Chapter 9, Regression and Correlation Analysis Video Solutions, Probability, Statistics, and Queuing Theory with Computer Science Applications, Second Edition (Computer Science and Scientific Computing)

Problem 1

[15] See Example 9.1.1. In the table below we show the number of pages and the price of 20 books reviewed in the November 1988 issue of Technometrics. (Two other books were reviewed with no price given.)
Books Reviewed Data
$$
\begin{aligned}
&\text { Books Reviewed Data }\\
&\begin{array}{cccccc}
\hline \text { Pages } & \text { Price } & \text { Pages } & \text { Price } & \text { Pages } & \text { Price } \\
\hline & & & & & \\
311 & 34.50 & 610 & 49.95 & 384 & 49.95 \\
415 & 80.00 & 278 & 34.95 & 232 & 32.50 \\
408 & 34.50 & 492 & 39.25 & 435 & 29.95 \\
699 & 33.95 & 687 & 42.50 & 540 & 99.75 \\
416 & 25.00 & 447 & 36.95 & 614 & 72.95 \\
307 & 34.95 & 429 & 69.75 & 260 & 34.95 \\
171 & 17.95 & 162 & 19.95 & - & - \\
& & & & & \\
\hline
\end{array}
\end{aligned}
$$
Let $X$ be the number of pages and $Y$ the price of the books.
(a) Draw the scatter diagram for the data. Does it look like a straight line would fit the data?
(b) Find the least squares regression line. How well does it seem to fit the data? Are there any outliers?
(c) Find the average number of pages per book and average price per book.
(d) Use the equation of the least squares regression line to estimate the price of a book with 390 pages. How does this compare to the estimate we made in Example 9.1.1?

Carolyn Behr-Jerome

Numerade Educator

01:12

Problem 2

Prove that the formula (9.18) for $\hat{\beta}_1$ is equivalent the formula (9.16).

Carson Merrill

Numerade Educator

03:04

Problem 3

Consider the proof of Theorem 9.1.1. Prove that $\sum_{i=1}^n c_i=0$ and $\sum_{i=1}^n c_i x_i=1$.

Farnood Ensan

Numerade Educator

Problem 4

Consider Example 9.1.2. Prove that, for this example, (9.34) yields $s_{\hat{\beta}_0}=6.367$ and $(9.35)$ yields $s_{\hat{\beta}_1}=0.01976$. You may assume that $s^2=189.3$ and that you are given the values calculated in Example 9.1.1.

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00:51

Problem 5

Show that the fitted linear regression line $\hat{y}=\hat{\beta}_0+\hat{\beta}_1 x$ goes through the point $(\bar{x}, \bar{y})$.

Victor Salazar

Numerade Educator

01:22

Problem 6

Analysts at Blithering Boats have constructed the linear regression line $\hat{y}=15+2.3 x$.
(a) What is the estimated change in the mean value of $y$ when $x$ increases by 2 ?
(b) If $\bar{y}=42.62$, determine $\bar{x}$.

Jerrah Biggerstaff

Numerade Educator

06:12

Problem 7

Suppose an analyst at Dogwood's Doughnuts is constructing a simple linear regression line with 41 points. She has calculated $S_{\mathrm{xy}}=-540$, $S S_{\mathrm{x}}=130, \bar{y}=35.7$, and $\bar{x}=12.3$. Find the least squares regression line.

Charles Carter

Numerade Educator

02:27

Problem 8

Wimpering Willie is constructing a simple linear regression line with 25 points. Willie has calculated the value of SSE to be 374.2.
(a) What is the standard error, $s_e$ ?
(b) For $x=75, \hat{y}=126.3$. If the observation $(75,140.2)$ is part of the original data set, should it be considered an outlier according to the oft quoted rule of thumb that any point for which $\left|y_i-\hat{y}\right| \geq$ $3 s_e$ is an outlier?
(c) If the observation $(75,140.2)$ has $h_i=0.07$, what is the standard residual of the point?
(d) Would MINITAB flag the observation of part (c) as "an observation with a large standard residual?"

Jon Southam

Numerade Educator

03:45

Problem 9

Consider Example 9.1.1. Using the results calculated in the example and the parameters calculated by the MINITAB command REGRESS and shown in Figure 9.1.2, do the following:
(a) Find the $95 \%$ confidence interval for $\beta_1$.
(b) Test the null hypothesis $H_0: \beta_1=0.07$ against $H_1: \beta_1 \neq 0.07$ at the $5 \%$ level of significance.

Raymond Matshanda

Numerade Educator

03:05

Problem 10

The sample correlation coefficient, $r$, between two random variables is 0.625 .
(a) If $n=21$, test the null hypothesis $H_0: \rho=0$ against the alternative hypothesis $H_1: \rho \neq 0$ at the $5 \%$ level of significance.
(b) What is the $p$-value of the test?

Jake Zanazzi

Numerade Educator

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Problem 11

If $n=17$ and the intermediate values for a simple linear regression model are
$$
\begin{gathered}
\sum x=136 \quad \sum y=552.212 \quad \sum x y=5081.33 \\
\sum x^2=1496 \quad \sum y^2=19031.7,
\end{gathered}
$$
do the following:
(a) Calculate Sxy, SSx, and SSy.
(b) Find the sample regression of $Y$ on $X$.
(c) Construct the ANOVA table.

Rashmi Sinha

Numerade Educator

02:37

Problem 12

Professor Ruddy Redback checks the chlorine residual in his swimming pool at the times shown in the following table, after his pool service has treated the pool with a water purifier.
$$
\begin{array}{cc}
\begin{array}{c}
\text { Hours } \\
\text { (after treatment) }
\end{array} & \begin{array}{c}
\text { Chlorine Residual } \\
\text { (parts per million) }
\end{array} \\
x & y \\
\hline & \\
06 & 1.86 \\
12 & 1.73 \\
18 & 1.61 \\
24 & 1.50 \\
30 & 1.39 \\
36 & 1.30 \\
& \\
\hline
\end{array}
$$
Use the method of least squares to fit an exponential curve of the form $y=\alpha \exp (\beta x)$ to the data for Professor Redback's pool.

Raymond Matshanda

Numerade Educator

06:38

Problem 13

The Perfect Product Prediction Company has collected the data shown in the table for the sales of Golden Jelly Belly jelly beans in ten market areas. The price is in cents for 5 beans and the sales are in thousands of cartons.
$$
\begin{array}{cc}
\text { Price } & \text { Sales } \\
x & y \\
\hline & \\
14 & 97.46 \\
15 & 90.56 \\
16 & 84.54 \\
17 & 79.26 \\
18 & 74.57 \\
19 & 70.40 \\
20 & 66.66 \\
21 & 63.28 \\
22 & 60.22 \\
23 & 57.44 \\
& \\
\hline
\end{array}
$$
Use the method of least squares to fit a curve of the form $y=\alpha x^\beta$ to the data.

Pawan Yadav

Numerade Educator

02:35

Problem 14

Henry Hunk, the head statistician at the Hunky Dory Boat Company, uses the multiple linear regression model
$$
Y=\beta_0+\beta_1 x_1+\cdots+\beta_5 x_5+\epsilon,
$$
that seems to satisfy the standard statistics model assumptions. His least squares calculations from 30 data points yield SSE $=0.42$ and $R^2=0.93$. Test the null hypothesis, $H_0: \beta_1=\beta_2=\cdots=\beta_5=0$, against the alternative hypothesis, $H_1$ : At least one of the parameters $\beta_1, \beta_2, \ldots, \beta_5$ is not zero. Use $\alpha=0.05$.

Nick Johnson

Numerade Educator

02:35

Problem 15

Sweet Suzy at Candy Canes has developed the linear multiple regression model
$$
\hat{y}=18.7+3.2 x_1+0.015 x_2,
$$
that seems to satisfy the standard statistics model assumptions. Her least squares calculations from 30 data points also yield
$$
s_{\hat{\beta}_2}=0.0036 \text { and } R^2=0.78 \text {. }
$$
(a) Test the null hypothesis, $H_0: \beta_1=\beta_2=0$, against the alternative hypothesis, $H_1$ : At least one of the parameters $\beta_1, \beta_2$ is not zero. Use $\alpha=0.05$.
(b) Test the null hypothesis, $H_0: \beta_2=0$, against the alternative hypothesis, $H_1: \beta_2 \neq 0$. Use $\alpha=0.05$.

Nick Johnson

Numerade Educator

Problem 16

Show that for the multiple linear regression model, the $F$-value from the analysis of variable table, and $R^2$ are related by the simple formula (9.88),
$$
F=\frac{R^2}{1-R^2}\left(\frac{n-k-1}{k}\right) .
$$

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Problem 17

In a multiple regression study at Carnal Cruises, part of the ANOVA table is as follows:
$$
\begin{array}{lrr}
\hline \text { SOURCE } & \text { DF } & \text { SS } \\
\hline \text { Regression } & 4 & 48 \\
\text { Error } & 36 & 36 \\
\hline
\end{array}
$$
(a) What is the sample size?
(b) How many predictor variables are there?
(c) Test the null hypothesis that all of the predictor variable coefficients are zero, against the alternative hypothesis that at least one of the coefficients is not zero. Use $\alpha=0.05$.
(d) Calculate the standard error of the estimate.

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Problem 18

George Grep found the following, partially filled in, ANOVA table from a multiple regression study with 3 predictor variables. Help George by filling in the five blank spaces in the table (don't forget MSE).
$$
\begin{array}{lrrr}
\hline \text { SOURCE } & \text { DF } & \text { SS } & \text { MS } \\
\hline \text { Regression } & & & 170.30 \\
\text { Error } & & & \\
\hline \text { Total } & 9 & 538.24 &
\end{array}
$$
(a) Calculate $R^2$.
(b) Test the null hypothesis that all of the predictor variable coefficients are zero, against the alternative hypothesis that at least one of the coefficients is not zero. Use $\alpha=0.05$.

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02:39

Problem 19

You will need a computer system, such as MINITAB, SAS/STAT, or EXPLORE to do this exercise. Sam Spade of Hilbert Hackers has discovered the following table that relates $x$ to $y$. He wants to fit a cubic to the data and is concerned about the numerical stability of the calculations. Help Sam by doing the following:
(a) Fit the regression function $y=\beta_0+\beta_1 x+\beta_2 x^2+\beta_3 x^3$ to the data. Record the values of $\hat{\beta}_0, \hat{\beta}_1, \hat{\beta}_2, \hat{\beta}_3$, the estimated standard deviation of $\hat{\beta}_3, R^2$, and $s_e$. Is $\beta_3$ significant at the five percent level?
(b) Standardize $x$ by the formula $x^{\prime}=(x-\bar{x}) / s_x$, and fit the regression function $y=\beta_0^*+\beta_1^* x^{\prime}+\beta_2^*\left(x^{\prime}\right)^2+\beta_3^*\left(x^{\prime}\right)^3$ to the data. Then record the same values that you recorded for (a) and make the same test.
$$
\begin{array}{crcccr}
{\text { Data }} \\
\hline x & y & x & y & x & y \\
10 & 7 & 10 & 8 & 10 & 6 \\
15 & 12 & 15 & 15 & 15 & 13 \\
20 & 10 & 20 & 11 & 20 & 7 \\
25 & 14 & 25 & 16 & 25 & 17 \\
\hline
\end{array}
$$

Dominador Tan

Numerade Educator

00:54

Problem 20

You will need a computer for this exercise and the next. Fit the regression equation $y=\beta_0+\beta_1 x_1+\beta_2 x_2$ to the data of Table 9.3.6. Compare the fit to that of the model fitted in Example 9.3.7.

Sheryl Ezze

Numerade Educator

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Problem 21

Consider Example 9.3.7. Consider the regressors $x_1, x_2$, and $x_3=x_1 x_2$ used in that example, plus the regressors $x_4=x_1^2$, and $x_5=x_2^2$. Then find the best subsets of regressors of size 1 , 2,3 , and 4 .

Nick Johnson

Numerade Educator