A fruit-packing company produced peaches last summer whose weights were normally distributed with mean 16 ounces and standard deviation 0.8 ounce. Among a sample of 1000 of those peaches, about how many could be expected to have weights of more than 13.2 ounces?
Click here to see page 1 of the table for areas under the standard normal curve. Click here to see page 2 of the table for areas under the standard normal curve.
The number of peaches expected to have weights of more than 13.2 ounces is (Round to the nearest whole number as needed.)
Answer
Final Answer: The number of peaches expected to have weights of more than 13.2 ounces is approximately \(\boxed{1000}\).
Step 1 ：The problem is asking for the number of peaches that weigh more than 13.2 ounces. This is a problem of normal distribution. We know that the mean weight of the peaches is 16 ounces and the standard deviation is 0.8 ounce.
Step 2 ：We can use the z-score formula to find the z-score for 13.2 ounces. The z-score is a measure of how many standard deviations an element is from the mean. The formula for the z-score is: \(z = \frac{X - \mu}{\sigma}\) where: \(X\) is the value we are interested in (13.2 ounces in this case), \(\mu\) is the mean (16 ounces), \(\sigma\) is the standard deviation (0.8 ounce).
Step 3 ：Substituting the given values into the z-score formula, we get: \(z = \frac{13.2 - 16}{0.8} = -3.5\)
Step 4 ：Once we have the z-score, we can use a z-table to find the proportion of peaches that weigh more than 13.2 ounces. The z-table gives us the area under the curve to the left of the given z-score. Since we want the proportion of peaches that weigh more than 13.2 ounces, we need to subtract the area under the curve to the left of the z-score from 1. This gives us a proportion of approximately 0.9997673709209645.
Step 5 ：Finally, we multiply this proportion by the total number of peaches (1000) to get the expected number of peaches that weigh more than 13.2 ounces. This gives us: \(1000 \times 0.9997673709209645 = 999.7673709209645\)
Step 6 ：Rounding to the nearest whole number, we get approximately 1000 peaches.
Step 7 ：Final Answer: The number of peaches expected to have weights of more than 13.2 ounces is approximately \(\boxed{1000}\).
