In: Statistics and Probability
It has been extensively demonstrated that adequate night sleep is crucial for cognitive performance. To investigate whether first year USC students are getting adequate sleep the night before a final exam, 81 randomly selected first year USC students were asked to record their number of hours of sleep the night before their final exams. The mean number of hours of sleep among the 81 students was 6.2 with an SD of 1.5 hours. (18 points)
Solution
Part (a)
Population: first year USC students Answer 1a
Sample:
a random sample 81 students from amongst first year USC students Answer 1b
Parameter of interest: mean number of hours first year USC students sleep the night before an exam. Answer 1c
Part (b)
95% CI for the mean number of hours first year USC students sleep the night before an exam is: [5.87, 6.53] hours Answer 2
Interpretation
If more samples of the same size are taken from the same population and 95% CI are calculated, 95% of those CI’s would contain the true population mean number of hours first year USC students sleep the night before an exam. Answer 3
Back-up Theory and Details of Calculations
Let X = number of hours first year USC students sleep the night before an exam.
Let μ and σ be the mean and standard deviation of X.
100(1 - α) % Confidence Interval for population mean μ, when σ is not known is: Xbar ± MoEwhere
MoE = (tn- 1, α /2)s/√n with
Xbar = sample mean,
tn – 1, α /2 = upper (α/2)% point of t-distribution with (n - 1) degrees of freedom,
s = sample standard deviation and
n = sample size.
Calculations
Given |
α |
0.05 |
1 - (α/2) = |
0.975 |
n |
81 |
SQRT(n) |
9 |
|
Xbar |
6.2 |
n - 1 |
80 |
|
s |
1.5 |
|||
tα/2 |
1.9901 |
|||
90% CI for μ |
6.2 |
± |
0.3317 |
|
Lower Bound |
5.8683 |
|||
Upper Bound |
6.5317 |
Part (c)
99% CI for the mean number of hours first year USC students sleep the night before an exam. [5.76, 6.64] hours Answer 4
Interpretation: same as in Part (b) Answer 5
Contrast with the 95% CI in part b
99% CI is wider than the 95% CI. This is natural since for a higher confidence level, a wider interval is required. Answer 6
Back-up Theory and Details of Calculations same as in Part (b), except that the tα/2 is larger at 2.6387
[Going beyond, 99% CI can be directly obtained from 95% CI using the following rule:
All other things remaining the same,
if M1 is the MoE for 100(1 - α) % Confidence Interval and M2 is the MoE for 100(1 - β) % Confidence Interval, then given M1, M2 = M1 x (tα/2)/(tβ/2) if population standard deviation is unknown]
Part (d)
If the 19 ‘non-responders’ were more likely to sleep more hours before an exam than a typical first year USC student, the mean of these 19 students will be larger and consequently, the mean of the combined sample of 100 students will be larger.
Further, when two samples of different means are pooled, the standard deviation of the pooled sample will always be greater than the standard deviation of both the individual samples.
With reference to the formula for CI, vide Back-up Theory under Part (b), both xbar and s would be larger resulting in a wider CI at the same confidence level. Thus,
effect on the estimate and CI for the mean number of hours first year USC students sleep the night before an exam would be
larger mean and wider CI Answer 7
DONE