Question

Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.

Linda's rats	Tuan's rats	Javier's rats
$43.5$	$47.0$	$51.2$
$39.4$	$40.5$	$40.9$
$41.3$	$38.9$	$37.9$
$46.0$	$46.3$	$45.0$
$38.2$	$44.2$	$48.6$

Determine whether or not the variance in weight gain is statistically the same among Javier’s and Linda’s rats. Test at a significance level of $10\%$

Short answer

There is no evidence that there is a difference between Javier's and Linda's rats.

Step-by-step solution

 Given information 

Given in the question that, Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat’s weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded.

Linda's rats	Tuan's rats	Javier's rats
$43.5$	$47.0$	$51.2$
$39.4$	$40.5$	$40.9$
$41.3$	$38.9$	$37.9$
$46.0$	$46.3$	$45.0$
$38.2$	$44.2$	$48.6$

Explanation

To find these results on the calculator:

Press STAT. Press 1 EDIT. Put the data into the lists$L_1,L_2,L_3$

Press STAT, and arrow over to TESTS, and arrow down to $2$Samp FTest. Select Data and press then selectl and press var then select $2$in list $1$and list$2$. Select the hypothesis as ${\sigma }_1\ne {\sigma }_2$.

The calculator displays :

$F=2.998578$

$p=0.312$

$S_{x1}=5.436635$

$S_{x2}=3.1396$

${\overline{x}}_1=44.72$

${\overline{x}}_2=41.68$

$n_1=5$

$n_2=5$

Hypothesis tests 

We use a solution sheet to conduct the hypothesis tests, and we have:

a) The null hypothesis that three mean commuting mileages are the same is:

$H_0:{\sigma }_1={\sigma }_2$

b) The alternate hypothesis is that at least any two of the means are different:

$H_1:{\sigma }_1\ne {\sigma }_2$

c) The degree of freedom in the numerator - $df($num) is $4$, and the degree of freedom in the denominator - df(denom) is $4$

d) We use the $F_{(4,4)}$distribution for the test.

e) The value of the test statistic (F-value) is$2.99857$

f) The $P$-value for the test is $0.312$

The graph of the distribution 

g) The graph of the distribution is

h) i. Level of significance $\alpha$is $0.1$

ii. Decision: We do not reject the null hypothesis

iii. Reason for decision: $P$-value is $0.312$which is greater than the $0.1$level of significance.

iv. Conclusion: There is no evidence that there is a difference between Javier's and Linda's rats.