Øving 8. STAT111 Sondre Hølleland Auditorium π April Oppgaver

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1 Øving 8 STAT111 Sondre Hølleland Auditorium π April 2016 Oppgaver Section 12.1: 9, 11 Section 12.2: 13, 23, 24 a. b. Fasit Section 12.1: 9. a)0.095 b) c) 0.83 og d) og e) a) og -0.1 b)3 og 2.5 c) og d) Section 12.2: 13. a) y = x b) c) d).956 e)y = x og r 2 = Utfyllende svar på gruppe. 24 a) Svaret gitt i oppgavetekst b) Svaret gitt i oppgavetekst Oppgaveteksten følger under.

2 624 CHAPTER 12 Regression and Correlation b. By how much can we expect 28-day strength to change when accelerated strength increases by 1 psi? c. Answer part (b) for an increase of 100 psi. d. Answer part (b) for a decrease of 100 psi. 8. Referring to Exercise 7, suppose that the standard deviation of the random deviation e is 350 psi. a. What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2000? b. Repeat part (a) with 2500 in place of c. Consider making two independent observations on 28-day strength, the first for an accelerated strength of 2000 and the second for x ¼ What is the probability that the second observation will exceed the first by more than 1000 psi? d. Let Y 1 and Y 2 denote observations on 28-day strength when x ¼ x 1 and x ¼ x 2, respectively. By how much would x 2 have to exceed x 1 in order that P(Y 2 > Y 1 ) ¼.95? 9. The flow rate y (m 3 /min) in a device used for airquality measurement depends on the pressure drop x (in. of water) across the device s filter. Suppose that for x values between 5 and 20, the two variables are related according to the simple linear regression model with true regression line y ¼ x. a. What is the expected change in flow rate associated with a 1-in. increase in pressure drop? Explain. b. What change in flow rate can be expected when pressure drop decreases by 5 in.? c. What is the expected flow rate for a pressure drop of 10 in.? A drop of 15 in.? d. Suppose s ¼.025 and consider a pressure drop of 10 in. What is the probability that the observed value of flow rate will exceed.835? That observed flow rate will exceed.840? e. What is the probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in.? 10. Suppose the expected cost of a production run is related to the size of the run by the equation y ¼ x. Let Y denote an observation on the cost of a run. If the variables size and cost are related according to the simple linear regression model, could it be the case that P(Y > 5500 when x ¼ 100) ¼.05 and P(Y > 6500 when x ¼ 200) ¼.10? Explain. 11. Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature ( F) in the chamber in which the reaction takes place according to the simple linear regression model with equation y ¼ x and s ¼.075. a. What is the expected change in reaction time for a 1 F increase in temperature? For a 10 F increase in temperature? b. What is the expected reaction time when temperature is 200 F? When temperature is 250 F? c. Suppose five observations are made independently on reaction time, each one for a temperature of 250 F. What is the probability that all five times are between 2.4 and 2.6 h? d. What is the probability that two independently observed reaction times for temperatures 1 apart are such that the time at the higher temperature exceeds the time at the lower temperature? 12. In Example 12.4 the probability of cancer metastasizing was pðxþ¼e 2þ:5x = 1 þ e 2þ:5x. a. Tabulate values of x, p(x), the odds pðxþ= ½1 pðxþš, and the log odds for x ¼ 0; 1; 2; 3;...; 10 b. Explain what happens to the odds when x is increased by 1. Your explanation should involve the.5 that appears in the formula for p(x). c. Support your answer to (b) algebraically, starting from the formula for p(x). d. For what value of x are the odds 1? 5? 10? 12.2 Estimating Model Parameters We will assume in this and the next several sections that the variables x and y are related according to the simple linear regression model. The values of b 0, b 1, and s 2 will almost never be known to an investigator. Instead, sample data consisting of n observed pairs (x 1, y 1 ),...,(x n, y n ) will be available, from which the model parameters and the true regression line itself can be estimated. These observations

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7 12.2 Estimating Model Parameters 637 Exercises Section 12.2 (13 30) 13. Exercise 4 gave data on x ¼ BOD mass loading and y ¼ BOD mass removal. Values of relevant summary quantities are X n ¼ 14 xi ¼ 517 X X yi ¼ 346 x 2 i ¼ 39;095 X X yi ¼ 17;454 xi y i ¼ 25;825 a. Obtain the equation of the least squares line. b. Predict the value of BOD mass removal for a single observation made when BOD mass loading is 35, and calculate the value of the corresponding residual. c. Calculate SSE and then a point estimate of s. d. What proportion of observed variation in removal can be explained by the approximate linear relationship between the two variables? e. The last two x values, 103 and 142, are much larger than the others. How are the equation of the least squares line and the value of r 2 affected by deletion of the two corresponding observations from the sample? Adjust the given values of the summary quantities, and use the fact that the new value of SSE is The accompanying data on x ¼ current density (ma/cm 2 ) and y ¼ rate of deposition (mm/min) appeared in the article Plating of 60/40 Tin/ Lead Solder for Head Termination Metallurgy (Plating and Surface Finishing, Jan. 1997: 38 40). Do you agree with the claim by the article s author that a linear relationship was obtained from the tin lead rate of deposition as a function of current density? Explain your reasoning. x y Refer to the data given in Exercise 1 on tank temperature and efficiency ratio. a. Determine the equation of the estimated regression line. b. Calculate a point estimate for true average efficiency ratio when tank temperature is 182. c. Calculate the values of the residuals from the least squares line for the four observations for which temperature is 182. Why do they not all have the same sign? d. What proportion of the observed variation in efficiency ratio can be attributed to the simple linear regression relationship between the two variables? 16. As an alternative to the use of father s height to predict son s height, Galton also used the midparent height, the average of the father s and mother s heights. Here are the heights of 11 female students along with their midparent heights in inches: Midparent Daughter Midparent Daughter a. Make a scatter plot of daughter s height against the midparent height and comment on the strength of the relationship. b. Is the daughter s height completely and uniquely determined by the midparent height? Explain. c. Use the accompanying MINITAB output to obtain the equation of the least squares line for predicting daughter height from midparent height, and then predict the height of a daughter whose midparent height is 70 in. Would you feel comfortable using the least squares line to predict daughter height when midparent height is 74 in.? Explain. Predictor Coef SE Coef T P Constant midparent S ¼ R-Sq ¼ 72.3% R-Sq(adj) ¼ 69.2% Analysis of Variance Source DF SS MS F P Regression Residual Error Total d. What are the values of SSE, SST, and the coefficient of determination? How well does the midparent height account for the variation in daughter height? e. Notice that for most of the families, the midparent height exceeds the daughter height. Is this what is meant by regression to the mean? Explain. 17. The article Characterization of Highway Runoff in Austin, Texas, Area (J. Environ. Engrg., 1998: ) gave a scatter plot, along with

8 12.2 Estimating Model Parameters 639 The regression equation is lichen N ¼ no3 depo Predictor Coef Stdev t-ratio P Constant no3 depo S ¼ R-sq ¼ 71.7% R-sq (adj) ¼ 69.2% Analysis of Variance Source DF SS MS F P Regression Error Total The article Effects of Bike Lanes on Driver and Bicyclist Behavior (ASCE Transportation Engrg. J., 1977: ) reports the results of a regression analysis with x ¼ available travel space in feet (a convenient measure of roadway width, defined as the distance between a cyclist and the roadway center line) and separation distance y between a bike and a passing car (determined by photography). The data, for ten streets with bike lanes, follows: x y x y P P a. Verify that xi ¼ 154:20, yi ¼ 80, P P x 2 i ¼ 2452:18, xi y i ¼ 1282:74, and P y 2 i ¼ 675:16. b. Derive the equation of the estimated regression line. c. What separation distance would you predict for another street that has 15.0 as its available travel space value? d. What would be the estimate of expected separation distance for all streets having available travel space value 15.0? 22. For the past decade rubber powder has been used in asphalt cement to improve performance. The article Experimental Study of Recycled Rubber- Filled High-Strength Concrete (Mag. Concrete Res., 2009: ) included on a regression of y ¼ axial strength (MPa) on x ¼ cube strength (MPa) based on the following sample data: x y x y a. Verify that a scatter plot supports the assumption that the two variables are related via the simple linear regression model. b. Obtain the equation of the least squares line, and interpret its slope. c. Calculate and interpret the coefficient of determination d. Calculate and interpret an estimate of the error standard deviation s in the simple linear regression model. e. The largest x value in the sample considerably exceeds the other x values. What is the effect on the equation of the least squares line of deleting the corresponding observation? 23. Show that the mle s of b 0 and b 1 are indeed the least squares estimates. [Hint: The pdf of Y i is normal with mean m i ¼ b 0 + b 1 x i and variance s 2 ; the likelihood is the product of the n pdf s.] 24. Denote the residuals P by e 1 ;...; e n ðe i ¼ y i ^y i Þ a. Show that ei ¼ 0 and P x i e i ¼ 0. [Hint: Examine the two normal equations.] b. Show that ^y i y ¼ ^b 1 ðx i xþ. c. Use (a) and (b) to derive the analysis of variance identity for regression, Equation (12.4), by showing that the cross-product term is 0. d. Use (b) and Equation (12.4) to verify the computational formula for SSE. 25. A regression analysis is carried out with y ¼ temperature, expressed in C. How do the resulting values of ^b 0 and ^b 1 relate to those obtained if y is reexpressed in F? Justify your assertion. [Hint: new y i ¼ y 0 i ¼ 1:8y i þ 32:] 26. Show that b 1 and b 0 of Expressions (12.2) and (12.3) satisfy the normal equations. 27. Show that the point of averages ðx; yþ lies on the estimated regression line. 28. Suppose an investigator has data on the amount of shelf space x devoted to display of a particular product and sales revenue y for that product. The investigator may wish to fit a model for which the true regression line passes through (0, 0). The appropriate model is Y ¼ b 1 x+e. Assume that (x 1, y 1 ),...,(x n, y n ) are observed pairs generated from this model, and derive the least squares estimator of b 1.[Hint: Write the sum of squared deviations as a function of b 1, a trial value, and use calculus to find the minimizing value of b 1.]

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13 La e 1,..., e n være residualene, definert som 24 a,b: e i = y i ŷ i a) Vis at e i = 0 og x i e i = 0. Hint: Se på normallikningene. Løsning: Normallikningene kan bli skrevet som (yi b 0 b 1 x i ) = 0 og (yi b 0 b 1 x i )x i = 0. Siden ŷ i = b 0 + b 1 x i og e i = y i ŷ i, kan disse liknigene skrives som b) Vis at ŷ i y = β 1 (x i x) Løsning: Vi bruker at b 0 = y b 1 x. 0 = (y i b 0 b 1 x i ) = (y i ŷ i ) = e i 0 = (y i b 0 b 1 x i )x i = (y i ŷ i )x i = x i e i ŷ i y = b 0 + b 1 x i y = (y b 1 x) + b 1 x i y = b 1 (x i x)