In a study of the legibility and visibility of highway signs, a Pennsylvania res

Question

In a study of the legibility and visibility of highway signs, a Pennsylvania research firm determined the maximum distance (in feet) at which each of 30 drivers could read a newly designed sign. The 30 participants were volunteers and ranged in age from 18 to 82 years old. The government agency that funded the research hoped to improve highway safety for older drivers and wanted to examine the relationship between age and the sign legibility distance. A scatterplot of the data is shown below. The sample correlation coefficient is r=0.801r=0.801 and the least-squares regression line is

y =576.683.01x

a). The p-value for testing the null hypothesis that the regression slope between age and maximum legibility distance in the population is zero (versus a two-sided alternative) is less than 0.001. Which of the following are correct interpretations of this p-value? Select all that apply.

In less than 0.1% of all possible random samples, we would see a sample regression slope of -3.01 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample regression y-intercept of 576.68 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample coefficient of determination of 0.642 or larger, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample correlation coefficient of -0.801 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

b). What is the scope of inference for this study?

We have evidence of an association between age and maximum distance at which a highway sign was read among the population of all Pennsylvania drivers.

We have evidence that an increase in age causes a decrease in average maximum distance at which a highway sign was read among drivers similar to those in the study.

We have evidence that an increase in age causes a decrease in average maximum distance at which a highway sign was read among the population of all Pennsylvania drivers.

We have evidence of an association between age and maximum distance at which a highway sign was read among drivers similar to those in the study.

Suppose a sample of 10 people had their foot lengths (cm) and heights (inches) measured. The results, along with the regression line, are shown in the scatterplot above. The equation of the regression line, correlation coefficient, and r2 are also shown.

c). Which one of the applet screenshots of null distributions below displays the appropriate way to test whether the true slope between people’s foot length and height is positive?

In less than 0.1% of all possible random samples, we would see a sample regression slope of -3.01 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample regression y-intercept of 576.68 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample coefficient of determination of 0.642 or larger, assuming there is no linear relationship between age and maximum legibility distance in the population.

In less than 0.1% of all possible random samples, we would see a sample correlation coefficient of -0.801 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

Explanation / Answer

a). The p-value for testing the null hypothesis that the regression slope between age and maximum legibility distance in the population is zero (versus a two-sided alternative) is less than 0.001. Which of the following are correct interpretations of this p-value? Select all that apply.

Ans:

In less than 0.1% of all possible random samples, we would see a sample regression slope of -3.01 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

b). What is the scope of inference for this study?

ANS: We have evidence that an increase in age causes a decrease in average maximum distance at which a highway sign was read among the population of all Pennsylvania drivers.

c). Which one of the applet screenshots of null distributions below displays the appropriate way to test whether the true slope between people’s foot length and height is positive?

Ans: a)

Because t=0.014/(0.517/sqrt(1000))=0.8563 is approximately equal to 0.7

In less than 0.1% of all possible random samples, we would see a sample regression slope of -3.01 or further away from zero, assuming there is no linear relationship between age and maximum legibility distance in the population.

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