Question

For a sample of nequals37​, find the probability of a sample mean being less than 12 comma 751 or greater than 12 comma 754 when muequals12 comma 751 and sigmaequals2.1.

Answer 1

50% probability of a sample mean being less than 12,751 or greater than 12,754

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
`\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 random variable X, with mean
`\mu` and standard deviation
`\sigma`, a large sample size can be approximated to a normal distribution with mean \mu and standard deviation, which is also called standard error
`s = (\sigma)/(√(n))`

In this problem, we have that:

`\mu = 12751, \sigma = 2.1, n = 37, s = (2.1)/(√(37)) = 0.3456`

Find the probability of a sample mean being less than 12,751 or greater than 12,754

Less than 12,751

pvalue of Z when X = 12751.

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

By the Central Limit Theorem

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

`Z = (12751 - 12751)/(0.3456)`

`Z = 0`

`Z = 0` has a pvalue of 0.5.

50% probability of the sample mean being less than 12,751.

Greater than 12,754

1 subtracted by the pvalue of Z when X = 12,754.

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

`Z = (12754 - 12751)/(0.3456)`

`Z = 8.68`

`Z = 8.68` has a pvalue of 1

1 - 1 = 0

0% probability of the sample mean being greater than 12754

Less than 12,751 or greater than 12,754

50 + 0 =50

50% probability of a sample mean being less than 12,751 or greater than 12,754