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------------------------------------------------ A population of values has a normal distribution with [tex]\mu = 58.9[/tex] and [tex]\sigma = 97.9[/tex]. You intend to draw a random sample of size [tex]n = 213[/tex].

1. Find the probability that a single randomly selected value is between 44.1 and 75.7.
[tex]
P(44.1 \ \textless \ X \ \textless \ 75.7) = \square
[/tex]

2. Find the probability that a sample of size [tex]n = 213[/tex] is randomly selected with a mean between 44.1 and 75.7.
[tex]
P(44.1 \ \textless \ M \ \textless \ 75.7) = \square
[/tex]

Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact [tex]z[/tex]-scores or [tex]z[/tex] scores rounded to 3 decimal places are accepted.

Answer :

Let's solve this problem step-by-step, focusing on both parts involving a normally distributed population with a mean ([tex]\(\mu\)[/tex]) and standard deviation ([tex]\(\sigma\)[/tex]).

### Part 1: Probability of a Single Randomly Selected Value

The population has a normal distribution with:
- Mean ([tex]\(\mu\)[/tex]) = 58.9
- Standard deviation ([tex]\(\sigma\)[/tex]) = 97.9

We are looking to find the probability that a single randomly selected value is between 44.1 and 75.7.

1. Find the z-scores for the bounds:
- For 44.1:
[tex]\[
z_{\text{lower}} = \frac{44.1 - 58.9}{97.9}
\][/tex]
- For 75.7:
[tex]\[
z_{\text{upper}} = \frac{75.7 - 58.9}{97.9}
\][/tex]

2. Calculate the probability:
- Use the standard normal distribution table (or a calculator) to find the cumulative probabilities for each [tex]\(z\)[/tex]-score.
- Subtract the cumulative probability of [tex]\(z_{\text{lower}}\)[/tex] from [tex]\(z_{\text{upper}}\)[/tex].

Probability that [tex]\(X\)[/tex] is between 44.1 and 75.7:
[tex]\[
P(44.1 < X < 75.7) = \text{CDF}(z_{\text{upper}}) - \text{CDF}(z_{\text{lower}})
\][/tex]

After computing these values, the probability is approximately 0.1282.

### Part 2: Probability of Sample Mean for Sample Size [tex]\(n = 213\)[/tex]

For this part, we want the probability that the mean of a sample of size 213 is between 44.1 and 75.7.

1. Calculate the standard error of the mean:
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{97.9}{\sqrt{213}}
\][/tex]

2. Find the z-scores using the standard error:
- For 44.1:
[tex]\[
z'_{\text{lower}} = \frac{44.1 - 58.9}{\text{Standard Error}}
\][/tex]
- For 75.7:
[tex]\[
z'_{\text{upper}} = \frac{75.7 - 58.9}{\text{Standard Error}}
\][/tex]

3. Calculate the probability for the sample mean:
- Again, use the standard normal distribution table (or a calculator) to find the cumulative probabilities of these z-scores.
- The probability that the sample mean [tex]\(M\)[/tex] falls between 44.1 and 75.7:
[tex]\[
P(44.1 < M < 75.7) = \text{CDF}(z'_{\text{upper}}) - \text{CDF}(z'_{\text{lower}})
\][/tex]

After computation, this probability is approximately 0.9802.

This approach enables us to find the probabilities for both the individual value and the sample mean within the given intervals.