According to a study published in the New England Journal of Medicine, overweight people on low-carbohydrate and Mediterranean diets lost more weight and got greater cardiovascular benefits than people on a conventional low-fat diet (The Boston Globe, July 17, 2008). A nutritionist wishes to verify these results and follows 30 dieters on the low-carbohydrate and Mediterranean diets and 30 dieters on the low-fat diet. These data (measured in pounds) can be found on the text website, labeled Different Diets. Let Low-carb or Mediterranean and Low-fat diets represent populations 1 and 2, respectively. Use Table 2.

Low-carb Low-carb/

Mediterrean Diets Low- Fat diet Mediterreand diet Low-fat Diet

9.5 6.5 6.8 5.9
8.1 5.8 9.1 6.9
10.4 9.9 9.4 9.1
11.9 5.1 10.2 8.0
11.8 8.0 9.5 8.9
12.6 6.3 9.5 3.4
6.7 6.3 9.4 4.6
9.6 4.4 12.0 6.2
11.6 5.7 9.9 4.6
8.4 5.9 9.7 6.7
9.0 6.8 9.2 4.6
7.5 5.1 13.0 7.1
7.2 6.3 11.3 11.0
8.5 5.5 13.6 4.5
8.8 5.5 9.0 3.9

a.Set up the hypotheses to test the claim that the mean weight loss for those on low-carbohydrate or Mediterranean diets is greater than the mean weight loss for those on a conventional low-fat diet.

H0: ?1 ? ?2 ? 0; HA: ?1 ? ?2 > 0
H0: ?1 ? ?2 = 0; HA: ?1 ? ?2 ? 0
H0: ?1 ? ?2 ? 0; HA: ?1 ? ?2 < 0

b-1.Using the appropriate commands in Excel, find the value of the test statistic. Assume that the population variances are equal and that the test is conducted at the 5% significance level. (Round your answer to 2 decimal places.)

Test statistic

b-2. What is the critical value and the rejection rule?
Reject H0 if tdf < -1.672.
Reject H0 if tdf < -2.002.
Reject H0 if tdf > 2.002.
Reject H0 if tdf > 1.672.

c. At the 5% significance level, can the nutritionist conclude that overweight people on low-carbohydrate or Mediterranean diets lost more weight than people on a conventional low-fat diet?
No
Yes

Low-carb/Mediterrean Diets Low-fat Diet
9.5 6.5
8.1 5.8
10.4 9.9
11.9 5.1
11.8 8
12.6 6.3
6.7 6.3
9.6 4.4
11.6 5.7
8.4 5.9
9 6.8
7.5 5.1
7.2 6.3
8.5 5.5
8.8 5.5
6.8 5.9
9.1 6.9
9.4 9.1
10. 2 8
9.5 8.9
9.5 3.4
9.4 4.6
12 6.2
9.9 4.6
9.7 6.7
9.2 4.6
13 7.1
11.3 11
13.6 4.5
9 3.9

Here researcher claims that the mean weight loss for those on low-carbohydrate or Mediterranean diets is greater than the mean weight loss for those on a conventional low-fat diet

Set the null and alternative hypotheses to test the above claim as below.

[tex]H_0:\mu_1\leq\mu_2\\\\=H_0:\mu_1-\mu_2\leq 0[/tex] versus [tex]H_A:\mu_1 > \mu_2\\\\H_A:\mu_1 -\mu_2>0[/tex]

b)

Assume that the variance of the two populations are equal.

Under the null hypothesis, the test statistics is defined as

[tex]t= \frac{\bar x_1 - \bar x_2}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}[/tex]

where [tex]s_p=\sqrt{\frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}}[/tex]

Use the foll owing Excel instructions and conduct the above test.

Step1: Enter the datainto two columns of spreadsheet.

Step2: Select Data Analysis from Data ribbon.

Step3: Select t-test. Two-sample Assuming equal variances.

Step4: Input the data range for variable 1 and variable 2.

Step5: Enter 0 as the Hypothesized Mean Difference.

Step6: Click OK.

Thus, the resultant outout is as follows:

Low-carb Low-fat

Mean 9.773333333 6.283333333

Variance 3.197885057 3.160747126

Observations 30 30

Pooled Variance 3.179316092

Hypothesized Mean Difference 0

df 58

t Stat 7.580610844

P(T < = t) one-tail 1.54916E - 10

t Critical one — tail 1.671552762

P(T <2) two-tail 3.09831E - 10

t Critical two - tail 2.001717484

1) From the above output, the test statistics is obtained as t = 7.5806

2) From the above output, the critical value for one-tailed testis obtained as [tex]t_{crit}=1.672[/tex]

Rejection rule: Reject [tex]H_0[/tex] if [tex]t_{crit}>1.672[/tex]

c)

At 5% level of significance, the calculated value of test statistics is greater than the critical value.

Therefore, reject the null hypothesis.

Hence, the nutritionist conclude that overweight people on low-carbohydrate or Mediterranean diets lost more weight than people on a conventional low-fat diet.