Question

Imagine a street on which there exist two sub shops: Big Mike's and Little John's. Each Saturday, Big Mike's draws an average of 500 people with a standard deviation of 20. Also on Saturdays, Little John's draws an average of only 400 people with a standard deviation of 50. If both distributions are normal, which shop is more likely to attract more than 600 people on a given Saturday?

Answer 1

Little John's sub shop, with a mean of 400 and a larger standard deviation of 50, has a slightly higher probability of attracting more than 600 people on a given Saturday compared to Big Mike's, which has a mean of 500 and a smaller standard deviation of 20.

Step-by-step explanation:

To determine which sub shop is more likely to attract more than 600 people on a Saturday, we look at the normal distribution of customers for both Big Mike's and Little John's. We know from the question that Big Mike's has an average (mean) of 500 people with a standard deviation of 20 and Little John's has an average of 400 with a standard deviation of 50.

The number 600 is more than 5 standard deviations above the mean for Big Mike's, and it is exactly 4 standard deviations above the mean for Little John's.

Since the number of standard deviations from the mean is an indicative measure of how unusual a value is in a normal distribution, and given the fact that beyond 3 standard deviations away from the mean are considered to be very rare in a normal distribution, the probability of both shops attracting more than 600 people is extremely low.

However, because Little John's mean is closer to 600 and has a larger standard deviation, it has a slightly higher probability than Big Mike's of drawing more than 600 customers on a given Saturday. This is because a larger standard deviation indicates a wider spread of the data, meaning there’s more variability and hence a higher chance of achieving an unusually high number of customers.