The manager of a computer shop is recording the time taken for customers to decide of which computer and accessories they buy from the time they enter the store. From the previous data, it is known that the average ‘decision’ time was 45 minutes. The manager assumes a normally distributed population with a standard deviation of 10 minutesa) What is the probability that a customer will take more than 60 minutes?b) What proportion of customers will take between 35 and 60 minutes?
Question
The manager of a computer shop is recording the time taken for customers to decide of which computer and accessories they buy from the time they enter the store. From the previous data, it is known that the average ‘decision’ time was 45 minutes. The manager assumes a normally distributed population with a standard deviation of 10 minutesa) What is the probability that a customer will take more than 60 minutes?b) What proportion of customers will take between 35 and 60 minutes?
Solution
a) To find the probability that a customer will take more than 60 minutes, we first need to convert the 60 minutes to a z-score. The z-score is calculated as follows:
Z = (X - μ) / σ
where: X = value for which we want to find the probability (60 minutes in this case) μ = mean of the distribution (45 minutes) σ = standard deviation of the distribution (10 minutes)
So, Z = (60 - 45) / 10 = 1.5
We then look up this z-score in a standard normal distribution table, or use a calculator that can calculate the area under the standard normal curve. The table or calculator gives us the probability that a value is less than X. But we want the probability that a customer will take more than 60 minutes, so we need to subtract the value we get from 1.
The area under the curve to the left of Z = 1.5 is approximately 0.9332. So, the probability that a customer will take more than 60 minutes is 1 - 0.9332 = 0.0668, or 6.68%.
b) To find the proportion of customers who will take between 35 and 60 minutes, we need to find the z-scores for both 35 and 60 minutes and find the area between these two values.
Z1 = (35 - 45) / 10 = -1 Z2 = (60 - 45) / 10 = 1.5
The area under the curve to the left of Z = -1 is approximately 0.1587, and the area to the left of Z = 1.5 is approximately 0.9332. The proportion of customers who will take between 35 and 60 minutes is the difference between these two areas, or 0.9332 - 0.1587 = 0.7745, or 77.45%.
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