Questions: The accompanying data represent the distance (in feet) traveled in a random sample of 500 home runs hit during the 2018 Major League baseball season. Complete parts (a) through (e). Click here to view the sample of 500 home runs. (d) Use the mean and standard deviation from part (b) as the parameters in the normal model to find the area under the normal curve between 350 and 390 feet. Compare this result to the relative frequency with which a home run distance between 350 and 389.9 feet, inclusively, is observed from the sample. The area under the normal curve between 350 and 390 feet is . The relative frequency with which a home run distance between 350 and 389.9 feet, inclusively, is observed from the sample is . These values close, which supports the conclusion that the normal model is a fit for the data. (Round to three decimal places as needed.) (e) Use the mean and standard deviation from part (b) as the parameters in the normal model to find the area under the normal curve to the right of 410 feet. Compare this result to the relative frequency with which a home run distance exceeds (or is equal to) 410 feet in the sample. The area under the normal curve to the right of 410 feet is . The relative frequency with which a home run distance exceeds (or is equal to) 410 feet is observed from the sample is . These values close, which supports the conclusion that the normal model is a fit for the data. (Round to three decimal places as needed.)

Solution Steps

To solve parts (d) and (e) of the question, we need to use the mean and standard deviation from part (b) to calculate the areas under the normal curve. We will use the cumulative distribution function (CDF) of the normal distribution to find these areas. For part (d), we will find the area between 350 and 390 feet, and for part (e), we will find the area to the right of 410 feet. We will then compare these areas to the relative frequencies observed in the sample data.

1. Use the mean and standard deviation from part (b) to define the normal distribution.
2. For part (d), calculate the area under the normal curve between 350 and 390 feet using the CDF.
3. For part (e), calculate the area under the normal curve to the right of 410 feet using the CDF.
4. Compare these areas to the relative frequencies observed in the sample data.

From part (b) of the problem, we need the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the home run distances. Let's assume the mean is \(\mu = 400\) feet and the standard deviation is \(\sigma = 20\) feet (these values are hypothetical as the actual values are not provided in the question).

To find the area under the normal curve between 350 and 390 feet, we first need to convert these values to their corresponding z-scores using the formula: \[ z = \frac{x - \mu}{\sigma} \]

For \(x = 350\) feet: \[ z_{350} = \frac{350 - 400}{20} = \frac{-50}{20} = -2.5 \]

For \(x = 390\) feet: \[ z_{390} = \frac{390 - 400}{20} = \frac{-10}{20} = -0.5 \]

Using the standard normal distribution table or a calculator, we find the area under the curve for \(z = -2.5\) and \(z = -0.5\).

\[ P(Z < -2.5) \approx 0.0062 \] \[ P(Z < -0.5) \approx 0.3085 \]

The area between \(z = -2.5\) and \(z = -0.5\) is: \[ P(-2.5 < Z < -0.5) = P(Z < -0.5) - P(Z < -2.5) = 0.3085 - 0.0062 = 0.3023 \]

Assume the relative frequency of home run distances between 350 and 389.9 feet from the sample is 0.300 (hypothetical value).

Convert 410 feet to its z-score: \[ z_{410} = \frac{410 - 400}{20} = \frac{10}{20} = 0.5 \]

Using the standard normal distribution table or a calculator, we find: \[ P(Z > 0.5) = 1 - P(Z < 0.5) \approx 1 - 0.6915 = 0.3085 \]

Assume the relative frequency of home run distances exceeding (or equal to) 410 feet from the sample is 0.310 (hypothetical value).

Final Answer