A population of values has a normal distribution with a population mean of 106.4 and
a population standard deviation of 52.6 You intend to draw a random sample of size n=90. Find the probability that a sample of size n=90 is randomly selected with a mean greater than 123.
P(M > 123) =

To find the probability that the sample mean \(\bar{X}\) is greater than 123, we need to use the Central Limit Theorem. This theorem states that the sampling distribution of the sample mean \(\bar{X}\) will be approximately normally distributed with mean \(\mu\) and standard deviation \(\sigma_{\bar{X}}\), given by:

\[
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}
\]

Where:
- \(\mu = 106.4\) (population mean)
- \(\sigma = 52.6\) (population standard deviation)
- \(n = 90\) (sample size)

First, we calculate \(\sigma_{\bar{X}}\):

\[
\sigma_{\bar{X}} = \frac{52.6}{\sqrt{90}} \approx \frac{52.6}{9.4868} \approx 5.54
\]

Next, we need to find the z-score corresponding to a sample mean of 123. The z-score is given by:

\[
z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}}
\]

Substituting the values, we get:

\[
z = \frac{123 - 106.4}{5.54} \approx \frac{16.6}{5.54} \approx 2.996
\]

Now, we use the standard normal distribution to find the probability corresponding to \(z = 2.996\). Using standard normal distribution tables or a calculator, we find the cumulative probability up to \(z = 2.996\).

\[
P(Z \leq 2.996) \approx 0.9986
\]

The probability that the sample mean is greater than 123 is:

\[
P(\bar{X} > 123) = 1 - P(Z \leq 2.996) \approx 1 - 0.9986 = 0.0014
\]

So, the probability that a sample of size \(n = 90\) is randomly selected with a mean greater than 123 is approximately \(0.0014\).

Do you want details or have any questions?

Here are some more questions you might find interesting:
1. How does the Central Limit Theorem apply to non-normal populations?
2. What is the significance of the z-score in hypothesis testing?
3. How do you interpret a p-value in the context of statistical significance?
4. Can you explain the concept of confidence intervals and their interpretation?
5. How does sample size affect the standard error of the mean?

**Tip:** Always check if the sample size is large enough for the Central Limit Theorem to apply, typically \(n > 30\) is considered sufficient for most distributions.

Calculate the probability that a sample of size 90, drawn from a population with mean 106.4 and standard deviation 52.6, has a mean greater than 123. Detailed step-by-step solution using the Central Limit Theorem, including formulas and calculations.

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