Sleep Researchers Know That Some People Are Early Birds (E), Preferring To Go To Bed By 10 P.M. And Arise
The answer is: A) The data support the researchers’ belief that night owls have better dream recall than early birds. B) we can be 92% confident that the true difference in mean number of dreams recalled per week between night owls and early birds is between 1.87 and 2.63.
A) These are the alternative and null hypotheses:
H0: μE = μN (the mean number of dreams recalled each week is the same for early birds and night owls) (the mean number of dreams recalled per week is the same for early birds and night owls)
Ha: μE < μN (the mean number of dreams recalled each week is smaller for early birds than for night owls) (the mean number of dreams recalled per week is lower for early birds than for night owls)
Using the following formula, we can run a two-sample t-test with unequal variances:
t = [(sN2 / nN) + (sE2 / nE)] / sqrt[(xN – xE)]
where nN and nE are the sample sizes for night owls and early birds, respectively, and xN and sN and xE and sE are the sample means and standard deviations for night owls and early birds, respectively.
When we enter the values, we obtain:
t = (9.55 – 7.26) / sqrt[(5.88^2 / 100) + (6.94^2 / 100)] = 5.01
The data are consistent with the researchers’ hypothesis that night owls are more capable of remembering their dreams than early birds.
B) We can use the following formula to determine the confidence interval:
CI is equal to (xN – xE) t/2 * sqrt[(sN / nN) + (sE / nE)].
where t/2 is the t-value for the required level of confidence and degrees of freedom, and xN, xE, sN, sE, nN, and nE are the same as previously (198 in this case).
With a t-value of 1.75 and a 92% confidence level (from a t-distribution with 198 degrees of freedom), we get:
CI is equal to (9.55 – 7.26) 1.75 * sqrt[(5.88 – + 6.94 / 100)] = (1.87, 2.63) (1.87, 2.63)
The genuine difference between night owls and early birds in terms of the average number of dreams recalled per week is therefore between 1.87 and 2.63, with a 92% confidence interval.
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