A population has a mean of 50 and a standard deviation of 18. If a random sample of 64 is taken, what is the probability that the sample mean is each of the following?

a. Greater than 54
b. Less than 54
c. Less than 49
d. Between 48.5 and 53.5
e. Between 50.1 and 51.7

Appendix A Statistical Tables
(Round the values of [tex]$z$[/tex] to 2 decimal places, e.g., 15.25. Round your answers to 4 decimal places, e.g., 0.2513.)

a. [tex]$P(\bar{x}\ \textgreater \ 54)=$[/tex] [tex]$\square$[/tex]
b. [tex]$P(\bar{x}\ \textless \ 54)=$[/tex] [tex]$\square$[/tex]
c. [tex]$P(\bar{x}\ \textless \ 49)=$[/tex] [tex]$\square$[/tex]
d. [tex]$P(48.5 \leq \bar{x} \leq 53.5)=$[/tex] [tex]$\square$[/tex]
e. [tex]$P(50.1 \leq \bar{x} \leq 51.7)=$[/tex] [tex]$\square$[/tex]

To solve this problem, we will use the properties of the sampling distribution of the sample mean. Here are the steps:

1. Identify the Known Values:
- Population mean ([tex]$ \mu $[/tex]) = 50
- Population standard deviation ([tex]$ \sigma $[/tex]) = 18
- Sample size ([tex]$ n $[/tex]) = 64

2. Calculate the Standard Error of the Mean (SEM):
The standard error is calculated using the formula:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\text{Standard Error} = \frac{18}{\sqrt{64}} = \frac{18}{8} = 2.25
\][/tex]

3. Find the Probability for Each Case:
We will use the Z-score formula to convert the sample mean to a Z-score, and then find the probabilities using the standard normal distribution table.

a. Probability that the sample mean is greater than 54 ([tex]$ P(\bar{x} > 54) $[/tex]):
- Calculate the Z-score for 54:
[tex]\[
z = \frac{54 - 50}{2.25} \approx 1.78
\][/tex]
- Find the probability using the Z-table or standard normal distribution:
[tex]\[
P(\bar{x} > 54) \approx 0.0377
\][/tex]

b. Probability that the sample mean is less than 54 ([tex]$ P(\bar{x} < 54) $[/tex]):
- This is simply:
[tex]\[
P(\bar{x} < 54) = 1 - P(\bar{x} > 54) \approx 0.9623
\][/tex]

c. Probability that the sample mean is less than 49 ([tex]$ P(\bar{x} < 49) $[/tex]):
- Calculate the Z-score for 49:
[tex]\[
z = \frac{49 - 50}{2.25} \approx -0.44
\][/tex]
- Find the probability:
[tex]\[
P(\bar{x} < 49) \approx 0.3284
\][/tex]

d. Probability that the sample mean is between 48.5 and 53.5:
- Calculate the Z-score for 48.5 and 53.5:
[tex]\[
z_1 = \frac{48.5 - 50}{2.25} \approx -0.67
\][/tex]
[tex]\[
z_2 = \frac{53.5 - 50}{2.25} \approx 1.56
\][/tex]
- Find the probability:
[tex]\[
P(48.5 \leq \bar{x} \leq 53.5) \approx 0.6876
\][/tex]

e. Probability that the sample mean is between 50.1 and 51.7:
- Calculate the Z-score for 50.1 and 51.7:
[tex]\[
z_1 = \frac{50.1 - 50}{2.25} \approx 0.04
\][/tex]
[tex]\[
z_2 = \frac{51.7 - 50}{2.25} \approx 0.76
\][/tex]
- Find the probability:
[tex]\[
P(50.1 \leq \bar{x} \leq 51.7) \approx 0.2573
\][/tex]

These results give the probabilities for each scenario regarding the sample mean.