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Chapter 10: Q5BSC (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.

For 50 randomly selected speed dates, attractiveness ratings by males of their

female date partners (x) are recorded along with the attractiveness ratings by females of their male date partners (y); the ratings are from Data Set 18 鈥淪peed Dating鈥� in Appendix B. The 50 paired ratings yield\(\bar x = 6.5\),\(\bar y = 5.9\), r= -0.277, P-value = 0.051, and\(\hat y = 8.18 - 0.345x\). Find the best predicted value of\(\hat y\)(attractiveness rating by female of male) for a date in which the attractiveness rating by the male of the female is x= 8.

Short Answer

Expert verified

The regression equation is\(\hat y = 8.18 - 0.345x\).

The predicted value of \(\hat y\) , that is, attractiveness rating by a female of male for a date in which the attractiveness rating by the male of the female is x= 8 is 5.9.

Step by step solution

01

Given information

The sample number of speed dates is\(n = 20\). Variable x represents theattractiveness ratings by males of their female date partners and y represents the attractiveness ratings by females of their male date partners.

The mean ratings are \(\bar x = 6.5\) and \(\bar y = 5.9\). The correlation coefficient is \(r = - 0.277\) and the P-value is 0.051. The regression equation is \(\hat y = 8.18 - 0.345x\).

02

Analyse the regression 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 (0.051) is greater than the level of significance (0.05), in this case, the null hypothesis fails to be rejected.

Therefore, there is enough evidence that the correlation coefficient is not significant.

Refer to Figure 10-5 for the description of a good model.

As the correlation coefficient is not significant, the regression model is a bad model.

Therefore, the regression equation cannot be used to predict the value of y.

03

Determine the predicted value

Resultant to conclusion, the bestpredicted valuefor a bad model is equal to its mean; that is \(\bar y = 5.9\).

Thus, the predicted value of the\(\hat y\)(attractiveness rating by the female of male) for a date in which the attractiveness rating by the male of the female is x= 8 is 5.9.

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Most popular questions from this chapter

In Exercises 5鈥�8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to the

StatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 鈥淔amily Heights鈥� in Appendix B.

Identify the following:

a. The P-value corresponding to the overall significance of the multiple regression equation

b. The value of the multiple coefficient of determination\({R^2}\).

c. The adjusted value of \({R^2}\)

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Bears r = 0.783 (x = head width of a bear, y = weight of a bear)

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a. What is the percentage of moviegoers who are younger than 30 years of age?

b. Find\({P_{25}}\), which is the 25th percentile.

c. Find the probability that a simple random sample of 25 moviegoers has a mean age that is less than 30 years.

d. Find the probability that for a simple random sample of 25 moviegoers, each of the moviegoers is younger than 30 years of age. For a particular movie and showtime, why might it not be unusual to have 25 moviegoers all under the age of 30?

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Internet and Nobel Laureates Listed below are numbers of Internet users per 100 people and numbers of Nobel Laureates per 10 million people (from Data Set 16 鈥淣obel Laureates and Chocolate鈥� in Appendix B) for different countries. Is there sufficient evidence to conclude that there is a linear correlation between Internet users and Nobel Laureates?

Internet Users

Nobel Laureates

79.5

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9

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Identifying a Model and\({R^2}\)Different samples are collected, and each sample consists of IQ scores of 25 statistics students. Let x represent the standard deviation of the 25 IQ scores in a sample, and let y represent the variance of the 25 IQ scores in a sample. What formula best describes the relationship between x and y? Which of the five models describes this relationship? What should be the value of\({R^2}\)?
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