Answered question

2022-03-22

Answer & Explanation

karton

karton

Expert2023-04-25Added 613 answers

Given the sample data:
x=[2.6,42.8,29.9,7.9,27,16.9,44.8,-3.7]

To estimate the population mean μ, we can use the sample mean x¯ as an unbiased estimator of μ.
x¯=2.6+42.8+29.9+7.9+27+16.9+44.8-3.78
x¯=20.975

Therefore, the estimated population mean μ is 20.975.

To find the 98% confidence interval about μ, we need to use the t-distribution with (n-1) degrees of freedom, where n is the sample size. Since n = 8, we have (n-1) = 7 degrees of freedom.

The formula for the confidence interval is:
x¯±tα2,n-1(sn)

where x¯ is the sample mean, s is the sample standard deviation, n is the sample size, tα2,n-1 is the t-score for the desired confidence level and degrees of freedom, and alpha is the complement of the confidence level.

First, we need to calculate the sample standard deviation s:
s=(1n-1)Sum[(ξ-x¯)2]
s=(17)((2.6-20.975)2+(42.8-20.975)2+(29.9-20.975)2+(7.9-20.975)2+(27-20.975)2+(16.9-20.975)2+(44.8-20.975)2+(-3.7-20.975)2)
s=19.1677

Next, we need to find the t-score for the 98% confidence level and 7 degrees of freedom. We can use a t-table or a calculator to find this value. Using a calculator, we have:
t0.012,7=2.998

Therefore, the 98% confidence interval about the population mean μ is:
20.975±2.998(19.16778)

Simplifying this expression gives:
(9.174,32.776)

Therefore, we can say with 98% confidence that the true population mean μ lies between 9.174 and 32.776.

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