While writing an article on the high cost of college education, a reporter took a random sample of the cost of new textbooks for a semester. The random variable x is the cost of one book. Her sample data can be summarized by the following. (Give your answers correct to two decimal places.)

n = 21, Óx = 3635.6, and Ó(x - x)2 = 9750.4

(a) Find the sample mean, x.
$ 173.12 .

(b) Find the sample standard deviation, s.
$ 22.08 .

(c) Find the 90% confidence interval to estimate the true mean textbook cost for the semester based on this sample.
Lower Limit $164.30 .
Upper Limit $ 181.94

To find the answers, the reporter used the following formulas:

(a) Sample mean (x̄):
The sample mean is calculated by dividing the sum of all the values in the sample by the total number of values in the sample.

x̄ = Σx / n

In this case, the sum of all the values (Σx) is given as 3635.6 and the total number of values (n) is given as 21. Plugging these values into the formula:

x̄ = 3635.6 / 21 = 173.12 (rounded to two decimal places)

Therefore, the sample mean is $173.12.

(b) Sample standard deviation (s):
The sample standard deviation is a measure of how spread out the values in the sample are. It is calculated using the formula:

s = sqrt( Σ(x - x̄)^2 / (n - 1) )

In this case, the sum of squares of differences (Σ(x - x̄)^2) is given as 9750.4 and the total number of values (n) is given as 21. Plugging these values into the formula:

s = sqrt(9750.4 / (21 - 1)) = sqrt(9750.4 / 20) = 22.08 (rounded to two decimal places)

Therefore, the sample standard deviation is $22.08.

(c) 90% confidence interval:
To estimate the true mean textbook cost for the semester based on this sample, the reporter needs to calculate the confidence interval. The formula for the confidence interval is:

x̄ ± (z * (s / sqrt(n)))

Where x̄ is the sample mean, z is the critical value corresponding to the desired confidence level (90% in this case), s is the sample standard deviation, and n is the total number of values in the sample.

Since the critical value for a 90% confidence level is 1.645, plugging in the values:

Lower Limit = 173.12 - (1.645 * (22.08 / sqrt(21))) = 164.30 (rounded to two decimal places)

Upper Limit = 173.12 + (1.645 * (22.08 / sqrt(21))) = 181.94 (rounded to two decimal places)

Therefore, the 90% confidence interval to estimate the true mean textbook cost for the semester based on this sample is $164.30 to $181.94.