Problem 2.
Using the central limit theorem and a standard normal table or calculator, what is the probability that a college basketball team wins between 60% and 70% of their games during an entire season (30 games in total)?
(Note that in this part of the question, you are asked about the winning percentage, not the exact number of wins.)
To solve this problem using the central limit theorem and a standard normal table or calculator, we need to follow these steps:
Step 1: Determine the mean and standard deviation of the winning percentage.
Let's denote X as the random variable representing the winning percentage of the team in a single game. Since the game outcomes can be considered independent and identically distributed, we have X ~ Binomial(30, p), where p is the probability of winning a single game.
The mean of the winning percentage is given by μ = 30p, and the standard deviation is σ = √(30p(1-p)).
Step 2: Normalize the problem.
To apply the central limit theorem, we need to convert the given winning percentage range into a z-score range. The z-score formula is z = (X - μ) / σ, where X is the winning percentage.
For the lower bound (60%):
z_lower = (60 - μ) / σ
For the upper bound (70%):
z_upper = (70 - μ) / σ
Step 3: Find the probabilities using the z-scores.
Using either a standard normal table or calculator, find the cumulative probabilities associated with the z-scores calculated in step 2.
P(X ≤ 60%) = P(z ≤ z_lower)
P(X ≤ 70%) = P(z ≤ z_upper)
Step 4: Calculate the desired probability.
To find the probability that the team wins between 60% and 70% of their games, we subtract the lower probability from the upper probability:
P(60% ≤ X ≤ 70%) = P(X ≤ 70%) - P(X ≤ 60%)
By following these steps, you can use the central limit theorem and a standard normal table or calculator to determine the probability that a college basketball team wins between 60% and 70% of their games during an entire season (30 games in total).