Conditions of the Problem

First, the given conditions can be summarized as follows.

Sample size:n = 51
Sample mean:m = 160
Sample standard deviation:s = 10
Confidence level:99%

Using these data, estimate the population mean μ and the population standard deviation σ by interval estimation.

Concept of Interval Estimation for the Population Mean

First, consider interval estimation for the population mean. When the sample size is sufficiently large and the sample mean and sample standard deviation are given, the confidence interval for the population mean is sample mean ± critical value × standard error can be calculated in this form.

The standard error is s / √n Substituting the values in this example gives

10 / √51 ≒ 1.40028

For a 99% confidence interval, 0.5% is excluded from each tail, so Excel’s normal distribution functions can be used to calculate the lower and upper limits.

How to Calculate a 99% Confidence Interval for a Population Mean in Excel

To calculate a 99% confidence interval for a population mean in Excel, the following formula can be used.

=NORM.INV((1-99/100)/2,160,10/SQRT(51))
from
=NORM.INV(1-(1-99/100)/2,160,10/SQRT(51))

Using this calculation, the population mean μ The 99% confidence interval is therefore as follows.

156.3931175 ≦ μ ≦ 163.6068825

Thus, based on this sample, the population mean height of female faculty members at a certain university, with a 99% confidence level, is approximately 156.39 cm to 163.61 cm can be estimated to lie within this interval.

How to Interpret the Result for the Population Mean

This result does not mean that “the population mean is definitely within this range.” More precisely, if sampling and interval estimation were repeated many times using the same method, about 99% of the resulting intervals would contain the true population mean This means that

In other words, the interval obtained here, 156.39～163.61 can be regarded as a range that contains the true population mean at the 99% confidence level.

Concept of Interval Estimation for the Population Standard Deviation

Next, consider interval estimation for the population standard deviation. Unlike the population mean, this calculation does not use a normal distribution function but instead uses chi-square distribution is used.

population variance σ2 The 99% confidence interval for the population variance, with degrees of freedom n-1 = 50 is calculated using

(n-1)s2 / χ2(0.995) ≦ σ2 ≦ (n-1)s2 / χ2(0.005)

which can be calculated in the following form. Substituting the values in this example gives

(50 × 102) / χ2(0.995) ≦ σ2 ≦ (50 × 102) / χ2(0.005)

and becomes

How to Calculate a 99% Confidence Interval for a Population Standard Deviation in Excel

In Excel, the inverse chi-square distribution function can be used to calculate the interval for the population variance and then the population standard deviation. First, the lower and upper limits for the population variance are obtained with the following formulas.

Lower bound: =(50*10^2)/CHISQ.INV.RT(0.005,50)
Upper bound: =(50*10^2)/CHISQ.INV.RT(0.995,50)

Because the population standard deviation is the square root of the population variance, the final step is to take the square root as follows.

Lower bound: =SQRT((50*10^2)/CHISQ.INV.RT(0.005,50))
Upper bound: =SQRT((50*10^2)/CHISQ.INV.RT(0.995,50))

Using this calculation, the population standard deviation σ The 99% confidence interval is therefore as follows.

7.9310158 ≦ σ ≦ 13.3652702

Thus, the population standard deviation, with a 99% confidence level, is approximately 7.93 to 13.37 can be estimated to lie within this interval.

How Can the Results Be Summarized?

Based on the above, the 99% confidence intervals derived from this sample data are as follows.

population mean： 156.3931175 ≦ μ ≦ 163.6068825
population standard deviation： 7.9310158 ≦ σ ≦ 13.3652702

This indicates that, for the heights of female faculty members at a certain university, the true mean is approximately 156.39 cm to 163.61 cm and the population standard deviation, which represents the true degree of dispersion, is approximately 7.93 to 13.37 is estimated to lie within this range.

Summary

In introductory statistics, interval estimation of the population mean and population standard deviation is a fundamental method for inferring population characteristics from a sample.

For interval estimation of the population mean, the sample mean and standard error is used, and in Excel NORM.INV can be calculated using a function. By contrast, interval estimation of the population standard deviation uses chi-square distribution is used, and in Excel CHISQ.INV.RT a function is used for the calculation.

For the present data, at the 99% confidence level, the population mean is 156.3931175～163.6068825 , while the population standard deviation is 7.9310158～13.3652702 was estimated as shown above.

Interval estimation is meaningful because it does not merely predict a single value, but instead shows the range in which the true value is considered likely to lie Understanding this concept is important when studying statistics.