As concrete cures, it gains strength. The following data represents the 7-day st

Question

As concrete cures, it gains strength. The following data represents the 7-day strength (x) and 28-day strength (y), in pounds per square inch, of a certain type of concrete. The equation of the least squares regression line is y =0.6764x + 2675.6 and the standard error of the estimate is 271.04. Complete parts (a) through (e). x 12300 3390 2430 2890 3330 2480 3380 2660 2620 3340 y 4070 5220 4640 4620 4850 4120 5020 4890 4190 4630 Click the icon to view the table of critical t-values. (a) Predict the mean 2-day strength of concrete wnose f-day strength is 2810 psI The mean strength is psi Round to one decimal place as needed.) (b) Construct a 95% confidence interval for the mean 28-day strength of concrete whose 7-day strength is 2810 psi Lower Bound Upper Bound- Round to one decimal place as needed.) (c) Predict the 28-day strength of concrete whose 7-day strength is 2810 psi The strength is psi. Round to one decimal place as needed.) (d) Construct a 95% prediction interval about the value of y if x = 2810 Lower Bound Upper Bound- The strength is psi.

Explanation / Answer

Solution

Given x = 7-day strength of concrete, y = 28-day strength of concrete, least square equation as:

y = 2675.6 + 0.6764x and standard error of estimate, say sE = 271.04.

Part (a)

Predicted value for the mean of 28-day strength of concrete for a given value of 7-day strength of concrete,

say x0 is given by ycapmean = 2675.6 + 0.6764x0

Given x0 = 2810, ycapmean = 2675.6 + (0.6764 x 2810) = 4576.3 psi ANSWER

Part (b)

100(1 - )% Confidence Interval (CI) for mean of 28-day strength of concrete for a given value of 7-day strength of concrete, say x0 is given by: ycapmean ± tn – 2,/2xsE , where tn – 2,/2= upper (/2)% point of t-distribution with degrees of freedom = n – 2, n being the sample size.

Using Excel Function, upper 2.5% point of t8[given n = 10] = 2.306,

So, 95% Confidence Interval (CI) for mean of 28-day strength at 7-day strength of 2810 psi is:

4576. 284 ± (2.306 x 271.04) = 4576. 284 ± 625.831

Lower bound = 3950.5 psi Upper bpound = 5202.1 psi ANSWER

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