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?
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?
Solution
To solve this problem, we need to use the concept of Z-score in statistics. The Z-score is a measure of how many standard deviations an element is from the mean.
Step 1: Identify the mean and standard deviation From the problem, we know that the mean (μ) is 45 minutes and the standard deviation (σ) is 10 minutes.
Step 2: Identify the X value The problem is asking for the probability that a customer will take more than 60 minutes. So, X = 60 minutes.
Step 3: Calculate the Z-score The formula for calculating the Z-score is: Z = (X - μ) / σ Substituting the given values into the formula, we get: Z = (60 - 45) / 10 = 1.5
Step 4: Find the probability Now, we need to find the probability that the Z-score is more than 1.5. We can look this up in a standard normal distribution table, or use a calculator that can calculate it.
The value for 1.5 in the Z-table is 0.9332. However, this value represents the probability that the decision time is less than 60 minutes. We want the probability that the decision time is more than 60 minutes.
So, we subtract the value from 1 (because the total probability is 1): P(Z > 1.5) = 1 - 0.9332 = 0.0668
So, the probability that a customer will take more than 60 minutes to decide is 0.0668, or 6.68%.
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