Chapter 10, Problem AP3.31CPT

To determine

To explain do the data provide convincing evidence that five-week crash dieters weight less, on average one year after finishing the diet.

Answer to Problem AP3.31CPT

There is no convincing evidence.

Explanation of Solution

It is given:

  $\begin{array}{l}n=15\\ \alpha =0.05\end{array}$

Let us now find out the difference the inside time and the drive through time for each visit:

Dieter	Before	After	Difference
1	158	163	-5
2	185	182	3
3	176	188	-12
4	172	150	22
5	164	161	3
6	234	220	14
7	258	235	23
8	200	191	9
9	228	228	0
10	246	237	9
11	198	209	-11
12	221	220	1
13	236	222	14
14	255	268	-13
15	231	234	-3

The mean of the difference is:

  $\begin{array}{l}{\overline{x}}_d=\frac{-5+3-12+22+3+14+23+9+0+9-11+1+14-13+3}{15}\\ =\frac{54}{15}\\ =3.6\end{array}$

The standard deviation of the difference is:

  $\begin{array}{l}s=\sqrt{\frac{\begin{array}{}$ {(-5-3.6)}^2+{(3-3.6)}^2+{(-12-3.6)}^2+{(22-3.6)}^2+{(3-3.6)}^2+$$ {(14-3.6)}^2+{(23-3.6)}^2+{(9-3.6)}^2+{(0-3.6)}^2+{(9-3.6)}^2+{(-11-3.6)}^2$$ +{(1-3.6)}^2+{(14-3.6)}^2+{(-13-3.6)}^2+{(3-3.6)}^2$\end{array}}{15-1}}\\ =11.5251\end{array}$

Now, we will test the hypotheses for the evidence:

So, given claim is: Dieters weight less after finishing the diet.

The claim will be either null hypothesis or the alternative hypothesis. Thus, the hypotheses will be defined as:

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

Thus, the value of test statistics will be:

  $\begin{array}{l}t=\frac{{\overline{x}}_d-{\mu }_d}{\frac{s_d}{\sqrt{n}}}\\ =\frac{3.6-0}{\frac{11.5251}{\sqrt{15}}}\\ =1.210\end{array}$

The P-value is the probability of obtaining the value of test statistic or a value more extreme assuming the null hypothesis is true.

The degree of freedom will be:

  $\begin{array}{l}df=n-1\\ =15-1\\ =14\end{array}$

And also, $0.10<P<0.15$ .

As we know that if the P-value is less than the significance level then reject the null hypothesis. So,

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

Thus, we conclude that we fail to provide the convincing evidence that five-week crash dieters weight loss, on average, one year after finishing the diet.