Question

The following information was obtained from two independent samples selected from two normally distributed populations with unknown but equal standard deviations.

n1=33;    x1¯=14.46;    s1=15.07.
n2=23;    x2¯=−4.75;    s2=14.65.
Find a point estimate and a 97.5% confidence interval for μ1−μ2.

For the following, round all answers to no fewer than 4 decimal places.

The point estimate of μ1−μ2 is:	Answer
The lower limit of the confidence interval is:	Answer
The upper limit of the confidence interval is:	Answer
The margin of error for this estimate is:	Answer
Test at the 0.025 significance level if μ1 is different than μ2
μ1 is AnswerGreater thanNot significantly different thanLess than μ2

Answer 1

Point estimate ( X̅1 - X̅2 ) = ( 14.46 - (-4.75) ) = 19.210

Confidence interval is :-
( X̅1 - X̅2 ) ± t( α/2 , n1+n2-2) SP √( (1/n1) + (1/n2))
t(α/2, n1 + n1 - 2) = t( 0.025/2, 33 + 23 - 2) = 2.306 ( From t table )
( 14.46 - -4.75 ) ± t(0.025/2 , 33 + 23 -2) 14.9003 √ ( (1/33) + (1/23))
Lower Limit = ( 14.46 - -4.75 ) - t(0.025/2 , 33 + 23 -2) 14.9003 √( (1/33) + (1/23))
Lower Limit = 9.8769
Upper Limit = ( 14.46 - -4.75 ) + t(0.025/2 , 33 + 23 -2) 14.9003 √( (1/33) + (1/23))
Upper Limit = 28.5431
97.5% Confidence Interval is ( 9.8769 , 28.5431 )

Margin of Error = t(α/2 , n1+n2-2) SP √( (1/n1) + (1/n2)) = 9.3331

To Test :-

H0 :- µ1 = µ2
H1 :- µ1 ≠ µ2

Test Statistic :-
t = (X̅1 - X̅2) / SP √ ( ( 1 / n1) + (1 / n2))



t = ( 14.46 - (-4.75) ) / 14.9003 √ ( ( 1 / 33) + (1 / 23 ))
t = 4.7463

Test Criteria :-
Reject null hypothesis if | t | > t(α/2, n1 + n2 - 2)
Critical value t(α/2, n1 + n1 - 2) = t(0.025 /2, 33 + 23 - 2) = 2.306
| t | > t(α/2, n1 + n2 - 2) = 4.7463 > 2.306
Result :- Reject Null Hypothesis

There is sufficient evidence to support the claim that μ1 is different than μ2.