Question 1190920
Here's how to solve this problem:

**(a) z-score of a 92.8 mph fastball:**

The z-score formula is:  z = (x - μ) / σ

Where:
* x is the value you're interested in (92.8 mph)
* μ is the mean (92.12 mph)
* σ is the standard deviation (2.43 mph)

z = (92.8 - 92.12) / 2.43
z = 0.68 / 2.43
z ≈ 0.28

Rounded to two decimal places, the z-score is 0.28.

**(b) Percentage of fastballs between 90.8 mph and 91.2 mph:**

1. **Calculate the z-scores:**
   * For 90.8 mph: z₁ = (90.8 - 92.12) / 2.43 ≈ -0.55
   * For 91.2 mph: z₂ = (91.2 - 92.12) / 2.43 ≈ -0.38

2. **Find the probabilities:** Use a z-table or calculator.
   * P(z < -0.38) ≈ 0.3520
   * P(z < -0.55) ≈ 0.2912

3. **Find the probability between the two speeds:**
   P(-0.55 < z < -0.38) = P(z < -0.38) - P(z < -0.55)
   P(-0.55 < z < -0.38) = 0.3520 - 0.2912 = 0.0608

Expressed as a decimal rounded to three decimal places, the answer is 0.061.

**(c) Percentage of fastballs below 90.8 mph:**

We already calculated the z-score for 90.8 mph in part (b) (z = -0.55).  We also found the probability associated with this z-score: P(z < -0.55) ≈ 0.2912.

Expressed as a decimal rounded to three decimal places, the answer is 0.291.

**(d) Speed for the fastest 7%:**

1. **Find the z-score:** The fastest 7% corresponds to a cumulative probability of 0.93 (1 - 0.07 = 0.93).  Look up the z-score closest to 0.93 in a z-table; it's approximately 1.48.

2. **Use the z-score formula to find the speed (x):**
   x = μ + zσ
   x = 92.12 + (1.48 * 2.43)
   x = 92.12 + 3.60
   x ≈ 95.72

Rounded to the nearest 0.1 mph, the speed must be at least 95.7 mph.