Question

The monthly utility bills in a city are normally distributed, with a mean of $132 and a standard deviation of $14. A random sample of 36 households is drawn from this population.

A. (10 pts) What would be the probability that the mean of the sample is less than $138? Show your work, including a sketch.
B. (3 pts) Would it be unusual to have a sample mean less than $138? Explain.

Answer 1

To find the probability, we use the z-score formula and calculate the z-score. From a standard normal distribution table or calculator, we find that the probability of getting a z-score less than 3 is 0.9987 or 99.87%.

Step-by-step explanation:

To find the probability that the mean of the sample is less than $138, we need to use the z-score formula. The z-score measures how many standard deviations a given value is away from the mean. We can use the formula:

z = (x - μ) / (σ / sqrt(n))

where:

μ is the mean of the population ($132)

n is the sample size (36)

Using the given values, we can calculate the z-score:

z = (138 - 132) / (14 / sqrt(36))

z = 6 / (14 / 6) = 6 / 2 = 3

From a standard normal distribution table or calculator, we can find that the probability of getting a z-score less than 3 is approximately 0.9987.

Therefore, the probability that the mean of the sample is less than $138 is about 0.9987 or 99.87%.