A study of college football games shows that the number of holding penalties assessed has a mean of 2.3 penalties per game and a standard deviation of 0.8 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 2.45 penalties per game or more?
Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

To find the probability that the mean number of holding penalties for a sample of 40 college games will be 2.45 penalties per game or more, we can use the **Central Limit Theorem (CLT)**. According to the CLT, the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (generally \(n \geq 30\)).

Given:
- Population mean, \(\mu = 2.3\)
- Population standard deviation, \(\sigma = 0.8\)
- Sample size, \(n = 40\)
- Sample mean, \(\bar{x} = 2.45\)

### Step 1: Determine the Standard Error of the Mean

The standard error (SE) of the mean is given by the formula:

\[
SE = \frac{\sigma}{\sqrt{n}}
\]

Substituting the given values:

\[
SE = \frac{0.8}{\sqrt{40}} = \frac{0.8}{6.3246} \approx 0.1265
\]

### Step 2: Calculate the Z-score

The Z-score is a measure of how many standard deviations an element is from the mean. The formula to calculate the Z-score for the sample mean is:

\[
Z = \frac{\bar{x} - \mu}{SE}
\]

Substitute the known values into the formula:

\[
Z = \frac{2.45 - 2.3}{0.1265} = \frac{0.15}{0.1265} \approx 1.1866
\]

### Step 3: Find the Probability

To find the probability that the sample mean is 2.45 or more, we need to find the area to the right of the Z-score calculated above. This is represented by \(P(Z \geq 1.1866)\).

Using a standard normal distribution table or a calculator, we can find \(P(Z < 1.1866)\), and then subtract it from 1 to get \(P(Z \geq 1.1866)\).

- The probability \(P(Z < 1.1866)\) is approximately 0.8810 (from Z-tables or calculator).

Thus,

\[
P(Z \geq 1.1866) = 1 - 0.8810 = 0.1190
\]

Therefore, the probability that the mean number of holding penalties will be 2.45 or more is approximately **0.119**.

Would you like more details or have any questions about this process? Here are some related questions you might find interesting:

1. What is the Central Limit Theorem, and why is it important?
2. How do you interpret a Z-score in practical terms?
3. What is the significance of the standard error in sampling?
4. How can we determine if a sample size is "large enough" for the CLT to apply?
5. How do we calculate probabilities for non-normal distributions?
6. Can the standard deviation of the population change the probability outcome?
7. What is the impact of a smaller sample size on the standard error?
8. How does the shape of the normal distribution relate to probabilities?

**Tip:** When working with probabilities and normal distributions, always sketch a quick graph of the distribution to visualize what you're calculating. It can help you understand whether you're finding the area to the left, right, or between certain values.

Calculate the probability that the mean number of holding penalties in a sample of 40 college football games will be 2.45 penalties per game or more using the Central Limit Theorem (CLT). Detailed step-by-step solution with formulas and calculations provided.

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