Question

1. each major league baseball (mlb) team is allowed to have 26 players on their roster for opening day. with 30 mlb teams, this means there are 780 players on the opening day rosters. an avid baseball fan wants to select a random sample of these players. a. describe a method for selecting a simple random sample of 100 players. b. describe a method that could be used to select a stratified random sample of 90 players. c. describe a method that could be used to select a cluster sample of 78 players. d. describe a method that could be used to select a systematic random sample of 78 players.

Explanation:

• a. Simple Random Sample:
• Assign each of the 780 players a unique number from 1 to 780.
• Use a random number generator (e.g., a computer - based random number generator or a table of random numbers) to generate 100 distinct random numbers within the range of 1 - 780.
• The players corresponding to these 100 numbers form the simple random sample.
• b. Stratified Random Sample:
• Stratify the population by teams. Since there are 30 teams, each team has 26 players.
• Calculate the proportion of players to be sampled from each team. For a sample of 90 players, the proportion is \(\frac{90}{780}=\frac{3}{26}\) per team.
• For each team, use a simple random sampling method (e.g., numbering the 26 players on each team from 1 - 26 and using a random number generator to select 3 players per team). Combine the selected players from all teams to form the stratified random sample.
• c. Cluster Sample:
• Consider each MLB team as a cluster.
• Use a simple random sampling method to select 3 clusters (teams) out of 30.
• Include all 26 players from each of the 3 selected teams. Since \(3\times26 = 78\), these 78 players form the cluster sample.
• d. Systematic Random Sample:
• Calculate the sampling interval \(k=\frac{780}{78}=10\).
• Randomly select a starting number \(i\) between 1 and 10.
• Select players with numbers \(i, i + 10,i + 20,\cdots,i+(77\times10)\). For example, if \(i = 3\), select players numbered \(3,13,23,\cdots,773\).

Answer:

• a. Assign numbers 1 - 780, use random number generator for 100 distinct numbers, select corresponding players.
• b. Stratify by teams, calculate proportion \(\frac{3}{26}\) per team, use simple random sampling per team, combine.
• c. Consider teams as clusters, randomly select 3 teams, include all 26 players from each.
• d. Sampling interval \(k = 10\), randomly select start \(i\) (1 - 10), select \(i,i + 10,\cdots,i+(77\times10)\).