Question: 3 Fill in the blank. 2 points Under current rules, the weight of a MLB baseball weighs between 142…

3 Fill in the blank. 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.

What is the probability that if we select 5 balls at random from this company, the average of those 5 balls would be too light (i.e., less than 142 grams) according to MLB standards?

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4 Numeric. 2 points

In question 3, you found the probability a baseball would be too light. Using this value, if we test 200 baseballs from this particular company, what is the probability that less than 6 of them would be too light?

Solution

The problem states: Under current rules, the weight of a MLB baseball must be between 142 and 149 grams. A company produces baseballs whose weights are normally distributed with a mean of 145 grams and a standard deviation of 1.50 grams. The question asks for the probability that if 5 balls are selected at random from this company, the average weight of those 5 balls would be less than 142 grams. To achieve this, we’ll follow these steps: First, calculate the standard error (SE) of the sample mean. This is given by the standard deviation divided by the square root of the sample size: \[ SE = \frac{1.5}{\sqrt{5}} \] Next, calculate the z-score for the sample mean of 142 grams. The z-score is given by subtracting the population mean from the sample mean, then dividing by the standard error: \[ z = \frac{142 - 145}{SE} \] Substitute the value of SE: \[ z = \frac{142 - 145}{0.6708} \] \[ z = \frac{-3}{0.6708} \] \[ z \approx -4.47 \] Now, find the probability corresponding to this z-score. A z-score of approximately -4.47 is extremely low, indicating that the probability is very small. Therefore, the probability that the average weight of these balls is less than 142 grams is nearly zero.