Answer:
Ninety-five percent of consumers in the U.S. consumed less than 63.59 gallons.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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:
The standard deviation is the square root of the variance, so ![\sigma = \sqrt{169} = 13](https://tex.z-dn.net/?f=%5Csigma%20%3D%20%5Csqrt%7B169%7D%20%3D%2013)
Also, the mean is 42.2, so ![\mu = 42.2](https://tex.z-dn.net/?f=%5Cmu%20%3D%2042.2)
Ninety-five percent of consumers in the U.S. consumed less than how many gallons?
The 95th percentile, which is the value of X when Z has a pvalue of 0.95. So X when ![Z = 1.645](https://tex.z-dn.net/?f=Z%20%3D%201.645)
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.645 = \frac{X - 42.2}{13}](https://tex.z-dn.net/?f=1.645%20%3D%20%5Cfrac%7BX%20-%2042.2%7D%7B13%7D)
![X - 42.2 = 13*1.645](https://tex.z-dn.net/?f=X%20-%2042.2%20%3D%2013%2A1.645)
![X = 63.59](https://tex.z-dn.net/?f=X%20%3D%2063.59)
Ninety-five percent of consumers in the U.S. consumed less than 63.59 gallons.