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DanielleElmas [232]
3 years ago
7

A company makes windows for use in homes and commercial buildings. The standards for glass thickness call for the glass to avera

ge 0.325 inch with a standard deviation equal to 0.065 inch. Suppose a random sample of n = 44 windows yields a sample mean of 0.337 inch.
Required:
What is the probability of x >= 20.337 if the windows meet the standards?
Mathematics
1 answer:
Hatshy [7]3 years ago
8 0

Answer:

0.1102 = 11.02% probability of x >= 0.337 if the windows meet the standards.

Step-by-step explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

The standards for glass thickness call for the glass to average 0.325 inch with a standard deviation equal to 0.065 inch.

This means \mu = 0.325, \sigma = 0.065

Sample of 44

This means that n = 44, s = \frac{0.065}{\sqrt{44}}

What is the probability of x >= 0.337 if the windows meet the standards?

This is 1 subtracted by the p-value of Z when X = 0.337. So

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{0.337 - 0.325}{\frac{0.065}{\sqrt{44}}}

Z = 1.225

Z = 1.225 has a p-value of 0.8898

1 - 0.8898 = 0.1102

0.1102 = 11.02% probability of x >= 0.337 if the windows meet the standards.

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