The average sales per customer at a home improvement store during the past year is $75 with a standard deviation of $12.The probability that the average sales per customer from a sample of 36 customers,taken at random from this population,exceeds $78 is:
A) 0.4332.
B) 0.0668.
C) 0.0987.
D) 0.9013.
E) 0.5000

Question

Quizplus · Accepted Answer

To solve this problem, we will use the central limit theorem, which states that the sample mean of a large enough sample (n > 30) will be normally distributed, regardless of the distribution of the population.

The sample mean, x̄, of a sample of 36 customers can be calculated as:

x̄ ~ N(μ, σ/√n)

Where: 
μ = population mean = $75 
σ = population standard deviation = $12 
n = sample size = 36

Substituting the values, we get: 
x̄ ~ N(75, 2)

To find the probability that the average sales per customer from a sample of 36 customers exceeds $78, we need to find the z-score and its associated probability.

z = (78 - 75) / 2 = 1.5

Using a standard normal table or calculator, the probability of z > 1.5 is 0.0668. Therefore, the probability that the average sales per customer from a sample of 36 customers taken at random from this population exceeds $78 is 0.0668 or 6.68%.

Thus, the correct answer is B.

The Average Sales Per Customer at a Home Improvement Store