The image contains a table with baseball statistics, including information about various teams such as home runs (HR), team salaries, attendance, and more. It also presents two sections labeled "PART I TASKS" and "PART II TASKS," which include several questions.

PART I TASKS:

You are asked to use the Poisson distribution to estimate probabilities for home runs per game. Here's a breakdown of how to approach these questions:

1. Compute the mean number of home runs per game:

   • The number of home runs (HR) per team over a season is listed.
   • Since a season has 162 games, we first calculate the mean home runs per game for each team: $\text{mean home runs per game} = \frac{\text{Total HRs}}{162}$
   • Since there are 2 teams per game, multiply the result by 2 to account for both teams in the match: $\lambda = \frac{\text{Total HRs}}{162} \times 2$

2. Find the probabilities using Poisson distribution:

   • The Poisson probability formula is $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$, where $\lambda$ is the mean home runs per game and $k$ is the number of home runs.
   • Questions:
     1. The probability of no home runs in a game $P(X = 0)$.
     2. The probability of two home runs in a game $P(X = 2)$.
     3. The probability of at least 4 home runs in a game $P(X \geq 4)$.

PART II TASKS:

1. Mean attendance:

   • The mean attendance is $2.439$ million, with a standard deviation of $0.618$ million.
   • The normal distribution can be used to estimate how many teams have an attendance greater than $3.5$ million. Use the formula for standard normal $Z$-score: $Z = \frac{x - \mu}{\sigma}$
   • Calculate $Z$-score for $x = 3.5$ million and use the cumulative distribution function (CDF) to find the proportion of teams.

2. Mean team salary:

   • The mean team salary is $121$ million, with a standard deviation of $40.0$ million.
   • Estimate the number of teams with a salary greater than $100$ million using the normal distribution, following the same process as the attendance question.

Would you like detailed calculations for either the Poisson or normal distribution tasks?

Example Expansion Questions:

1. How would changing the number of games in a season affect the Poisson probability results?
2. What would happen if the standard deviation of the team's salary were doubled?
3. Can you explain the connection between the Poisson and normal distributions in certain contexts?
4. How would the estimation change if you had more accurate attendance data?
5. Can you find a way to apply a different distribution to model home runs?

Tip:

Understanding the difference between discrete (Poisson) and continuous (normal) distributions is key to applying them correctly in real-world scenarios.