## Related questions with answers

In a study designed to investigate the effects of a strong magnetic field on the early development of mice, 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 Oe/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.

$\scriptstyle \begin{array}{c} \hline \text { In Magnetic Field } & \text { Not in Magnetic Field } \\ \hline\end{array}$

$\scriptstyle\begin{array}{cccc} \text { Cage } & \text { Weight Gain (g) } & \text { Cage } & \text { Weight Gain (g) } \\ \hline 1 & 22.8 & 11 & 23.5 \\ 2 & 10.2 & 12 & 31.0 \\ 3 & 20.8 & 13 & 19.5 \\ 4 & 27.0 & 14 & 26.2 \\ 5 & 19.2 & 15 & 26.5 \\ 6 & 9.0 & 16 & 25.2 \\ 7 & 14.2 & 17 & 24.5 \\ 8 & 19.8 & 18 & 23.8 \\ 9 & 14.5 & 19 & 27.8 \\ 10 & 14.8 & 20 & 22.0 \\ \hline \end{array}$

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, $s_{Y}=3.18$.

Solution

VerifiedGiven:

$s_1=5.67$

$s_2=3.18$

$n_1=n_2=10$

$\alpha=0.05$

Claim: Different

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain an equality.

$H_0: \sigma_1^2=\sigma_2^2$

$H_a: \sigma_1^2\neq \sigma_2^2$

Compute the value of the test statistic:

$F=\dfrac{s_1^2}{s_2^2}=\dfrac{5.67^2}{3.18^2}\approx 3.1792$

The critical value is given in the F-distribution table in the appendix in the row with $dfd=n_2-1=10-1=9$ and in the column with $dfn=n_1-1=10-1=9$:

$f_{0.025,9,9}=0.248$

$f_{0.975,9,9}=4.03$

The rejection region contains all values smaller than 0.248 and all values larger than 4.03.

If the value of the test statistic is in the rejection region, then reject the null hypothesis:

$0.248<3.1792<4.03\Rightarrow \text{ Fail to reject } H_0$

There is not sufficient evidence to support the claim that the variances are significantly different.

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