Answer :
To determine which confidence level will produce the widest confidence interval, we need to analyze the standard errors associated with different confidence levels. Given a sample proportion of 0.6, we can proceed through the following steps:
1. Understanding Confidence Intervals and Standard Errors:
Confidence intervals provide a range within which we can expect the true population parameter to lie, based on our sample data. The width of the confidence interval depends on the standard error and the critical value associated with the desired confidence level.
2. Sample Proportion:
The sample proportion (p) is given as 0.6.
3. Standard Error Calculation:
The standard error for a proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
where [tex]\( n \)[/tex] is the sample size. Since we are given sample proportion directly, let's ignore the denominator of sample size for simplicity.
4. Critical Values:
Different confidence levels have different critical values:
- For a 90% confidence level, the critical value is approximately 1.645.
- For a 95% confidence level, the critical value is approximately 1.96.
- For a 99% confidence level, the critical value is approximately 2.576.
5. Compute Standard Errors for Each Confidence Level:
We multiply the standard error by the corresponding critical value to get the margin of error. The observed standard errors for each confidence level are:
- For 90% confidence level, the standard error is:
[tex]\[ SE_{90} \approx 0.8059 \][/tex]
- For 95% confidence level, the standard error is:
[tex]\[ SE_{95} \approx 0.9602 \][/tex]
- For 99% confidence level, the standard error is:
[tex]\[ SE_{99} \approx 1.2620 \][/tex]
6. Determine the Widest Confidence Interval:
The confidence interval which is the widest will have the largest standard error. Comparing the standard errors, we see:
[tex]\[ SE_{90} \approx 0.8059,\ SE_{95} \approx 0.9602,\ SE_{99} \approx 1.2620 \][/tex]
In conclusion, the 99% confidence level will produce the widest confidence interval, as it has the largest standard error of approximately 1.2620.
1. Understanding Confidence Intervals and Standard Errors:
Confidence intervals provide a range within which we can expect the true population parameter to lie, based on our sample data. The width of the confidence interval depends on the standard error and the critical value associated with the desired confidence level.
2. Sample Proportion:
The sample proportion (p) is given as 0.6.
3. Standard Error Calculation:
The standard error for a proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{p(1 - p)}{n}} \][/tex]
where [tex]\( n \)[/tex] is the sample size. Since we are given sample proportion directly, let's ignore the denominator of sample size for simplicity.
4. Critical Values:
Different confidence levels have different critical values:
- For a 90% confidence level, the critical value is approximately 1.645.
- For a 95% confidence level, the critical value is approximately 1.96.
- For a 99% confidence level, the critical value is approximately 2.576.
5. Compute Standard Errors for Each Confidence Level:
We multiply the standard error by the corresponding critical value to get the margin of error. The observed standard errors for each confidence level are:
- For 90% confidence level, the standard error is:
[tex]\[ SE_{90} \approx 0.8059 \][/tex]
- For 95% confidence level, the standard error is:
[tex]\[ SE_{95} \approx 0.9602 \][/tex]
- For 99% confidence level, the standard error is:
[tex]\[ SE_{99} \approx 1.2620 \][/tex]
6. Determine the Widest Confidence Interval:
The confidence interval which is the widest will have the largest standard error. Comparing the standard errors, we see:
[tex]\[ SE_{90} \approx 0.8059,\ SE_{95} \approx 0.9602,\ SE_{99} \approx 1.2620 \][/tex]
In conclusion, the 99% confidence level will produce the widest confidence interval, as it has the largest standard error of approximately 1.2620.
