In: Statistics and Probability
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]
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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]
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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.