Question

2. Hoaglin, Mosteller, and Tukey (1983) present data on blood levels of beta-endorphin as a function of stress. They took beta-endorphin levels for 19 patients 12 hours before surgery (12h), and again 10 minutes before surgery (10m). The data are presented in the table below, in fmol/ml ID 1 12h 10.0 6.5 8.0 12.0 5.011.5 5.0 3.5 7.5 5.8 10m 6.514.0 13.5 18.0 14.59.0 18.0 42.0 7.5 6.0 2 4 6 8 9 10 ID | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18119 12h 4.7 8.0 7.0 17.0 8.817.0 15.0 4.4 2.0 10m 25.0 12.0 52.0 20.0 16.0 15.0 11.5 2.5 2.0 The table below includes summary statistics for the sample: Mean 8.35 16.05 7.7 Standard Deviation 4.397 12.509 13.519 12h 10m Difference (10m-12h) Based on these data, what effect does increased stress have on endorphin levels? When addressing this question, use a significance level of .05. (a) Why would this problem most likely be addressed by using a dependent t as the statistical test? (b) Given the following null and alternative hypotheses, compute the appropriate test statistic. (c) what is the decision rule for making a decision about the null hypothesis? i.e., what is the critical value(s) that would be used for this problem? (d) What is the decision regarding the null hypothesis? (e) Based on the results, what practical conclusion might we come to with regard to stress and endorphin levels?

Answer 1

(a) Since the data was collected from the same patients over two time periods, we must use a dependent t- test

(b)

Data:     

n = n1 = n2 = 19    

d-bar = 7.7    

s (of d) = 13.519    

Hypotheses:     

Ho: μ1 = μ2     

Ha: μ1 ≠ μ2     

Test Statistic:     

SE = s/√n = 13.519/√19 = 3.101471306   

t = d-bar/SE = 7.7/3.10147130619612 = 2.482692645   

(c)

Decision Rule:    

α = 0.05    

Degrees of freedom = 19 - 1 = 18

Lower Critical t- score = -2.100922037   

Upper Critical t- score = 2.100922037   

Reject Ho if |t| > 2.100922037

(d)

Decision (in terms of the hypotheses):    

Since 2.482692645 > 2.101 we reject Ho and accept Ha. There is sufficient evidence that the mean difference is different from 0.

(e)The endorphin levels and hence stress are significantly different 10 minutes before surgery and 12 hours before surgery.