Question

19. A health educator was interested in determining whether college students at her college really do gain weight during their freshman year. A random sample of 5 college students was chosen and the weight for each student was recorded in August (the beginning of the freshman year) and May (the end of the freshman year). 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)	Weight (Pounds)
Student	August	May
1	180	182
2	170	175
3	135	140
4	162	160
5	200	200

(a) What is the appropriate hypothesis test to use for this analysis? Please identify and explain why it is appropriate.

(b) Let μ1 = mean weight in May. Let μ2 = mean weight in August. Which of the following statements correctly defines the null hypothesis? (i) μ1 - μ2 > 0 (μd > 0) (ii) μ1 - μ2 = 0 (μd = 0) (iii) μ1 - μ2 < 0 (μd < 0)

(c) Let μ1 = mean weight in May. Let μ2 = mean weight in August. Which of the following statements correctly defines the alternative hypothesis? (i) μ1 - μ2 > 0 (μd > 0) (ii) μ1 - μ2 = 0 (μd = 0) (iii) μ1 - μ2 < 0 (μd < 0)

(d) Determine the test statistic. Round your answer to three decimal places. Describe method used for obtaining the test statistic.

(e) Determine the p-value. Round your answer to three decimal places. Describe method used for obtaining the p-value.

(f) Compare p-value and significance level α. 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.

Answer 1

(a) appropriate hypothesis test to use for this analysis : Paired t-test:

As Two measurements i.e Weight in August and Weight in May on each student are being collected and tested to test the claim that college students gain weight during their freshman year. Therefore, paired t-test to test the mean difference between these dependent observations is appropriate

(b) Let = mean weight in May (end of the freshman). Let = mean weight in August(beginning of freshman);

null hypothesis : - = 0 ( )

(c) Let = mean weight in May (end of the freshman). Let = mean weight in August(beginning of freshman);

Claim : college students gain weight during their freshman year i.e.

mean weight in May (end of the freshman) >  mean weight in August(beginning of freshman) > 0

i.e > i.e

Alternate hypothesis :; ( )

(d) Determine the test statistic

Student	Weight (Pounds):August	Weight (Pounds):May	d: May weight - August Weight	(d-)	(d-)2
1	180	182	2	0	0
2	170	175	5	3	9
3	135	140	5	3	9
4	162	160	-2	-4	16
5	200	200	0	-2	4
		Total	10		38
		= 10/5	2

d : sample difference in weight (Weight in may - weight in August)

n : sample size = 5

  : Sample Mean of differences
s_d : Sample Standard Deviation of the differences

test statistic = 1.451

(e) Determine the p-value

Degrees of freedom = n-1 = 5-1 =4

For right tailed test :

p-value is computed using excel function : T.DIST.RT

T.DIST.RT function
Returns the right-tailed Student's t-distribution.
The t-distribution is used in the hypothesis testing of small sample data sets. Use this function in place of a table of critical values for the t-distribution.
Syntax
T.DIST.RT(x,deg_freedom)
The T.DIST.RT function syntax has the following arguments:
• X Required. The numeric value at which to evaluate the distribution.
• Deg_freedom Required. An integer indicating the number of degrees of freedom.

(f) As P-Value i.e. is greater than Level of significance i.e (P-value:0.1102 > 0.05:Level of significance); Fail to Reject Null Hypothesis

(g), There is no sufficient evidence to conclude that that college students gain weight during their freshman year at 0.05 significance level.