To test the effect of a physical fitness course on one's physical ability, the number of sit-ups that a person could do in one minute, both before and after the course, was recorded. Ten individuals are randomly selected to participate in the course. The results are displayed in the following table. Using this data, find the 90% confidence interval for the true difference in the number of sit-ups each person can do before and after the course. Assume that the numbers of sit-ups are normally distributed for the population both before and after completing the course. Sit-ups before56272328273454502821Sit-ups after60453139393658603829Step 1 of 4:Find the point estimate for the population mean of the paired differences. Let x1x1 be the number of sit-ups before taking the course and x2 be the number of sit-ups after taking the course and use the formula d=x2−x1 to calculate the paired differences. Round your answer to one decimal place.

Step-by-step explanation:

For before,

Mean, x1 = (56 + 27 + 23 + 28 + 27 + 34 + 54 + 50 + 28 + 21)/10 = 34.8

Standard deviation = √(summation(x - mean)²/n

n = 10

Summation(x - mean)² = (56 - 34.8)^2 + (27 - 34.8)^2 + (23 - 34.8)^2 + (28 - 34.8)^2 + (27 - 34.8)^2 + (34 - 34.8)^2 + (54 - 34.8)^2 + (50 - 34.8)^2 + (28 - 34.8)^2 + (21 - 34.8)^2 = 1593.6

Standard deviation = √(1593.6/10

s1 = 12.62

For after,

Mean, x1 = (60 + 45 + 31 + 39 + 39 + 36 + 58 + 60 + 38 + 29)/10 = 43.5

Summation(x - mean)² = (60 - 43.5)^2 + (45 - 43.5)^2 + (31 - 43.5)^2 + (39 - 43.5)^2 + (39 - 43.5)^2 + (36 - 43.5)^2 + (58 - 43.5)^2 + (60 - 43.5)^2 + (38 - 43.5)^2 + (29 - 43.5)^2 = 1250.5

Standard deviation = √(1250.5/10

s2 = 11.18

The formula for determining the confidence interval for the difference of two population means is expressed as

Confidence interval = (x1 - x2) ± z√(s²/n1 + s2²/n2)

For a 90% confidence interval, we would determine the z score from the t distribution table because the number of samples are small

Degree of freedom =

(n1 - 1) + (n2 - 1) = (10 - 1) + (10 - 1) = 18

z = 1.734

The point estimate is

x1 - x2 = 34.8 - 43.5 = - 8.7

Margin of error = z√(s1²/n1 + s2²/n2) = 1.734√(12.62²/10 + 11.18²/10) = 8.5

Confidence interval = - 8.7 ± 8.5

if a = 1,   and b = 2, then the answer is 47.99999999. if you round that to the nearest whole number, the answer is 48.