You are given the following results of a paired difference

\" You are given the following results of a paired- difference test:

\"a. Construct and interpret a 99% confidence interval estimate for the paired difference in mean values.

b. Construct and interpret a 90% confidence interval estimate for the paired difference in mean values.

Please explain equations. Thanks.

Solution

a)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    -4.6          
t(alpha/2) = critical t for the confidence interval =    2.946712883          
s = sample standard deviation =    0.25          
n = sample size =    16          
df = n - 1 =    15          
Thus,              
Margin of Error E =    0.184169555          
Lower bound =    -4.784169555          
Upper bound =    -4.415830445          
              
Thus, the confidence interval is              
              
(   -4.784169555   ,   -4.415830445   ) [ANSWER]

Note that 0 is not inside this interval. Thus, at 0.1 level, there is significant evidence that the means differ. [answer]

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b)

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.05          
X = sample mean =    -4.6          
t(alpha/2) = critical t for the confidence interval =    1.753050356          
s = sample standard deviation =    0.25          
n = sample size =    16          
df = n - 1 =    15          
Thus,              
Margin of Error E =    0.109565647          
Lower bound =    -4.709565647          
Upper bound =    -4.490434353          
              
Thus, the confidence interval is              
              
(   -4.709565647   ,   -4.490434353   ) [ANSWER]

Note that 0 is not inside this interval. Thus, at 0.1 level, there is significant evidence that the means differ. [answer]