Question

A research was done to see if the firs- born twins have a higher IQ than the second-born twins. In the study, 10 pairs of twins are randomly selected, and their data are summarized as below: Group Mean Stand Deviation Min Max First Born 108 7.2 75 120 Second Born 103 8.1 72 115 Diff First - Second 5 7.3 -3 12 (1) (8pts) Perform an appropriate statistical inference to answer the research question. (if this is a two independent sample problem, use pooled variance t test) (No points will be given without necessary steps) (2) (2pts)Based on your decision in (1), what type of error could be made here (Type 1 or Type ID? Which of the following best describe the error you indicated? A. The first born twins actually have higher IQ scores than second born, but we concluded they don't B. The first born twins actually don't have higher IQ scores than second born, but we concluded they do.

Answer 1

Given that,
mean(x)=108
standard deviation , s.d1=7.2
number(n1)=10
y(mean)=103
standard deviation, s.d2 =8.1
number(n2)=10
null, Ho: u1 = u2
alternate, H1: u1 > u2
level of significance, α = 0.05
from standard normal table,right tailed t α/2 =1.734
since our test is right-tailed
reject Ho, if to > 1.734
calculate pooled variance s^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
s^2 = (9*51.84 + 9*65.61) / (20- 2 )
s^2 = 58.725
we use test statistic (t) = (x-y)/sqrt(s^2(1/n1+1/n2))
to=108-103/sqrt((58.725( 1 /10+ 1/10 ))
to=5/3.427
to=1.459
| to | =1.459
critical value
the value of |t α| with (n1+n2-2) i.e 18 d.f is 1.734
we got |to| = 1.459 & | t α | = 1.734
make decision
hence value of |to | < | t α | and here we do not reject Ho
p-value: right tail -ha : ( p > 1.459 ) = 0.0809
hence value of p0.05 < 0.0809,here we do not reject Ho
ANSWERS
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null, Ho: u1 = u2
alternate, H1: u1 > u2
test statistic: 1.459
critical value: 1.734
decision: do not reject Ho
p-value: 0.0809
we do not have enough evidence to support the claim that if the first born twins have a higher IQ than second born twins.
option:A
Type 2 error is possible when its fails to reject the null hypothesis.