Current learning objective: Using the Normal Distribution Question \( 20 \quad \) Practice similar questions Score: 0 of 5 points Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 257 feet and a stand deviation of 56 feet. Note: All calculations should be rounded to four decimal places. a. If \( X= \) distance in feet for a fly ball, then \( X \sim N \) 3136 (For numerical answers, enter values as integers.) b. One fly ball is randomly chosen from this distribution. Find the probability that this ball traveled fewer than 228 feet. o b.1. Select the graph where the shaded region corresponds to the probability:

To solve the problem, we will follow these steps: ### Step 1: Define the Normal Distribution Given that the distance of fly balls hit to the outfield is normally distributed, we can express this as: - Mean (\( \mu \)) = 257 feet - Standard Deviation (\( \sigma \)) = 56 feet Thus, we can write: \[ X \sim N(257, 56^2) \] ### Step 2: Calculate the Probability We need to find the probability that a randomly chosen fly ball traveled fewer than 228 feet, which can be expressed as: \[ P(X < 228) \] To find this probability, we will first standardize the value using the Z-score formula: \[ Z = \frac{X - \mu}{\sigma} \] Substituting the values: \[ Z = \frac{228 - 257}{56} \] ### Step 3: Calculate the Z-score Let's calculate the Z-score. \[ Z = \frac{228 - 257}{56} = \frac{-29}{56} \] Now, we will calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(228-257\right)}{56}\) - step1: Remove the parentheses: \(\frac{228-257}{56}\) - step2: Subtract the numbers: \(\frac{-29}{56}\) - step3: Rewrite the fraction: \(-\frac{29}{56}\) It seems that I cannot directly calculate the probability from the Z-score using the tool. However, I can guide you on how to find it. ### Step 5: Using Z-tables or Calculator To find \( P(Z < -0.5179) \), you can use a Z-table or a statistical calculator. The Z-table provides the area to the left of a given Z-score in a standard normal distribution. For \( Z = -0.5179 \), you would look up this value in the Z-table or use a calculator that provides cumulative probabilities for the standard normal distribution. ### Step 6: Interpretation Once you find the probability from the Z-table or calculator, that value will represent the probability that a randomly chosen fly ball traveled fewer than 228 feet. ### Step 7: Graphical Representation For part b.1, you would select a graph that shows the area to the left of the Z-score corresponding to 228 feet. This area represents the probability \( P(X < 228) \). If you have access to a Z-table or a calculator, you can find the probability and then select the appropriate graph based on the shaded region representing this probability. If you need further assistance with the Z-table or any specific calculations, please let me know!