Answer to Question #254471 in Statistics and Probability for Goldie

Question

A sleep researcher conducts an experiment to determine whether sleep loss affects the ability to maintain sustained attention. Fifteen individuals are randomly divided into the following three groups of five subjects each: group 1, which gets the normal amount of sleep (7–8 hours); group 2, which is sleep-deprived for 24 hours; and group 3, which is sleep-deprived for 48 hours. All three groups are tested on the same auditory vigilance task. Subjects are presented with half-second tones spaced at irregular intervals over 1-hour duration. Occasionally, one of the tones is slightly shorter than the rest. The subject’s task is to detect the shorter tones. The following percentages of correct detection were observed: 

Normal Sleep: 85 83 76 64 75

Sleep-Deprived for 24-hours: 60 58 76 52 63

Sleep-Deprived for 48-hours: 60 48 38 47 50

Determine whether there is an overall effect for sleep deprivation, using the conceptual equations of the one-way ANOVA. Use ∝= 0.05.

Answer 1

totals

Normal Sleep: 85 83 76 64 75 383

Sleep-Deprived for 24-hours: 60 58 76 52 63 309

Sleep-Deprived for 48-hours: 60 48 38 47 50 243

Total 935

$n=15$

The correction factor, $CF={935^2\over 15}=58281.67$

Total sum of squares.

$TSS=(85^2+83^2+76^2+.......+47^2+50^2)- 58281.67= 61081-58281.67=2799.33$

Treatment sum of squares.

$TRSS=({383^2+309^2+243^2\over4})-58281.67=60243.8-58281.67=1962.13$

Error sum of squares

$ESS=TSS-TRSS=2799.33-1962.13=837.2$

Degrees of freedom

For this case, $n=15,k=3$

For the treatments, degrees of freedom is k-1=3-1=2

For the error, degrees of freedom is n-k=15-3=12

Hypotheses

$H_0:\mu_1=\mu_2=\mu_3\\vs\\ H_1: At\space least \space one \space is \space different$

ANOVA table

source degrees of freedom sum of squares mean squares F

Treatments 2 1962.13 ${1962.13\over2}=981.065$ 14.062

Error 12 837.2 ${837.2\over12}=69.77$

Total 14 2799.33

Critical value,

$F_{0.05, 2,12}=3.885294$

Decision criteria

Reject $H_0$ if $F\gt F_{0.05,2,12}$

Since, $F=14.062\gt F_{0.05,2,12}=3.885294$, the null hypothesis is rejected and conclude that, there is sufficient evidence to support the claim that sleep deprivation appears to affect sustained attention at 5% level of significance.