Quiz #3 (Unit 4, modules 11, 12 and 13) Please Show Work, when possible! (If you don’t, I can’t give you partial credit!) 1. (15 pts.) Researchers at the University of Denver Infant Study Center investigated whether babies take longer to learn to crawl in cold months (when they are often bundled in clothes that restrict their movement) than in warmer months. The study sought an association between babies’ first crawling age (in weeks) and the average temperature during the month when the baby is 6 months old. Parents reported the birth month and age at which their child was first able to crawl a distance of 4 feet in one minute. The graph below plots average crawling ages (in weeks) against the mean temperatures when the babies were 6 months old. The accompanying chart describes the data. Mean Standard Deviation Correlation Temperature Age 48 32 17.42 1.93 -0.72 a) (3 pts.) Describe the association. Strength: Moderately strong (r = -0.72, closer to -1) Direction: Negative Form: Linear b) (5 pts.) Find the equation of the regression line. Keep AT LEAST 4 decimals in your approximations!!! Your final answer should be approximately: yˆ 35.8290 0.0798 x . Since I’m giving you the answer, I expect to see your work. No work = no credit. rs (0.72)(1.93) Slope = b = y 0.079770379... is about -0.0798 sx 17.42 Y-intercept = y bx = 32 – (-0.0798)(48) = 35.8304 Regression Line: ( AGˆ E ) 0.0798(Temperature) 35.8304 c) (4 pts.) Explain (in context) what the slope of the line means. The slope is -0.0798. This means that for an increase of 1 degree in temperature, the baby is predicted to start crawling 0.0798 weeks earlier. d) (3 pts.) Use the regression line to predict the average crawling age if the average temperature was 56 when a baby was 6 months old. ( AGˆ E ) 0.0798(Temperature) 35.8290 = -0.0798(56)+35.8290 = 31.3602 If the average temperature is 56 when a baby is 6 months old, they are predicted to start crawling at 31.4 weeks old. 2. (15 pts.) Here we have measurements from 507 men and women. Weight is measured in kilograms. Height is measured in centimeters. The waist girth (narrowest part of torso below the rib cage, average of contracted and relaxed positions) is measured in centimeters. Which measurement, height or waist girth, is a better predictor of a person’s weight? Answer the following questions below to answer that question completely. Simple linear regression results: Dependent Variable: Height Independent Variable: Weight Weight = -105 + 1.0176 Height Sample size: 507 r = 0.7173 Estimate of error standard deviation: 9.31 Simple linear regression results: Dependent Variable: Waist Girth Independent Variable: Weight Weight = -15.2 + 1.0955 Waist Girth Sample size: 507 r = 0.9040 Estimate of error standard deviation: 5.71 a) (5 pts.) Using the correlation, explain which variable is a better predictor of Weight. Be sure to explain why (what about the correlation indicates to you that the variable you chose is a better predictor?) In your description, to get full credit, you need to make it clear to me that you know what r measures. r = 0.7173 versus 0.9040, 0.9040 is better, as it is closer to 1. Waist girth is a better predictor of one’s weight since the correlation for waist girth and weight is 0.904, which is closer to 1 than the correlation between height and weight. b) (5 pts.) Calculate r 2 for each scatterplot. Explain what it means about the relationship between the explanatory variable and response variable. Use this to explain which variable is a better predictor. In your description, to get full credit, you need to make it clear to me that you know what r 2 measures. r^2 = 51.45% versus 81.72%. 81.72% is better, as it is closer to 100% The calculation for r^2 indicates that waist girth is a better predictor of weight, since that value is closer to 100%. This means that waist girth accounts for about 81.7% of the variation in weight, leaving only about 18.3% explained by other factors (lurking variables). c) (5 pts.) Using the standard error, explain which variable is a better predictor of Weight. Be sure to explain why (what about the standard error indicates to you that the variable you chose is a better predictor?) In your description, to get full credit, you need to make it clear to me that you know what the standard error measures. We are comparing 9.31 kg versus 5.71 kg. 5.71 kg is better, as that value reveals a smaller estimate of error in using the regression line as a predictor. The standard error implies that waist girth (again) is a better predictor of weight than height is. This is because the estimated amount of error in using the regression line is 5.71 kg, as compared to 9.31 kg. This means that when we use the regression line that compares waist girth and height, we can expect that our approximations will be within about 5.71 kg of the true weight.