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A health educator was interested in determining whether college students at her college really do gain weigh during their freshman year. A random sample of 5 college students was chosen and the weight for each student was recorded in August and May. Does the data below suggest that college students gain weight during their freshman year? The health educator wants to use a 0.05 significance level to test the claim.
Weight (pounds) 

Student 
August May 
1 2 3 4 5 
175 180 170 164 135 142 160 166 200 208 
(a) What is the appropriate hypothesis test to use for this analysis? Please identify and explain why it is appropriate.
(b) Let ?1 = mean weight in May. Let ?2 = mean weight in August. Which of the following statements correctly defines the null hypothesis?
(i) ?1  ?2 > 0 (?d > 0)
(ii) ?1  ?2 = 0 (?d = 0)
(iii) ?1  ?2 < 0 (?d < 0)
(c) Let ?1 = mean weight in May. Let ?2 = mean weight in August. Which of the following statements correctly defines the alternative hypothesis?
(i) ?1  ?2 > 0 (?d > 0)
(ii) ?1  ?2 = 0 (?d = 0)
(iii) ?1  ?2 < 0 (?d < 0)
(d) Determine the test statistic. Round your answer to three decimal places. Describe method used for obtaining the test statistic.
(e) Determine the pvalue. Round your answer to three decimal places. Describe method used for obtaining the pvalue.
(f) Compare pvalue and significance level ?. What decision should be made regarding the null hypothesis (e.g., reject or fail to reject) and why?
(g) What do the results of this study tell us about freshman college student weight gain? Justify your conclusion.
(a)
Here we have paired data so paired t test will be used.
Assumption: The difference in weights (August  May) are normally distributed.
(b)
Let d= August  May
The null hypothesis is
(c)
(d)
Following table shows the calculations:
August  May  d=AugustMay  (dmean)^2 
175  180  5  1 
170  164  6  100 
135  142  7  9 
160  166  6  4 
200  208  8  16 
Total  20  130 
Sample size n=5
So
So test statistics will be
(e)
Degree of freedom: df=4
The pvalue is: 0.0958
Excel function used for pvalue: "=TDIST(1.57,4,1)"
(f)
Since pvalue is greater than 0.05 so we fail to reject the null hypothesis.
(g)
There is no evidence to conclude that college students gain weight during their freshman year.
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