Answer :
To construct a 99% confidence interval for the proportion of voters against the fire department bond measures based on a sample of 1000 voters, follow these steps:
1. Identify the sample proportion (p-hat):
- Given that 9 voters out of 1000 are against the bond measures, the sample proportion (p-hat) is:
[tex]\[ \hat{p} = \frac{9}{1000} = 0.009 \][/tex]
2. Determine the z-value (z):
- For a 99% confidence interval, the z-value (z) is:
[tex]\[ z^* = 2.576 \][/tex]
3. Calculate the standard error (SE):
- The standard error is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
- Where [tex]\( n \)[/tex] is the sample size (1000 voters):
[tex]\[ SE = \sqrt{\frac{0.009 \cdot (1 - 0.009)}{1000}} \approx 0.002986 \][/tex]
4. Find the margin of error (ME):
- The margin of error is calculated using the formula:
[tex]\[ ME = z^* \cdot SE \][/tex]
- Substituting the known values:
[tex]\[ ME = 2.576 \cdot 0.002986 \approx 0.007693 \][/tex]
5. Calculate the confidence interval:
- The 99% confidence interval is found by adding and subtracting the margin of error from the sample proportion:
[tex]\[ \text{Lower bound} = \hat{p} - ME = 0.009 - 0.007693 \approx 0.001307 \][/tex]
[tex]\[ \text{Upper bound} = \hat{p} + ME = 0.009 + 0.007693 \approx 0.016693 \][/tex]
Hence, the 99% confidence interval for the proportion of voters against the fire department bond measures is approximately:
[tex]\[ [0.001307, 0.016693] \][/tex]
1. Identify the sample proportion (p-hat):
- Given that 9 voters out of 1000 are against the bond measures, the sample proportion (p-hat) is:
[tex]\[ \hat{p} = \frac{9}{1000} = 0.009 \][/tex]
2. Determine the z-value (z):
- For a 99% confidence interval, the z-value (z) is:
[tex]\[ z^* = 2.576 \][/tex]
3. Calculate the standard error (SE):
- The standard error is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
- Where [tex]\( n \)[/tex] is the sample size (1000 voters):
[tex]\[ SE = \sqrt{\frac{0.009 \cdot (1 - 0.009)}{1000}} \approx 0.002986 \][/tex]
4. Find the margin of error (ME):
- The margin of error is calculated using the formula:
[tex]\[ ME = z^* \cdot SE \][/tex]
- Substituting the known values:
[tex]\[ ME = 2.576 \cdot 0.002986 \approx 0.007693 \][/tex]
5. Calculate the confidence interval:
- The 99% confidence interval is found by adding and subtracting the margin of error from the sample proportion:
[tex]\[ \text{Lower bound} = \hat{p} - ME = 0.009 - 0.007693 \approx 0.001307 \][/tex]
[tex]\[ \text{Upper bound} = \hat{p} + ME = 0.009 + 0.007693 \approx 0.016693 \][/tex]
Hence, the 99% confidence interval for the proportion of voters against the fire department bond measures is approximately:
[tex]\[ [0.001307, 0.016693] \][/tex]
