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Sagot :
To construct a 90% confidence interval for the proportion of all air travelers who favor tighter security procedures, we need to follow these steps:
1. Determine the sample proportion (\( \hat{p} \)):
- Given that 370 out of 500 travelers favor tighter security procedures, the sample proportion is
[tex]\[ \hat{p} = \frac{370}{500} = 0.74 \][/tex]
2. Calculate the complement of the sample proportion (\( \hat{q} \)):
- The complement of \( \hat{p} \) is
[tex]\[ \hat{q} = 1 - \hat{p} = 1 - 0.74 = 0.26 \][/tex]
3. Find the Z-score associated with the 90% confidence level:
- For a 90% confidence level, the Z-score is typically found in Z-tables, which is 1.645.
4. Compute the standard error (SE):
- The standard error for the proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}} \][/tex]
where \( n \) is the sample size.
- Plugging in our values:
[tex]\[ SE = \sqrt{\frac{0.74 \cdot 0.26}{500}} \approx 0.01962 \][/tex]
5. Calculate the margin of error (ME):
- The margin of error is given by multiplying the Z-score by the standard error:
[tex]\[ ME = Z \cdot SE = 1.645 \cdot 0.01962 \approx 0.03227 \][/tex]
6. Determine the confidence interval:
- The confidence interval is calculated by adding and subtracting the margin of error from the sample proportion:
[tex]\[ CI = \hat{p} \pm ME \][/tex]
So, the lower bound of the confidence interval is:
[tex]\[ 0.74 - 0.03227 \approx 0.7077 \][/tex]
The upper bound of the confidence interval is:
[tex]\[ 0.74 + 0.03227 \approx 0.7723 \][/tex]
Therefore, the 90% confidence interval for the proportion of all air travelers who are in favor of tighter security procedures is approximately \([0.7077, 0.7723]\).
However, matching this to the multiple-choice options given:
- The correct formula used would be:
[tex]\[ 0.74 \pm 1.645 \sqrt{\frac{(.74)(.26)}{500}} \][/tex]
Thus, the correct answer from the options is:
[tex]\[ 0.74 \pm 1.645 \sqrt{\frac{(.74)(.26)}{500}} \][/tex]
1. Determine the sample proportion (\( \hat{p} \)):
- Given that 370 out of 500 travelers favor tighter security procedures, the sample proportion is
[tex]\[ \hat{p} = \frac{370}{500} = 0.74 \][/tex]
2. Calculate the complement of the sample proportion (\( \hat{q} \)):
- The complement of \( \hat{p} \) is
[tex]\[ \hat{q} = 1 - \hat{p} = 1 - 0.74 = 0.26 \][/tex]
3. Find the Z-score associated with the 90% confidence level:
- For a 90% confidence level, the Z-score is typically found in Z-tables, which is 1.645.
4. Compute the standard error (SE):
- The standard error for the proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}} \][/tex]
where \( n \) is the sample size.
- Plugging in our values:
[tex]\[ SE = \sqrt{\frac{0.74 \cdot 0.26}{500}} \approx 0.01962 \][/tex]
5. Calculate the margin of error (ME):
- The margin of error is given by multiplying the Z-score by the standard error:
[tex]\[ ME = Z \cdot SE = 1.645 \cdot 0.01962 \approx 0.03227 \][/tex]
6. Determine the confidence interval:
- The confidence interval is calculated by adding and subtracting the margin of error from the sample proportion:
[tex]\[ CI = \hat{p} \pm ME \][/tex]
So, the lower bound of the confidence interval is:
[tex]\[ 0.74 - 0.03227 \approx 0.7077 \][/tex]
The upper bound of the confidence interval is:
[tex]\[ 0.74 + 0.03227 \approx 0.7723 \][/tex]
Therefore, the 90% confidence interval for the proportion of all air travelers who are in favor of tighter security procedures is approximately \([0.7077, 0.7723]\).
However, matching this to the multiple-choice options given:
- The correct formula used would be:
[tex]\[ 0.74 \pm 1.645 \sqrt{\frac{(.74)(.26)}{500}} \][/tex]
Thus, the correct answer from the options is:
[tex]\[ 0.74 \pm 1.645 \sqrt{\frac{(.74)(.26)}{500}} \][/tex]
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