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.

You Explain It! CO2 and Energy Production

The least-squares regression equation ŷ = 1.3491x - 11.2847 relates the carbon dioxide emissions (in millions of tonnes), y, and electricity produced (terawatt-hours), x, for all countries in the world.

Source: www.bp.com

a. Interpret the slope.

Problems 17–22 use the results from Problems 27–32 in Section 4.1.

"[NW] [DATA] An Unhealthy Commute (Refer to Problem 27, Section 4.1.) The following data represent commute times (in minutes) and score on a well-being survey.

b. Explain the meaning of the slope and y-intercept in this context, if possible.

Problems 17–22 use the results from Problems 27–32 in Section 4.1.

[DATA] Hurricanes (Refer to Problem 32, Section 4.1.) The following data represent the maximum wind speed (in knots) and atmospheric pressure (in millibars) for a random sample of hurricanes that originated in the Atlantic Ocean.

b. Explain what the slope represents.

Problems 17–22 use the results from Problems 27–32 in Section 4.1.

[DATA] Credit Scores (Refer to Problem 28, Section 4.1.) An economist wants to determine the relation between one’s FICO score, x, and the interest rate of a 36-month auto loan, y. The given data represent the interest rate (in percent) a bank would offer on a 36-month auto loan for various FICO scores.

b. Describe what the slope and y-intercept represent in this context. Note: Credit scores range from 300 to 850.

Problems 17–22 use the results from Problems 27–32 in Section 4.1.

[DATA] Height versus Head Circumference (Refer to Problem 29, Section 4.1.) A pediatrician wants to determine the relation that exists between a child’s height, x, and head circumference, y. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the following data.

b. Explain the meaning of the slope and y-intercept in this context, if applicable.

Problems 17–22 use the results from Problems 27–32 in Section 4.1.

[DATA] American Black Bears (Refer to Problem 30, Section 4.1.) The American black bear (Ursus americanus) is one of eight bear species in the world. It is the smallest North American bear and the most common bear species on the planet. In 1969, Dr. Michael R. Pelton of the University of Tennessee initiated a long-term study of the population in Great Smoky Mountains National Park. One aspect of the study was to develop a model that could be used to predict a bear’s weight (since it is not practical to weigh bears in the field). One variable thought to be related to weight is the length of the bear. The data below represent the lengths of 12 American black bears.

b. Explain the significance of the slope and y-intercept in this situation, if applicable.

Cola Consumption vs. Bone Density Example 5 in Section 4.1 discussed the effect of cola consumption on bone mineral density in the femoral neck of women.

b. Explain the meaning of the slope.

Using the data and results from Problem 2, complete the following:

c. Interpret the slope and y-intercept, if applicable.