Question

Exercise IX Two groups of rats are put on a diet while growing up, and their weight gain between day 28 and day 84is recorded. 10 rats are put on a diet with a high protein conten, while 7 rats are put on a diet with a low protein content. The collected data (weight gain in grams)is shown in the table belo, with the total weight gain in each group given in the last row High protein content Low protein content 134 46 104 119 124 161 107 83 113 120 1220 101 85 107 132 94 Total 707 Using the numbers in the table, the sample variances in the two groups are calculated to be s, = 495 and s = 425, where H and L indicate the groups with high and low protein (ontent. respectively. It is further given that the usual test, for whether the expected weight gain is the same for rats on high and low protein diets, has 13.7 degrees of freedom. Quest ion IX.1(18 Which of theflwing choices is correct (both statements need to be correct)? 1口Rats in the low protein diet group gain nxre weight than rats in the high protein diet group. However, te difference is not statistically significant at the significance level = 0.05 2Rats in the high protein diet group gain more weight than rats in the low protein diet group . The difference is statistically significant at the significance level a 0.05 3 □ Rats in the high protein diet group gain more weight than rats in the low protein diet group. The difference is statistically significant at the significanevel a 0.01

Answer 1

We have the following information from the samples

there are rats in the the high protein content group. the total weight gain of the 10 rats is 1220 gms.

The sample mean weight gain of high protein content group is

grams

The sample variance of the weight gains for high protein sample is

there are $n_L=7$ rats in the the low protein content group. the total weight gain of the 7 rats is 707 gms.

The sample mean weight gain of low protein content group is

grams

The sample variance of the weight gains for low protein sample is

From the sample information we can see that the mean weight gain for the rats in high protein group is more than the rats in low protein group. But this is only in this sample of size 10 and 7. We want to test if this difference in the mean weight gain is true in the population.

Let $mu_H,mu_L$ be the true values of mean weight gains for the rats in high protein and low protein diets.

We want to test the following hypotheses

The sample sizes are less than 30 and we do not know the population standard deviations. Hence we will use t-test and t statistics to test the hypotheses.

Since we assume that the population variances of weight gains in the 2 groups are not the same, the standard error of the difference between the means is

The hypothesized difference in the means from null hypothesis is

The test statistics is

We have been given the degrees of freedom for t statistics = 13.7

This is a right tail test (The alternative hypothesis has ">"). The p-value is P(T>2.00).

Using excel function =T.DIST.RT(2,13.7), we get a p-value = 0.033

We will reject the null hypothesis if the p-value is less than the significance level.

Let us set the significance level at .

We can see that the p-value is greater than the significance level  . Hence we can not reject the null hypothesis.

Let us set the significance level at .

We can see that the p-value is less than the significance level . Hence we reject the null hypothesis.

We can conclude that at