Answer to Question #346755 in Statistics and Probability for Bethsheba Kiap

Question #346755

A random sample is drawn from a population of known standard deviation 11.3.


Construct a 90% confidence interval for the population mean based on the information


given (not all of the information given need be used).


a. n = 36, š‘„Ģ…= 105.2, š‘  = 11.2


b. n = 100, š‘„Ģ…= 105.2, š‘  = 11.2



1
Expert's answer
2022-06-01T13:14:37-0400

a.

The critical value forĀ "\\alpha = 0.1"Ā isĀ "z_c = z_{1-\\alpha\/2} = 1.6449."

The corresponding confidence interval is computed as shown below:

"CI=(\\bar{X}-z_c\\times\\dfrac{\\sigma}{\\sqrt{n}}, \\bar{X}+z_c\\times\\dfrac{\\sigma}{\\sqrt{n}})"

"=(105.2-1.6449\\times\\dfrac{11.3}{\\sqrt{36}}, 105.2+1.6449\\times\\dfrac{11.3}{\\sqrt{36}})"

"=(102.1021, 108.2979)"

Therefore, based on the data provided, theĀ 90%Ā confidence interval for the population mean isĀ "102.1021 < \\mu < 108.2979," which indicates that we areĀ 90%Ā confident that the true population meanĀ "\\mu" is contained by the intervalĀ "(102.1021, 108.2979)."


b.

The critical value forĀ "\\alpha = 0.1"Ā isĀ "z_c = z_{1-\\alpha\/2} = 1.6449."

The corresponding confidence interval is computed as shown below:

"CI=(\\bar{X}-z_c\\times\\dfrac{\\sigma}{\\sqrt{n}}, \\bar{X}+z_c\\times\\dfrac{\\sigma}{\\sqrt{n}})"

"=(105.2-1.6449\\times\\dfrac{11.3}{\\sqrt{100}}, 105.2+1.6449\\times\\dfrac{11.3}{\\sqrt{100}})"

"=(103.3413, 107.0587)"

Therefore, based on the data provided, theĀ 90%Ā confidence interval for the population mean isĀ "103.3413 < \\mu < 107.0587," which indicates that we areĀ 90%Ā confident that the true population meanĀ "\\mu" is contained by the intervalĀ "(103.3413, 107.0587)."



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