Question

The weight for crates of apples is normally distributed with a mean weight of 34.6 pounds and a standard deviation of 2.8 pounds. What is the probability that the weight is between 31 and 35 pounds

Answer 1

42.22% probability that the weight is between 31 and 35 pounds

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 34.6, \sigma = 2.8`

What is the probability that the weight is between 31 and 35 pounds

This is the pvalue of Z when X = 35 subtracted by the pvalue of Z when X = 31. So

X = 35

`Z = (X - \mu)/(\sigma)`

`Z = (35 - 34.6)/(2.8)`

`Z = 0.14`

`Z = 0.14` has a pvalue of 0.5557

X = 31

`Z = (X - \mu)/(\sigma)`

`Z = (31 - 34.6)/(2.8)`

`Z = -1.11`

`Z = -1.11` has a pvalue of 0.1335

0.5557 - 0.1335 = 0.4222

42.22% probability that the weight is between 31 and 35 pounds