Question

Water permeability of concrete can be measured by letting water flow across the surface and determining the amount lost (in inches per hour). Suppose that the permeability index \(x\) for a randomly selected concrete specimen of a particular type is normally distributed with mean value 1000 and standard deviation 150 . a. How likely is it that a single randomly selected specimen will have a permeability index between 850 and \(1300 ?\) b. If the permeability index is to be determined for each specimen in a random sample of size 10 , how likely is it that the sample average permeability index will be between 950 and \(1100 ?\) between 850 and 1300 ?

Short answer

a. The probability that a single randomly selected specimen will have a permeability index between 850 and 1300 is the calculated probability from step 1. \n b. The probability that the sample average permeability index will be between 950 and 1100, and between 850 and 1300 in a sample size of 10 are the calculated probabilities from step 2.

Step-by-step solution

Find Probability for Part a

First, calculate the z-scores for the values 850 and 1300. The formula for z-score is \( z = (X - \mu) / \sigma \). Using this formula, calculate the z-score for 850 (\( z_1 \)) and for 1300 (\( z_2 \)). Next, refer to the standard normal distribution table to find the probabilities corresponding to the \( z_1 \) and \( z_2 \). The area between these two z-scores in the standard normal distribution curve represents the desired probability.

Find Probability for Part b

Repeat the processes in step 1 by considering a sample size of 10. However, keep in mind that when you are dealing with the average of a sample, the standard deviation is divided by the square root of the sample size (this is the standard error). Now, calculate the z-scores for the values 950 and 1100, in the same way as in step 1, using the formula for z-score and standard error. Moreover, make identical calculation for the range 850 to 1300. Refer to the standard normal distribution table to determine the probabilities.

Interpret the Results

The probabilities obtained from the z-scores indicate the likelihood of the permeability index falling within the given ranges. Therefore, the results from step 1 represent answer for part a of the question, and the results from step 2 represent answer for part b.