Question

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)

1. Identify the population, sample, and parameter of interest in this study.  (3 points)
2. Construct a 95% CI for the mean number of hours first year USC students sleep the night before an exam.  Explain the meaning of this CI (you are expected to explain what this confidence interval says about the parameter of interest)    (5 points)
3. Now construct a 99% CI for the mean number of hours first year USC students sleep the night before an exam.  Explain the meaning of this CI and contrast with the 95% CI in part b.                          (5 points)
4. Suppose the 81 students represent a subset of a larger pool of 100 students randomly selected for the study.  The 81 students provided the investigators with their number of hours of sleep, while the remaining 19 students did not respond when contacted by the investigators.  If the 19 ‘non-responders’ were more likely to sleep more hours before an exam than a typical first year USC student, how would this affect the estimate and CI for the mean number of hours first year USC students sleep the night before an exam?                                                                                                                (5 points)

Answer 1

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 = (t_{n- 1, α /2})s/√n with

Xbar = sample mean,

t_{n – 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 (t_α/2)/(t_β/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