Answer :
Final answer:
To the confidence interval formula (CI = ( s(bar{x})) - Z*( s(sigma)/ s(sqrt(n))), ( s(bar{x})) + Z*( s(sigma)/ s(sqrt(n)))) is used. The Z-score varies based on the confidence level (e.g., 1.96 for 95% CI), and by substituting the sample size, sample mean, and Z-score into the formula, different confidence intervals are obtained.
Explanation:
Calculating Confidence Intervals for a Population Mean
To calculate confidence intervals for the population mean ( s(mu)), with a known population standard deviation ( s(sigma)), and a sample mean ( s(bar{x})), we use the formula:
CI = ( s(bar{x}}) - Z*( s(sigma)/ s(sqrt(n))), ( s(bar{x}}) + Z*( s(sigma)/ s(sqrt(n))))
where CI is the confidence interval, Z is the Z-score associated with the confidence level, s(sigma) is the population standard deviation, n is the sample size, and s(bar{x}) is the sample mean.
For the given data points we have:
n = 10, s(bar{x}) = 99.8, s(sigma) = 15, and a 95% confidence level; the Z-score for 95% is approximately 1.96.
n = 10, s(bar{x}) = 99.8, s(sigma) = 15, and a 99% confidence level; the Z-score for 99% is approximately 2.58.
n = 50, s(bar{x}) = 100.3, s(sigma) = 15, and a 95% confidence level; Z-score is 1.96 for 95%.
n = 50, s(bar{x}) = 100.3, s(sigma) = 15, and a 90% confidence level; the Z-score for 90% is approximately 1.645.
n = 300, s(bar{x}) = 98.6, s(sigma) = 15, and a 95% confidence level; again, the Z-score is 1.96 for 95%.
By substituting these values into the CI formula for each scenario, we can compute the various confidence intervals.