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The amount of time that customers wait in line during peak hours at one bank is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. What is the probability that a customer will wait under 14 minutes?

Question

The amount of time that customers wait in line during peak hours at one bank is normally distributed with a mean of 12 minutes and a standard deviation of 2 minutes. What is the probability that a customer will wait under 14 minutes?

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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, standard deviation, and the value for which we need to find the probability. In this case, the mean (μ) is 12 minutes, the standard deviation (σ) is 2 minutes, and the value (X) is 14 minutes.

Step 2: Calculate the Z-score. The formula for the Z-score is:

Z = (X - μ) / σ

Substituting the given values into the formula, we get:

Z = (14 - 12) / 2 = 1

Step 3: Look up the Z-score in the Z-table to find the probability. The Z-table shows the cumulative probability from the mean up to a given Z-score. For a Z-score of 1, the cumulative probability is 0.8413.

This means that the probability that a customer will wait under 14 minutes is 0.8413 or 84.13%.

This problem has been solved

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