Confidence Interval for Ratio of Variances
of Independent Normal Populations

 RATVARCI.83p RATVARCI.86p ratvarci.89p

Consider two independent normal populations X and Y having unknown variances VarX and VarY respectively. We wish to construct a confidence interval for the ratio of variances VarX / VarY. This ratio can be initially estimated by the ratio of sample variances (Sx)^2 / (Sy)^2 obtained from independent random samples from X and Y having sizes n and m respectively.

Under these circumstances, the expression F = [ (Sy)^2 VarX ] / [(Sx)^2 VarY ] follows an F distribution with m - 1 and n - 1 degress of freedom. Thus, if our desired level of confidence is r, then we find two f-scores Q and T such that P(F < Q) = (1 - r) / 2 and P(F > T) = (1 - r) / 2 (or equivalently, P(F <= T) = (r + 1) / 2 ). The confidence interval for VarX / VarY is then [Q (Sx)^2 / (Sy)^2, T (Sx)^2 / (Sy)^2].

To find a confidence interval for the ratio of standard deviations, we simply take square roots.

Using the RATVARCI Program

To execute the program, we need only enter the sample size and sample variance for each population and the desired level of confidence. The program displays the f-scores and the confidence intervals for the ratio of the true variances VarX / VarY and the ratio of the true standard deviations.

Caution: Be patient. It may take a minute or so for the program to find the two f-scores.

Example. Boxes of Cheer detergent have the following statement: "Individual packages of Cheer may weigh slightly more or less than the marked weight. This is due to normal variations incurred with high speed packaging machines. However, each day's production of Cheer will average slightly above the marked weight."

Thus, we can assume that the packaged weights are normally distributed. Suppose the ratio of the variances of weights of two different sizes of boxes is to be estimated. A sample of 35 "98 oz." packages yielded weights with a sample variance of 1.45 ounces. A sample of 30 "42 oz." packages yielded weights with a sample variance of 1.05 ounces. Find a 95% confidence interval for the true ratio of the variances.

Solution. After calling up the RATVARCI program, enter 35 for X SAMPLE SIZE, enter 1.45 for X SAMPLE VAR., enter 30 for Y SAMPLE SIZE, enter 1.05 for Y SAMPLE VAR., and enter .95 for CONF. LEVEL.

We obtain a 95% confidence interval of [0.6696, 2.7905] for VarX / VarY and an interval of that [0.8183, 1.6705] for sx / sy, where sx is the true standard deviation of the "98 oz." packages and sy is the true standard deviation of the "42 oz." packages.

Exercises

1. With the data from the example above, find a 98% confidence interval for the ratios VarY / VarX and sy / sx.

2. Consider the random collections of boys' and girls' Math SAT scores below. Find a 99% confidence interval for the ratio of true variances VarX / VarY of boys' scores to girls' scores. Is it possible that in fact VarX = VarY?

A Random Collection of Boys' Math SAT Scores
 450 540 500 580 620 580 650 520 610 710 640 520 520 600 650 600 550 410 580 500 620 500 570 590 450 520 540 660 490 570 460 410

A Random Collection of Girls' Math SAT Scores
 590 620 540 520 490 530 520 570 570 350 320 420 380 470 590 480 380 520 540 570 510 490 510 510 500 490 410 500 520 550 580 630

Solutions

1. Since the ratios are reversed, we simply reverse the roles of x and y. Enter 30 for X SAMPLE SIZE, enter 1.05 for X SAMPLE VAR., enter 35 for Y SAMPLE SIZE, enter 1.45 for Y SAMPLE VAR., and enter .98 for CONF. LEVEL.

We obtain a 98% confidence interval for VarY / VarX of [0.3134, 1.7174] and [0.5598,1.3105] for sy / sx.

2. Enter the data into the STAT Edit screen (LIST EDIT on TI-86, APPS 6 on TI-89), then use the 2-Var Stats command (TwoVar on the TI-86 and 89) in order to compute the statistics Xbar, Sx, Ybar, and Sy.

Next, we can just type in the numerical values of the statistics n = 32, Sx^2 = 74.297^2, and Sy^2 = 75.348^2 into the RATVARCI program, or we can access the statistics to enter these values into the program as follows:

On the TI-83: For X SAMPLE SIZE, press VARS, press 5, press 1, press ENTER. For X SAMPLE VAR., press VARS, press 5, press 3, press x^2, press ENTER. For Y SAMPLE SIZE, press VARS, press 5, press 1, press ENTER. For Y SAMPLE VAR., press VARS, press 5, press 6, press x^2, press ENTER. For, CONF. LEVEL, enter .99.

On the TI-86: For X SAMPLE SIZE, type 2nd ALPHA 9 to obtain n, press ENTER. For X SAMPLE VAR., press STAT (2nd +), press F5, press F3, press x^2, press ENTER. For Y SAMPLE SIZE, type 2nd ALPHA 9 to obtain n, press ENTER. For Y SAMPLE VAR., press STAT (2nd +), press F5, press MORE, then press F1, press x^2, press ENTER. For CONF. LEVEL, enter .99.

On the TI-89: For X SAMPLE SIZE, type and enter nStat. For X SAMPLE VAR., type and enter Sx^2. For Y SAMPLE SIZE, type and enter nStat. For Y SAMPLE VAR., type and enter Sy^2. For CONF. LEVEL, enter .99.

We obtain a 99% confidence interval for VarX / VarY of [0.3762, 2.5131]. Because VarX / VarY could equal 1, we conclude that it is in fact possible for VarX to be equal to VarY.

Note: If the data sets had different sizes, then we first would use the 1-Var Stats command (OneVar on the TI-86 and 89) to compute the statistics on each list separately. Then enter the appropriate values into the RATVARCI program as in the original example above.