Question

1. a sample of full - time workers in the u.s. were surveyed and asked their age as well as how many vacation days they used last year. a scatterplot of the data is shown below.

a) describe the relationship between age and vacation days used for these workers.
b) would the value of the correlation r be positive or negative? how do you know?

Explanation:

Part (a)

To describe the relationship, we analyze the scatterplot's direction, form, and strength. The points show a general trend: as age (x - axis) increases, vacation days used (y - axis) tend to decrease. The relationship is negative (downward trend), linear (roughly follows a straight - line pattern), and moderately strong (points cluster around a line but with some scatter).

The correlation coefficient \( r \) measures the strength and direction of a linear relationship. A positive \( r \) indicates a positive linear relationship (as one variable increases, the other increases), and a negative \( r \) indicates a negative linear relationship (as one variable increases, the other decreases). From the scatterplot, as age (the explanatory variable) increases, vacation days used (the response variable) decreases, so the linear relationship is negative. Thus, \( r \) should be negative.

Answer:

The relationship between age and vacation days used is a negative, linear, and moderately strong association. As age increases, the number of vacation days used tends to decrease.

Part (b)