Perform the following steps.
Faculty and Students The number of faculty and the number of students are shown for a random selection of small colleges. Is there a significant relationship between the two variables? Switch $x$ and $y$ and repeat the process. Which do you think is really the independent variable?

$$\begin{array}{l|ccccccc} \text { Faculty } & 99 & 110 & 113 & 116 & 138 & 174 & 220 \\ \hline \text { Students } & 1353 & 1290 & 1091 & 1213 & 1384 & 1283 & 2075 \end{array}$$

c. State the hypotheses.

Perform the following steps.
Faculty and Students The number of faculty and the number of students are shown for a random selection of small colleges. Is there a significant relationship between the two variables? Switch $x$ and $y$ and repeat the process. Which do you think is really the independent variable?

$$\begin{array}{l|ccccccc} \text { Faculty } & 99 & 110 & 113 & 116 & 138 & 174 & 220 \\ \hline \text { Students } & 1353 & 1290 & 1091 & 1213 & 1384 & 1283 & 2075 \end{array}$$

d. Test the significance of the correlation coefficient at $\alpha=0.05$. Using Table I or the $P$-value method.

For each exercise, find the equation of the regression line and find the $y^{\prime}$ value for the specified $x$ value. Remember that no regression should be done when $r$ is not significant.

The number of faculty and the number of students in a random selection of small colleges are shown.

$$\begin{array}{l|ccccccc} \text{Faculty} & 99 & 110 & 113 & 116 & 138 & 174 & 220 \\ \hline \text{Students} & 1353 & 1290 & 1091 & 1213 & 1384 & 1283 & 2075 \end{array}$$

Now find the equation of the regression line when $x$ is the variable of the number of students.

Perform the following steps.
Faculty and Students The number of faculty and the number of students are shown for a random selection of small colleges. Is there a significant relationship between the two variables? Switch $x$ and $y$ and repeat the process. Which do you think is really the independent variable?

$$\begin{array}{l|ccccccc} \text { Faculty } & 99 & 110 & 113 & 116 & 138 & 174 & 220 \\ \hline \text { Students } & 1353 & 1290 & 1091 & 1213 & 1384 & 1283 & 2075 \end{array}$$

e. Give a brief explanation of the type of relationship. Assume all assumptions have been met.

A student-faculty government committee of 4 people is to be formed from 20 student volunteers and 5 faculty volunteers. Find the probability that the committee will consist of the following, assuming the selection is made at random: (a) All faculty members. (b) Two students and two faculty members. (c) All students. (d) One faculty member and three students.

There are many interesting relationships among the various nutrients found in fruits and vegetables. Listed below are the number of grams of carbohydrates and the number of kilocalories for a 100-gram sample of various raw foods. Is there a linear relationship between the variables?

Perform the following steps.

a. Draw the scatter plot for the variables. b. Compute the value of the correlation coefficient. c. State the hypotheses. d. Test the significance of the correlation coefficient at $\alpha=0.05$, using Table I. e. Give a brief explanation of the type of relationship. Assume all assumptions have been met.

The table shows which school subjects are favored by a group of students. Predict the number of students out of 400 that would pick science as their favorite subject.

The data below shows the number of times that 50 students have lost a pencil this week: $5,9,10,5,9,9,8,4,9,8,5,7,3,10,7,7,8,7,6,6,9,6,4,4,10,5,6,6,3,8,7,8,3$ $4,6,6,5,7,5,4,3,5,2,4,2,8,1,0,3,5$

Construct a frequency table for this data.

Perform the following steps.
Alumni Contributions The director of an alumni association for a small college wants to determine whether there is any type of relationship between the amount of an alumnus's contribution (in dollars) and the number of years the alumnus has been out of school. The data follow.

$$\begin{array}{l|cccccc} \text { Years } x & 1 & 5 & 3 & 10 & 7 & 6 \\ \hline \text { Contribution } y & 500 & 100 & 300 & 50 & 75 & 80 \end{array}$$

b. Compute the value of the correlation coefficient.

Use the same data as for the corresponding exercises in what discussed before. For each exercise, find the equation of the regression line and finc the $y^{\prime}$ value for the specified $x$ value. Remember that no regression should be done when $r$ is not significant.
Water and Carbohydrates Here are the number of grams of water and the number of grams of carbohydrates for a random selection of raw foods ( $100 \mathrm{~g}$ each).

$$\begin{array}{c|lllllll} \text { Water } & 83.93 & 80.76 & 87.66 & 85.20 & 72.85 & 84.61 & 83.81 \\ \hline \text { Carbs } & 15.25 & 16.55 & 11.10 & 13.01 & 24.27 & 14.13 & 15.11 \end{array}$$

Find $y'$ for $x=75.$

Explain how the motors and appliances in a building affect the (a) design cooling load, (b) annual energy consumption for cooling, and (c) annual energy consumption for heating of the building.

Perform the following steps.
Water and Carbohydrates Here are the number of grams of water and the number of grams of carbohydrates for a random selection of raw foods ( $100 \mathrm{~g}$ each). Is there a linear relationship between the variables?

$$\begin{array}{l|lllllll} \text { Water } & 83.93 & 80.76 & 87.66 & 85.20 & 72.85 & 84.61 & 83.81 \\ \hline \text { Carbs } & 15.25 & 16.55 & 11.10 & 13.01 & 24.27 & 14.13 & 15.11 \end{array}$$

c. State the hypotheses.

Perform the following steps.
Water and Carbohydrates Here are the number of grams of water and the number of grams of carbohydrates for a random selection of raw foods ( $100 \mathrm{~g}$ each). Is there a linear relationship between the variables?

$$\begin{array}{l|lllllll} \text { Water } & 83.93 & 80.76 & 87.66 & 85.20 & 72.85 & 84.61 & 83.81 \\ \hline \text { Carbs } & 15.25 & 16.55 & 11.10 & 13.01 & 24.27 & 14.13 & 15.11 \end{array}$$

b. Compute the value of the correlation coefficient.

Perform the following steps.
Alumni Contributions The director of an alumni association for a small college wants to determine whether there is any type of relationship between the amount of an alumnus's contribution (in dollars) and the number of years the alumnus has been out of school. The data follow.

$$\begin{array}{l|cccccc} \text { Years } x & 1 & 5 & 3 & 10 & 7 & 6 \\ \hline \text { Contribution } y & 500 & 100 & 300 & 50 & 75 & 80 \end{array}$$

a. Draw the scatter plot for the variables.