Questions: Susan has been on a bowling team for 12 years. After examining all of her scores over that period of time, she finds that they follow a normal distribution. Her average score is 237, with a standard deviation of 12. What is the probability that in a one-game playoff, her score is more than 239? Multiple Choice 0.5578 0.7306 0.4226 0.2694

Solution Steps

To determine the probability that Susan's score exceeds 239, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

Substituting the values:

\[ z = \frac{239 - 237}{12} = \frac{2}{12} = 0.1667 \]

Thus, the Z-score for a score of 239 is \( z = 0.1667 \).

Next, we need to find the probability that Susan's score is greater than 239. This can be expressed as:

\[ P(X > 239) = P(Z > 0.1667) = 1 - P(Z \leq 0.1667) \]

Using the cumulative distribution function \( \Phi \), we have:

\[ P(X > 239) = \Phi(\infty) - \Phi(0.1667) \]

From the calculations, we find:

\[ P(X > 239) = 0.4338 \]

Final Answer