Question

A company makes windows for use in homes and commercial buildings. The standards for glass thickness call for the glass to average inch with a standard deviation equal to inch. Suppose a random sample of n windows yields a sample mean of inch. Complete parts a and b below. a. What is the probability of if the windows meet the​ standards? nothing

Answer 1

The question is incomplete. The complete question follows.

A company makes windows for use in homes and commercial builidings. The standards for glass thickness call for the glass to average 0.325 inch with a standard deviation equal to 0.065 inch. Suppose a random sample of n=44 windows yield a sample mean of 0.337 inch. Complete parts a and b.

a. What is the probability of x ≥ 0.337 if the windows meet the standards?

b. Based on your answer to part a, what would you conclude about the population of windows? Is it meeting the standards? (A result is unusual if it has a probability less than 0.05)

Answer and Step-by-step explanation: To answer this question, use Central Limit Theorem: it states regardless of the original population distribution, if the sample size is large enough, the sample mean distribution will approach a normal distribution.

The calculations for the CLT is given by normalizing the distribution, i.e.:

$z=\frac{x-\mu}{\sigma/\sqrt{n} }$

where

z is z-score

x is the sample mean

μ is population mean

σ is standard deviation of the population

n is the number of individuals in the sample

Calculating z-score for the window maker company:

$z=\frac{0.337-0.325}{0.065/\sqrt{44} }$

$z=\frac{0.012}{0.0098}$

z = 1.22

Using the z-score table, we found the probability:

P = 0.8888

As it is a "more than situation":

P(x≥0.337) = 1 - 0.8888

P(x≥0.337) = 0.1112  or 11.12%

a. Probability of x≥0.337 if the windows meet the standards is 11.12%.

b. Comparing results, probability of x≥0.337 is bigger than 0.05:

0.1112 > 0.05

So, we can conclude that the random sample of n=44 is meeting the standards estipulated.