Question 1190920: Statistician Jim Albert compiled data about every pitch thrown by 20 starting pitchers during the 2009 MLB season. The data set included the type of pitch thrown (curveball, changeup, slider, etc.) as well as the speed of the ball as it left the pitcher's hand. The data showed that the speeds of the fastballs are normally distributed with mean 92.12 mph and standard deviation 2.43 mph.
(a) Compute the z-score of the fastball with speed 92.8 mph. Round your answer to two decimal places.
(b) What percentage of these fastballs would you expect to have speeds between 90.8 mph and 91.2 mph? Express your answer as a decimal. Round to three decimal places.
(c) What percentage of these fastballs would you expect to have speeds below 90.8 mph? Express your answer as a decimal. Round to three decimal places.
(d) Finding Values: A baseball fan wishes to identify the fastballs among the fastest 7% of all such pitches. Above what speed must a fastball be in order to be included in the fastest 7%? Round your answer to the nearest 0.1 mph.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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