Question

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.

Answer 1

The question involves comparing the mean weight loss of two different groups of people who were on different diets. The goal is to determine if the liquid diet yields a higher mean weight loss than the powder diet. To compare the mean weight loss, we can use a hypothesis test and calculate the p-value.

Step-by-step explanation:

The subject of this question is Mathematics, specifically statistics. The question involves comparing the mean weight loss of two different groups of people who were on different diets. The powder diet group had a mean weight loss of 42 pounds with a standard deviation of 12 pounds, while the liquid diet group had a mean weight loss of 45 pounds with a standard deviation of 14 pounds. The goal is to determine if the liquid diet yields a higher mean weight loss than the powder diet.

To compare the mean weight loss of these two groups, we can use a hypothesis test. We will formulate the null and alternative hypotheses:

Null Hypothesis (H0): The mean weight loss from the liquid diet is equal to or less than the mean weight loss from the powder diet.

Alternative Hypothesis (Ha): The mean weight loss from the liquid diet is greater than the mean weight loss from the powder diet.

We can then use a t-test to analyze the data and determine if there is enough evidence to reject the null hypothesis and conclude that the liquid diet yields a higher mean weight loss. We would need information on the significance level and the degrees of freedom to conduct the t-test and calculate the p-value.

Answer 2

There is not enough evidence to support the claim that the liquid diet yields a higher mean weight loss than the powder diet (P-value = 0.15).

Step-by-step explanation:

This is a hypothesis test for the difference between populations means.

The claim is that the liquid diet yields a higher mean weight loss than the powder diet.

Then, the null and alternative hypothesis are:

`H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2< 0`

The significance level is 0.05.

The sample 1 (powder diet group), of size n1=49 has a mean of 42 and a standard deviation of 12.

The sample 2 (liquid diet group), of size n2=36 has a mean of 45 and a standard deviation of 14.

The difference between sample means is Md=-3.

`M_d=M_1-M_2=42-45=-3`

The estimated standard error of the difference between means is computed using the formula:

`s_(M_d)=\sqrt{(\sigma_1^2)/(n_1)+(\sigma_2^2)/(n_2)}=\sqrt{(12^2)/(49)+(14^2)/(36)}\\\\\\s_(M_d)=√(2.939+5.444)=√(8.383)=2.895`

Then, we can calculate the t-statistic as:

`t=(M_d-(\mu_1-\mu_2))/(s_(M_d))=(-3-0)/(2.895)=(-3)/(2.895)=-1.04`

The degrees of freedom for this test are:

`df=n_1+n_2-1=49+36-2=83`

This test is a left-tailed test, with 83 degrees of freedom and t=-1.04, so the P-value for this test is calculated as (using a t-table):

`\text{P-value}=P(t<-1.04)=0.15`

As the P-value (0.15) is greater than the significance level (0.05), the effect is not significant.

The null hypothesis failed to be rejected.

There is not enough evidence to support the claim that the liquid diet yields a higher mean weight loss than the powder diet.