Philosophical and health issues are prompting an increasing number of Taiwanese
ID: 3223607 • Letter: P
Question
Philosophical and health issues are prompting an increasing number of Taiwanese to switch to a vegetarian lifestyle. A study published in the Journal of Nutrition compared the daily intake of nutrients by vegetarians and omnivores living in Taiwan. Among the nutrients considered was protein. Too little protein stunts growth and interferes with all bodily functions; too much protein puts a strain on the kidneys, can cause diarrhea and dehydration, and can leach calcium from bones and teeth. Independent random samples of 51 females vegetarians and 53 female omnivores yielded the following summary statistics, in grams, on daily protein intake. Do the data provide sufficient evidence to conclude that the mean daily protein intakes of female vegetarians and female omnivores differ? Perform the required hypothesis test at the 1% significance level. Assume equality of variances. Refer to exercise 1 in this assignment. a. Determine a 99% confidence interval for the difference between the mean daily protein intakes of female vegetarians and female omnivores. b. Interpret your answer in part (a).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 0.01. 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 = 3.71
DF = 51 + 53 - 2
D.F = 102
t = [ (x1 - x2) - d ] / SE
t = - 2.93
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 102 degrees of freedom is more extreme than - 2.93; that is, less than - 2.93 or greater than 2.93.
Thus, the P-value = 0.004183
Interpret results. Since the P-value (0.004) is less than the significance level (0.01), we cannot accept the null hypothesis.
From the above test we have sufficient evidence in the favor of the claim that there is difference between female vegetarian and female omnivores.