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Chapter 9: Problem 24

In a study designed to investigate the effects of a strong magnetic field on the early development of mice (7), ten cages, each containing three 30 -day- old albino female mice, were subjected for a period of 12 days to a magnetic field having an average strength of \(80 \mathrm{Oe} / \mathrm{cm}\). Thirty other mice, housed in ten similar cages, were not put in the magnetic field and served as controls. Listed in the table are the weight gains, in grams, for each of the twenty sets of mice. Test whether the variances of the two sets of weight gains are significantly different. Let \(\alpha=0.05\). For the mice in the magnetic field, \(s_{X}=5.67\); for the other mice, \(s_{Y}=3.18\).

Short Answer

Expert verified
The difference in variance of weight gains between two groups of mice is statistically tested using F-test on provided variances. If the calculated F-value is higher than the critical F-value for \(\alpha=0.05\), with degree of freedom of 29 for both groups, then variances are significantly different and null hypothesis is rejected, otherwise not. Exact conclusion will depend on exact calculated F value and value from F-table.

Step by step solution

01

- Definition of Hypotheses

Start by defining the null hypothesis \(H_0\) and the alternative hypothesis \(H_1\). The null hypothesis is that the variances are the same for both groups. The alternative hypothesis is that the variances are not the same. So, \( H_0 : \sigma^2_{X}=\sigma^2_{Y} \) and \( H_1 : \sigma^2_{X} \neq \sigma^2_{Y} \)
02

- Calculation of F-Ratio

Calculate the F-ratio, which is the ratio of the higher variance to the smaller one. As per convention, the larger sample variance should always go in the numerator to keep the ratio greater than or equal to 1. Hence,\[ F = \frac{s^2_{X}}{s^2_{Y}} = \frac{5.67^2}{3.18^2} \]
03

- Finding Critical F Value

The critical F value, for determining the hypothesis, can be found using an F-distribution table for \(\alpha=0.05\). The degrees of freedom for each group will be the number of subjects (30) minus 1, so we have \(df_1=df_2=29\). Look up the critical F value in an F-table corresponding to alpha level \(0.05\) and degree of freedom \(29\) for both groups.
04

- Concluding the test

Compare the calculated F value to the critical F value from the table. If the calculated F value is higher than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis. If the calculated F is less than the critical F value, the null hypothesis is not rejected.

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Key Concepts

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

Null Hypothesis
The null hypothesis, denoted as H0, is a fundamental concept in statistical hypothesis testing. It represents a default statement or position that indicates no effect or no difference. In the context of variance comparison, the null hypothesis asserts that there are no significant differences between the variances of two or more groups.

For example, in the study of the effects of a strong magnetic field on the early development of mice, the null hypothesis would claim that the exposure to the magnetic field does not significantly change the variance in weight gains of the mice compared to the control group. Mathematically, this hypothesis is expressed as:
H0: \(\sigma^2_X = \sigma^2_Y\), where \(\sigma^2_X\) and \(\sigma^2_Y\) represent the variances of the weight gains for mice in the magnetic field and mice in the control group respectively.
Alternative Hypothesis
The alternative hypothesis is indicated by H1 or Ha and is the hypothesis that researchers want to test against the null hypothesis. It is essentially what one would conclude if they find sufficient evidence to reject the null hypothesis.

In the mouse development study, the alternative hypothesis posits that there is indeed a significant difference in variance between the weight gains of mice exposed to the magnetic field and those that were not. This is formally written as:
H1: \(\sigma^2_X eq \sigma^2_Y\). Accepting the alternative hypothesis implies that the exposure to the magnetic field does affect the variation in weight gains, leading researchers to consider further investigation into the magnetic field's impact.
F-distribution
The F-distribution is a probability distribution that arises prominently in analyses relating to variance comparison, especially in ANOVA (Analysis of Variance) and the F-test. It is used to compare the variances of two different populations to assess if they come from distributions with the same variance.

The shape of the F-distribution is skewed to the right, and it is specified by two sets of degrees of freedom: df1 for the numerator and df2 for the denominator. In hypothesis testing, the calculated F-ratio, which compares the variances of the twogroups, is compared against a value from the F-distribution 鈥� the critical F-value. This critical value acts as a threshold: if the calculated F-ratio exceeds it, the null hypothesis is in doubt.
Variance Comparison
Variance comparison is used to determine whether there are significant differences between the variances of different datasets, which in turn can inform us about the consistency, spread, and reliability of data. This process often employs the F-test, which involves calculating the F-ratio of variances and using the F-distribution to interpret the results.

In the given exercise with the mice, the variance comparison was conducted by calculating the F-ratio using the sample variances: F = \(\frac{s^2_X}{s^2_Y}\), then comparing this to a critical value from the F-distribution table. A larger F-ratio suggests a higher likelihood that the variances do indeed differ significantly, which might lead us to reject the null hypothesis in favor of the alternative hypothesis - indicating a potential effect of the magnetic field on the mice.

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