Concept explainers
If there is at least one x value at which more than one observation has been made, there is a formal test procedure for testing
Versus
Ha: H0 is not true (the true regression function is not linear)
Suppose observations are made at x1, x2, …, xc. Let Y11, Y12, …,
The ni observations at xi contribute ni – 1 df to SSPE, so the number of degrees of freedom for SSPE is
The test statistic is F = MSLF/MSPE, and the corresponding P-value is the area under the
x | 110 | 110 | 110 | 230 | 230 | 230 | 360 |
y | 235 | 198 | 173 | 174 | 149 | 124 | 115 |
x | 360 | 360 | 360 | 505 | 505 | 505 | 505 |
y | 130 | 102 | 95 | 122 | 112 | 98 | 96 |
(So c = 4, n1 = n2 = 3, n3 = n4 = 4.)
a. Test H0 versus Ha at level .05 using the lack-of-fit test just described.
b. Does a
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Chapter 13 Solutions
Student Solutions Manual for Devore's Probability and Statistics for Engineering and the Sciences, 9th
- Find the equation of the regression line for the following data set. x 1 2 3 y 0 3 4arrow_forwardRespiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.arrow_forwardOlympic Pole Vault The graph in Figure 7 indicates that in recent years the winning Olympic men’s pole vault height has fallen below the value predicted by the regression line in Example 2. This might have occurred because when the pole vault was a new event there was much room for improvement in vaulters’ performances, whereas now even the best training can produce only incremental advances. Let’s see whether concentrating on more recent results gives a better predictor of future records. (a) Use the data in Table 2 (page 176) to complete the table of winning pole vault heights shown in the margin. (Note that we are using x=0 to correspond to the year 1972, where this restricted data set begins.) (b) Find the regression line for the data in part ‚(a). (c) Plot the data and the regression line on the same axes. Does the regression line seem to provide a good model for the data? (d) What does the regression line predict as the winning pole vault height for the 2012 Olympics? Compare this predicted value to the actual 2012 winning height of 5.97 m, as described on page 177. Has this new regression line provided a better prediction than the line in Example 2?arrow_forward
- bThe average rate of change of the linear function f(x)=3x+5 between any two points is ________.arrow_forwardThe following fictitious table shows kryptonite price, in dollar per gram, t years after 2006. t= Years since 2006 0 1 2 3 4 5 6 7 8 9 10 K= Price 56 51 50 55 58 52 45 43 44 48 51 Make a quartic model of these data. Round the regression parameters to two decimal places.arrow_forwardFor the following exercises, consider this scenario: The profit of a company decreased steadily overa ten-year spam.The following ordered pairs shows dollars and the number of units sold in hundreds and the profit in thousands ofover the ten-year span, (number of units sold, profit) for specific recorded years: (46,600),(48,550),(50,505),(52,540),(54,495). Use linear regression to determine a function Pwhere the profit in thousands of dollars depends onthe number of units sold in hundreds.arrow_forward
- The relationship between yield of maize, date of planting, and planting density was investigated in an article. Let the variables be defined as follows. y = percent maize yield x = planting date (days after April 20) z = planting density (plants/ha) The following regression model with both quadratic terms where x₁ = x, X₂ = Z, X3 = x² and x4 = 2² provides a good description of the relationship between y and the independent variables. y =a +B₁x₁ + B₂X₂ + B3X3+B₁x₁ + e (a) If a = 21.07, B₁ = 0.653, B₂ = 0.0022, B3 = -0.0207, and B4 = 0.00002, what is the population regression function? y = 509 X (b) Use the regression function in Part (a) to determine the mean yield for a plot planted on May 7 with a density of 41,182 plants/ha. (Give the exact answer.) (c) Would the mean yield be higher for a planting date of May 7 or May 23 (for the same density)? The mean yield would be higher for [May 7 You may need to use the appropriate table in Appendix A to answer this question.arrow_forward3b. A linear regression yields R2 = 0. Does this imply that βˆ1 = 0?arrow_forwardThe following data on x= maternal age in years of the young birth mothers and y weight of baby born in grams summarizes the result of a study. Assume that a simple linear regression model y = Bo + B1x + e is an appropriate model for the study. x-bar 17 (avg of y-bar 3004.1 x's) (avg of y's) SSXX = 20 SS= 4903 SSyw = 1539182.9 n- 10 Calculate the estimated for Bo and B correct to T WO decimal places. Using your results, predict the average weight of a baby born for a value of x = 18. Enter your answer correct to TWO DECIMAL PLACES.arrow_forward
- Let y = sales at a fast-food outlet (1000s of $), x, = number of competing outlets within a 1-mile radius, x, = population within a 1-mile radius (1000s of people), and x, be an indicator variable that equals 1 if the outlet has a drive-up window and o otherwise. Suppose that the true regression model is Y = 12.00 - 1.1x, + 6.8x, + 15.3x, + E (a) What is the mean value of sales when the number of competing outlets is 3, there are 8000 people within a 1-mile radius, and the outlet has a drive-up window? (Round your answer to one decimal place.) thousand dollars (b) What is the mean value of sales for an outlet without a drive-up window that has four competing outlets and 4000 people within a 1-mile radius? (Round your answer to one decimal place.) thousand dollars (c) Interpret B. O There are on average 15.3 fast-food outlets with drive-ups in a 1 mile radius. O It takes $15,300 to build a drive-up. O Approximately 1 in 15.3 fast-food outlets have drive-ups. O A drive-up will add…arrow_forwardFor these (x,y) pairs of data points: 1,5 3,7 4,6 5,8 7,9 Compute b1. Compute b0. What is the equation of the regression line?arrow_forward| Find the regression lines of Y on X and X on Y for the following data. EX = 70, EY = 83, EX? = 590, EY² = 755, EXY= 640, n= 10.arrow_forward
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