Question

A researcher wants to determine whether or not a five-week crash diet is effective over a long period of time. A random sample of 15 dieters is selected. Each person’s weight is recorded before starting the diet and one year after it is concluded. Based on the data shown below (weight in pounds), can we conclude that the diet has a long-term effect, that is, that dieters managed to not regain the weight they lost? Include appropriate statistical evidence to justify your answer.

Short answer

Critical value of t at $0.05$level of significance $=1.76131$

As t$<1.76131$, we do not reject the null hypothesis and conclude that the five -week crash diet is not effective over a long period of time.

Step-by-step solution

Step 1. given information

We have given the weight of the person before diet and after diet in pounds.

Step 2. Simplify

The null and alternative hypothesis is given by :

$$\begin{gathered} H_0:{\mu }_d=0 \\ {H}_1:{\mu }_d>0 \end{gathered}$$

Where is the mean of the deviation.

The test static is given by,

$t=\frac{\overline{d}}{Sd\left(\overline{d}\right)}~t_{n-1}$

Where d is the mean of deviation in the paired sample.

We have to reject the null hypothesis if$t>t_{\alpha },n-1$

now,

And,

The above table is obtained from excel.

The value of test static t$=0.312361$

Critical value of t at $0.05$level of significance =$1.76131$

As $t<1.76131,$we do not reject the null hypothesis and conclude that the five -week crash diet is not effective over a long period of time.