Counselors at a summer camp for averweight and obese adolescents develop a weigh
ID: 3045581 • Letter: C
Question
Counselors at a summer camp for averweight and obese adolescents develop a weight-management program for each camper that centers on nutrition oducation and physical activity. In a recent study, the body mass index (BMI) was measured for each of 72 campers both at the start and end of camp. Summary stafistics on BR measurements are shown in the table to the right Complete parts athrough h Mean Deviation Starting BMI 336 62 Ending BMI 30.0 59 Paired Oifflerences 3.6 18 a Give the null and alternative hypotheses for determining whether the mean BMI at the and of camp is less than the mean BMI at the start of camp Lot bethe population mean BMI at the start of camp, P2 be te population mean BMI atthe endofan, and A-P1-2. b. How should the data be analyzed, as an independent-samples t-test or as a paired-diference t-Hest? Explain. Choose the correct answer below d A. Use an independent-samples Moet because the sample of dneronces was chosen randomly O B. Use a paired-dference ttest because, although the data were not paired originally, pairing them after sampling significanty reduoes variation within the sampies c. Use a pared diflerence t test because te assumpton of independet sampos is malid O D. Use an independent-campies -test because the sample size is greater than 30. By the Central Limit Thecrem, this condtion guarantoes the tost statistic will be approximatsly normal e. Caloulate the test statistio using the formula for an independent-samples t-test y nomal t(Round to two decimal places as needed) d. Calculate the test statste using mua for pared-diferene West t- (Roundto two deomal places as needed.) e. Compare the tost staristics parts e and d. Which teat statistic provides more evidence in support of the atemaive hypothesis? OA The est statistic using the fomula for an independent-sampies t-testExplanation / Answer
Solution:-
A) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: d< 0
Alternative hypothesis: d > 0
Note that these hypotheses constitute a one-tailed test.
b) C) Use a pair difference t-test, becuase the assumption of independent samles is invalid.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).
c)
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 1.00864
DF = 142
t = [ (x1 - x2) - d ] / SE
t = 3.57
d)
s = sqrt [ ((di - d)2 / (n - 1) ]
s = 1.8
SE = s / sqrt(n)
S.E = 0.2121
DF = n - 1 = 72 -1
D.F = 71
t = [ (x1 - x2) - D ] / SE
t = 16.97
where di is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.
Since we have a one-tailed test, the P-value is the probability that a t statistic having 71 degrees of freedom is more than 16.97.
Thus, the P-value = less than 0.0001
Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.
Reject H0. The mean difference appears to differ from zero.