Question

SeeClear Windows makes windows for use in homes and commercial buildings. The standards for glass thickness call for the glass to average 0.375 inches with a standard deviation equal to 0.050 inches.

90% of the standard glasses will have a thickness between what two values symmetrically distributed around the mean?(round to 3 decimal digits, e.g. 0.123)

1.The smaller number is ?

2.The Larger number is?

3. Suppose we randomly select 64 pieces of windows, what is the mean thickness of all possible samples?

4. Suppose we randomly select 64 pieces of windows, what is the standard deviation of all possible sample means (standard error)?

Answer 1

Given that, mean (μ) = 0.375 inches and

standard deviation = 0.050 inches

We want to find the values of x1 and x2 such that, P(x1 < X < x2) = 0.90

First we find, the z-score such that, P(-z < Z < z) = 0.90

=> 2 * P(Z < z) - 1 = 0.90

=> 2 * P(Z < z) = 1.90

=> P(Z < z) = 0.95

Using standard normal z-table we get, z-score corresponding probability of 0.95 is, z = 1.645

=> P(-1.645 < Z < 1.645) = 0.90

For z = -1.645

x1 = (-1.645 * 0.050) + 0.375 = -0.082 + 0.375 = 0.293

For z = 1.645

x2 = (1.645 * 0.050) + 0.375 = 0.082 + 0.375 = 0.457

Therefore, the 90% of the standard glasses will have a thickness between 0.293 inches and 0.457 inches.

1) The smaller number is 0.293

2) The Larger number is 0.457

3) Given that, sample size is n = 64

The mean and standard deviation of the sampling distribution of the sample mean are,

The mean thickness of all possible samples is 0.375 inches

4) The standard deviation of all possible sample means is 0.00625 inches