Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve this question, we need to compute confidence intervals (CI) for the true average yield point of the modified steel-reinforcing bars. Here are the detailed steps:
### Part (a)
We are given the following information:
- The population standard deviation (\(\sigma\)) is 100.
- The sample size (\(n\)) is 64.
- The sample average yield point (\(\bar{x}\)) is 8442 lb.
- We need to find a 90% confidence interval for the true average yield point.
Step 1: Determine the z-score for a 90% confidence level
For a 90% confidence interval, the z-score corresponding to the 90% confidence level is 1.645.
Step 2: Calculate the standard error of the mean (SE)
The standard error is calculated using the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
Given \(\sigma = 100\) and \(n = 64\),
[tex]\[ SE = \frac{100}{\sqrt{64}} = \frac{100}{8} = 12.5 \][/tex]
Step 3: Calculate the margin of error
The margin of error (ME) is obtained by multiplying the z-score by the standard error:
[tex]\[ ME = z \times SE \][/tex]
For a 90% confidence level:
[tex]\[ ME = 1.645 \times 12.5 = 20.5625 \][/tex]
Step 4: Compute the confidence interval (CI) limits
The 90% confidence interval is calculated as:
[tex]\[ \text{CI} = \left(\bar{x} - ME, \bar{x} + ME\right) \][/tex]
[tex]\[ \text{CI} = \left(8442 - 20.5625, 8442 + 20.5625\right) \][/tex]
[tex]\[ \text{CI} = \left(8421.4375, 8462.5625\right) \][/tex]
Thus, the 90% confidence interval for the true average yield point is:
[tex]\[ (8421.4, 8462.6) \ \text{lb} \][/tex]
### Part (b)
To obtain a 96% confidence interval, we need to make two changes: updating the z-score and recalculating the margin of error and confidence interval limits accordingly.
Step 1: Determine the z-score for a 96% confidence level
For a 96% confidence interval, the z-score is 2.05.
Step 2: Calculate the standard error (as previously calculated)
[tex]\[ SE = 12.5 \][/tex]
Step 3: Calculate the new margin of error
[tex]\[ ME = z \times SE \][/tex]
For a 96% confidence level:
[tex]\[ ME = 2.05 \times 12.5 = 25.625 \][/tex]
Step 4: Compute the confidence interval limits
The 96% confidence interval is calculated as:
[tex]\[ \text{CI} = \left(\bar{x} - ME, \bar{x} + ME\right) \][/tex]
[tex]\[ \text{CI} = \left(8442 - 25.625, 8442 + 25.625\right) \][/tex]
[tex]\[ \text{CI} = \left(8416.375, 8467.625\right) \][/tex]
Thus, the 96% confidence interval for the true average yield point is:
[tex]\[ (8416.38, 8467.63)\ \text{lb} \][/tex]
Finally:
- The 90% confidence interval is (8421.4, 8462.6) lb.
- The 96% confidence interval is (8416.38, 8467.63) lb.
- For a 96% confidence interval, the appropriate z-score is 2.05.
### Part (a)
We are given the following information:
- The population standard deviation (\(\sigma\)) is 100.
- The sample size (\(n\)) is 64.
- The sample average yield point (\(\bar{x}\)) is 8442 lb.
- We need to find a 90% confidence interval for the true average yield point.
Step 1: Determine the z-score for a 90% confidence level
For a 90% confidence interval, the z-score corresponding to the 90% confidence level is 1.645.
Step 2: Calculate the standard error of the mean (SE)
The standard error is calculated using the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
Given \(\sigma = 100\) and \(n = 64\),
[tex]\[ SE = \frac{100}{\sqrt{64}} = \frac{100}{8} = 12.5 \][/tex]
Step 3: Calculate the margin of error
The margin of error (ME) is obtained by multiplying the z-score by the standard error:
[tex]\[ ME = z \times SE \][/tex]
For a 90% confidence level:
[tex]\[ ME = 1.645 \times 12.5 = 20.5625 \][/tex]
Step 4: Compute the confidence interval (CI) limits
The 90% confidence interval is calculated as:
[tex]\[ \text{CI} = \left(\bar{x} - ME, \bar{x} + ME\right) \][/tex]
[tex]\[ \text{CI} = \left(8442 - 20.5625, 8442 + 20.5625\right) \][/tex]
[tex]\[ \text{CI} = \left(8421.4375, 8462.5625\right) \][/tex]
Thus, the 90% confidence interval for the true average yield point is:
[tex]\[ (8421.4, 8462.6) \ \text{lb} \][/tex]
### Part (b)
To obtain a 96% confidence interval, we need to make two changes: updating the z-score and recalculating the margin of error and confidence interval limits accordingly.
Step 1: Determine the z-score for a 96% confidence level
For a 96% confidence interval, the z-score is 2.05.
Step 2: Calculate the standard error (as previously calculated)
[tex]\[ SE = 12.5 \][/tex]
Step 3: Calculate the new margin of error
[tex]\[ ME = z \times SE \][/tex]
For a 96% confidence level:
[tex]\[ ME = 2.05 \times 12.5 = 25.625 \][/tex]
Step 4: Compute the confidence interval limits
The 96% confidence interval is calculated as:
[tex]\[ \text{CI} = \left(\bar{x} - ME, \bar{x} + ME\right) \][/tex]
[tex]\[ \text{CI} = \left(8442 - 25.625, 8442 + 25.625\right) \][/tex]
[tex]\[ \text{CI} = \left(8416.375, 8467.625\right) \][/tex]
Thus, the 96% confidence interval for the true average yield point is:
[tex]\[ (8416.38, 8467.63)\ \text{lb} \][/tex]
Finally:
- The 90% confidence interval is (8421.4, 8462.6) lb.
- The 96% confidence interval is (8416.38, 8467.63) lb.
- For a 96% confidence interval, the appropriate z-score is 2.05.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
