THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A dependent random sample from two normally distributed populations gives the following results:
n = 20,
= 26.5,s 2 = 3.2
-Find the width of the 98% confidence interval.
A)5.46
B)3.64
C)0.91
D)0.60

Question

THE NEXT QUESTIONS ARE BASED ON THE FOLLOWING INFORMATION:
A dependent random sample from two normally distributed populations gives the following results:
n = 20,  
= 26.5,s 2 = 3.2
-Find the width of the 98% confidence interval.
A)5.46 
B)3.64 
C)0.91 
D)0.60

Quizplus · Accepted Answer

The 98% confidence interval for the difference between two means, based on a dependent sample, is given by:
(d-bar - t(alpha/2)*s.d./sqrt(n), d-bar + t(alpha/2)*s.d./sqrt(n))
where d-bar is the sample mean of the differences, s.d. is the standard deviation of the differences, n is the sample size, and t(alpha/2) is the t-value with (n-1) degrees of freedom and alpha/2 level of significance.

From the given information, we have:
n = 20, d-bar = 0 (since the null hypothesis of no difference is usually assumed), and s.d. = s1/SQRT(n) = 3.2/SQRT(20) = 0.716

To find the t-value, we need to use the t-table or calculator with (n-1) = 19 degrees of freedom and alpha/2 = 0.01 (since the confidence level is 98%). 
The t-value is approximately 2.539.

Thus, the 98% confidence interval for the difference in means is:
(0 - 2.539*0.716/sqrt(20), 0 + 2.539*0.716/sqrt(20))
= (-1.59, 1.59)

The width of this interval is 2*1.59 = 3.18, which is closest to choice B (3.64).