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

Answer:

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

$\mu = 32.3, \sigma = 1.2$

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

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

$Z = \frac{32 - 32.3}{1.2}$

$Z = -0.25$

$Z = -0.25$ 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.