Assume the sample is from a normally distributed population...

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Interpret the results. The waiting times (in minutes) of a random sample of 22 people at a bank have a sample standard deviation of 3.6 minutes. Use a $98 \%$ level of confidence.

Assume the sample is from a normally distributed population and construct the indicated confidence intervals for $(a)$ the population variance $\sigma^{2}$ and $(b)$ the population standard deviation $\sigma .$ Interpret the results.
The waiting times (in minutes) of a random sample of 22 people at a bank have a sample standard deviation of 3.6 minutes. Use a $98 \%$ level of confidence.

Key Concepts

Interpretation of Confidence Intervals

The interpretation of a confidence interval for the population variance or standard deviation is that, with a given level of confidence (such as 98%), one can assert that the true parameter lies within the calculated interval. This means that if the same sampling procedure were repeated numerous times, approximately 98% of the computed intervals would contain the true variance or standard deviation.

Chi-Square Distribution

The chi-square distribution plays a key role in constructing confidence intervals for the population variance and standard deviation when the underlying population is normally distributed. It is used because the sample variance, when properly scaled, follows a chi-square distribution with n-1 degrees of freedom, allowing for the establishment of probability-based bounds for the true variance.

Degrees of Freedom

Degrees of freedom in this context refer to the number of independent values that can vary in the calculation of a statistic, which, for a sample variance from a normal population, is n - 1 (where n is the sample size). This concept is crucial because it determines the specific chi-square distribution that is used to calculate the confidence interval.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, within which the parameter of interest (such as the population variance or standard deviation) is expected to lie with a certain probability (confidence level). In the context of variance or standard deviation, the interval is constructed using distributions that account for sample variability and the degree of uncertainty inherent in the estimation process.

Population Variance and Standard Deviation

The population variance (?²) and population standard deviation (?) are fundamental measures of dispersion that indicate the variability or spread of a normally distributed population. Variance quantifies the average squared deviations from the mean, while the standard deviation is the square root of the variance, providing a measure in the original units of the data.

Transcript

00:01 So we have some data, a sample size of n is 22, and we're told the sample standard deviation is 3 .6.

00:08 And we're going to go ahead and assume that the data comes from a normal distribution, so it's normal data.

00:16 And we want to make a 98 % confidence interval for sigma, and we also want to make a 98 % confidence interval for sigma squared.

00:28 So the population standard deviation and the population variance.

00:32 And we're going to do the variance first, because to get the standard deviation, you just take the square root of the variance.

00:37 And so we're going to do the variance first, and we use the following formula, n minus 1 times the sample variance divided by chi squared alpha over 2.

00:49 And we're going to note the degrees of freedom as well.

00:53 And that's the lower bound of sigma squared, and then the upper bound will be n minus 1 times the sample variance, which is the standard deviation squared, divided by the chi squared value of 1 minus alpha over 2.

01:08 And of course, the degrees of freedom.

01:10 So degrees of freedom, that's probably straightforward, n minus 1, so it's going to be 22 minus 1, so 21 degrees of freedom.

01:22 Good.

01:23 The alpha, the alpha is the value that makes this confidence interval 100%.

01:27 That means the alpha is 0 .02.

01:30 So we put 0 .02 for alpha.

01:33 Good.

01:38 So we can go ahead and start filling some of this stuff in.

01:41 And we'll get the chi squared values in a moment.

01:43 Oh, here they are.

01:44 I'll show you how we got those.

01:48 So n minus 1 is 21.