Question

In: Statistics and Probability

Suppose that we are interested in factors influencing young children’s attention to television. Specifically, we want...

  1. Suppose that we are interested in factors influencing young children’s attention to television. Specifically, we want to test the amount of attention that children pay to television depends upon the degree of their understanding of the programming. We randomly sample 50 young children and have them watch a 60-minute program judged “easy” to understand, and have another random sample of 40 young children watch a 60-minute program judged “difficult” to understand. On average, children who watched the easy program attended for 40 minutes, with a sample standard deviation of 10 minutes. The children who watched the difficult program attended for an average of 30 minutes, with a sample standard deviation of 8 minutes.
    1. Estimate the magnitude of the effect of the manipulation of the comprehensibility of the TV program using Cohen’s d. State what the number means in English.
    2. Construct the 99% confidence interval to estimate the size of the effect of program comprehensibility on attention. State what the CI means in English. Also, based on the results of your CI, how do you evaluate the null hypothesis that there is no effect of the comprehensibility of the TV show? What is alpha?

Solutions

Expert Solution

a.
cohen's d size = modulus of (mean 1 -mean 2)/S.D pooled
pooled variance s^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
s^2 = (49*100 + 39*64) / (90- 2 )
s^2 = 84.0455
S.D pooled = sqrt (84.0455)
S.D pooled = 9.1676
cohen's d size = modulus of (40-30)/9.1676
cohen's d size = 1.0907
large effect
magnitude of the effect of the manipulation of the comprehensibility of the TV program =1.0907

b.
i.
TRADITIONAL METHOD
given that,
mean(x)=40
standard deviation , s.d1=10
number(n1)=50
y(mean)=30
standard deviation, s.d2 =8
number(n2)=40
I.
standard error = sqrt(s.d1^2/n1)+(s.d2^2/n2)
where,
sd1, sd2 = standard deviation of both
n1, n2 = sample size
standard error = sqrt((100/50)+(64/40))
= 1.897
II.
margin of error = t a/2 * (standard error)
where,
t a/2 = t -table value
level of significance, α = 0.01
from standard normal table, two tailed and
value of |t α| with min (n1-1, n2-1) i.e 39 d.f is 2.708
margin of error = 2.708 * 1.897
= 5.138
III.
CI = (x1-x2) ± margin of error
confidence interval = [ (40-30) ± 5.138 ]
= [4.862 , 15.138]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
mean(x)=40
standard deviation , s.d1=10
sample size, n1=50
y(mean)=30
standard deviation, s.d2 =8
sample size,n2 =40
CI = x1 - x2 ± t a/2 * Sqrt ( sd1 ^2 / n1 + sd2 ^2 /n2 )
where,
x1,x2 = mean of populations
sd1,sd2 = standard deviations
n1,n2 = size of both
a = 1 - (confidence Level/100)
ta/2 = t-table value
CI = confidence interval
CI = [( 40-30) ± t a/2 * sqrt((100/50)+(64/40)]
= [ (10) ± t a/2 * 1.897]
= [4.862 , 15.138]
-----------------------------------------------------------------------------------------------
interpretations:
1. we are 99% sure that the interval [4.862 , 15.138] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 99% of these intervals will contains the true population proportion
ii.
Given that,
mean(x)=40
standard deviation , s.d1=10
number(n1)=50
y(mean)=30
standard deviation, s.d2 =8
number(n2)=40
null, Ho: u1 = u2
alternate, H1: u1 != u2
level of significance, α = 0.01
from standard normal table, two tailed t α/2 =2.708
since our test is two-tailed
reject Ho, if to < -2.708 OR if to > 2.708
we use test statistic (t) = (x-y)/sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =40-30/sqrt((100/50)+(64/40))
to =5.2705
| to | =5.2705
critical value
the value of |t α| with min (n1-1, n2-1) i.e 39 d.f is 2.708
we got |to| = 5.27046 & | t α | = 2.708
make decision
hence value of | to | > | t α| and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 5.2705 ) = 0
hence value of p0.01 > 0,here we reject Ho
ANSWERS
---------------
null, Ho: u1 = u2
alternate, H1: u1 != u2
test statistic: 5.2705
critical value: -2.708 , 2.708
decision: reject Ho
p-value: 0
we have enough evidence to support the claim that difference of means of watch a 60-minute program judged easy to understand
and watch a 60-minute program judged difficult to understand.


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