Question

Given a normal population whose mean is 675 and whose standard deviation is 44, find each of the following: A. The probability that a random sample of 5 has a mean between 677 and 693. Probability

Answer 1

27.88% probability that a random sample of 5 has a mean between 677 and 693.

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 = (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 = (\sigma)/(√(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 = 675, \sigma = 44, n = 5, s = (44)/(√(5)) = 19.6774`

The probability that a random sample of 5 has a mean between 677 and 693.

This is the pvalue of Z when X = 693 subtracted by the pvalue of Z when X = 677. So

X = 693

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (693 - 675)/(19.6774)`

`Z = 0.91`

`Z = 0.91` has a pvalue of 0.8186

X = 677

`Z = (X - \mu)/(s)`

`Z = (677 - 675)/(19.6774)`

`Z = 0.1`

`Z = 0.1` has a pvalue of 0.5398

0.8186 - 0.5398 = 0.2788

27.88% probability that a random sample of 5 has a mean between 677 and 693.