Chapter 25, Problem 21E

To determine

To explain is there a significant difference in calories between serving of strawberry and vanilla yogurt and test an appropriate hypothesis and state your conclusion and also check the assumptions and conditions.

Answer to Problem 21E

There is no sufficient evidence to support the claim that there a significant difference in calories between serving of strawberry and vanilla yogurt.

Explanation of Solution

In the question, the table of different brands of ice cream and calories of the two types of servings are given. Thus, first we will check the assumptions and conditions for inference such as:

Random condition: It is satisfied as we are assuming that the servings were randomly selected.

Independent condition: It is satisfied because we are assuming that the brands are independent.

Normal condition: It is satisfied because the normal quantile plot is roughly linear.

Thus, all the conditions and assumptions for inference are satisfied.

Now, it is given that,

  $n=12$

And we will find the difference between the two samples as:

  

Now, the sample mean and the standard deviation of the difference are as follows:

  $\begin{array}{l}\overline{d}=\frac{10+0+\mathrm{....}+0+0}{12}\\ =12.5\\ s_d=\sqrt{\frac{{(10-12.5)}^2+{(0-12.5)}^2+\mathrm{....}+{(0-12.5)}^2}{12-1}}\\ =32.5087\end{array}$

Let us now define the hypotheses as:

  $\begin{array}{l}{H}_0:{\mu }_d=0\\ H_a:{\mu }_d\ne 0\end{array}$

Now the degree of freedom is then as:

  $\begin{array}{l}df=n-1\\ =12-1\\ =11\end{array}$

Thus, the value of the test statistics is then as:

  $\begin{array}{l}t=\frac{\overline{d}}{s_d/\sqrt{n}}\\ =\frac{12.5}{32.5087/\sqrt{12}}\\ =1.33\end{array}$

Thus, the P-value can be calculated using table T of appendix F as:

  $P>0.20$

As we know that the P-value is less than the significance level then the null hypothesis is rejected. Thus, we have,

  $P>0.05\Rightarrow \text{Fail to Reject }{H}_0$

Thus, we conclude that there is no sufficient evidence to support the claim that there a significant difference in calories between serving of strawberry and vanilla yogurt.