Question

The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz?

Answer 1

40.13% probability that the volume of soda in a randomly selected bottle will be less than 32 oz.

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 = 32.3, \sigma = 1.2`

What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz?

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

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

`Z = (32 - 32.3)/(1.2)`

`Z = -0.25`

`Z = -0.25` has a pvalue of 0.4013

40.13% probability that the volume of soda in a randomly selected bottle will be less than 32 oz.