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Suppose twenty-two communities have an average of = 123.6 reported cases of larceny per year. assume that σ is known to be 36.8 cases per year. find a 90%, 95%, and 98% confidence interval for the population mean annual number of reported larceny cases in such communities. compare the lengths of the confidence intervals. as the confidence levels increase, do the confidence intervals increase in length?

Respuesta :

We are given the following data:

Average = m = 123.6
Population standard deviation = σ= psd = 36.8
Sample Size = n = 22

We are to find the confidence intervals for 90%, 95% and 98% confidence level.

Since the population standard deviation is known, and sample size is not too small, we can use standard normal distribution to find the confidence intervals.

Part 1) 90% Confidence Interval
z value for 90% confidence interval = 1.645

Lower end of confidence interval = [tex]m-z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Lower end of confidence interval=[tex]123.6-1.645* \frac{36.8}{ \sqrt{22}}=110.69 [/tex]

Upper end of confidence interval = [tex]m+z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Upper end of confidence interval=[tex]123.6+1.645* \frac{36.8}{ \sqrt{22}}=136.51[/tex]

Thus the 90% confidence interval will be (110.69, 136.51)

Part 2) 95% Confidence Interval
z value for 95% confidence interval = 1.96

Lower end of confidence interval = [tex]m-z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Lower end of confidence interval=[tex]123.6-1.96* \frac{36.8}{ \sqrt{22}}=108.22 [/tex]

Upper end of confidence interval = [tex]m+z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Upper end of confidence interval=[tex]123.6+1.96* \frac{36.8}{ \sqrt{22}}=138.98[/tex]

Thus the 95% confidence interval will be (108.22, 138.98)

Part 3) 98% Confidence Interval
z value for 98% confidence interval = 2.327

Lower end of confidence interval = [tex]m-z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Lower end of confidence interval=[tex]123.6-2.327* \frac{36.8}{ \sqrt{22}}=105.34 [/tex]
Upper end of confidence interval = [tex]m+z *\frac{psd}{ \sqrt{n} } [/tex]
Using the values, we get:
Upper end of confidence interval=[tex]123.6+2.327* \frac{36.8}{ \sqrt{22}}=141.86[/tex]

Thus the 98% confidence interval will be (105.34, 141.86)

Part 4) Comparison of Confidence Intervals
The 90% confidence interval is: (110.69, 136.51)
The 95% confidence interval is: (108.22, 138.98)
The 98% confidence interval is: (105.34, 141.86)

As the level of confidence is increasing, the width of confidence interval is also increasing. So we can conclude that increasing the confidence level increases the width of confidence intervals.