Confidence Intervals for Variance and Standard Deviation (Chapter 6.4 Study Notes)

Confidence Intervals

Introduction to Confidence Intervals for Variance and Standard Deviation

Confidence intervals provide a range of plausible values for a population parameter, based on sample data. In this section, we focus on constructing and interpreting confidence intervals for a population variance (σ2) and standard deviation (σ), using the chi-square distribution.

Chi-Square Distribution

Definition and Application

The chi-square distribution is a family of distributions used primarily for hypothesis testing and constructing confidence intervals for variance and standard deviation. If a random variable x is normally distributed, then the distribution of the statistic

$\chi^2 = \frac{(n-1)s^2}{\sigma^2}$

follows a chi-square distribution with n - 1 degrees of freedom, where n is the sample size and s2 is the sample variance.

Properties of the Chi-Square Distribution

• All chi-square values are greater than or equal to zero.

• The chi-square distribution is a family of curves, each determined by its degrees of freedom (df = n - 1).

• The total area under each chi-square curve is 1.

• The distribution is positively skewed (not symmetric).

• As degrees of freedom increase, the chi-square distribution approaches a normal distribution.

Critical Values for Chi-Square

Finding Critical Values

For a given confidence level c, there are two critical values from the chi-square distribution:

• Right-tail critical value: $\chi^2_{R}$

• Left-tail critical value: $\chi^2_{L}$

The area between these two values corresponds to the confidence level c. The critical values are found using a chi-square distribution table, based on the desired confidence level and degrees of freedom.

Example: Finding Critical Values

Find the critical values for a 95% confidence interval when the sample size is 18 (df = 17):

• Right-tail critical value: $\chi^2_{R} = 30.191$

• Left-tail critical value: $\chi^2_{L} = 7.564$

Thus, 95% of the area under the chi-square curve with 17 degrees of freedom lies between 7.564 and 30.191.

Constructing Confidence Intervals for Variance and Standard Deviation

Formulas for Confidence Intervals

The confidence interval for the population variance σ2 is given by:

$\left( \frac{(n-1)s^2}{\chi^2_{R}}, \frac{(n-1)s^2}{\chi^2_{L}} \right)$

The confidence interval for the population standard deviation σ is:

$\left( \sqrt{\frac{(n-1)s^2}{\chi^2_{R}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{L}}} \right)$

where:

• n = sample size

• s = sample standard deviation

• $\chi^2_{R}$ = right-tail critical value

• $\chi^2_{L}$ = left-tail critical value

Interpretation

With confidence level c, we can say that the interval calculated contains the true population variance or standard deviation, assuming the estimation process is repeated many times.

Example: Constructing Confidence Intervals

You randomly select and weigh 30 samples of an allergy medicine. The sample standard deviation is 1.20 milligrams. Assuming the weights are normally distributed, construct 99% confidence intervals for the population variance and standard deviation.

• n = 30, df = 29, c = 0.99

• Critical values: $\chi^2_{R}$ and $\chi^2_{L}$ (from table)

Confidence Interval for Variance:

Left endpoint: 0.80 mg2 Right endpoint: 3.18 mg2

With 99% confidence, the interval (0.80, 3.18) mg2 contains the population variance.

Confidence Interval for Standard Deviation:

Left endpoint: 0.89 mg Right endpoint: 1.78 mg

With 99% confidence, the interval (0.89, 1.78) mg contains the population standard deviation.

Summary Table: Confidence Intervals for Variance and Standard Deviation

Parameter	Confidence Interval Formula	Interpretation
Variance ($\sigma^2$)	$\left( \frac{(n-1)s^2}{\chi^2_{R}}, \frac{(n-1)s^2}{\chi^2_{L}} \right)$	Interval likely to contain the true population variance
Standard Deviation ($\sigma$)	$\left( \sqrt{\frac{(n-1)s^2}{\chi^2_{R}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{L}}} \right)$	Interval likely to contain the true population standard deviation

Additional info: The chi-square distribution is only appropriate for normally distributed populations. For non-normal populations, these confidence intervals may not be valid.