A researcher believes a new diet should improve weight gain. To test his hypothesis a random sample of 10 people on the old diet and an independent random sample of 10 people on the new diet were selected. The selected people on the old diet gain an average of 5 pounds with a standard deviation of 2 pounds, while the average gain for selected people on the new diet was 8 pounds with a standard deviation of 1.5 pounds. Assume that the values are normally distributed in each population and that the population variances are approximately equal. Using $\alpha$= 0.05, the observed t value for this test is ___.
A) -1.96
B) -1.645
C) -2.100
D) -3.79
E) -1.734

Question

Quizplus · Accepted Answer

The null hypothesis is that the old diet and the new diet lead to the same weight gain. The alternative hypothesis is that the new diet leads to a greater weight gain than the old diet.
Since the population variances are assumed to be equal, we can use the pooled standard deviation:
s_p = sqrt(((n1-1)s1^2 + (n2-1)s2^2)/(n1+n2-2))
s_p = sqrt(((10-1)2^2 + (10-1)1.5^2)/(10+10-2)) = 1.75
Now, we can calculate the t-value:
t = (x̄1 - x̄2) / (s_p * sqrt(1/n1 + 1/n2))
t = (5 - 8) / (1.75 * sqrt(1/10 + 1/10)) = -3.79
With a significance level of 0.05 and a two-tailed test, the critical t-value is ±2.093. Since the observed t-value (-3.79) is outside of the critical region, we reject the null hypothesis and conclude that the new diet leads to a greater weight gain than the old diet. Therefore, the answer is D.

A Researcher Believes a New Diet Should Improve Weight Gain α\alphaα

A Researcher Believes a New Diet Should Improve Weight Gain $\alpha$