MTH 131 Ch 4 Text HW Solutions

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Northwestern Michigan College *

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Statistics

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Jan 9, 2024

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Chapter 4 1. The correlation coefficient, r, is a number between . a. -1 and 1 2. The coefficient of determination is a number between . e. 0 and 1 3. Which of the following is not a valid linear regression equation? How could it be fixed to be valid? d. y = 5 + 0.4 x This could be fixed by adding the “hat” over the y to designate it as a predicted value. 4. Body frame size is determined by a person's wrist circumference in relation to height. A researcher measures the wrist circumference and height of a random sample of individuals. The output from SALT is recreated in the table below. a. What is the value of the correlation coefficient? 0.7693 b. Find and interpret the coefficient of determination using context from the problem. 0.7693 2 = 0.5918 . About 59% of the variation in height is explained by the model using wrist circumference.

5. Bone mineral density and cola consumption has been recorded for a sample of patients. Let x represent the number of colas consumed per week and y the bone mineral density in grams per cubic centimeter. Assume the conditions are met. Calculate the correlation coefficient using technology. -0.8241 6. A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let x represent the third homework score and y the midterm exam score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed. Calculate the correlation coefficient using technology. 0.9828 7. An object is thrown from the top of a building. The following data measure the height of the object from the ground for a five-second period. Calculate the correlation coefficient using technology. -0.9422 8. A teacher believes that the third homework assignment is a key predictor in how well students will do on the midterm. Let x represent the third homework score and y the midterm exam score. A random sample of last terms students were selected and their grades are shown in question 6. Assume all conditions are met. a. Find the regression equation. ^ y = 2.82 x + 26.08 b. Find the predicted midterm score when the homework 3 score is 15. ^ y = 2.82 ∗ 15 + 26.08 = 68.38 9. Body frame size is determined by a person's wrist circumference in relation to height. A researcher measures the wrist circumference and height of a random sample of individuals. Output from SALT is displayed below. Variables Number of Observations Standard Deviation Mean Correlation Slope Intercept Wrist Circumference (in) 33 0.9707 6.9788 0.7693 5.0321 34.443 Height (in) 6.3501 69.5606 a. Write the regression equation? ^ y = 34.443 + 5.0321 x b. What is the predicted height (in inches) for a person with a wrist circumference of 7 inches? ^ y = 34.443 + 5.0321 ∗ 7 = 69.6677 . Someone with a wrist circumference of 7 inches is predicted to be about 69.7 inches tall.

10. Bone mineral density and cola consumption has been recorded for a sample of patients. Let x represent the number of colas consumed per week and y the bone mineral density in grams per cubic centimeter. Assume the data is normally distributed. Calculate and interpret the coefficient of determination using technology. Correlation coefficient is -0.8241, so the coefficient of determination is ( − 0.8241 ) 2 = 0.6791 . About 68% of the variation in bone density is explained by the model using number of colas consumed per week. 11. Bone mineral density and cola consumption has been recorded for a sample of patients. Let x represent the number of colas consumed per week and y the bone mineral density in grams per cubic centimeter. Assume the data is normally distributed. A regression equation for the following data is ^ y = 0.8893 − 0.0031 x . Which is the best interpretation of the slope coefficient? b. For every additional average weekly soda consumption, a person’s bone density decreases by 0.0031 grams per cubic centimeter. 12. The following data represent the leaching rates (percent of lead extracted vs. time in minutes) for lead in solutions of magnesium chloride (MgCl 2 ). a. Find the correlation coefficient. 0.9403 b. Find the coefficient of determination. 0.9403 2 = 0.8841 c. Find the regression equation. ^ y = 0.0257 x + 1.6307 d. Find the predicted value for 100 minutes. ^ y = 0.0257 ∗ 100 + 1.6307 = 4.2 e. Write a sentence interpreting the predicted value using units and context. After 100 minutes, we expect about 4.2 percent of lead extracted in a solution of magnesium chloride. 13. A study was conducted to determine if there was a linear relationship between a person's age and their peak heart rate (beats per minute). Age (x) 16 26 32 37 42 53 48 21 Peak Heart Rate (y) 220 194 193 178 172 160 174 214 a. What is the estimated regression equation that relates a person’s age and their peak heart rate. ^ y = 241.8127 − 1.5618 x b. Interpret the slope coefficient for this problem. When aging one year, we expect the peak heart rate to decrease by about 1.6 beats per minute c. Find and interpret the coefficient of determination. − 0.9681 2 = 0.9372 . About 93.7% of the variation in peak heart rate is explained by the model using age. d. Find and interpret the predicted peak heart rate for someone that is 25 years old. ^ y = 241.8127 − 1.5618 ∗ 25 = 202.8 beats per minute.

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14. The following data represent the weight of a child riding a bike and the rolling distance achieved after going down a hill without pedaling. a. Find the correlation coefficient. Based on the correlation coefficient, should we use the model with this data? 0.9568, yes we should use this model. b. Find the regression line for this problem and use it to find the approximate rolling distance for a child on a bike that weighs 150 lbs. Is this a reasonable approximation? Explain. ^ y = 0.6117 x − 8.5647 ^ y = 0.6117 ∗ 150 − 8.5647 = 83.19 meters. This is not a reasonable approximate as this value is far from all other data we have and is an extrapolation. 15. In a sample of 20 football players for a college team, their weight and 40-yard-dash time was recorded in seconds. The table below shows the results. A strong linear correlation was found between the two variables. Find the predicted 40-yard dash value for a football player that is 200 lbs. ^ y = 3.7223 + 0.0064 x ^ y = 3.7223 + 0.0064 ∗ 200 = 5.0023 . A football player weighting 200 pounds is expected to take about 5 seconds to run a 40-yard dash. 16. The following data represent the age of a car and the average month cost spent on repairs. A significant linear correlation is found between the two variables. Use the data to find a prediction for the monthly cost of a vehicle that is 15 years old. ^ y = 18.1333 + 7.9576 x ^ y = 18.1333 + 7.9576 ∗ 15 = 137.4973 . We expect the monthly cost of a vehicle that is 15 years old to be about $137.50. (Note: While not super far from our data, it is a bit away from the rest of the data and should be used with caution. It is possible that after a car is 10 years that the monthly cost changes from the pattern presented here.) 17. The following data represent the enrollment at a small college during its first ten years of existence. A significant linear relationship is found between the two variables. Find a prediction for the enrollment after the college has been open for 14 years. ^ y = 826.2 + 25.4909 x ^ y = 826.2 + 25.4909 ∗ 14 = 1182.0726 . The predicted enrollment for a college that has been open for 14 years is 1182 students. 18. The table below shows the percentage of adults in the United States who were married before age 24 for several years. A significant linear relationship was found between the two variables. Find the prediction for the percentage of adults who married before age 24 in the United States in 2015. ^ y = 1259.9691 − 0.6156 x ^ y = 1259.9691 − 0.6156 ∗ 2015 = 19.5351 . We expect about 19.5% of people to be married before 24 years old in 2015.

19. It has long been thought that the length of one’s femur is positively correlated to the length of one’s tibia. The following are data for a classroom of students who measured each (approximately) in inches. A significant linear correlation was found between the two variables. Find the prediction for the length of someone’s tibia when it is known that their femur is 23 inches long. Is this a reasonable approximation? Explain. ^ y = 3.585 + 0.5899 x ^ y = 3.585 + 0.5899 ∗ 23 = 17.1527 . We expect a person’s tibia to be about 17 inches long when their femur is 23 inches long. 20. A new fad diet called Trim-to-the-MAX is running some tests that they can use in advertisements. They sample 25 of their users and record the number of days each has been on the diet along with how much weight they have lost in pounds. The data is below. A significant linear correlation was found between the two variables. Find the prediction for the weight lost when a person has been on the diet for 60 days. ^ y = 0.4912 + 0.6947 x ^ y = 0.4912 + 0.6947 ∗ 60 = 42.1732 . We expect someone who has been on the diet for 60 days to have lost about 42 pounds. 21. The data below represent the driving speed (mph) of a vehicle and the corresponding gas mileage (mpg) for several recorded instances. Find the prediction for gas mileage when a vehicle is driving at 77 mph. ^ y = 32.4031 − 0.1662 x ^ y = 32.4031 − 0.1662 ∗ 77 = 19.6057 . We expect the vehicle to get about 19.6 mpg for gas mileage when driving at 77 mph. 22. An elementary school uses the same system to test math skills at their school throughout the course of the 5 grades at their school. The age and score (out of 100) of several students is displayed below. Find a prediction for the score a student would earn given that he/she is 5 years old. ^ y = 15.2059 + 5.5809 x ^ y = 15.2059 + 5.5809 ∗ 5 = 43.1104 . We would expect a 5 year old to score about 43 points on the test. 23. A nutritionist feels that what mothers eat during the months they are nursing their babies is important for healthy weight gain of their babies. She samples several of her clients and records their average daily caloric intake for the first three months of their babies’ lives and also records the amount of weight the babies gained in those three months. The data are below. A significant linear correlation is found between the two variables. Find the prediction for the weight gain of a baby whose mother gets 2500 calories per day. Is this a reasonable approximation? Explain. ^ y = 1.7627 + 0.0019 x ^ y = 1.7627 + 0.0019 ∗ 2500 = 6.5127 . A baby who’s mother get 2500 calories per day is expected to gain about 6.5 points in the first 3 months of their life. This is a reasonable approximation as 2500 is reasonably close to the data and the correlation is 0.8864 suggesting a strong relationship.

24. The data below show the predicted average high temperature ( o F) per month by the Farmer’s Almanac in Portland, OR alongside the actual high temperature per month that occurred. A significant positive linear correlation is found between the two temperatures. Find a prediction for the actual high temperature in the coming year, given that the Farmer’s Almanac is predicting the high to be 58 o F. ^ y =− 3.2853 + 1.0944 x ^ y =− 3.2853 + 1.0944 ∗ 58 = 60.1899 . The predicted actual high temperature is about 60.2 o F when the Farmer’s Almanac predicts 58 o F

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MTH 131 Ch 4 Text HW Solutions

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