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Chapter 9: Problem 5

Toxaphene is an insecticide that has been identified as a pollutant in the Great Lakes ecosystem. To investigate the effect of toxaphene exposure on animals, groups of rats were given toxaphene in their diet. The article "Reproduction Study of Toxaphene in the Rat"" \((J\). of Emiron. Sei. Health, 1988: 101-126) reports weight gains (in grams) for rats given a low dose (4 ppm) and for control rats whose diet did not include the insecticide. The sample standard deviation for 23 female control rats was \(32 \mathrm{~g}\) and for 20 female low-dose rats was \(54 \mathrm{~g}\). Does this data suggest that there is more variability in low-dose weight gains than in control weight gains? Assuming normality, carry out a test of hypotheses at significance level .05.

Short Answer

Expert verified
Yes, there is more variability in low-dose weight gains.

Step by step solution

01

Define Hypotheses

We need to test whether there is more variability, i.e., larger variance, in the low-dose group than in the control group. The null hypothesis \(H_0\) is that the variances are equal: \(\sigma^2_1 = \sigma^2_2\). The alternative hypothesis \(H_a\) is that the low-dose group has greater variance: \(\sigma^2_1 > \sigma^2_2\).
02

Identify Known Values

From the problem, we know the standard deviations: \(s_1 = 54\) g for low-dose rats and \(s_2 = 32\) g for control rats. The sample sizes are \(n_1 = 20\) for low-dose rats and \(n_2 = 23\) for control rats.
03

Calculate Test Statistic

The test statistic for comparing two variances is the F-ratio:\[ F = \frac{s_1^2}{s_2^2} = \frac{54^2}{32^2} = \frac{2916}{1024} \approx 2.85. \]
04

Determine Critical Value

Using \(\alpha = 0.05\) and the degrees of freedom \(df_1 = n_1 - 1 = 19\) and \(df_2 = n_2 - 1 = 22\), we look up the critical value \(F_{0.05, 19, 22}\) in the F-distribution table. The critical value is approximately 2.13.
05

Compare Test Statistic to Critical Value

We compare the calculated F-statistic \(2.85\) to the critical value \(2.13\). Since \(2.85 > 2.13\), we reject the null hypothesis.
06

Draw Conclusion

Since we rejected the null hypothesis, there is sufficient evidence to suggest that there is more variability in low-dose weight gains than in control weight gains at the 0.05 significance level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

F-distribution
The F-distribution is a continuous probability distribution that arises frequently in statistical scenarios, especially when comparing group variances. It is characterized by two parameters: degrees of freedom in the numerator and the denominator. In the context of this problem, these degrees of freedom correspond to the sample sizes of the two groups minus one. For example, given the two groups (low-dose and control) of rats, each with their own sample size, we calculate degrees of freedom as follows: for the low-dose group, it's 19 (since there are 20 rats minus 1), and for the control group, it's 22 (since there are 23 rats minus 1).

The F-distribution is particularly helpful for testing if two population variances are equal. This is because it forms the basis for the F-test, where you calculate the ratio of the variances. If the variances are equal, the F-ratio should be close to 1. In cases where the F-ratio significantly deviates from one, you may conclude that the variances differ.
Variance Comparison
Variance comparison in hypothesis testing is essential for understanding whether two groups differ in how much their data spreads out. Variance measures the extent to which data points in each group deviate from the mean. When using the F-test for comparing variances, you calculate the F-ratio by taking the square of the standard deviations (squared values of deviations from the mean) of the two groups and forming a ratio. With the exercise's standard deviations of 54 g for the low-dose group and 32 g for the control group, the variances are calculated as follows:
  • Low-dose group variance: 54虏
  • Control group variance: 32虏
Thus, you compare these variances by forming the F-ratio, which helps in determining if one group's variance significantly deviates from the other. This comparison is why the test statistic is compared to a critical value from the F-distribution table.
Significance Level
The significance level, often denoted by \(\alpha\), is a threshold used in hypothesis testing to decide whether to reject the null hypothesis. It is expressed as a probability and represents the risk of committing a Type I error, which is rejecting a true null hypothesis. In the exercise, the significance level is set at 0.05, meaning there is a 5% risk of concluding that the variances are different when they actually are not.Choosing a significance level involves balancing the risk of false positives and negatives. A lower \(\alpha\) might reduce the risk of false positives but increase false negatives. In hypothesis testing, once the test statistic is calculated, it is compared to the critical value derived from the significance level and degrees of freedom. If the test statistic exceeds this critical value, the null hypothesis is rejected. In this case, because 2.85 is greater than the critical F-value of 2.13, the null hypothesis is rejected at the \(\alpha = 0.05\) level, indicating the low-dose group variance is likely higher.

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