Question

You are interested in finding out the mean number of customers entering a 24-hour convenience store every 10-minutes. You suspect this can be modeled by the Poisson distribution with a a mean of λ = 3.76 customers. You are to randomly pick n = 71 10-minute time frames, and observe the number of customers who enter the convenience store in each. After which, you are to average the 71 counts you have. That is, compute the value of X. (a) What can you expect the value of X to be? Enter your answer using all the decimals you can. (b) Find the value of the standard deviation of X. Enter your answer using all the decimals you can. σχ =

Answer 1

The expected value of X, the mean number of customers in a 10-minute interval, is 3.76. The standard deviation of X is about 1.9391 customers.

Step-by-step explanation:

When you suspect a quantity is modeled by the Poisson distribution with a mean (λ) of 3.76 customers per 10-minute interval, and you collect data over n = 71 such intervals, the average number of customers observed (X) should be equal to the mean of the Poisson distribution. Therefore, in this scenario:

a. Expected value of X (mean) = λ = 3.76 customers (since the mean of the sample should approximate the mean of the distribution)

b. To find the standard deviation of X, you would typically use the formula σ = √(λ/n). However, since the sample size n does not affect the standard deviation of a Poisson distribution (it only affects the standard deviation of the sample mean), the standard deviation of X is the square root of λ:

σX = √(λ) = √(3.76) = approximately 1.9391 customers