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Chapter 10: Q6BSC (page 468)

let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5 on page 493. Use a 0.05 significance level.

Head widths (in.) and weights (lb) were measured for 20 randomly selected bears (from Data Set 9 鈥淏ear Measurements鈥� in Appendix B). The 20 pairs of measurements yield\(\bar x = 6.9\)in.,\(\bar y = 214.3\)lb, r= 0.879, P-value = 0.000, and\(\hat y = - 212 + 61.9x\). Find the best predicted value of\(\hat y\)(weight) given a bear with a head width of 6.5 in.

Short Answer

Expert verified

The predicted value of the \(\hat y\)(weight) given a bear with a head width of 6.5 in is 190.35 lb.

Step by step solution

01

Given information

The sample number of bears is\(n = 20\). x represents thehead widths and y represents head weights.

The mean head width and weight are \(\bar x = 6.9\) and \(\bar y = 214.3\). The correlation coefficient is \(r = 0.879\) and the P-value is 0.000. The regression equation is \(\hat y = - 212 + 61.9x\).

02

Analyse the model

The statistical hypotheses are formed as,

\({H_0}:\)The correlation coefficient is not significant.

\({H_1}:\)The correlation coefficient is significant.

Since the P-value (0.000) is less than the level of significance (0.05). In this case, the null hypothesis is rejected.

Therefore,the correlation coefficient is significant.

Referring to Figure 10-5, the regression model is a good model and thus the regression equation can be used to predict the value of y.

03

Determine the predicted value

Thepredicted value is computed as,

\(\begin{array}{c}\hat y = - 212 + \left( {61.9 \times 6.5} \right)\\ = - 212 + 402.35\\ = 190.35\end{array}\).

Thus, the predicted value of the \(\hat y\)(weight) for a bear with a head width of 6.5 in is 190.35 lb.

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Most popular questions from this chapter

Stocks and Sunspots. Listed below are annual high values of the Dow Jones Industrial Average (DJIA) and annual mean sunspot numbers for eight recent years. Use the data for Exercises 1鈥�5. A sunspot number is a measure of sunspots or groups of sunspots on the surface of the sun. The DJIA is a commonly used index that is a weighted mean calculated from different stock values.

DJIA

14,198

13,338

10,606

11,625

12,929

13,589

16,577

18,054

Sunspot

Number

7.5

2.9

3.1

16.5

55.7

57.6

64.7

79.3

Correlation Use a 0.05 significance level to test for a linear correlation between the DJIA values and the sunspot numbers. Is the result as you expected? Should anyone consider investing in stocks based on sunspot numbers?

Notation Twenty different statistics students are randomly selected. For each of them, their body temperature (掳C) is measured and their head circumference (cm) is measured.

a. For this sample of paired data, what does r represent, and what does \(\rho \)represent?

b. Without doing any research or calculations, estimate the value of r.

c. Does r change if the body temperatures are converted to Fahrenheit degrees

Cigarette Nicotine and Carbon Monoxide Refer to the table of data given in Exercise 1 and use the amounts of nicotine and carbon monoxide (CO).

a. Construct a scatterplot using nicotine for the xscale, or horizontal axis. What does the scatterplot suggest about a linear correlation between amounts of nicotine and carbon monoxide?

b. Find the value of the linear correlation coefficient and determine whether there is sufficient evidence to support a claim of a linear correlation between amounts of nicotine and carbon monoxide.

c. Letting yrepresent the amount of carbon monoxide and letting xrepresent the amount of nicotine, find the regression equation.

d. The Raleigh brand king size cigarette is not included in the table, and it has 1.3 mg of nicotine. What is the best predicted amount of carbon monoxide?

Tar

25

27

20

24

20

20

21

24

CO

18

16

16

16

16

16

14

17

Nicotine

1.5

1.7

1.1

1.6

1.1

1.0

1.2

1.4

Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2

1.3

1.5

1.6

8

3.4

3.5

2.8

2.6

2.2

3

7.1

2.3

2.1

3.4

6.4

5

4.2

2.8

3.9

5.2

6.9

6.9

5

5.5

6

5.5

8.6

9.4

10

7.6










Pain Intensity After Duragesic Treatment

0.4

1.4

1.8

2.9

6

1.4

0.7

3.9

0.9

1.8

0.9

9.3

8

6.8

2.3

0.4

0.7

1.2

4.5

2

1.6

2

2

6.8

6.6

4.1

4.6

2.9

5.4

4.8

4.1










Two Independent Samples The methods of Section 9-2 can be used to test the claim that two populations have the same mean. Identify the specific claim that the treatment is effective, then use the methods of Section 9-2 to test that claim. The methods of Section 9-2 are based on the requirement that the samples are independent. Are they independent in this case?

let the predictor variable x be the first variable given. Use the given data to find the regression equation and the best predicted value of the response variable. Be sure to follow the prediction procedure summarized in Figure 10-5 on page 493. Use a 0.05 significance level.

For 30 recent Academy Award ceremonies, ages of Best Supporting Actors (x) and ages of Best Supporting Actresses (y) are recorded. The 30 paired ages yield\(\bar x = 52.1\)years,\(\bar y = 37.3\)years, r= 0.076, P-value = 0.691, and

\(\hat y = 34.4 + 0.0547x\). Find the best predicted value of\(\hat y\)(age of Best Supporting Actress) in 1982, when the age of the Best Supporting Actor (x) was 46 years.
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