The following is part of an abstract from an article in the New England Journal
ID: 3317531 • Letter: T
Question
The following is part of an abstract from an article in the New England Journal of Medicine.1 This article described an experiment in which each participating child was assigned to drink either a sugar-free or a sugar-containing beverage each day. (You answered a question about boxplots from this study on the out-of-class part of the first midterm.) Background The consumption of beverages that contain sugar is associated with overweight, possibly because liquid sugars do not lead to a sense of satiety, so the consumption of other foods is not reduced. However, data are lacking to show that the replacement of sugarcontaining beverages with noncaloric beverages diminishes weight gain. Methods We conducted an 18-month trial involving 641 primarily normal-weight children from 4 years 10 months to 11 years 11 months of age. Participants were randomly assigned to receive 250 ml (8 oz) per day of a sugar-free, artificially sweetened beverage (sugar-free group) or a similar sugar-containing beverage that provided 104 kcal (sugar group). Beverages were distributed through schools. At 18 months, 26% of the children had stopped consuming the beverages; the data from children who did not complete the study were imputed. Several variables were measured at the beginning of the study and at the end of the study; of interest is whether change in these variables over the study period differed between children who drank the sugar-sweetened and sugar-free drinks. Information about four variables is provided below for children who completed the study; the size of each sample (N) is also shown. (Don’t worry about “imputation” of data for children who did not complete the study; just analyze the data in this table.)
Means and standard deviations of change over 18 months in four variables (Standard deviations are shown in parentheses)
is there good evidence that drink (sugar-sweetened vs. sugarfree) makes a difference in the change of that variable over the 18 months of the study? (i) Conduct a two-sample t test to calculate a t statistic; (ii) find the P-value associated with the obtained value of t, saying how many degrees of freedom the t has (using Option 2 from Moore et al., The Basic Practice of Statistics, p. 490) and saying whether the P value is one-sided or two-sided; and (iii) with = .005, say whether there is good evidence that drink makes a difference
Drink Drink Sugar-sweetened (N = 252) Sugar-free (N = 227) Increase in sum of thicknesses of four skinfolds (mm) 5.7 (10.0) 3.2 (8.1)Explanation / Answer
Solution:-
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.005. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.8282
DF = 477
t = [ (x1 - x2) - d ] / SE
t = 3.02
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a t statistic having 477 degrees of freedom is more extreme than -3.02; that is, less than - 3.02 or greater than 3.02.
Thus, the P-value = 0.0027
Interpret results. Since the P-value (0.0027) is less than the significance level (0.005), we cannot accept the null hypothesis.
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