Questions: The speeds of all pitches thrown by the IRSC softball team are normally distributed with a mean of 63.5 miles per hour and a standard deviation of 4 miles per hour. Determine the probability that a randomly selected pitch has a speed of more than 64 miles per hour. Round the solution to four decimal places, if necessary. P(x>64)=

The speeds of all pitches thrown by the IRSC softball team are normally distributed with a mean of 63.5 miles per hour and a standard deviation of 4 miles per hour. Determine the probability that a randomly selected pitch has a speed of more than 64 miles per hour. Round the solution to four decimal places, if necessary.
P(x>64)=

Transcript text: The speeds of all pitches thrown by the IRSC softball team are normally distributed with a mean of 63.5 miles per hour and a standard deviation of 4 miles per hour. Determine the probability that a randomly selected pitch has a speed of more than 64 miles per hour. Round the solution to four decimal places, if necessary. \[ P(x>64)= \]

Solution

Solution Steps

Step 1: Determine the Z-score

To find the probability that a randomly selected pitch has a speed greater than 64 miles per hour, we first calculate the Z-score for \( x = 64 \) mph using the formula:

\[ Z = \frac{x - \mu}{\sigma} \]

Substituting the values:

\[ Z = \frac{64 - 63.5}{4} = \frac{0.5}{4} = 0.125 \]

Step 2: Calculate the Probability

Next, we need to find the probability \( P(X > 64) \). This can be expressed using the cumulative distribution function \( \Phi \):

\[ P(X > 64) = 1 - P(X \leq 64) = 1 - \Phi(Z_{start}) = \Phi(Z_{end}) - \Phi(Z_{start}) \]

Where \( Z_{end} = \infty \) and \( Z_{start} = 0.125 \). Thus, we have:

\[ P(X > 64) = \Phi(\infty) - \Phi(0.125) \]

From the output, we find:

\[ P(X > 64) = 0.4503 \]