Question

Use the sample information x¯ = 41, σ = 4, n = 20 to calculate the following confidence intervals for μ assuming the sample is from a normal population. (a) 90 percent confidence. (Round your answers to 4 decimal places.) The 90% confidence interval is from to (b) 95 percent confidence. (Round your answers to 4 decimal places.) The 95% confidence interval is from to (c) 99 percent confidence. (Round your answers to 4 decimal places.) The 99% confidence interval is from to (d) Describe how the intervals change as you increase the confidence level. The interval gets narrower as the confidence level increases. The interval gets wider as the confidence level decreases. The interval gets wider as the confidence level increases. The interval stays the same as the confidence level increases.

Answer 1

The formula to construct the confidence interval when population standard deviation is known is-

where, sample mean, sample size

Given:

(a) 90% confidence interval for population mean :

For 90% confidence,

Critical value:

                             

                             

So, the 90% confidence interval for population mean is calculated as (39.5287, 42.4713) Or

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(b) 95% confidence interval for population mean :

Critical value:                   

                            

                            

                            

                            

                             

So, the 95% confidence interval for population mean is calculated as (39.2469, 42.7531) Or

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(c) 99% confidence interval for population mean :

Critical value:

                             

                             

                             

                              

So, the 99% confidence interval for the population mean is calculated as (38.6960, 43.3040) Or

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(d) The correct answer is , "The interval gets wider as the confidence level increases"

The length of a confidence interval is given by-

For 90% confidence:

For 95% confidence :

For 99% confidence:

So, we can see that as the confidence level increases the length of the confidence interval is getting larger. Hence, we say that as the confidence level increase the confidence interval gets wider.