Do Freshman Get More Sleep than Juniors? – By Jeremy Asuncion, Jack Vang, Jay Calunsag, David Escalante – Hypothesis

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Do Freshman Get More Sleep than Juniors? – By Jeremy Asuncion, Jack Vang, Jay Calunsag, David Escalante – Hypothesis

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Do Freshman Get More Sleep than Juniors?

By Jeremy Asuncion, Jack Vang, Jay Calunsag, David Escalante

Hypothesis

H0: Freshmen get the same amount of sleep as Juniors.

Ha: Freshmen get more sleep than Juniors.

Experimental Design

  • Survey with two questions: Grade Level and Mean Hours of Sleep

  • Four populations were sampled: Freshmen, Sophomores, Juniors, and Seniors

  • Sampling Method:

    • Cluster sample; 9 classes selected randomly using true random number generator

    • Only data from Freshmen and Juniors were kept

    • 50 data were required from both Freshmen and Juniors

  • Surveys with obviously unrealistic results were not considered — for example, no hours of sleep

Difference of Means

μf= mean hours of sleep for Freshmen.

μj= mean hours of sleep for Juniors.

H0:μf−μj=0⟹μf=μj

Ha:μf−μj>0⟹μf>μj

Assumptions

  • Randomization: Classes were randomly selected using TRG

  • 10% Conditions: The freshmen sample satisfy this conditon if a randomly selected subset of n=50 data is selected; the Juniors satisfy this condition

  • Nearly Normal:

  • Independence Conditions: Assume that Juniors and Sophomores are independent, and that each student from each class is independent(That's right! No one has friends!)

t= ˉxf−ˉxj√s2fnf+s2jnj =6.71−6.39√2.1649+1.9948 =1.53

df= (s2fnf+s2jnj)21nf−1(s2fnf)2+1nj−1(s2jnj)2 =(2.3649+1.9948)2149−1(2.3649)2+148−1(1.9948)2 =95.168

p =P(t95.168>1.53)

=1−∫t−∞(Γ(df+12)√df⋅πΓ(df2)(1+x2df)−df+12)dx
= 1 - 0.935 = 0.065
Graph

χ2 Test of Homogeneity

H0: The distribution of average sleep times for Freshmen is the same as the distribution of average sleep times for Juniors

Ha: The distribution of average sleep times are not the same

If Ha is true, we'd have to check the standardized residuals, C=O−E√E, to see who got more sleep.

  • Data is categorized into two categories:

    • μ≥8 — required amount of sleep for teenager

    • μ<8 — less than the required amount

  • There are two populations(Freshmen and Juniors) and one variable(Average sleep time), therefore χ2 test of homogeneity is used

df=(R−1)(C−1)=1⋅1=1

Assumptions

  • Counted Data: Since the data was categorized, we have counted data

Randomization: A TRG was used to randomly select classes

  • Expected Cell Frequency:

1112(3181)(6448)

=(17.17413.28646.28634.714)

χ2= 4∑i=1(Oi−Ei)2Ei =(21−17.174)217.174 +(10−13.286)213.286 +(43−46.286)246.286 +(38−34.714)234.714 =1.967

p =2P(χ2>1.967)

=1−∫x0(xdf/2−1exp(−x2)2df/2Γ(df2))dx
= 1 - 0.839 = 0.161
Graph

Conclusions

  • We used an α=0.05

pt=0.065>α=0.05 and pχ2=0.161>α=0.05, so we fail to reject the null hypothesis

WHAT DOES IT MEAN?

  • There isn't enough evidence to suggest Freshmen get more sleep than juniors

It appears that both Freshmen and Juniors get the same amount of sleep

Related Studies

  • A study found that approximately 20% of Seniors had more sleep deficits than Freshmen

This may suggest that there were errors in our study

THIS IS THE LINK ⇒http://goo.gl/1DgejY

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