A study of college football games shows that the number of holding penalties assessed has a mean of 2.2 penalties per game and a standard deviation of 0.9 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2.5 penalties per game or more?2 points0.0160.0180.0220.024
Question
A study of college football games shows that the number of holding penalties assessed has a mean of 2.2 penalties per game and a standard deviation of 0.9 penalties per game. What is the probability that, for a sample of 40 college games to be played next week, the mean number of holding penalties will be 2.5 penalties per game or more?2 points0.0160.0180.0220.024
Solution
To solve this problem, we need to use the concept of z-score in statistics. The z-score is a measure of how many standard deviations an element is from the mean.
Step 1: Identify the given values. The population mean (μ) = 2.2 penalties per game The standard deviation (σ) = 0.9 penalties per game The sample size (n) = 40 games The sample mean (x̄) = 2.5 penalties per game
Step 2: Calculate the standard error of the mean (σx̄). The standard error of the mean is the standard deviation of the sampling distribution of the mean. It can be calculated using the formula:
σx̄ = σ/√n
Substituting the given values:
σx̄ = 0.9/√40 ≈ 0.142
Step 3: Calculate the z-score. The z-score can be calculated using the formula:
z = (x̄ - μ) / σx̄
Substituting the given values:
z = (2.5 - 2.2) / 0.142 ≈ 2.11
Step 4: Look up the z-score in the z-table to find the probability. The z-table shows the probability that the mean is less than 2.5. But we want the probability that the mean is more than 2.5, so we subtract the value from 1.
P(x̄ > 2.5) = 1 - P(z < 2.11)
Looking up 2.11 in the z-table gives a probability of 0.9826. So,
P(x̄ > 2.5) = 1 - 0.9826 = 0.0174
So, the probability that the mean number of holding penalties will be 2.5 penalties per game or more is approximately 0.0174 or 1.74%. The closest answer among the options given is 0.018.
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