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.
### 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.
