In a new experimental teaching method, when students have questions in class, they send them to the instructor via a laptop computer. From time to time, the instructor pauses to read the questions and to provide the answers, without revealing the identities of the students who are asking the questions. This new method is supposed to eliminate the reluctance of students to ask questions for fear of revealing their lack of understanding in public. In a study of this new method, $67$ male students and $72$ female students participated. Treat these as if they were simple random samples. Of the male students, $38$ said that they liked the new method better than the traditional one in which students raise their hands to ask questions. Of the female students, only $22$ said they liked the new method better.

a. Construct a $95\%$ confidence interval for the difference between the proportions of male and female students who like the new method better.

b. An educator claims that the proportion of students who like the new method better is the same for males and females. Does the confidence interval contradict this claim?

In a study of birth order and intelligence, IQ tests were given to $18$- and $19$-year-old men to estimate the size of the difference, if any, between the mean IQs of first born sons and second born sons. The following data for $10$ firstborn sons and $10$ second born sons are consistent with the means and standard deviations reported in the article. It is reasonable to assume that the samples come from populations that are approximately normal.

$$\begin{array}{c} \hline \text{Firstborn}\\ \hline \begin{array}{rrrrr} 104 & 82 & 102 & 96 & 129\\ 89 & 114 & 107 & 89 & 103\\ \end{array}\\ \hline \end{array}$$

$$\begin{array}{c} \hline \text{Secondborn}\\ \hline \begin{array}{rrrrr} 103 & 103 & 91 & 113 & 102\\ 103 & 92 & 90 & 114 & 113\\ \end{array}\\ \hline \end{array}$$

Can you conclude that there is a difference in mean IQ between firstborn and secondborn sons? Use the $\alpha = 0.01$ level.

In a study of birth order and intelligence, IQ tests were given to $18$- and $19$-year-old men to estimate the size of the difference, if any, between the mean IQs of firstborn sons and of secondborn sons. The following data for $10$ firstborn sons and $10$ secondborn sons are consistent with the means and standard deviations reported in the article. It is reasonable to assume that the samples come from populations that are approximately normal.

$$\begin{array}{c} \hline \textbf{Firstborn}\\ \hline \begin{array}{rrrrr} 104 & 82 & 102 & 96 & 129\\ 89 & 114 & 107 & 89 & 103\\ \end{array}\\ \hline \end{array}$$

$$\begin{array}{c} \hline \textbf{Secondborn}\\ \hline \begin{array}{rrrrr} 103 & 103 & 91 & 113 & 102\\ 103 & 92 & 90 & 114 & 113\\ \end{array}\\ \hline \end{array}$$

Can you conclude that the standard deviation of IQ differs between firstborn and secondborn sons? Use the $\alpha = 0.05$ level.

Construct the confidence interval for the difference $\mu_1 − \mu_2$ for the given level and values of $\overline{x}_1, \overline{x}_2, s_1, s_2, n_1,$ and $n_2$.

$$\text{Level}\ 99.5\%: \overline{x}_1 = 82.9, \overline{x}_2 = 64.1, s_1 = 9.8, s_2 = 6.2, n_1 = 19, n_2 = 10$$

The following display presents a $95\%$ confidence interval for the mean difference between matched pairs.

$$\begin{array}{|c|} \hline \text{TInterval}\\ (7.7632, 9.3028)\\ \begin{align*} \overline{x} &= 8.533\\ Sx & = 1.548\\ n & = 18 &\\ \end{align*}\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ Find the point estimate of $\mu_d$.

$\textbf{b.}\hspace{10pt}$ How many degrees of freedom are there?

$\textbf{c.}\hspace{10pt}$ Fill in the blanks: We are $95\%$ confident that the mean difference is between $\_\_\_\_\_\_\_\_\_\_\_\_\_$ and $\_\_\_\_\_\_\_\_\_\_\_\_\_$.

Traffic engineers compared rates of traffic accidents at intersections with raised medians with rates at intersections with two-way left-turn lanes. They found that out of $4644$ accidents at intersections with raised medians, $2280$ were rear-end accidents, and out of $4584$ accidents at two-way left-turn lanes, $1982$ were rear-end accidents.

a. Assuming these to be random samples of accidents from the two types of intersection, construct a 95% confidence interval for the difference between the proportions of accidents that are of the rear-end type at the two types of intersection.

b. Does the confidence interval contradict the claim that the proportion of rear-end accidents is the same at both types of intersection?

Construct the confidence interval for the difference $\mu_1 − \mu_2$ for the given level and values of $\overline{x}_1, \overline{x}_2, s_1, s_2, n_1,$ and $n_2$.

$$\text{Level}\ 95\%: \overline{x}_1 = 47.7, \overline{x}_2 = 42.6, s_1 = 33.9, s_2 = 17.6, n_1 = 13, n_2 = 19$$

It is generally known that boys grow at an unusually fast rate between the ages of about $12$ and $14$. Following are heights, in inches, of $40$ boys measured at age $12$ and again at age $14$.

$$\begin{array}{cccc} \hline \text{Age 12} & \text{Age 14} \hspace{25pt} & \text{Age 12} & \text{Age 14}\\ \hline 57.9 & 62.9 \hspace{25pt} & 55.4 & 61.8\\ 61.1 & 65.9 \hspace{25pt} & 58.7 & 64.2\\ 62.7 & 67.5 \hspace{25pt} & 64.3 & 69.8\\ 67.5 & 73.7 \hspace{25pt} & 58.1 & 63.3\\ 59.2 & 64.9 \hspace{25pt} & 63.3 & 69.6\\ 61.4 & 67.0 \hspace{25pt} & 61.2 & 66.4\\ 60.7 & 65.5 \hspace{25pt} & 64.5 & 69.2\\ 55.9 & 62.1 \hspace{25pt} & 55.9 & 62.0\\ 59.7 & 65.4 \hspace{25pt} & 60.4 & 65.7\\ 56.3 & 61.5 \hspace{25pt} & 57.8 & 62.8\\ 63.0 & 68.5 \hspace{25pt} & 68.3 & 73.6\\ 58.6 & 63.9 \hspace{25pt} & 63.0 & 67.7\\ 61.1 & 65.8 \hspace{25pt} & 64.4 & 69.2\\ 59.5 & 64.5 \hspace{25pt} & 58.2 & 64.6\\ 61.6 & 66.3 \hspace{25pt} & 59.7 & 66.1\\ 59.3 & 64.2 \hspace{25pt} & 60.2 & 65.9\\ 62.1 & 67.1 \hspace{25pt} & 63.7 & 68.3\\ 62.8 & 68.1 \hspace{25pt} & 60.2 & 65.2\\ 65.3 & 69.9 \hspace{25pt} & 62.7 & 67.9\\ 60.4 & 66.7 \hspace{25pt} & 55.6 & 61.7\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ Construct a $95\%$ confidence interval for the mean increase in height for boys between the ages of $12$ and $14$.

$\textbf{b.}\hspace{10pt}$ A pediatrician claims that the mean increase in height is $5.5$ inches. Does the confidence interval contradict this claim?

Refer to the earlier exercise. Compute the margin of error for a $90\%$ confidence interval for the difference $\mu_1 − \mu_2$.

The following output presents a confidence interval for the difference between two proportions.

$$\begin{array}{l} \hline \\ \begin{array}{lrrrr} \text{Sample} & \text{X} & \text{N} & \text{Sample p}\\ 1 & 85 & 103 & 0.825243\\ 2 & 72 & 125 & 0.576000\\ \end{array}\\ \\ \begin{array}{l} \text{Difference = p (1) - p (2)}\\ \text{Estimate for difference: 0.249243}\\ \text{95\\\% CI for difference: (0.135735, 0.362751)}\\ \end{array}\\ \hline \end{array}$$

a. Find the point estimate of $p_1 - p_2$.

b. Fill in the blanks: We are $\_\_\_\_\_\_\_\_\_\_\_\_\_$ confident that the difference between the proportions is between $\_\_\_\_\_\_\_\_\_\_\_\_\_$ and $\_\_\_\_\_\_\_\_\_\_\_\_\_$.

The following display presents a $95\%$ confidence interval for the difference between two proportions.

$$\begin{array}{|c|} \hline \text{2-PropZInt}\\ (.12307, .5092)\\ \begin{align*} \hat{p}_1 &= .7307692308\\ \hat{p}_2 &= .4146341463\\ n_1 & = 52\\ n_2 &= 41\\ &\\ \end{align*}\\ \hline \end{array}$$

a. Find the point estimate of $p_1 - p_2$.

b. Fill in the blanks: We are $95\%$ confident that the difference between the proportions is between $\_\_\_\_\_\_\_\_\_\_\_\_\_$ and $\_\_\_\_\_\_\_\_\_\_\_\_\_$.

Construct the confidence interval for the difference $p_1 − p_2$ for the given level and values of $x_1, n_1, x_2,$ and $n_2$.

$$\text{Level}\ 99\%: x_1 = 24, n_1 = 53, x_2 = 17, n_2 = 41$$

Refer to the earlier exercise. Find the critical value for a $90\%$ confidence interval for the difference $\mu_1 − \mu_2$.

For a given population of high school seniors, the Scholastic Aptitude Test (SAT) in mathematics has a mean score of 500 with a standard deviation of 100. Another widely used test is the American College Testing (ACT) exam. The mathematics portion of the ACT has a mean of 18 and a standard deviation of 6. Both SAT and ACT scores are normally distributed. What is the probability that a randomly selected high school senior's score on the mathematics part of the SAT will be (a) more than 675? (b) less than 450? (c) between 450 and 675?