Question

A company produces packets of soap powder labeled "Giant Size 32 Ounces." The actual weight of soap powder in such a box has a Normal distribution with a mean of 33 oz and a standard deviation of 0.7 oz. To avoid having dissatisfied customers, the company says a box of soap is considered underweight if it weighs less than 32 oz. To avoid losing money, it labels the top 5% (the heaviest 5%) overweight. What proportion of boxes is underweight (i.e., weigh less than 32 oz)? .9236 .0764 .2420 .7580

Answer 1

0.0764

Explanation:

Problems of normally distributed samples can be 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 = 33, \sigma = 0.7`

What proportion of boxes is underweight (i.e., weigh less than 32 oz)?

This is the pvalue of Z when X = 32. So

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

`Z = (32 - 33)/(0.7)`

`Z = -1.43`

`Z = -1.43` has a pvalue of 0.0764, which is the correct answr