n =5 sd = 12.0208 xbar = 57 The ages of a random sample of five university profe

Question

n =5

sd = 12.0208

xbar = 57

The ages of a random sample of five university professors are 39, 54, 61, 72, and 59. Assuming that the ages are normally distributed, using this information, find a 99% confidence interval for: the population standard deviation of the ages of all professors at the university,. the average age of all professors at the university. the predicted age of a new professor hired at the university. Set up a 99% tolerance estimate for the age of all professors at the university that will include at least 90% of all professors at the university. Interpret your results. Does this random sample of five university professors, yield "meaningful" results for using a 99% level of confidence Explain.

Explanation / Answer

Part a) Answer:

Here, we have to find the confidence interval for population standard deviation.

The formula for confidence interval is given as below:

Lower limit = sqrt[(n – 1)S^2 / R]

Upper limit = sqrt[(n – 1)S^2 / L]

By plugging all values in the above formula we get the following results

Confidence Interval Estimate for the Population Variance

Data

Sample Size

5

Sample Standard Deviation

12.02082

Confidence Level

99%

Intermediate Calculations

Degrees of Freedom

4

Sum of Squares

578.0004539

Single Tail Area

0.005

Lower Chi-Square Value

0.2070

Upper Chi-Square Value

14.8603

Results

Interval Lower Limit for Variance

38.8957

Interval Upper Limit for Variance

2792.4199

Interval Lower Limit for Standard Deviation

6.2366

Interval Upper Limit for Standard Deviation

52.8434

Assumption:

Population from which sample was drawn has an approximate normal distribution.

We are 99% sure that the population standard deviation will lies between the two values 6.2366 and 52.8434.

Part b) answer:

The formula for the confidence interval for mean is given as below:

Confidence interval = mean -/+ t*Sample SD/ sqrt(n)

By plugging all values in the above formula we get the following results

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

12.02081528

Sample Mean

57

Sample Size

5

Confidence Level

99%

Intermediate Calculations

Standard Error of the Mean

5.375872022

Degrees of Freedom

4

t Value

4.6041

Interval Half Width

24.7510

Confidence Interval

Interval Lower Limit

32.25

Interval Upper Limit

81.75

Part c)

We are predicted that the population average for age of professors will be lies between the 32.25 years to 81.75 years.

Part d)

Tolerance estimates for confidence interval

Data

Sample Standard Deviation

12.02081528

Sample Mean

57

Sample Size

5

Confidence Level

90%

Intermediate Calculations

Standard Error of the Mean

5.375872022

Degrees of Freedom

4

t Value

2.1318

Interval Half Width

11.4605

Confidence Interval

Interval Lower Limit

45.54

Interval Upper Limit

68.46

Answer part e)

The confidence interval using the sample size as 5 is not appropriate for the estimation purpose because the results based on the confidence intervals are not unbiased at all and it is not helpful for prediction of the professors’ age for entire population. The samples size should be large for getting the unbiased results.

Confidence Interval Estimate for the Population Variance

Data

Sample Size

5

Sample Standard Deviation

12.02082

Confidence Level

99%

Intermediate Calculations

Degrees of Freedom

4

Sum of Squares

578.0004539

Single Tail Area

0.005

Lower Chi-Square Value

0.2070

Upper Chi-Square Value

14.8603

Results

Interval Lower Limit for Variance

38.8957

Interval Upper Limit for Variance

2792.4199

Interval Lower Limit for Standard Deviation

6.2366

Interval Upper Limit for Standard Deviation

52.8434

Assumption:

Population from which sample was drawn has an approximate normal distribution.

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