Question: 5 Numeric 2 points Under current rules, the weight of a MLB baseball weighs between 142 and 149…

5 Numeric 2 points

Under current rules, the weight of a MLB baseball weighs between 142 and 149 grams. One particular company which makes baseballs produces baseballs whose weights are approximately normally distributed, with mean of 145 grams and standard deviation of 1.38 grams.

If the company wanted to identify the heaviest 3.5% of balls to remove prior to packaging, what is the most a ball could weigh to not be removed?

Draw a picture that represents the problem. Show your work.

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Solution

The problem asks for the maximum weight a baseball could have to not be removed if the company wants to remove the heaviest 3.5% of baseballs. First, we recognize that this is a problem involving a normal distribution where we want to find a certain percentile. Given: - Mean (\(\mu\)) = 145 grams - Standard deviation (\(\sigma\)) = 1.38 grams - Remove the heaviest 3.5% of baseballs We need to find the 96.5th percentile of this distribution since 100% - 3.5% = 96.5%. Step 1: Find the Z-score for the 96.5th percentile. Using a Z-table or calculator, the Z-score for the 96.5th percentile is approximately: \[ Z = 1.81 \] Step 2: Use the Z-score formula to find the weight. \[ Z = \frac{X - \mu}{\sigma} \] Plug in the known values: \[ 1.81 = \frac{X - 145}{1.38} \] Step 3: Solve for \(X\). \[ 1.81 \times 1.38 = X - 145 \] \[ 2.4978 = X - 145 \] \[ X = 145 + 2.4978 \] \[ X = 147.4978 \] Therefore, the maximum weight a baseball could have to not be removed is approximately 147.5 grams.