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 plans to conduct a survey about concessions that are sold during games. Tickets are most expensive for seats near the field and on the sideline. The cheapest seats are high up in the end zones (where one of the authors sat as a student). A map of the stadium is shown.$100,000$

(a) The athletic department is considering a stratified random sample. What would you recommend as the strata? Why?

(b) Explain why a cluster sample might be easier to obtain. What would you recommend for the clusters? Why?

Short answer

Part (a) Sideline, Corner and End zone.

Part (b) The single people in the simple random samples as they are spread over randomly overall boxes.

Step-by-step solution

Part (a) Step 1: Given information

No. of fan seats for a football game, $n=100,000$

Also, tickets on the sidelines are the most expensive, while the cheapest seats are in the end zones.

Part (a) Step 2: Concept

A simple random sample (SRS) of size n is made up of $n$ people chosen from the population with an equal chance of being the sample that is actually chosen.

Part (a) Step 3: Explanation

The University of Michigan sports department intends to undertake a poll about game-day concessions. Seats near the pitch and on the sidelines are the most expensive. Simple random samples are obtained from independent subgroups using stratified random sampling. The strata are separate subgroups. The price of deception is the same for all three sections: sideline, corner, and the end zone. Furthermore, each person will be assigned to a specific section. As a result, the following strata should be used: sideline, corner, and the end zone.

Part (b) Step 1: Explanation

Simple random samples are obtained from independent subgroups using stratified random sampling. Cluster sampling separates the population into non-overlapping subgroups, which are subsequently included in the sample. All of the sitting boxes $(1-44)$ are divided into subgroups, with some subgroups being chosen. It will be easy to pick individuals because all persons from a few boxes will be included in the sample, allowing us to simply place people at the entrances of those boxes to conduct surveys of the people who live there. Part (a) will be much more challenging because we will need to draw a simple random sample from the sideline/corner/end zone strata, and it will be much more difficult to discover and interview people.