R Exercise Requirements Complete the assignment in R and submit your R file In your R file, please clearly indicate which question you are answering with your R code Problem 7 The mean preparation fee H R Block charged retail customers last year was $183 Use this price as the population mean and assume the population standard deviation of preparation fees is $50 Now we randomly select 30 H R Block retail customers What are the values of the mean and the standard deviation of the sampling distribution of the sample mean What is the probability that the mean price for a sample of 30 H R Block retail customers is within $8 (this value is generally called margin of error) of population mean What is the probability that the mean price for a sample of 50 H R Block retail customers is within $8 of population mean What is the probability that the mean price for a sample of 100 H R Block retail customers is within $8 of population mean What is the impact of sample size based on b) above What sample size would you recommend to have at least a 95 probability that the sample mean is within $8 of population mean Now, let's assume we don't know the population mean but we still know the population standard deviation to be $50 We randomly sampled 40 H R Block retail customers and the mean price is $183 What is the probability that the population mean is within $5 of the sample mean We randomly sampled 100 H R Block retail customers and the mean price is $183, assuming the population standard deviation is still $50 Construct a 90 , 95 , and 99 confidence interval of the population mean, respectively Provide a practical interpretation of the above 90 confidence interval What conclusions can you draw based on the 90 , 95 , and 99 confidence intervals you constructed above When the population standard deviation is unknown, the best we can do is to replace it with the sample standard deviation, s Just like the sample mean is a random variable, so is the sample standard deviation s The replacement of with s adds more variability Some adjustment to the Central Limit Theorem is thus necessary It turns out that when the population standard deviation is unknown and the sample size n is sufficiently large, the sample statistic t (X ) (s n) approximately follows a t distribution with a degree of freedom n 1 As a result, when we look for a confidence interval with unknown, we will replace normal distributions with t distribution Use this result to find a 92 confidence interval for the population mean price that the retail customers pay for, given the sample mean is $183, the sample standard deviation is $50, and the sample size is 36 Comparing this result to that in question g), you should notice that the margin of error is slightly larger when the population standard deviation is unknown

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R Exercise Requirements Complete the assignment in R and submit your R file. In your R file, please clearly indicate which question you are answering

R Exercise

Requirements

Complete the assignment in R and submit your R file. In your R file, please clearly indicate which question you are answering with your R code.

Problem 7

The mean preparation fee H&R Block charged retail customers last year was $183. Use this price as the population mean and assume the population standard deviation of preparation fees is $50.

Now we randomly select 30 H&R Block retail customers. What are the values of the mean and the standard deviation of the sampling distribution of the sample mean?
What is the probability that the mean price for a sample of 30 H&R Block retail customers is within $8 (this value is generally called margin of error) of population mean? What is the probability that the mean price for a sample of 50 H&R Block retail customers is within $8 of population mean? What is the probability that the mean price for a sample of 100 H&R Block retail customers is within $8 of population mean?
What is the impact of sample size based on b) above?
What sample size would you recommend to have at least a .95 probability that the sample mean is within $8 of population mean?
Now, let's assume we don't know the population mean; but we still know the population standard deviation to be =$50. We randomly sampled 40 H&R Block retail customers and the mean price is $183. What is the probability that the population mean is within $5 of the sample mean?
We randomly sampled 100 H&R Block retail customers and the mean price is $183, assuming the population standard deviation is still $50. Construct a 90%, 95%, and 99% confidence interval of the population mean, respectively.
Provide a practical interpretation of the above 90% confidence interval. What conclusions can you draw based on the 90%, 95%, and 99% confidence intervals you constructed above?
When the population standard deviation is unknown, the best we can do is to replace it with the sample standard deviation, s. Just like the sample mean is a random variable, so is the sample standard deviation s. The replacement of with s adds more variability. Some adjustment to the Central Limit Theorem is thus necessary. It turns out that when the population standard deviation is unknown and the sample size n is sufficiently large, the sample statistic t=(X -)/(s/n) approximately follows a t distribution with a degree of freedom n-1. As a result, when we look for a confidence interval with unknown, we will replace normal distributions with t distribution. Use this result to find a 92% confidence interval for the population mean price that the retail customers pay for, given the sample mean is $183, the sample standard deviation is $50, and the sample size is 36. Comparing this result to that in question g), you should notice that the margin of error is slightly larger when the population standard deviation is unknown.