91Ó°ÊÓ

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 Environ. Sci. 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.

Step by step solution

01

State the hypotheses

We are testing whether there is more variability in weight gains for low-dose rats compared to control rats. We will use the null hypothesis \( H_0: \sigma_1^2 = \sigma_2^2 \) (the variances are equal) and the alternative hypothesis \( H_a: \sigma_1^2 > \sigma_2^2 \) (the variance of low-dose weight gains is greater).

02

Identify the test statistic

Since we are comparing the variances of two groups, we will use the F-test for comparing two variances. The test statistic is given by:\[ F = \frac{s_2^2}{s_1^2} \]where \( s_2 = 54 \) g is the sample standard deviation of the low-dose group and \( s_1 = 32 \) g is the sample standard deviation of the control group.

03

Calculate the test statistic

Compute the test statistic using the given standard deviations:\[ F = \frac{54^2}{32^2} = \frac{2916}{1024} \approx 2.85 \]

04

Determine the critical value

The critical value for an F-test at a 0.05 significance level, with degrees of freedom \(df_2 = n_2 - 1 = 19\) for the low-dose group and \(df_1 = n_1 - 1 = 22\) for the control group, can be found in F-distribution tables or by using statistical software. The critical value for \( F(19, 22) \) at \( \alpha = 0.05 \) is approximately 2.18.

05

Make the decision

Compare the calculated F-statistic \( F = 2.85 \) to the critical value \( 2.18 \). Since \( F = 2.85 > 2.18 \), we reject the null hypothesis \( H_0 \).

06

Conclusion

Since the calculated F-statistic is greater than the critical value, there is enough evidence at the \( \alpha = 0.05 \) significance level to conclude that there is more variability in the weight gains of the low-dose rats compared to the control rats.

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Key Concepts

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

F-test

The F-test is a statistical test used to compare the variances of two populations. It helps determine if a dataset shows more variability than another. This test is vital when studying two groups and you want to see if one group is more varied or dispersed than the other.

In the context of our exercise, we are comparing weight gains of rats subjected to a low dose of toxaphene against those not exposed. We use the F-test because it conveniently handles hypotheses testing related to ratio of variances:

• The formula for the F-test statistic is: \[F = \frac{s_2^2}{s_1^2}\] where \(s_2\) is the standard deviation of the group showing potentially higher variability and \(s_1\) of the other group.


• In our analysis, \(s_2\) is 54 grams and \(s_1\) is 32 grams, which are the sample standard deviations for the low-dose and control groups respectively.

Using the F-test, we place these values in the formula to get the F-statistic, which will be compared to a critical value to make decisions about our hypotheses.

Variance Comparison

Variance comparison involves determining the extent to which numerical values in data sets deviate from their average value. It is paramount in statistics, especially when assessing whether two groups fluctuate similarly or differently.

In our scenario with the rats, we are checking whether their weight gains show a statistically significant difference in variability. The core principle here is:

• Variance is calculated as the square of the standard deviation. Thus, for the low-dose group, variance is \(54^2\) and for control, \(32^2\).


• Higher variance in the low-dose group suggests greater dispersion in this group's weight gains as compared to the control group.


• The F-test statistic derived from the variance comparison guides our decision whether the observed differences are significant or occurred by random chance.

Understanding variance is indispensable for making informed conclusions about data variability and whether these conclusions are statistically meaningful.

Significance Level

The significance level, often represented by \(\alpha\), determines the threshold for rejecting our null hypothesis. It's a probability value that defines the risk of concluding that a difference exists when there is none.

Commonly set at 0.05, the significance level in hypothesis testing indicates a 5% risk of erroneously claiming a difference in variability between groups.

In our exercise, with \(\alpha = 0.05\):

• The critical value corresponding to this significance level and degree of freedom helps us decide whether our calculated F-statistic is significant.


• If our F-statistic exceeds this critical value, we reject the null hypothesis, indicating that the groups indeed differ in variability.


• It frames how confident we are that our results are not due to random variation.

Choosing an appropriate significance level is integral for hypothesis testing, balancing between Type I error risk and decision accuracy.

Critical Value

The critical value is a key component in hypothesis testing and helps determine whether the test statistic is extreme enough to reject the null hypothesis. It represents the threshold above which we state the differences observed in our data are statistically significant.

For our rat experiment, we calculate an F-statistic of approximately 2.85, and we need to compare this to our critical value derived from F-distribution tables at a 0.05 significance level with specific degrees of freedom:

• Degrees of freedom for the numerator \((df_2 = 19)\) and denominator \((df_1 = 22)\) are based on sample sizes minus one.


• The critical value for these parameters is approximately 2.18.


• If our test statistic \(F = 2.85\) is greater than 2.18, we reject the null hypothesis, concluding increased variability in the low-dose group.

Comparing the computed statistic to the critical value is crucial, as it dictates our acceptance or rejection of the null hypothesis, confirming or denying significant differences in data spread.