To study the effectiveness of possible treatments for insomnia, a sleep research
ID: 3322135 • Letter: T
Question
To study the effectiveness of possible treatments for insomnia, a sleep researcher conducted a study in which four participants were instructed to count sheep (the Sheep Condition), four were told to concentrate on their breathing (the Breathing Condition), and four were not given any special instructions (the Control Condition). The average number of minutes taken for participants to fall asleep over the next seven days were M = 38, S2 = 14 minutes for the Sheep condition; M = 29, S2 = 13.33 minutes for the Breathing condition; and M = 41, S2 = 6 for the Control condition. Using the .01 significance level, did the different techniques have different effects?
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 = 2.614
DF = 6
t = [ (x1 - x2) - d ] / SE
t = - 1.15
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 6 degrees of freedom is more extreme than -1.15; that is, less than -1.15 or greater than 1.15.
Thus, the P-value = 0.314
Interpret results. Since the P-value (0.314) is greater than the significance level (0.01), we have to accept the null hypothesis.
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