Question

The distribution of the amount of water used per wash cycle, in gallons, by a particular brand of dishwasher during a wash cycle is approximately normal with mean 4 gallons and standard deviation 0.2 gallon. Which of the following is closest to the probability that more than 3.75 gallons are used for any wash cycle?

Answer 1

Answer:

The probability that more than 3.75 gallons are used for any wash cycle is 0.8944 or 89.44%

Explanation:

Normal Distribution

Some probabilistic events are modeled by using the so-called bell curve or normal distribution $N(\mu , \sigma)$ where $\mu$ is the mean and $\sigma$ is the standar deviation. The cummulative values for the normal distribution are usually given as 'left tail' of the sum of all values less than certain maximum value. In other words, the value of $P(X\leq X_o)$ is given by tables or any digital means because the cannot be directly computed by formulas. We used the Excel's formula called NORM.DIST(x,mean,standard_dev,cumulative).

The study case of the question provides the following data

$\mu=4,\ \sigma=0.2$

And we are required to compute $P(X>3.75)$. Since the Excel formula gives us the left-tail value, we use the negation of that value to find our desired probability, that is:

$P(X>3.75)=1-P(X\leq 3.75)$

We now find the value of NORM.DIST(3.75,4,0.2,TRUE)=0.1056. Thus

$P(X>3.75)=1-0.1056=0.8944$

Thus the probability that more than 3.75 gallons are used for any wash cycle is 0.8944 or 89.44%