13 Corrosion of steel reinforcing bars is the most important durability problem for rein-

Example 12.13 Corrosion of steel reinforcing bars is the most important durability problem for rein-

  forced concrete structures. Carbonation of concrete results from a chemical reaction that lowers the pH value by enough to initiate corrosion of the rebar. Representative data on x carbonation depth (mm) and y strength (MPa) for a sample of core specimens taken from a particular building follows (read from a plot in the article “The Carbonation of Concrete Structures in the Tropical Environment of Singapore,” Magazine of Concrete Res., 1996: 293–300).

  CHAPTER 12 Simple Linear Regression and Correlation

  Figure 12.17 Minitab scatter plot with confidence intervals and prediction intervals for the data of Example 12.13

  A scatter plot of the data (see Figure 12.17) gives strong support for use of the sim- ple linear regression model. Relevant quantities are as follows:

  g x i 5 659.0

  g x 2 i 5 28,967.50 x 5 36.6111 xx S 5 4840.7778

  g y i 5 293.2

  g x i y i 5 9293.95

  g y 2 i 5 5335.76

  Let’s now calculate a confidence interval, using a 95 confidence level, for the mean strength for all core specimens having a carbonation depth of 45 mm—that is,

  a confidence interval for b 0 1b 1 (45) . The interval is centered at yˆ 5 bˆ 0 1 bˆ 1 (45) 5 27.18 2 .2976(45) 5 13.79 The estimated standard deviation of the statistic is Yˆ

  s Y

  ˆ

  1 5 2.8640B 5 .7582

  12.4 Inferences Concerning m Y • x and the Prediction of Future Y Values

  The 16 df t critical value for a 95 confidence level is 2.120, from which we deter- mine the desired interval to be

  The narrowness of this interval suggests that we have reasonably precise information about the mean value being estimated. Remember that if we recalculated this inter- val for sample after sample, in the long run about 95 of the calculated intervals

  would include b 0 1b 1 (45) . We can only hope that this mean value lies in the single

  interval that we have calculated.

  Figure 12.18 shows Minitab output resulting from a request to fit the simple linear regression model and calculate confidence intervals for the mean value of strength at depths of 45 mm and 35 mm. The intervals are at the bottom of the out- put; note that the second interval is narrower than the first, because 35 is much closer to than is 45. Figure 12.17 shows (1) curves corresponding to the confidence lim- its for each different x value and (2) prediction limits, to be discussed shortly. Notice how the curves get farther and farther apart as x moves away from . x

  The regression equation is strength 5 27.2 2 0.298 depth

  R-sq 76.6 R-sq(adj) 75.1

  Analysis of Variance SOURCE

  Stdev.Fit

  95.0 C.I.

  95.0 P.I.

  Stdev.Fit

  95.0 C.I.

  95.0 P.I.

  ■ In some situations, a CI is desired not just for a single x value but for two or more

  Figure 12.18 Minitab regression output for the data of Example 12.13

  x values. Suppose an investigator wishes a CI both for m Y v and for m Y w , where v and

  w are two different values of the independent variable. It is tempting to compute the interval (12.6) first for x5v and then for x5w . Suppose we use a 5 .05 in each computation to get two 95 intervals. Then if the variables involved in computing the two intervals were independent of one another, the joint confidence coefficient would

  (.95) be . (.95) < .90

  However, the intervals are not independent because the same bˆ 0 , bˆ 1 , and S are

  used in each. We therefore cannot assert that the joint confidence level for the two intervals is exactly 90. It can be shown, though, that if the 100(1 2 a) CI (12.6)

  is computed both for x5v and x5w to obtain joint CIs for m Y v and m Y w , then

  the joint confidence level on the resulting pair of intervals is at least 100(1 2 2a) . In particular, using a 5 .05 results in a joint confidence level of at least 90, whereas using a 5 .01 results in at least 98 confidence. For example, in Example

  12.13 a 95 CI for m Y 45 was (12.185, 15.401) and a 95 CI for m Y35 was (15.330, 18.207). The simultaneous or joint confidence level for the two statements 12.185 , m Y 45 , 15.401 and 15.330 , m Y 35 , 18.207 is at least 90.

  CHAPTER 12 Simple Linear Regression and Correlation

  The validity of these joint or simultaneous CIs rests on a probability result called the Bonferroni inequality, so the joint CIs are referred to as Bonferroni intervals. The method is easily generalized to yield joint intervals for k different m

  Yx ’s . Using the interval (12.6) separately first for x5x 1 , then for

  x5x 2 ,c , and finally for x5x k yields a set of k CIs for which the joint or simul- taneous confidence level is guaranteed to be at least 100(1 2 ka) .

  Tests of hypotheses about b 0 1b 1 x are based on the test statistic T obtained by replacing b 0 1b 1 x in the numerator of (12.5) by the null value m 0 . For exam-

  ple, H 0 :b 0 1b 1 (45) 5 15 in Example 12.13 says that when carbonation depth is

  45, expected (i.e., true average) strength is 15. The test statistic value is then

  t 5 [bˆ 0 1 bˆ 1 (45) 2 15]s bˆ 0 1bˆ 1 (45) , and the test is upper-, lower-, or two-tailed

  according to the inequality in H a .