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For normally distributed populations, find the following confidence intervals for μ (assume σ=15 ):
a. n=10,x-bar =99.8,95%CI
b. n=10,x-bar =99.8,99%CI
c. n=50,x-bar =100.3,95%CI
d. n=50,x-bar =100.3,90%CI
e. n=300,x-bar =98.6,95%CI

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.

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Rewritten by : Jeany

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