HW 8: Problem 3 Previous Problem Problem List Next Problem (1 point) Two random
ID: 3312431 • Letter: H
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HW 8: Problem 3 Previous Problem Problem List Next Problem (1 point) Two random samples are selected from two independent populations. A summary of the samples sizes, sample means, and sample standard deviations is given below n 39, 59.7, 81 5.9 n2= 38, z2=70.7, 82 10.0 Find a 97.5% conndence interval for the dmerence 1-2 of the rneans, assuming equal population variances. Confidence Interval Preview My Answers Submit Answers You have attempted this problem 4 times Your overall recorded score is 0% You have unlimited attempts remaining. Email instructorExplanation / Answer
3.
TRADITIONAL METHOD
given that,
mean(x)=59.7
standard deviation , s.d1=5.9
number(n1)=39
y(mean)=70.7
standard deviation, s.d2 =10.9
number(n2)=38
I.
calculate pooled variance s^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
s^2 = (38*34.81 + 37*118.81) / (77- 2 )
s^2 = 76.25
II.
standard error = sqrt(S^2(1/n1+1/n2))
=sqrt( 76.25 * (1/39+1/38) )
=1.99
III.
margin of error = t a/2 * (stanadard error)
where,
t a/2 = t -table value
level of significance, = 0.025
from standard normal table, two tailed and value of |t | with (n1+n2-2) i.e 75 d.f is 2.287
margin of error = 2.287 * 1.99
= 4.552
IV.
CI = (x1-x2) ± margin of error
confidence interval = [ (59.7-70.7) ± 4.552 ]
= [-15.552 , -6.448]
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DIRECT METHOD
given that,
mean(x)=59.7
standard deviation , s.d1=5.9
sample size, n1=39
y(mean)=70.7
standard deviation, s.d2 =10.9
sample size,n2 =38
CI = x1 - x2 ± t a/2 * sqrt ( s^2 ( 1 / n1 + 1 /n2 ) )
where,
x1,x2 = mean of populations
s^2 = pooled variance
n1,n2 = size of both
a = 1 - (confidence Level/100)
ta/2 = t-table value
CI = confidence interval
CI = [( 59.7-70.7) ± t a/2 * sqrt( 76.25 * (1/39+1/38) ]
= [ (-11) ± 4.552 ]
= [-15.552 , -6.448]
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interpretations:
1. we are 97.5% sure that the interval [-15.552 , -6.448]contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 97.5% of these intervals will contains the true population proportion
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