Essential Statistics
Essential Statistics
2nd Edition
ISBN: 9781259570643
Author: Navidi
Publisher: MCG
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Chapter 11.4, Problem 10E

a.

To determine

Find b0 and b1.

a.

Expert Solution
Check Mark

Answer to Problem 10E

The intercept b0 is 31.53.

The slope b1 is 0.691.

Explanation of Solution

Calculation:

The given information is that the sample data consists of 8 values for x and y.

Slope or b1:

b1=rsysx

where,

r represents the correlation coefficient between x and y.

sy represents the standard deviation of y.

sx represents the standard deviation of x.

Software procedure:

Step-by-step procedure to find the mean, standard deviation for x and y values using MINITAB is given below:

  • Choose Stat > Basic Statistics > Display Descriptive Statistics.
  • In Variables enter the columns of y and x.
  • Choose Options Statistics, select Mean and Standard deviation.
  • Click OK.

Output obtained from MINITAB is given below:

Essential Statistics, Chapter 11.4, Problem 10E , additional homework tip  1

Correlation:

r=1n1(xx¯sx)(yy¯sy)

Where,

y¯ represents the mean of y values.

x¯ represents the mean of x values.

sx represents the standard deviation of x.

sy represents the standard deviation of y.

n represents the sample size.

The table shows the calculation of correlation:

xyxx¯yy¯xx¯sxyy¯sy(xx¯sx)(yy¯sy)
23511.874.870.3695650.2897090.107066
1622–5.13–24.13–1.01383–1.435461.455313
1756–4.139.87–0.816210.587151–0.47924
1934–2.13–12.13–0.42095–0.721590.303754
30678.8720.871.7529641.2415232.176345
1959–2.1312.87–0.420950.765616–0.32228
1855–3.138.87–0.618580.527662–0.3264
27255.87–21.131.160079–1.25699–1.45821
Total     1.456

Thus, the correlation is

r=1.45681=1.4567=0.208

b1=rsysx

Substitute r as 0.208, sy as 16.81 and sx5.06

b1=0.208(16.815.06)=0.208(3.32)=0.691

Thus, the slope b1 is 0.691.

Intercept or b0:

b0=y¯b1x¯

y¯ represents the mean of y values.

x¯ represents the mean of x values.

b1 represents the slope coefficient.

Substitute y¯ as 46.13, x¯ as 21.13 and b1 as 0.691.

b0=46.130.691(21.13)=46.1314.60=31.53

Thus, the intercept b0 is 31.53.

b.

To determine

Find the predicted value y^ using the given value of x.

b.

Expert Solution
Check Mark

Answer to Problem 10E

The predicted value y^ is 48.805.

Explanation of Solution

Calculation:

The given value of x is 25.

The estimated regression equation is y^=31.53+0.691x

Substitute x as 25,

y^=31.53+0.691(25)=31.53+17.275=48.805

Thus, the predicted value y^ is 48.805.

c.

To determine

Find the residual standard deviation se.

c.

Expert Solution
Check Mark

Answer to Problem 10E

The residual standard deviation se is 17.753.

Explanation of Solution

Calculation:

The residual standard deviation se is calculated using the formula,

se=(yy^)2n2

Where,

(yy^)2 represents the sum of squares due to error

n represents the sample size.

Use the estimated regression equation to find the predicted value of y for each value of x.

yy^  Essential Statistics, Chapter 11.4, Problem 10E , additional homework tip  2yy^(yy^)2
5147.4233.57712.79493
2242.586–20.586423.7834
5643.27712.723161.8747
3444.659–10.659113.6143
6752.2614.74217.2676
5944.65914.341205.6643
5543.96811.032121.705
2550.187–25.187634.385
Total  1,891.09

Substitute (yy^)2 as 1,891.09 and n as 8.

se=1,891.0982=1,891.096=315.18=17.753

Thus, the residual standard deviation se is 17.753.

d.

To determine

Find the sum of squares for x.

d.

Expert Solution
Check Mark

Answer to Problem 10E

The sum of squares for x is 178.8752.

Explanation of Solution

Calculation:

The table shows the calculation of sum of squares for x:

xxx¯(xx¯)2
231.873.4969
16-5.1326.3169
17-4.1317.0569
19-2.134.5369
308.8778.6769
19-2.134.5369
18-3.139.7969
275.8734.4569
Total 178.8752

Thus, the sum of squares for x is 178.8752.

e.

To determine

Find the critical value for a 95% confidence or prediction interval.

e.

Expert Solution
Check Mark

Answer to Problem 10E

The critical value for a 95% confidence or prediction interval is 2.447.

Explanation of Solution

Calculation:

Critical value:

Software procedure:

Step-by-step procedure to find the critical value using MINITAB is given below:

  • Choose Graph > Probability Distribution Plot choose View Probability > OK.
  • From Distribution, choose ‘t’ distribution.
  • In Degrees of freedom, enter 4.
  • Click the Shaded Area tab.
  • Choose Probability and Two tail for the region of the curve to shade.
  • Enter the Probability value as 0.05.
  • Click OK.

Output obtained from MINITAB is given below:

Essential Statistics, Chapter 11.4, Problem 10E , additional homework tip  3

Thus, the critical value for a 95% confidence or prediction interval is 2.447.

f.

To determine

Construct the 95% confidence interval for the mean response for the given value of x.

f.

Expert Solution
Check Mark

Answer to Problem 10E

The 95% confidence interval for the mean response for the given value of x is (28.9515,68.6656).

Explanation of Solution

Calculation:

The given value of x is 25.

Software procedure:

Step-by-step procedure to construct the 95% confidence interval for the mean response for the given value of x is given below:

  • Choose Stat > Regression > Regression.
  • In Response, enter the column containing the y.
  • In Predictors, enter the columns containing the x.
  • Click OK.
  • Choose Stat > Regression > Regression>Predict.
  • Choose Enter the individual values.
  • Enter the x as 25.
  • Click OK.

Output obtained from MINITAB is given below:

Essential Statistics, Chapter 11.4, Problem 10E , additional homework tip  4

Interpretation:

Thus, the 95% confidence interval for the mean response for the given value of x is (28.9515,68.6656)

g.

To determine

Construct the 95% prediction interval for the individual response for the given value of x.

g.

Expert Solution
Check Mark

Answer to Problem 10E

The 95% prediction interval for the individual response for the given value of x is (1.04439, 96.5727).

Explanation of Solution

From the MINITAB output obtained in the previous part (f) it can be observed that the 95% prediction interval for the individual response for the given value of x is (1.04439, 96.5727)

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Chapter 11 Solutions

Essential Statistics

Ch. 11.1 - Prob. 11ECh. 11.1 - Prob. 12ECh. 11.1 - Prob. 13ECh. 11.1 - Prob. 14ECh. 11.1 - Prob. 15ECh. 11.1 - Prob. 16ECh. 11.1 - Prob. 17ECh. 11.1 - Prob. 18ECh. 11.1 - Prob. 19ECh. 11.1 - Prob. 20ECh. 11.1 - Prob. 21ECh. 11.1 - Prob. 22ECh. 11.1 - Prob. 23ECh. 11.1 - Prob. 24ECh. 11.1 - Prob. 25ECh. 11.1 - Prob. 26ECh. 11.1 - Prob. 27ECh. 11.1 - In Exercises 25–30, determine whether the...Ch. 11.1 - Prob. 29ECh. 11.1 - Prob. 30ECh. 11.1 - Prob. 31ECh. 11.1 - Prob. 32ECh. 11.1 - 33. Pass the ball: The NFL Scouting Combine is an...Ch. 11.1 - 34. Carbon footprint: Carbon dioxide (CO2) is...Ch. 11.1 - 35. Foot temperatures: Foot ulcers are a common...Ch. 11.1 - Prob. 36ECh. 11.1 - Prob. 37ECh. 11.1 - Prob. 38ECh. 11.1 - Prob. 39ECh. 11.1 - Prob. 40ECh. 11.1 - Prob. 41ECh. 11.1 - Prob. 42ECh. 11.1 - Prob. 43ECh. 11.2 - 1. The following table presents the percentage of...Ch. 11.2 - 2. At the final exam in a statistics class, the...Ch. 11.2 - 3. For each of the following plots, interpret the...Ch. 11.2 - Prob. 4CYUCh. 11.2 - Prob. 5ECh. 11.2 - In Exercises 5–7, fill in each blank with the...Ch. 11.2 - Prob. 7ECh. 11.2 - Prob. 8ECh. 11.2 - In Exercises 8–12, determine whether the statement...Ch. 11.2 - Prob. 10ECh. 11.2 - Prob. 11ECh. 11.2 - Prob. 12ECh. 11.2 - Prob. 13ECh. 11.2 - Prob. 14ECh. 11.2 - Prob. 15ECh. 11.2 - Prob. 16ECh. 11.2 - Prob. 17ECh. 11.2 - Prob. 18ECh. 11.2 - Prob. 19ECh. 11.2 - Prob. 20ECh. 11.2 - Prob. 21ECh. 11.2 - Prob. 22ECh. 11.2 - Prob. 23ECh. 11.2 - Prob. 24ECh. 11.2 - Prob. 25ECh. 11.2 - Prob. 26ECh. 11.2 - 27. Blood pressure: A blood pressure measurement...Ch. 11.2 - Prob. 28ECh. 11.2 - 29. Interpreting technology: The following display...Ch. 11.2 - Prob. 30ECh. 11.2 - Prob. 31ECh. 11.2 - Prob. 32ECh. 11.2 - Prob. 33ECh. 11.2 - Prob. 34ECh. 11.2 - Prob. 35ECh. 11.3 - Prob. 1CYUCh. 11.3 - Prob. 2CYUCh. 11.3 - Prob. 3CYUCh. 11.3 - Prob. 4CYUCh. 11.3 - Prob. 5CYUCh. 11.3 - Prob. 6CYUCh. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - Prob. 11ECh. 11.3 - Prob. 12ECh. 11.3 - Prob. 13ECh. 11.3 - Prob. 14ECh. 11.3 - Prob. 15ECh. 11.3 - Prob. 16ECh. 11.3 - Prob. 17ECh. 11.3 - Prob. 18ECh. 11.3 - Calories and protein: The following table presents...Ch. 11.3 - Prob. 20ECh. 11.3 - Butterfly wings: Do larger butterflies live...Ch. 11.3 - Blood pressure: A blood pressure measurement...Ch. 11.3 - Prob. 23ECh. 11.3 - Prob. 24ECh. 11.3 - Getting bigger: Concrete expands both horizontally...Ch. 11.3 - Prob. 26ECh. 11.3 - Prob. 27ECh. 11.3 - Prob. 28ECh. 11.3 - Prob. 29ECh. 11.3 - Prob. 30ECh. 11.3 - Prob. 31ECh. 11.4 - Prob. 1CYUCh. 11.4 - Prob. 2CYUCh. 11.4 - Prob. 3ECh. 11.4 - Prob. 4ECh. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 10ECh. 11.4 - Calories and protein: Use the data in Exercise 19...Ch. 11.4 - Prob. 12ECh. 11.4 - Butterfly wings: Use the data in Exercise 21 in...Ch. 11.4 - Prob. 14ECh. 11.4 - Prob. 15ECh. 11.4 - Prob. 16ECh. 11.4 - Prob. 17ECh. 11.4 - Prob. 18ECh. 11.4 - Prob. 19ECh. 11.4 - Prob. 20ECh. 11.4 - Prob. 21ECh. 11 - Prob. 1CQCh. 11 - Prob. 2CQCh. 11 - Prob. 3CQCh. 11 - Prob. 4CQCh. 11 - Prob. 5CQCh. 11 - Prob. 6CQCh. 11 - Prob. 7CQCh. 11 - Prob. 8CQCh. 11 - Prob. 9CQCh. 11 - Prob. 10CQCh. 11 - Prob. 11CQCh. 11 - Prob. 12CQCh. 11 - Prob. 13CQCh. 11 - Prob. 14CQCh. 11 - Prob. 15CQCh. 11 - Prob. 1RECh. 11 - Prob. 2RECh. 11 - Prob. 3RECh. 11 - Prob. 4RECh. 11 - Prob. 5RECh. 11 - Prob. 6RECh. 11 - Prob. 7RECh. 11 - Prob. 8RECh. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Prob. 13RECh. 11 - Interpret technology: The following TI-84 Plus...Ch. 11 - Prob. 15RECh. 11 - Prob. 1WAICh. 11 - Prob. 2WAICh. 11 - Prob. 3WAICh. 11 - Prob. 4WAICh. 11 - Prob. 5WAICh. 11 - Prob. 6WAICh. 11 - Prob. 7WAICh. 11 - Prob. 1CSCh. 11 - Prob. 2CSCh. 11 - Prob. 3CSCh. 11 - Prob. 4CSCh. 11 - Prob. 5CSCh. 11 - Prob. 6CSCh. 11 - Prob. 7CSCh. 11 - Prob. 8CSCh. 11 - Prob. 9CSCh. 11 - Prob. 10CSCh. 11 - Prob. 11CS
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