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

Answer:

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

Step-by-step 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:

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

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 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 $s = \frac{\sigma}{\sqrt{n}}$

In this problem, we have that:

$\mu = 12751, \sigma = 2.1, n = 37, s = \frac{2.1}{\sqrt{37}} = 0.3456$

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.

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

By the Central Limit Theorem

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

$Z = \frac{12751 - 12751}{0.3456}$

$Z = 0$

$Z = 0$ 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.

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

$Z = \frac{12754 - 12751}{0.3456}$

$Z = 8.68$

$Z = 8.68$ 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