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 documents the weight loss (in pounds) of 30 dieters on
the low-carbohydrate and Mediterranean diets and 30 dieters on the
low-fat diet. Let Low-carb or Mediterranean and Low-fat diets
represent populations 1 and 2, respectively. (You may find
it useful to reference the appropriate table: z table
or t table)
Low-carb/ Mediterranean Diets |
Low-fat Diet | Low-carb/ Mediterranean Diets |
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 | ||||||||
Click here for the Excel Data File
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.
H_{0}: μ_{1} − μ_{2} = 0; H_{A}: μ_{1} − μ_{2} ≠ 0
H_{0}: μ_{1} − μ_{2} ≥ 0; H_{A}: μ_{1} − μ_{2} < 0
H_{0}: μ_{1} − μ_{2} ≤ 0; H_{A}: μ_{1} − μ_{2} > 0
b-1. Calculate the value of the test statistic.
Assume that the population variances are unknown but equal.
(Round your answer to 3 decimal places.)
b-1. Find the p-value
p-value < 0.01
0.01 ≤ p-value < 0.025
0.025 ≤ p-value < 0.05
0.05 ≤ p-value < 0.10
p-value ≥ 0.10
c. At the 5% significance level, can the
nutritionist conclude that people on low-carbohydrate or
Mediterranean diets lose more weight than people on a conventional
low-fat diet?
Yes
No
a) H_{0}: μ_{1} − μ_{2} ≤ 0; H_{A}: μ_{1} − μ_{2} > 0
===========================
b-1) t = 7.581
===========================
p-value < 0.01
==============================
c) Yes, because p-value < 0.05 so, we Reject the null hypothesis
==========================
I have used Excel and the command
Data > Data Analysis > T -test with equal variances > select the data > ok
below is the output
t-Test: Two-Sample Assuming Equal Variances | ||
Variable 1 | Variable 2 | |
Mean | 9.773333 | 6.283333333 |
Variance | 3.197885 | 3.160747126 |
Observations | 30 | 30 |
Pooled Variance | 3.179316 | |
Hypothesized Mean Difference | 0 | |
df | 58 | |
t Stat | 7.581 | |
P(T<=t) one-tail | 1.55E-10 | |
t Critical one-tail | 1.671553 | |
P(T<=t) two-tail | 3.1E-10 | |
t Critical two-tail | 2.001717 |
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