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Solution11 - Stat 350 Solution to Homework 11 1(a The slope...

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Stat 350 Solution to Homework 11 1. (a) The slope of the estimated regression line ( β = .095) is the expected change of in the response variable y for each one-unit increase in the x variable. This, of course, is just the usual interpretation of the slope of a straight line. Since x is measured in inches, a one-unit increase in x corresponds to a one-inch increase in pressure drop. Therefore, the expected change in flow rate is .095 m 3 /min. (b) When the pressure drop, x, changes from 10 inches to 15 inches, then a 5 unit increase in x has occurred. Therefore, using the definition of the slope from (a), we expect about a 5(.095) = .475 m 3 /min. increase in flow rate (it is an increase since the sign of β = .095 is positive ). (c) For x = 10, μ y.10 = -.12 + .095(10) = .830. For x = 15, μ y.15 = -.12 + .095(15) = 1.305. (d) When x = 10, the flow rate y is normally distributed with a mean value of μ y.10 = .830 and a standard deviation of σ y.10 = σ = .025. Therefore, we standardize and use the z table to find: P(y > .835) = P(z > ) = P(z > .20) = 1 - P(z .20) = 1- .5793 = .4207 (using Table I). 2. (a) The slope of the estimated regression line ( = -.01) is the expected change in reaction time for a one degree Fahrenheit increase in the temperature of the chamber. So, with a one degree Fahrenheit increase in temperature, the true average reaction time will decrease by .01 hours. With a 10 degree increase in temperature, the true average reaction time will decrease by (10)(.01) = 1 hour. (b) When x = 200, When x = 250, (c) Next, the probability that all five observed reaction times are between 2.4 and 2.6 is (.8164) 5 = .3627 5. (a) Using the formulas for the various sums of squares, we find: SS xy = - = 40.968 - (12.6)(27.68)/9 = 2.216 SS xx = - = 18.24 - (12.6) 2 /9 = .600. Therefore, the estimated slope is: b = SS xy /SS xx = 2.216/.600 = 3.6933. The estimated intercept is a = - b = (27.68)/9 - (3.6933)(12.6)/9 = -2.0951. The estimated regression line is then: = a + bx = -2.0951 + 3.6933x.
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(b) For x = 1.5, the point estimate of the average y value is: μ y.1.5 -2.0951 + 3.6933(1.5) = 3.445. For another measurement made when x = 1.5, the point estimate of the y value (for this x value) would be the same; i.e., = 3.445.
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