Question

People who sustain a concussion or a traumatic brain injury (TBI) are likely to experience sleep problems. Do these problems persist? Researchers recruited a sample of 31 patients who had their first ever TBI 18 months earlier and a control group of 42 otherwise similar individuals without prior brain trauma. They compared sleep duration per 24 hours (recorded electronically over a two‑week period) for the two groups. A finding was reported in a scientific journal: "In the chronic state after traumatic brain injury, sleep need per 24 hours was persistently increased in trauma patients (8.1±0.5 hours) as compared to healthy controls (7.1±0.7 hours) ." The values cited are group mean plus or minus standard deviation. (a) The researchers performed a two‑sided two‑sample ? test and reported a highly statistically significant ? ‑value (<0.00005) .

Compute the test statistic using the citied summary values. (Enter your answer rounded to two decimal places.)

Calculate the p-value.

Obtain a 95%95% confidence interval for ?1−?2μ1−μ2 , the difference in mean sleep need per 2424 hours for the population of subjects with a past TBI and the control, trauma‑free population. (Enter your answers rounded to two decimal places.) Calculate the upper and lower limit.

Answer 1

Solution-A

t test assuming unequal variance

t=x1ar-x2bar/sqrt(s1^2/n1+s2^2/n2)

=(8.1-7.1)/sqrt(0.5^2/31+0.7^2/42)

t=7.119073

test statistic=7.12

Solution-B

Calculate the p-value.

df=n1+n2-2=31+42-2=71

p value for two tail

=T.DIST.2T(7.119073,71)

=7.09953E-10

=0.0000000007

Solutiin-c:

alpha=0.05

alpha/2=0.05/2=0.025

df=(s1^2/n1+s2^2/n2)^2/ (s1^2/n1)^2/(n1-1)+(s2^2/n2)^2/(n2-1)=70.944501

t crit==T.INV(0.025,70.944501)=1.994437112

95% confidece interval for difference in means

(x1-x2)+-tc*sqrt(s1^2/n1+s2^2/n2)

(8.1-7.1)+-1.994437112*sqrt(0.5^2/31+0.7^2/42)

0.7198459,1.280154

0.72.1.28

0.72<mu1-mu2<1.28

lower limit=0.72

upper limit=1.28