12. Regression

Using the sample data below, run a hypothesis test on $\beta$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $\alpha =0.01$.

[DATA] Weight of a Car versus Miles per Gallon (Refer to Problem 37, Section 4.1.) An engineer wants to determine how the weight of a car, x, affects gas mileage, y. The following data represent the weights of various domestic cars and their miles per gallon in the city for the 2015 model year.

b. Explain how to interpret the slope and y-intercept, if it makes sense to do so.

Using the sample data below, create a confidence interval for $\beta$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $\alpha =0.01$.

You Explain It! Study Time and Exam Scores

After the first exam in a statistics course, Professor Katula surveyed 14 randomly selected students to determine the relation between the amount of time they spent studying for the exam and exam score. She found that a linear relation exists between the two variables. The least-squares regression line that describes this relation is:

ŷ = 6.3333x + 53.0298

b. Interpret the slope.

Bull Markets A bull market is defined as a market condition in which the price of a security rises for an extended period of time. A bull market in the stock market is often defined as a condition in which a market rises by 20% or more without a 20% decline. The data to the right represent the number of months and percentage change in the S&P 500 (a group of 500 stocks) during the 25 bull markets dating back to 1929 (the year of the famous market crash).

e. Interpret the slope.

Income and Education

In Problem 15 from Section 4.1, a scatter diagram and correlation coefficient suggested there is a linear relation between the percentage of individuals who have at least a bachelor's degree and median income in the states. In fact, the least-squares regression equation is ŷ = 1103x + 31,955 where y is the median income and x is the percentage of individuals 25 years and older with at least a bachelor's degree in the state.

c. Interpret the slope.

Age Gap at Marriage

Is there a relation between the age difference between husband and wife and the percent of a country that is literate? Researchers found the least-squares regression between age difference (husband age minus wife age), y, and literacy rate (percent of the population that is literate), x, is

ŷ = -0.0527x + 7.1

The model applied for 18 ≤ x ≤ 100.

Source: Xu Zhang and Solomon W. Polachek, State University of New York at Binghamton “The Husband-Wife Age Gap at First Marriage: A Cross-Country Analysis.”

a. Interpret the slope.