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Answer :
To solve this problem, we need to find two probabilities related to a normally distributed population with a mean ([tex]\(\mu\)[/tex]) of 58.9 and a standard deviation ([tex]\(\sigma\)[/tex]) of 97.9.
### Part 1: Probability for a Single Randomly Selected Value
We want to find the probability that a single randomly selected value from this population is between 44.1 and 75.7.
1. Calculate the Z-scores for the lower bound (44.1) and the upper bound (75.7):
[tex]\[
z_{\text{lower}} = \frac{44.1 - 58.9}{97.9}
\][/tex]
[tex]\[
z_{\text{upper}} = \frac{75.7 - 58.9}{97.9}
\][/tex]
2. Use the standard normal distribution (Z-distribution) to find the probabilities corresponding to these Z-scores. This involves calculating the cumulative distribution function (CDF) values:
- The CDF value for [tex]\(z_{\text{upper}}\)[/tex] gives the probability that a value is less than 75.7.
- The CDF value for [tex]\(z_{\text{lower}}\)[/tex] gives the probability that a value is less than 44.1.
3. Subtract the lower bound probability from the upper bound probability to find the probability that a value 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]
The result is approximately 0.1282.
### Part 2: Probability for the Sample Mean
Now, we need to find the probability that the mean of a sample of 213 observations is between 44.1 and 75.7.
1. Calculate the standard deviation of the sample mean (often called the standard error) using:
[tex]\[
\sigma_{\text{sample mean}} = \frac{\sigma}{\sqrt{n}} = \frac{97.9}{\sqrt{213}}
\][/tex]
2. Calculate the Z-scores for the sample mean with the lower and upper bounds:
[tex]\[
z_{\text{lower sample}} = \frac{44.1 - 58.9}{\sigma_{\text{sample mean}}}
\][/tex]
[tex]\[
z_{\text{upper sample}} = \frac{75.7 - 58.9}{\sigma_{\text{sample mean}}}
\][/tex]
3. Use the standard normal distribution (Z-distribution) to find the probabilities corresponding to these sample mean Z-scores:
- The CDF value for [tex]\(z_{\text{upper sample}}\)[/tex] gives the probability that the sample mean is less than 75.7.
- The CDF value for [tex]\(z_{\text{lower sample}}\)[/tex] gives the probability that the sample mean is less than 44.1.
4. Subtract the lower bound probability from the upper bound probability to find the probability that the sample mean is between 44.1 and 75.7:
[tex]\[
P(44.1 < M < 75.7) = \text{CDF}(z_{\text{upper sample}}) - \text{CDF}(z_{\text{lower sample}})
\][/tex]
The result is approximately 0.9802.
These calculations give us the probabilities required for both single values and sample means.
### Part 1: Probability for a Single Randomly Selected Value
We want to find the probability that a single randomly selected value from this population is between 44.1 and 75.7.
1. Calculate the Z-scores for the lower bound (44.1) and the upper bound (75.7):
[tex]\[
z_{\text{lower}} = \frac{44.1 - 58.9}{97.9}
\][/tex]
[tex]\[
z_{\text{upper}} = \frac{75.7 - 58.9}{97.9}
\][/tex]
2. Use the standard normal distribution (Z-distribution) to find the probabilities corresponding to these Z-scores. This involves calculating the cumulative distribution function (CDF) values:
- The CDF value for [tex]\(z_{\text{upper}}\)[/tex] gives the probability that a value is less than 75.7.
- The CDF value for [tex]\(z_{\text{lower}}\)[/tex] gives the probability that a value is less than 44.1.
3. Subtract the lower bound probability from the upper bound probability to find the probability that a value 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]
The result is approximately 0.1282.
### Part 2: Probability for the Sample Mean
Now, we need to find the probability that the mean of a sample of 213 observations is between 44.1 and 75.7.
1. Calculate the standard deviation of the sample mean (often called the standard error) using:
[tex]\[
\sigma_{\text{sample mean}} = \frac{\sigma}{\sqrt{n}} = \frac{97.9}{\sqrt{213}}
\][/tex]
2. Calculate the Z-scores for the sample mean with the lower and upper bounds:
[tex]\[
z_{\text{lower sample}} = \frac{44.1 - 58.9}{\sigma_{\text{sample mean}}}
\][/tex]
[tex]\[
z_{\text{upper sample}} = \frac{75.7 - 58.9}{\sigma_{\text{sample mean}}}
\][/tex]
3. Use the standard normal distribution (Z-distribution) to find the probabilities corresponding to these sample mean Z-scores:
- The CDF value for [tex]\(z_{\text{upper sample}}\)[/tex] gives the probability that the sample mean is less than 75.7.
- The CDF value for [tex]\(z_{\text{lower sample}}\)[/tex] gives the probability that the sample mean is less than 44.1.
4. Subtract the lower bound probability from the upper bound probability to find the probability that the sample mean is between 44.1 and 75.7:
[tex]\[
P(44.1 < M < 75.7) = \text{CDF}(z_{\text{upper sample}}) - \text{CDF}(z_{\text{lower sample}})
\][/tex]
The result is approximately 0.9802.
These calculations give us the probabilities required for both single values and sample means.
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