91. A powder diet is tested on $49$ people, and a liquid diet is tested on $36$ different people. Of interest is whether the liquid diet yields a higher mean weight loss than the powder diet. The powder diet group had a mean weight loss of $42$ pounds with a standard deviation of $12$ pounds. The liquid diet group had a mean weight loss of $45$ pounds with a standard deviation of $14$ pounds.

The powder diet group had a mean weight loss of $42$ pounds with a standard deviation of $12$ pounds. The liquid diet group had a mean weight loss of $45$ pounds with a standard deviation of $14$ pounds.

(a) The null hypothesis is described as follows:
$H_0:{\mu }_1\le {\mu }_2$
(b)The alternate hypothesis is described as follows:
$H_a:{\mu }_1>{\mu }_2$
(c) The random variable is the difference in mean weight loss between the liquid and powder diets.
(d) Student's t distribution.
(e) To determine test static as follows:
To enter the stat list editor, press STAT followed by $1$.

Then, ENTER all the values the OUTPUT will be:

Hence the test statistics is $-1.036$.

(f) The $p$value is determined from the output as $0.15189$.
(g) Using the data from the last exercise, create a mental picture of the situation.
The horizontal axis should be clearly labelled and scaled, and the region(s) matching to the $p$-value should be shaded on the graph.

(i) $\alpha =0.05$
(ii) Decision: Null hypothesis is not rejected.
(iii) $\alpha <p$-value
(iv) As a result, there is not enough proof to conclude that the liquid diet yields a more weight loss than the powder diet, at the $5\%$level of significance.