Answer :
By using mean and standard deviation, the results obtained are
a) The probability that the mean price for a sample of 30 federal income tax returns is within $16 of the population mean = 0.6212
b) The probability that the mean price for a sample of 50 federal income tax returns is within $16 of the population mean = 0.7416
c) )The probability that the mean price for a sample of 100 federal income tax returns is within $16 of the population mean = 0.8904
What is mean and Standard Deviation?
Suppose there is a data set. Mean is the average of the values of the data set.
Variance is the sum of the square of deviation of the data values from the mean
Square root of variance is the standard deviation.
The Mean Preparation Fee ( [tex]\mu[/tex] ) for 2017 Federal Income Tax Returns= $273
Standard deviation [tex]\sigma[/tex] of preparation fees = $100
$16 of mean = 273-16 to 273+16
= $257 to $289
(a) The probability that the mean price for a sample of 30 federal income tax returns is within $16 of the population mean
The formula used for finding the required probability
[tex]\ P(\$257 < \bar{X} < \$289) = P(\frac{\bar{x}_1-\mu}{\sigma/\sqrt{n}} < z < \frac{\bar{x}_2-\mu}{\sigma/\sqrt{n}}) \\ \ \\ = P(\frac{257-273}{100/\sqrt{30}} < z < \frac{289-273}{100/\sqrt{30}}) \\ \ \\ = P(-0.88 < z < 0.88 ) \\ \ \\ = P(z < 0.88) -P(z < -0.88)\\ \ \\ =0.8106 -0.1894[/tex]
=0.6212
(b) The probability that the mean price for a sample of 50 federal income tax returns is within $16 of the population mean
[tex]P(\frac{257-273}{100/\sqrt{50}} < z < \frac{289-273}{100/\sqrt{50}}) \\ \\\ = P(-1.13 < z < 1.13 ) \\ \ \\ = P(z < 1.13 ) -P(z < -1.13 )\\ \ \\ =0.8708 -0.1292 \\ \ \\ =0.7416[/tex]
(c)The probability that the mean price for a sample of 100 federal income tax returns is within $16 of the population mean
[tex]P(\frac{257-273}{100/\sqrt{100}} < z < \frac{289-273}{100/\sqrt{100}}) \\ \ \ \\ = P(-1.60 < z < 1.60 ) \\ \ \\ = P(z < 1.60 ) -P(z < -1.60 )\\ \ \\ =0.9452 -0.0548 \\ \ \\ =0.8904[/tex]
To learn more about mean and standard deviation, refer to the link -
https://brainly.com/question/28225633
#SPJ4
Complete Question
The cpa practice advisor reports that the mean preparation fee for federal income tax returns was $273 use this price as the population mean and assume the population standard deviation of preparation fees is $100 use z-table. round your answers to four decimal places. a. what is the probability that the mean price for a sample of 30 federal income tax returns is within $16 of the population mean? b. what is the probability that the mean price for a sample of 50 federal income tax returns is within $16 of the population mean? c. what is the probability that the mean price for a sample of $100 federal income tax returns is within $16 of the population mean? d. which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a probability that the sample mean is within of the population mean?
