Question

Suppose that a random sample of size 36 is to be selected from a population with mean 50 and standard deviation 7. What is the approximate probability that will be within 0.5 of the population mean?

Answer 1

Answer:

The probability that the sample mean will be within 0.5 of the population mean is 0.3328.

Step-by-step explanation:

It is provided that a random variable X has mean, μ = 50 and standard deviation, σ = 7.

A  random sample of size, n = 36 is selected.

According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.

Then, the mean of the distribution of sample mean is given by,

$\mu_{\bar x}=\mu=50$

And the standard deviation of the distribution of sample mean is given by,

$\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}=\frac{7}{\sqrt{36}}=1.167$

So, the distribution of the sample mean of X is N (50, 1.167²).

Compute the probability that the sample mean will be within 0.5 of the population mean as follows:

$P(|\bar X-\mu_{\bar x}|\leq 0.50)=P(-0.50$

                               $=P(\frac{-0.50}{1167}$

Thus, the probability that the sample mean will be within 0.5 of the population mean is 0.3328.