Question

twenty different statistics students are randomly selected. for each of them, their body temperature (°c) is measured and their head circumference (cm) is measured.
a. for this sample of paired data, what does r represent, and what does ρ represent?
b. without doing any research or calculations, estimate the value of r.
c. does r change if body temperatures are converted to fahrenheit degrees?

a. choose the correct answer below.

a. r is a statistic that represents the value of the linear correlation coefficient computed from the paired sample data, and ρ is a parameter that represents the proportion of the variation in head circumference that can be explained by variation in body temperature.
b. r is a statistic that represents the value of the linear correlation coefficient computed from the paired sample data, and ρ is a parameter that represents the value of the linear correlation coefficient that would be computed by using all of the paired data in the population of all statistics students.
c. r is a statistic that represents the proportion of the variation in head circumference that can be explained by variation in body temperature, and ρ is a parameter that represents the value of the linear correlation coefficient that would be computed by using all of the paired data in the population of all statistics students.

Explanation:

To solve part a, we analyze the definitions of \( r \) (sample linear correlation coefficient) and \(
ho \) (population linear correlation coefficient).

• Option A: Incorrect. The proportion of variation explained is \( r^2 \), not \(

ho \).

• Option B: Correct. \( r \) is a sample statistic (from sample data), and \(

ho \) is the population parameter (from all statistics students’ data) for the linear correlation coefficient.

• Option C: Incorrect. The proportion of variation explained is \( r^2 \), not \( r \), and \(

ho \) is not defined as that proportion.

Answer:

B. \( r \) is a statistic that represents the value of the linear correlation coefficient computed from the paired sample data, and \(
ho \) is a parameter that represents the value of the linear correlation coefficient that would be computed by using all of the paired data in the population of all statistics students.