Question

Question: Summer weight-loss camp. Camp Jump Start is an 8-week summer camp for overweight and obese adolescents. Counselors develop a weight-management program for each camper that centers on nutrition education and physical activity. To justify the cost of the camp, counselors must provide empirical evidence that the weight-management program is effective. In a study published in Paediatrics (April 2010), the body mass index (BMI) was measured for each of 76 campers both at the start and end of camp. Summary statistics on BMI measurements are shown in the table.

Source: Based on J. Huelsing, N. Kanafani, J. Mao, and N. H. White, "Camp Jump Start: Effects of a Residential Summer Weight-Loss Camp for Older Children and Adolescents," Pediatrics, Vol. 125, No. 4, April 2010 (Table 3).

a. Give the null and alternative hypotheses for determining whether the mean BMI at the end of camp is less than the mean BMI at the start of camp.

b. How should the data be analyzed, as an independent samples test or as a paired difference test? Explain.

c. Calculate the test statistic using the formula for an independent samples test. (Note: This is not how the test should be conducted.)

d. Calculate the test statistic using the formula for a paired difference test.

e. Compare the test statistics, parts c and d. Which test statistic provides more evidence in support of the alternative hypothesis?

f. The p-value of the test, part d, was reported as p 6 .0001. Interpret this result, assuming a = .01.

g. Do the differences in BMI values need to be normally distributed in order for the inference, part f, to be valid? Explain.

h. Find a 99% confidence interval for the true mean change in BMI for Camp Jump Start campers. Interpret the result.

Short answer

a.

b. The samples are not independent.

c. 3.10

d. T-value = 19.18 and P-value = 0.000

e. The test statistics in part c are much lower than in part d. The p-value smaller than

f. The sample size is sufficiently large n_d = 76

g. 2.845, 3.755

h. As a result, we are 99.9% certain that the genuine change in mean BMI between the beginning and the completion of camp is between 2.85 and 3.76.

Step-by-step solution

(a) Null and alternative hypothesis

Let µ₁ = mean BMI at the beginning of the camp µ₂ = mean BMI at the finish of the camp. The parameter of the interest is µ₁ - µ₂

To determine if the mean BMI at the finish of camp is less than the mean BMI at the beginning of camp. We test:

(b) Independent samples

A paired difference t-test should be used to analyze it. Each adolescent's BMI was assessed at the start and conclusion of the program. The samples are not different.

(c) Test statistics formula

(d) Paired difference method

The test statistics using the formula for paired difference method is:

Using MINITAB, they find the paired-different t-statistics as follows:

Paired T-test and CI

(e) Support the alternative hypothesis

Part c's test statistics are substantially smaller than Part d's test statistics. Part d's test statistics give more proof in favor of the alternative hypothesis.

(f) Interpret the result

H₀ is rejected because the p-value is less than an (p < 0.0001 0.01) . At α = 0.01 there is sufficient data to suggest that the mean BMI after camp is less than the mean BMI at the beginning of camp.

(g) BMI needs values

No, because the sample size is large enough (n = 76), the Central Limit Theorem applies.

(h) Interpret the result

The mean BMI change for camp jump-start campers has a 99% confidence interval. The 99% confidence interval for the genuine mean change in BMI for camp jump-start campers is obtained from the MINITAB result in section (d) (2.845,3.755). As a result, we are 99.9% certain that the genuine change in mean BMI between the beginning and the completion of camp is between 2.85 and 3.76.