AP Statistics Solutions to Packet 14

Transcription

1 AP Statistics Solutions to Packet 4 Inference for Regression Inference about the Model Predictions and Conditions

2 HW #,, 6, 7 4. AN ETINCT BEAST, I Archaeopteryx is an extinct beast having feathers like a bird but teeth and a long bony tail like a reptile. Here are the lengths in centimeters of the femur (a leg bone) and the humerus (a bone in the upper arm) for the five fossil specimens that preserve both bones: Femur: Humerus: The strong linear relationship between the lengths of the two bones helped persuade scientists that all five specimens belong to the same species. (a) Examine the data. Make a scatterplot with femur length as the explanatory variable. Use your calculator to obtain the correlation r and the equation of the least-squares regression line. Do you think the femur length will allow good prediction of humerus length? The correlation is r = 0.994, and linear regression gives yˆ = x The scatterplot below shows a strong, positive, linear relationship, which is confirmed by r. (b) Explain in words what the slope β of the true regression line says about Archaeopteryx. What is the estimate of β from the data? What is your estimate of the intercept α of the true regression line? β represents how much we can expect the humerus length to increase when femur length increases by cm, b (the estimate of β ) is.969, and the estimate of α is a =.660. (c) Calculate the residuals for the five data points. Check that their sum is 0 (up to roundoff error.) Use the residuals to estimate the standard deviation σ in the regression model. You have now estimated all three parameters in the model. The residuals are 0.86, 0.668,.045, 0.940, and 0.90; the sum is (but carrying a different number of digits might change this). Squaring and summing the residuals gives.79, so that s =.79/=.98.

3 4. BACKPACKS Body weights and backpack weights were collected for eight students. Weight (lbs): Backpack weight (lbs): These data were entered into a statistics package and least-squares regression of backpack weight on body weight was requested. Here are the results: Predictor Constant BodyWT Coef Stdev t-ratio 4.. P s =.70 R-sq = 6.% R-sq(adj) = 57.0% (a) What is the equation of the least-squares line? (Hint: Look for the column Coef. What is the intercept? What is the slope? backpack weight = (body weight). The intercept is 6.65 and the slope is (b) The model for regression inference has three parameters, which we call α, β, and σ. Can you determine the estimates for α and β from the computer output? What are they? The estimate for α is the intercept of the least-squares line, that is, The estimate for β is the slope of the least-squares line, that is, (c) The computer output reports that s =.70. This is an estimate of the parameter σ. Use the formula for s to verify the computer s value of s. The estimate for σ is s resid s = = =.695 n 6

4 4.6 AN ETINCT BEAST, II Refer to exercise 4.. Below is part of the output from the S-PLUS statistical software when we regress the length y of the humerus on the length x of the femur. Coefficients: (Intercept) Femur Value Std. Error t value Pr(> t ) (a) What is the equation of the least-squares regression line? yˆ = x (b) We left out the t statistic for testing H 0 : β = 0 and its P-value. Use the output to find t. b.969 t = = = SE b (c) How many degrees of freedom does t have? Use Table C to approximate the P-value of t against the one-sided alternative H a : β > 0. df = ; since t >.9, we know that p < (d) Write a sentence to describe your conclusions about the slope of the true regression line. There is very strong evidence that β > 0, that is, that the line is useful for predicting the length of the humerus given the length of the femur. (e) Determine a 99% confidence interval for the true slope of the regression line. (Show your calculation.) Interpret the interval. For df =, the critical value for a 99% confidence interval is t * = The interval is.969 ± (5.84)(0.075) or.969 ± 0.49, that is, to.659. We are 99% confident that the true slope of the LSRL of the length of humerus on the length of femur is between and.66. 4

5 4.7 JET SKIS, I Data for the number of jet skis in use and number of fatalities for the years 987 to 000 are given below. Year Number in use Accidents Fatalities ,756 6,88 78,50 4,76 05,95 7,8 454, , , , ,6,5,650,6,00 4,08 4,00 (a) Formulate null and alternative hypotheses about the slope of the true regression line. State a onesided alternative hypothesis. H0: β = 0 (there is no association between number of jet skis in use and number of fatalities). Ha: β > 0 (there is a positive association between number of jet skis in use and number of fatalities) (b) What conditions or assumptions are necessary in order to perform a linear regression test of significance? Are these reasonable assumptions in this situation? y responses are independent not given, proceed with caution. True relationship is linear yes σ is constant yes y varies normally - yes (c) Perform a linear regression t test. Report the t statistic, the degrees of freedom, and the P-value. Write your conclusion in plain language. LinRegTTest (TI-84) reports that t = 7.6 with df = 8. The P-value is With the earlier caveat, there is sufficient evidence to reject H0 and conclude that there is an association between year and number of fatalities. As the number of jet skis in use increases, the number of fatalities increases. (d) Determine a 98% confidence interval for the true slope of the regression line. (Show your calculation.) Write your conclusion in plain language. The confidence interval takes the form b ± t * SEb. With t * =.84, and SEb = , the 98% confidence interval is approximately ( , ). We are 98% confident that the true slope of the LSRL of fatalities on number of jet skis in use in thousands is between 0.04 and

6 HW # 9, DOES FAST DRIVING WASTE FUEL? The table below gives data on the fuel consumption of a small car at various speeds from 0 to 50 kilometers per hour. Is there evidence of straight-line dependence between speed and fuel use? Make a scatterplot and use it to explain the result of your test. Speed (km/h) Fuel used (liters/00km) Speed (km/h) Fuel used (liters/00km) Regression of fuel consumption on speed gives b = , SEb = 0.04, and t = 0.6. With df =, we see that p > (0.5) = 0.50 (software reports 0.54), so we have no evidence to suggest a straight-line relationship. While the relationship between these two variables is very strong, it is definitely not linear. 6

7 4.0 The table below presents data on the relationship between the speed of runners (x, in feet per second) and the number of steps y that they take in a second. Speed (ft/s): Steps per second: Here is part of the Data Desk regression output for these data: R-squared = 99.8% s = with 7 = 5 degrees of freedom Variable Constant Speed Coefficient s.e. of coefficient t-ratio Prob < < (a) How can you tell from this output, even without the scatterplot, that there is a very strong straightline relationship between running speed and steps per second? r is very close to, which means that nearly all the variation in steps per second is accounted for by foot speed. Also, the P-value for β is small. (b) What parameter in the regression model gives the rate at which steps per second increase as running speed increases? Find and interpret a 99% confidence interval for this rate. β (the slope) is this rate; the estimate is listed as the coefficient of Speed, Using a t(5) distribution the confidence interval is ± (4.0)(0.006) = to We are 99% confident that the true slope of the LSRL of steps per second on running speed is between 0.07 and

8 4. THE LEANING TOWER OF PISA The Leaning Tower of Pisa leans more as time passes. Here are measurements of the lean of the tower of the years 975 to 987. The lean is the distance between where a point on the tower would be if the tower were straight and where it actually is. The distances are tenths of a millimeter is excess of.9 meters. For example, the 975 lean, which was.964 meters, appears in the table as 64. We use only the last two digits of the year as our time variable. Year: Lean: Here is part of the output from the Data Desk regression procedure with year as the explanatory variable and lean as the response variable: Variable Constant year Coefficient s.e. of coefficient t-ratio prob 0.0 < (a) Plot the data. Briefly describe the shape, strength, and direction of the relationship. The tower is tilting at a steady rate. The plot (below) shows a strong positive linear relationship. (b) The main purpose of the study is to estimate how fast the tower is tilting. What parameter in the regression model gives the rate at which the tilt is increasing, in tenths of a millimeter per year? β (the slope) is this rate; the estimate is listed as the coefficient of year : (c) We want a 95% confidence interval for this rate. How many degrees of freedom does t have? Find the critical value t* and the confidence interval. Interpret the interval. df = ; t * =.0; ± (.0)(0.099) = to We are 95% confident that the true slope of the LSRL of tilt on year is between 8.6 and 0.0 8

9 4. THE GENTLE MANATEE The relationship between the number of powerboats registered and the number of manatees killed each year was explored in Chapter. We will revisit the data below: Year Powerboat registrations (000) Manatees killed Year Powerboat registrations (000) Manatees killed We conducted inference on the manatee data earlier, but was this prudent? Check the conditions, and report your interpretations. The major difficulty is that the observations are not independent. The number of powerboat registrations for any year is related to the number of registrations for the previous year. The other conditions can be assumed to be satisfied. The true relationship is linear The standard deviation of the response about the true line is the sam everywhere The response varies normally about the true regression line PISA, PISA! In Exercise 4. we regressed the lean of the Leaning Tower of Pisa on year to estimate the rate at which the tower is tilting. Here are the residuals from that regression, in order by years across the rows: Use the residuals to check the regression conditions, and describe your findings. Is the regression in Exercise 4. trustworthy? The number of points is so small that it is hard to judge much from the stemplot. The scatterplot of residuals vs. year does not suggest any problems. The regression in Exercise 4. should be fairly reliable. 9

10 4.5 DO HEAVIER PEOPLE BURN MORE ENERGY? Metabolic rate, the rate at which the body consumes energy, is important in studies of weight gain, dieting, and exercise. Lean body mass is an important influence on metabolic rate. Men and women show a similar pattern, so we will ignore gender. Here are the data on mass (in kilograms) and metabolic rate (in calories): Mass: Rate: Mass: Rate: Use your calculator or software to analyze these data. Make a scatterplot and find the least-squares line. Give a 90% confidence interval for the slope β and explain clearly what your interval says about the relationship between lean body mass and metabolic rate. Find the residuals and examine them. Are the conditions for regression inference met? The scatterplot (below) shows a positive association. The regression line is yˆ = x the linear relationship with body mass accounts for r = 74.8% of the variation in metabolic rate. Minitab output (on the next page) reports b = and SEb =.786; with df = 7, the critical value is t * =.740, so the 90% confidence interval for β is ± (.740)(.786) = 0.9 to.47 cal/kg. For each additional kilogram of mass, metabolic rate increases by about 0 to calories. The residuals are listed on the next page (in order, down the columns). A stemplot (on the next page) suggests that the distribution of residuals is right-skewed, and the largest residual may be an outlier. A scatterplot (on the next page) of the residuals against the explanatory variable gives some hint that the variation about the line is not constant (in violation of the regression assumptions). However, the three highest residuals account for most of that impression (as well as the skewness of the distribution), so these three individuals may need to be examined further. 0

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12 HW #4 9, 4.9 BEAVERS AND BEETLES Ecologists sometimes find rather strange relationships in our environment. One study seems to show that beavers benefit beetles. The researchers laid out circular plots, each four meters in diameter, in an area where beavers were cutting down cottonwood trees. In each plot, they measured the number of stumps from trees cut by beavers and the number of cluster of beetle larvae. Here are the data: Stumps: Beetle Larvae: Stumps: Beetle Larvae: (a) Make a scatterplot that shows how the number of beaver-caused stumps influences the number of beetle larvae clusters. What does your plot show? Stumps (the explanatory variable) should be on the horizontal axis; the plot shows a positive linear association. (b) Here is the Minitab regression output for these data: Predictor Constant Stumps Coef Stdev.85.6 T P s = 6.49 R-sq = 8.9% Find the least-squares regression line and draw in on your plot. What percent of the observed variation in beetle larvae counts can be explained by straight-line dependence on beaver stump counts? The regression line is yˆ = x. Regression on stump counts explains 8.9% of the variation in the number of beetle larvae. (c) Is there strong evidence that beaver stumps help explain beetle larvae counts? Give appropriate statistical evidence to support your conclusion. Our hypotheses are H0: β = 0 versus Ha: β 0, and the test statistic is t = 0.47 (df = ). The output shows p = 0.000, so we know that p < ; we have strong evidence that beaver stump counts help explain beetle larvae counts.

13 4. WEEDS AMONG THE CORN Lamb s quarter is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in 6 small plots of ground, then weeded the plots by hand to allow a fixed number of lamb s quarter plants to grow in each meter of corn row. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) in each of the plots: Weeds per meter Corn yield Weeds per meter Corn yield Weeds per meter Corn yield Weeds per meter Corn yield Use your calculator or software to analyze these data. (a) Make a scatterplot and find the least-squares line. What percent of the observed variation in corn yield can be explained by a linear relationship between yield and weeds per meter? Scatterplot below. Regression gives yˆ = x; the linear relationship explains about r = 0.9% of the variation in yield. (b) Is there good evidence that more weeds reduce corn yield? The t statistic for testing H0: β = 0 vs. Ha : β < 0 is t =.9; with df = 4, the P-value is We have some evidence that weeds influence corn yields, but it is not strong enough to meet the usual standards of statistical significance. (c) Explain from your findings in (a) and (b) why you expect predictions based on this regression to be quite imprecise. Predict the mean corn yield under these experimental conditions when there are 6 weeds per meter of row. The small value of r and the lack of significance of the t test indicate that this regression has little predictive use. When x = 6, y ˆ = 59.9 bu/acre; the 95% confidence interval with t * =.45 and : SE µ ˆ =.54 is 59.9 ± (.45) (.54). The width of this interval is another indication that the model has little practical use.