Question

Go Blue! Michigan Stadium, also known as "The Big House," seats over $100,000$fans for a football game. The University of Michigan Athletic Department wants to survey fans about concessions that are sold during games. Tickets are most expensive for seats on the sidelines. The cheapest seats are in the end zones (where one of the authors sat as a student). A map of the stadium is shown.

a. Describe how to obtain a sample using stratified random sampling. Explain your choice of strata and why this method might be preferred to simple random sampling.

b. Describe how to obtain a sample using cluster sampling. Explain your choice of clusters and why this method might be preferred to simple random sampling.

Short answer

(a) The strata is a section and SRS is preferred as it contains fans from each section.

(b) The cluster are seating boxes and SRS is preferred as all individuals have a few boxes only.

Step-by-step solution

Part (a) Step 1: Given information 

We need to find the strata and the reason for SRS being a better option

Part (a) Step 2: Explanation 

Simple random samples are obtained from independent subgroups using stratified random sampling (stratum).

The price of a ticket is the same for all three sections: sideline, corner, and the end zone. Furthermore, because each player will be assigned to only one part, it is best to use the sections as strata: Sideline, Corner, and the End zone.

This method may be preferable to a simple random sample because the sample will include fans from each section and hence from each ticket price category, allowing us to consider the views of fans from each economic class.

Part (b) Step 1: Given information 

We need to find the cluster of the sample

Part (b) Step 2: Explanation 

Cluster sampling separates the population into non-overlapping subgroups (clusters), which are subsequently included in the sample.

All of the seating boxes ($1-44$) could be considered subgroups (clusters), from which some subgroups will be chosen.

It will be easy to choose the individuals because all of the people in a few boxes will be included in the sample, and we can simply place people at the entrances of those boxes to conduct surveys of the people who live there.

Because the participants in the sample will be dispersed throughout the stadium, basic random sampling will be significantly more challenging.