Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
### 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 this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
