AP Statistics Help: Linear Regression Models

Question 1

1. For each additional year of age, the predicted weekly streaming time decreases by about 0.15 hours, on average, for users like those sampled.
2. If a user gets 1 year older, that causes their weekly streaming time to drop by exactly 0.15 hours.
3. At age 0, the model predicts 18 hours per week, so newborns would typically stream about 18 hours weekly.
4. A 120-year-old is predicted to stream $18.0-0.15(120)$ hours per week, and this prediction is as reliable as those for ages 13–62.
5. The intercept 18.0 means that 18% of users stream each week.

Question 2

1. For each additional gram of fertilizer, the predicted plant height increases by about 0.48 cm, on average, for plots like those observed.
2. Applying one more gram of fertilizer causes every plant to grow 0.48 cm taller than it otherwise would.
3. If 0 grams of fertilizer are applied, the plant height will be exactly 12.4 cm.
4. At 200 grams of fertilizer, the model can be used to predict plant height accurately because regression lines work for any $x$.
5. The intercept 12.4 means fertilizer explains 12.4% of plant height.

Question 3

1. For each 1°C increase in water temperature, the predicted dissolved oxygen decreases by about 0.18 mg/L, on average, for sites like those measured.
2. Raising the temperature by 1°C causes dissolved oxygen to decrease by exactly 0.18 mg/L at every site.
3. At 0°C, dissolved oxygen will be exactly 12.1 mg/L, so the model is accurate for freezing conditions.
4. Because the slope is negative, higher temperatures and dissolved oxygen are independent.
5. The intercept 12.1 means 12.1% of the sites had 0 dissolved oxygen.

Question 4

1. If a student studies 0 hours, the model guarantees the student will score exactly 58.2 points.
2. For each additional hour studied, the predicted exam score increases by about 4.1 points, on average, for students similar to those in the data.
3. Because the slope is positive, studying an extra hour causes every student’s score to increase by 4.1 points.
4. A student who studies 20 hours is predicted to score $58.2+4.1(20)$ points, so the model is accurate for any study time.
5. About 58.2% of the variation in exam scores is explained by study hours.

Question 5

1. A house that is 0 square feet would be predicted to sell for $95{,}000, so the model is unrealistic and cannot be used at all.
2. For each additional 100 square feet of size, the predicted selling price increases by about $8{,}700$, on average, for houses like those observed.
3. Increasing a home’s size by 100 square feet causes the selling price to increase by $8{,}700$ for every home.
4. A 3500-square-foot house (x = 35) is predicted to sell for $95+8.7(35)$ thousand dollars, so the model should be used for any home size.
5. The intercept 95 means most houses in the neighborhood are about $95{,}000$.

Question 6

1. Each additional mile from the city center is associated with a decrease of about \42$ in the predicted monthly rent, on average, for apartments like those observed.
2. Moving an apartment 1 mile farther from the city center causes the rent to drop by \42$.
3. At 0 miles from the city center, every apartment will rent for exactly \1850$.
4. An apartment 40 miles away is predicted to rent for $1850-42(40)$, so the model is valid far beyond the observed distances.
5. The intercept 1850 means the average rent of all apartments in the city is \1850$.

Question 7

1. Increasing a school’s average class size by 1 student will cause the school’s average test score to drop by 3.5 points.
2. For each additional student in average class size, the predicted average test score decreases by about 3.5 points, on average, for schools like those studied.
3. Because the intercept is 610, a school with 0 students per class would score 610 points, and this is a reliable prediction.
4. The equation shows that smaller classes are the only reason some schools have higher scores.
5. Since the slope is negative, the correlation must be $r=0$.

Question 8

1. For each additional 1,000 cars per day (within 10–60 thousand), the predicted ozone level increases by about 1.1 ppb, on average.
2. If traffic volume is 0, then the ozone level will be 18 ppb.
3. An increase of 1.1 ppb in ozone causes traffic volume to increase by 1,000 cars per day.
4. Because the slope is positive, increasing traffic causes ozone to increase by 1.1 ppb for every additional 1,000 cars.
5. At 100 thousand cars per day, the model predicts 128 ppb, so it is appropriate to use the model at 100 thousand cars per day.

Question 9

1. For each additional 1,000 pounds of vehicle weight, the predicted highway fuel economy decreases by about 5.2 mpg, on average, for cars like those in the study.
2. Reducing a car’s weight by 1,000 pounds will cause its highway mpg to increase by exactly 5.2 for every car.
3. A car that weighs 0 pounds would be predicted to get 46.0 mpg, and that prediction is meaningful because it comes from the model.
4. Because the slope is negative, there is no relationship between weight and mpg.
5. The intercept 46.0 means that 46% of cars get 0 mpg when weight is 0.

Question 10

1. Each additional customer served is associated with an increase of about $0.62 in the predicted total tips for that hour, on average, for shifts like those observed.
2. Serving one more customer causes tips to increase by exactly $0.62 every hour.
3. If 0 customers are served, the server will earn exactly $8.50 in tips.
4. A shift with 120 customers can be predicted accurately using the model since the relationship is linear.
5. The intercept 8.5 means most hours have about 8.5 customers.