Mathematics, 07.04.2020 23:23 xxYingxYangxx7311

The heights of men in the USA are normally distributed with a mean of 68 inches and a standard deviation of 4 inches. A random sample of five men is selected. What is the probability that the sample mean is greater than 69.5 inches?

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Answer from: hunterwilliams375

20.05% probability that the sample mean is greater than 69.5 inches

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

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore 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 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 $\mu$ 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.

In this problem, we have that:

$\mu = 68, \sigma = 4, n = 5, s = \frac{4}{\sqrt{5}} = 1.79$

What is the probability that the sample mean is greater than 69.5 inches?

This is 1 subtracted by the pvalue of Z when X = 69.5. So

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

By the Central Limit Theorem

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

$Z = \frac{69.5 - 68}{1.79}$

Z = 0.84

Z = 0.84 has a pvalue of 0.7995

1 - 0.7995 = 0.2005

20.05% probability that the sample mean is greater than 69.5 inches

Answer from: Quest

They sought new allies to germany tried to get mexico to join and put pressure on the us. the us joined the allies because of the lucitania

Answer from: Quest

hello i am doing good for you that it's summer in your country or state ; )

btw in my country we just have sun and rain as (because i live in the caribbean lol)

also, thx a lot for the points : )

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The heights of men in the USA are normally distributed with a mean of 68 inches and a standard devia...