of Independent Normal Populations

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.

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.

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?

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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

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.

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