(6 points) Wasserman (1989) studied a process for the manufacturing of steel bolts. Historically, these bolts have a mean thickness of 10.0 mm and a standard deviation of 1.6 mm. In a quality check the engineer has a sample of 5 randomly selected and measured. (a) (2 points) Assuming a near normal distribution what are the mean and standard error (standard deviation of the sample mean) of these quality checks? (b) (2 points) A recent sample of five wafers yielded a sample mean of 10.4 mm. Find the probability of observing such a mean of something larger based on the historic mean and standard deviation. (c) (2 points) 90% of the means taken from samples of 5 should be smaller than what value?

a) Mean = 10, standard error = 0.7155

b) 28.77% probability of observing such a mean of something larger based on the historic mean and standard deviation.

c) 12.048 mm

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the central limit theorem.

Normal probability distribution:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sample means with size n can be approximated to a normal distribution with mean and standard deviation

In this problem, we have that:

(a) (2 points) Assuming a near normal distribution what are the mean and standard error (standard deviation of the sample mean) of these quality checks?

Mean = 10, standard error = 0.7155

(b) (2 points) A recent sample of five wafers yielded a sample mean of 10.4 mm. Find the probability of observing such a mean of something larger based on the historic mean and standard deviation.

This is 1 subtracted by the pvalue of Z when Z = 10.4. So

By the Central Limit Theorem

has a pvalue of 0.7123.

1 - 0.7123 = 0.2877

28.77% probability of observing such a mean of something larger based on the historic mean and standard deviation.

(c) (2 points) 90% of the means taken from samples of 5 should be smaller than what value?

This is the value of X when Z has a pvalue of 0.90. So it is X when Z = 1.28.