.

Consider the following set of salary data: Men (1) Women (2) Sample Size 100 80 Mean \ 12,850 \ 13,000 Standard Deviation \ 345 \ 500 To determine if women have a higher mean salary than men, we would test:

Which of the following represents the ratio of variances?

The data for a random sample of five paired observations are shown below. Pair Observation 1 Observation 2 1 3 5 2 4 4 3 3 4 4 2 5 5 5 6 a. Calculate the difference between each pair of observations by subtracting observation 2 from observation 1. Use the differences to calculate $\bar { d }$ and $s _ { d }$ . b. Calculate the means $\bar { x } _ { 1 }$ and $\bar { x } _ { 2 }$ of each column of observations. Show that $\bar { d } = \bar { x } _ { 1 } - \bar { x } _ { 2 }$ . c. Form a 90\% confidence interval for $\mu _ { D }$ .

Which of the following represents the difference in two population proportions?

A consumer protection agency is comparing the work of two electrical contractors. The agency plans to inspect residences in which each of these contractors has done the wiring in order toestimate the difference in the proportions of residences that are electrically deficient. Suppose the proportions of residences with deficient work are expected to be about .1 for both contractors. How many homes should be sampled in order to estimate the difference in proportions using a 90% confidence interval of width .1?

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?

Independent random samples, each containing 500 observations were selected from two binomial populations. The samples from populations 1 and 2 produced 210 and 320 successes, respectively $\text { Test } H _ { 0 } : \left( p _ { 1 } - p _ { 2 } \right) = 0 \text { against } H _ { \mathrm { a } } : \left( p _ { 1 } - p _ { 2 } \right) < 0 \text {. Use } \alpha = .05$

A new weight-reducing technique, consisting of a liquid protein diet, is currently undergoing tests by the Food and Drug Administration (FDA) before its introduction into the market. The weights of a random sample of five people are recorded before they are introduced to the liquid protein diet. The five individuals are then instructed to follow the liquid protein diet for 3 weeks. At the end of this period, their weights (in pounds) are again recorded. The results are listed in the table. Let $\mu _ { 1 }$ be the true mean weight of individuals before starting the diet and let $\mu _ { 2 }$ be the true mean weight of individuals after 3 weeks on the diet. Person Weight Before Diet Weight After Diet 1 149 142 2 194 189 3 187 184 4 196 190 5 203 199 Summary information is as follows: $\bar { d } = 5 , s _ { d } = 1.58$ . Test to determine if the diet is effective at reducing weight. Use $\alpha = .10$ .

Specify the appropriate rejection region for testing $\mathrm { H } _ { 0 } : \sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ in the given situation. - $\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 } ^ { 2 } < \sigma { } _ { 2 } ^ { 2 } ; \alpha = 0.05 , \mathrm { n } _ { 1 } = 21 , \mathrm { n } _ { 2 } = 8$

Consider the following set of salary data: Men (1) Women (2) Sample Size 100 80 Mean \ 12,850 \ 13,000 Standard Deviation \ 345 \ 500 Calculate the appropriate test statistic for a test about $\mu _ { 1 } - \mu _ { 2 }$ .

Consider the following set of salary data: Men (1) Women (2) Sample Size 100 80 Mean \ 12,850 \ 13,000 Standard Deviation \ 345 \ 500 What assumptions are necessary to perform a test for the difference in population means?

A paired difference experiment yielded the following results. $n _ { d } = 50 , \quad \sum ^ { d } = 967 , \quad \sum { d ^ { 2 } = 19,201 }$ Test $H _ { 0 } : \mu _ { d } = 20$ against $H _ { \mathrm { a } } : \mu _ { d } \neq 20$ , where $\mu _ { d } = \mu _ { 1 } - \mu _ { 2 }$ , using $\alpha = .05$ .

Specify the appropriate rejection region for testing $\mathrm { H } _ { 0 } : \sigma { } _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 }$ in the given situation. - $\mathrm { H } _ { \mathrm { a } } : \sigma _ { 1 } ^ { 2 } \neq \sigma _ { 2 } ^ { 2 } ; \alpha = 0.10 , \mathrm { n } _ { 1 } = 10 , \mathrm { n } _ { 2 } = 16$ Assume that $\mathrm { s } _ { 2 } ^ { 2 } > \mathrm { s } _ { 1 } ^ { 2 }$ .

A paired difference experiment has 75 pairs of observations. What is the rejection region for testing $H _ { a } : \mu _ { d } > 0 ? \text { Use } \alpha = .01$

Assume that $\sigma _ { 1 } ^ { 2 } = \sigma _ { 2 } ^ { 2 } = \sigma ^ { 2 }$ . Calculate the pooled estimator of $\sigma ^ { 2 }$ for $s _ { 1 } ^ { 2 } = .88$ , $s _ { 2 } ^ { 2 } = 1.01 , n _ { 1 } = 10$ , and $n _ { 2 } = 12$ .

A paired difference experiment yielded $\mathrm { n } _ { \mathrm { d } }$ pairs of observations. For the given case, what is the rejection region for testing $\mathrm { H } _ { 0 } : \mu _ { \mathrm { d } } = 9$ against Ha: $\mu _ { \mathrm { d } } \neq 9$ ? $\mathrm { n } _ { \mathrm { d } } = 15 , \alpha = 0.10$