Answered

At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

A population of values has a normal distribution with [tex]\(\mu=218.2\)[/tex] and [tex]\(\sigma=81.6\)[/tex].

a. Find the probability that a single randomly selected value is between 207.3 and 208.5. Round your answer to four decimal places.
[tex]\[ P(207.3 \ \textless \ X \ \textless \ 208.5) = \][/tex]
[tex]\(.0058 \ \square\)[/tex]

b. Find the probability that a randomly selected sample of size [tex]\(n=183\)[/tex] has a mean between 207.3 and 208.5. Round your answer to four decimal places.
[tex]\[ P(207.3 \ \textless \ M \ \textless \ 208.5) = \][/tex]
[tex]\(\square\)[/tex]

Sagot :

GinnyAnswer
To find the probabilities, we'll start by calculating the corresponding Z-scores for the given values.

### Part a

#### Given:
- Population mean ([tex]\(\mu\)[/tex]) = 218.2
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 81.6
- Lower bound ([tex]\(X_{lower}\)[/tex]) = 207.3
- Upper bound ([tex]\(X_{upper}\)[/tex]) = 208.5

We need to find the Z-scores for both bounds using the formula:
[tex]\[ Z = \frac{X - \mu}{\sigma} \][/tex]

#### Steps:
1. Calculate the Z-score for the lower bound ([tex]\(X_{lower} = 207.3\)[/tex]):
[tex]\[ Z_{lower} = \frac{207.3 - 218.2}{81.6} = -0.1336 \][/tex]

2. Calculate the Z-score for the upper bound ([tex]\(X_{upper} = 208.5\)[/tex]):
[tex]\[ Z_{upper} = \frac{208.5 - 218.2}{81.6} = -0.1189 \][/tex]

3. Use the standard normal distribution (Z-table) to find the probabilities corresponding to [tex]\( Z_{lower} \)[/tex] and [tex]\( Z_{upper} \)[/tex].

4. The desired probability is the difference between these probabilities:
[tex]\[ P(207.3 < X < 208.5) = P(Z_{upper}) - P(Z_{lower}) \][/tex]

After calculating, we get:
[tex]\[ P(207.3 < X < 208.5) = 0.0058 \][/tex]

### Part b

#### Given:
- Sample size ([tex]\(n\)[/tex]) = 183
- Population mean ([tex]\(\mu\)[/tex]) = 218.2
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 81.6
- Lower bound for mean ([tex]\(M_{lower}\)[/tex]) = 207.3
- Upper bound for mean ([tex]\(M_{upper}\)[/tex]) = 208.5

Since we are dealing with a sample mean, we need to use the standard error of the mean ([tex]\(\text{SE}\)[/tex]) instead of the population standard deviation. The standard error is calculated as:
[tex]\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \][/tex]

#### Steps:
1. Calculate the standard error:
[tex]\[ \text{SE} = \frac{81.6}{\sqrt{183}} \approx 6.026 \][/tex]

2. Convert the lower bound and upper bound of the sample mean to Z-scores:
[tex]\[ Z_{lower} = \frac{207.3 - 218.2}{6.026} = -1.807 \][/tex]
[tex]\[ Z_{upper} = \frac{208.5 - 218.2}{6.026} = -1.608 \][/tex]

3. Use the standard normal distribution (Z-table) to find the probabilities corresponding to [tex]\( Z_{lower} \)[/tex] and [tex]\( Z_{upper} \)[/tex].

4. The desired probability is the difference between these probabilities:
[tex]\[ P(207.3 < M < 208.5) = P(Z_{upper}) - P(Z_{lower}) \][/tex]

After calculating, we get:
[tex]\[ P(207.3 < M < 208.5) = 0.0185 \][/tex]

### Conclusion

- Part a: [tex]\( P(207.3 < X < 208.5) = 0.0058 \)[/tex]
- Part b: [tex]\( P(207.3 < M < 208.5) = 0.0185 \)[/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.