Question

g 293 households were surveyed as part of a study on gas consumption. Water usage per day per household follows a normal distribution. The mean water usage per day per household in this survey was 40 gallons, and a 95.44% of the data falls within the interval (32.5, 47.5). What is the variance?

Answer 1

Answer:

The variance is 14.0625 square gallons.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 40 gallons

We are given that the distribution of gas consumption is a bell shaped distribution that is a normal distribution.

About 95.44% of the data falls within the interval (32.5, 47.5).

Empirical Rule:

• According to this rule almost all the data lies within three standard deviations of the mean.
• About 68% of data lies within one standard deviation of mean.
• About 95.44% of data lies within two standard deviation from the mean.

Thus, we can write,

$\mu + 2(\sigma) = 47.5\\\mu - 2(\sigma) = 32.5$

Putting the values, we get,

$40 + 2\sigma= 47.5\\ 40 -2\sigma = 32.5$

Solving, we get

$40 + 2\sigma - (40 - 2\sigma) = 47.5 - 32.5\\4\sigma = 15\\\sigma = 3.75$

Thus, the standard deviation of water usage per day per household is 7.5

The variance can be calculated as:

$\sigma^2 = (3.75)^2 = 14.0625$

The variance is 14.0625 square gallons.