Part I - Experimental Design

This section is about the design of experiments and statistical sampling. While you won't be required to simulate random sampling using technology, you should understand how it works and be able to design a theoretical experiment with random sampling.

1. Suppose you are tasked with analyzing customer loyalty for Wishbone, an online discount retail company. The company has over 10,000 customers, with over 400,000 orders placed on the site.

a. Comment on the difficulties involved in collecting information and statistically analyzing the customers' data.

b. Would a simple random sample be feasible in this situation? Why?

2. One idea suggested by the Wishbone representative is to simply add an optional survey asking about customer satisfaction that customers can choose whether or not to fill out after placing an order.

a. What sampling method would this be considered?

b. Would this be an effective way to collect customer data? Explain any challenges that it may create.

3. Another suggestion for collecting data is to send an email to each customer who has made an order in the past asking them to fill out a brief survey.

a. Would this be an effective form of data collection? Why or why not?

b. What type of bias would this method produce?

4. If you were in charge of collecting data on Wishbone's customers, what method would you use to produce a randomized sample of a reasonable side for statistical analysis?

Part II - Data Processing & Analysis

This section is about processing raw population data and sample data, and taking a first look at finding the probability of producing an observed set of results based on estimated or computer population parameters.

5.The population data for all 12,515 of Wishbone's previous customers.The data shows a value of 1 if the customer placed an order in the last month, and a 0 if the customer did not place an order in the last month. The total number of orders placed or (1) is 10,343 and the total number of orders not placed or (0) is 2,172.

a. What type of distribution would describe this data? Be sure to check all the requirements!

b. Count the number of values of 1 and divide by the total number of values to find the population proportion,p, of customers who placed an order in the last month.

c. Use the formula for the appropriate distribution to calculate the standard deviation of the population.

Sample 1 was selected using systematic sampling of every 120^th customer; Sample 2 was taken using stratified sampling of new customers vs previous customers; Sample 3 was taken using stratified sampling of high value customers (>$20) vs low value customers (<$20); Sample 4 was taken using cluster sampling; and Sample 5 is a simple random sample.

6. For each Sample, count the number of customers who did place an order in the past month and divide by the sample size of 100 to find , the sample proportion.

a. Sample 1:

b. Sample 2:

c. Sample 3:

d. Sample 4:

e. Sample 5:

7. Now, use the 5 sample proportions you found in the previous problem as your data; you may either enter them in a new dataset or you may find it easier to simply enter the values in L1 on your calculator and compute 1-Var Stats.

a. Calculate the mean value of the sample proportions.

b. Calculate the population standard deviation, , for the sample proportions. How does this compare to the standard deviation of the original population?

8. Use the value ofp computed in problem 5 as the population parameterp.

a. What is the mean value of the sampling distribution, ?

b. Calculate the value of the standard error of the sampling distributions with a sample size of , that is, . How does this compare to the standard deviation of the different sample proportions you calculated in question 7b?

9. Now, examine Sample 1, which was computed using systematic sampling.

a. Does this sample meet the requirements of the Central Limit Theorem for approximation by the normal distribution? List each requirement, and whether it is met by this sample.

b. Now, using the mean and standard error for the sampling distribution you found in question 8, calculate the probability of finding a value as far from the mean as the sample proportion from Sample 1. (Hint: if , calculate ; if , calculate )

c. Does this result make sense, given the sample size and the population parameters you calculated?

10. The second file, approval.xlsx , contains customers' selective approval rating when ordering on Wishbone, rated using a sliding scale from 0-100. The first tab contains population data for all 12,515 previous customers.

a. Calculate the mean approval rating, , for the entire population.

b. Calculate the standard deviation, , for the population.

11. The second tab contains results for a sample of 100 customers, selected at random to conduct a survey that is required for customers to their checkout.

a. Calculate the mean rating, , of the sample.

b. Does this sample meet the requirements of the Central Limit Theorem?

12. Now, using the population mean and standard deviation found in question 10, compute the parameters of the sampling distribution.

a. What is the mean of the sampling distribution, ?

b. Use the appropriate formula for sampling distribution of a sample mean to compute the standard error of the sampling distribution, . How does this compare to the population standard deviation you found in question 10b?

13. Using the mean and standard deviation values from question 12 as your population parameters, calculate the probability of finding a more extreme value for the sample mean than the value from 11a, . (Hint: if , find ; if , find )

Part III - Analysis and Conclusion

For this section, you will use the statistics computed in Part II to draw conclusions about the data and sampling methods used.

14. For each data set, you computed the probability of observing a sample mean when selecting a random sample from a population with known parameters.

a. The probability you computed in 9b represents the likelihood of selecting a sample with a similar proportion of repeat customers from the overall population. Assuming a probability of 5% or greater is reasonably likely, are you surprised by the observed results?

b. The probability you computed in 13 represents the likelihood of selecting a sample with a similar average customer rating from the overall population.

Assuming a probability of 5% or greater is reasonably likely, are you surprised by the observed results?

15. Suppose for the sample approval ratings in the approval.xlsx file, the sample was collected only from customers placing an order within the last month, while the population mean was computed using a customer history going back two years. What conclusions can you draw about customer approval ratings in the previous month compared to those in the past two years?