Question 1202706
answer to your questions:


a. What is the probability that the mean price for a sample of 30 H&R Block retail customers is within $8 of the population mean?


that probability is equal to 0.61916352


b. What is the probability that the mean price for a sample of 50 H&R
Block retail customers is within $8 of the population mean?


that probability is equal to 0.742100965


c. What is the probability that the mean price for a sample of 100 H&R Block retail customers is within $8 of the population mean?


that probability is equal to 0.890401417


d. Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a .95 probability that the sample mean is within
$8 of the population mean?


none of them.
to have at least .95 probability  that the sample mean is within 8 dollars of the population mean, the sample size would have to be 151 or greater.


the excel spreadsheet shown below contains the calculations necessary for resolutuion of this problem.


<img src = "http://theo.x10hosting.com/2023/061002.jpg">


graphical confirmation of these results are shown below:


<img src = "http://theo.x10hosting.com/2023/061003.jpg">


<img src = "http://theo.x10hosting.com/2023/061004.jpg">


<img src = "http://theo.x10hosting.com/2023/061005.jpg">


<img src = "http://theo.x10hosting.com/2023/061006.jpg">


to calculate the minimum sample size required so that the probability of getting a score within 8 dollars of the mean, you would do the following.


critical z-score at 95% two tail confidence interval is equal to plus or minus z = 1.959963985.
standard error is equal to standard deviation divided by square root of sample size.
when standard deviation is 50, this becomes:
standard error is equal to 50 / square root of sample size.


z-score formula is:
z = (x-m)/s
z is the z-score
(x-m) is the margin of error
s is the standared error, which is equal to 50/sqrt(n), where n is the sample size.
the formula becomes z = 8/s
when s = 50/sqrt(n), the frmula becomes z = 8/(50/sqrt(n))
this is simplified to z = 8/50 * sqrt(n)
solve for sqrt(n) to get sqrt(n) = z*50/8
when z = 1.959963985, the formula becomes:
sqrt(n) = 1.959963985 * 50 / 8 = 12.24977491
n is equal to sqrt(n)^2 = 150.0569853.
n has to be an integers, so round up to the next integer to get a sample size of at least 151 so that the probasbility of getting a margin of error of less than 8 is greater than 95%.


let me know if you have any questions.
theo