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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.
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