The article “Mix Design for Optimal Strength Development of Fly Ash Concrete” (Cement and Concrete Research, Vol. 19, No. 4, 1989, pp. 634–640) investigates the compressive strength of concrete when mixed with fly ash (a mixture of silica, alumina, iron, magnesium oxide, and other ingredients). The compressive strength for nine samples in dry conditions on the twenty-eighth day are as follows (in Mpa):

(a) Given the probability plot of the data in Fig. 4-25, what is a logical assumption about the underlying distribution of the data?

(b) Find a 99% one-sided lower confidence bound on mean compressive strength. Provide a practical interpretation of this bound.

(c) Find a 98% two-sided CI on mean compressive strength. Provide a practical interpretation of this interval and explain why the lower endpoint of the interval is or is not the same as in part (b).

(d) Find a 99% one-sided upper confidence bound on the variance of compressive strength. Provide a practical interpretation of this bound.

(e) Find a 98% two-sided CI on the variance of compression strength. Provide a practical interpretation of this interval and explain why the upper endpoint of the interval is or is not the same as in part (d).

(f) Suppose it was discovered that the largest observation 40.2 was misrecorded and should actually be 20.4. Now the sample mean and  = 22.9 the sample variance s² = 39.83. Use these new values and repeat parts (c) and (e).

Compare the original computed intervals and the newly computed intervals with the corrected observation value. How does this mistake affect the values of the sample mean, the sample variance, and the width of the two-sided CIs?

(g) Suppose, instead, it was discovered that the largest observation 40.2 is correct, but that the observation 25.8 is incorrect and should actually be 24.8. Now the sample mean and  = 25.0 the sample variance s² = 70.84. Use these new values and repeat parts (c) and (e). Compare the original computed intervals and the newly computed intervals with the corrected observation value. How does this mistake affect the values of the sample mean, the sample variance, and the width of the two-sided CIs?

(h) Use the results from parts (f) and (g) to explain the effect of mistakenly recorded values on sample estimates. Comment on the effect when the mistaken values are near the sample mean and when they are not.

(i) Using the original data, construct a 99% PI on the compressive strength of a single sample in dry conditions.

(j) Find a tolerance interval for the compressive strength that includes 95% of the concrete in the population with 99% confidence.