Exam 20: Comparing Groups

A researcher was interested in comparing the salaries of female and male employees of a particular company. Independent random samples of female employees (sample 1) and male employees (sample 2) were taken to calculate the mean salary, in dollars per week, for each group. A $95 \%$ confidence interval for the difference, $\mu _ { 1 } - \mu _ { 2 }$ , between the mean weekly salary of all female employees and the mean weekly salary of all male employees was determined to be $( - \$ 180 , \$ 60 )$ .

Data were collected on annual personal time (in hours) taken by a random sample of 16 women (group1) and 7 men (group 2) employed by a medium sized company. The women took an average of $24.75$ hours of personal time per year with a standard deviation of $2.84$ hours. The men took an average of $21.89$ hours of personal time per year with a standard deviation of $3.29$ hours. The Human Resources Department believes that women tend to take more personal time than men because they tend to be the primary child care givers in the family. The correct null and alternative hypotheses to test this belief are

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Use the sample data below to construct a $99 \%$ confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ where $\mu _ { 1 }$ and $\mu _ { 2 }$ represent the mean for the treatment group and the control group respectively. Treatment Group: $n _ { 1 } = 85 , \bar { Y } _ { 1 } = 189.1 , s _ { 1 } = 38.7$ Control Group: $n _ { 2 } = 75 , \bar { Y } _ { 2 } = 203.7 , s _ { 2 } = 39.2$

A two-sample $z$ -test for two population proportions is to be performed using the $P$ -value approach. The null hypothesis is $H _ { 0 } : p _ { 1 } = p _ { 2 }$ and the alternative is $H _ { A } : p _ { 1 } \neq p _ { 2 }$ . Use the given sample data to find the $p$ -value for the hypothesis test. Give an interpretation of the $p$ -value. $n _ { 1 } = 200 , x _ { 1 } = 11 , n _ { 2 } = 100 \text {, and } x _ { 2 } = 8$

Results of a small experiment show that people are likely to offer a different amount for used exercise equipment when bargaining with a friend than when bargaining with a stranger. The $p$ -value from testing the difference in mean offers was equal to $0.00162$ . At an $\alpha = 0.05$ , the correct conclusion is to

A consumer advocate decided to investigate the average wait time for a table for one at two local restaurants. Eighteen customers were sent to each restaurant at the same randomly selected times and the time they waited for a table was recorded in minutes. The following sample data was obtained. A 10 15 7 9 20 5 15 5 7 0 10 12 19 6 0 5 22 18 B 12 17 5 12 8 2 10 25 6 35 10 14 9 22 20 18 5 13 Find a $90 \%$ confidence interval for the difference, $\mu _ { A } - \mu _ { B }$ , between the mean wait time for restaurant $\mathrm { A }$ and the mean wait time for restaurant $\mathrm { B }$ .

Suppose the proportion of women who favor stricter gun control legislation is $p _ { w }$ and the proportion of men who favor stricter gun control legislation is $p _ { m }$ . The survey found a $90 \%$ confidence interval for $p _ { w } - p _ { m }$ is $( - 0.08,0.24 )$ . Give an interpretation of this confidence interval.

Suppose the proportion of women who follow a regular exercise program is $p _ { w }$ and the proportion of men who follow a regular exercise program is $p _ { m }$ . A study found a $98 \%$ confidence interval for $p _ { w } - p _ { m }$ is $( - 0.024,0.115 )$ . Give an interpretation of this confidence interval.

Suppose the proportion of sophomores at a particular college who purchased used textbooks in the past year is $p _ { s }$ and the proportion of freshmen at the college who purchased used textbooks in the past year is $p _ { f }$ . A study found a $90 \%$ confidence interval for $p _ { s } - p _ { f }$ is $( 0.234,0.421 )$ . Give an interpretation of this confidence interval.

A survey of randomly selected college students found that 45 of the 99 freshmen and 57 of the 100 sophomores surveyed had purchased used textbooks in the past year. Construct a $98 \%$ confidence interval for the difference in the proportions of college freshmen and sophomores who purchased used textbooks.

A researcher is interested in the academic performance differences between individuals using an optimistic versus a pessimistic approach to their studies.If the researcher claims a Significant difference between groups, when in fact none exists:

Two types of flares are tested for their burning times (in minutes) and sample results are given below. Brand A: $n = 35 , \bar { Y } = 19.4 , s = 1.4$ Brand B: $n = 40 , \bar { Y } = 15.1 , s = 0.8$ Construct a $95 \%$ confidence interval for the difference $\mu _ { A } - \mu _ { B }$ based on the sample data.

A two-sample $z$ -test for two population proportions is to be performed using the $P$ -value approach. The null hypothesis is $H _ { 0 } : p _ { 1 } = p _ { 2 }$ and the alternative is $H _ { A } : p _ { 1 } \neq p _ { 2 }$ . Use the given sample data to find the $p$ -value for the hypothesis test. Give an interpretation of the $p$ -value. $n _ { 1 } = 50 , x _ { 1 } = 20 , n _ { 2 } = 75 , \text { and } x _ { 2 } = 15$

A survey of randomly chosen adults found that 38 of the 61 women and 46 of the 83 men follow regular exercise programs.Construct a 95% confidence interval for the difference in The proportions of women and men who have regular exercise programs.

A study was conducted to determine if patients recovering from knee surgery should receive physical therapy two or three times per week. Suppose $p _ { 1 }$ represents the proportion of patients who showed improvement after one month of therapy three times a week and $p _ { 2 }$ represents the proportion of patients who showed improvement after one month of therapy twice a week. A $90 \%$ confidence interval for $p _ { 1 } - p _ { 2 }$ is $( 0.12,0.38 )$ . Give an interpretation of this confidence interval.

A grocery store is interested in determining whether or not a difference exists between the shelf life of Tasty Choice doughnuts and Sugar Twist doughnuts. A random sample of 100 boxes of each brand was selected and the mean shelf life in days was determined for each brand. A $90 \%$ confidence interval for the difference of the means, $\mu _ { T C } - \mu _ { S T }$ , was determined to be $( 1.3,2.5 )$ .

A researcher is interested in the academic performance differences between individuals using an optimistic versus a pessimistic approach to their studies.If the researcher fails to find a Significant difference, when in fact one exists in the population:

A researcher was interested in comparing the number of hours of television watched each day by two-year-olds and three-year-olds. A random sample of 18 two-year-olds and 18 three-year-olds yielded the follow data. 2y 0.5 1.5 1.5 2.0 1.5 0.0 1.0 1.0 1.0 0.0 2.0 1.5 2.5 2.0 0.5 0.0 1.5 2.5 3y 2.0 3.0 1.5 1.5 1.5 2.0 1.0 0.0 0.0 1.5 1.5 2.0 2.5 2.0 3.0 1.0 1.5 0.5 Find a $98 \%$ confidence interval for the difference, $\mu _ { 2 } - \mu _ { 3 }$ , between the mean number of hours for two-year-olds $( 2 y )$ and the mean number of hours for three-year-olds ( $3 y$ ).