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.
Reference: Ref 3-5

How heavy does a box have to be for it to be labeled overweight?
A. 31.60 oz
B. 31.85 oz
C. 34.15 oz
D. 34.40 oz

Answer 1

Answer:

C. 34.15 oz

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu= 33, \sigma = 0.7$

How heavy does a box have to be for it to be labeled overweight?

Top 5%, so X when Z has a pvalue of 1-0.05 = 0.95.

So X when Z = 1.645.

$Z = \frac{X - \mu}{\sigma}$

$1.645 = \frac{X - 33}{0.7}$

$X - 33 = 0.7*1.645$

$X = 34.15$

So the correct answer is:

C. 34.15 oz