The Length Of Time Bank Customers Must Wait For A Teller Are Normally
The length of time bank customers must wait for a teller is typically modeled as a normal distribution, characterized by a mean of 3 minutes and a standard deviation of 1 minute. To determine the percentage of customers waiting between 2 and 3.5 minutes, we utilize the properties of the standard normal distribution.
First, we convert the raw waiting times to z-scores, which measure how many standard deviations a value is from the mean. The z-score is calculated using the formula:
z = (X - µ) / σ
where:
X is the value of interest
µ is the mean (3 minutes)
σ is the standard deviation (1 minute)
Calculating the z-scores:
For X = 2 minutes:
z = (2 - 3) / 1 = -1
For X = 3.5 minutes:
z = (3.5 - 3) / 1 = 0.5
Next, we consult standard normal distribution tables or use statistical software to find the cumulative probability corresponding to each z-score.
From standard normal tables:
P(z = -1) ≈ 0.1587
P(z = 0.5) ≈ 0.6915
The probability that a customer's waiting time falls between 2 and 3.5 minutes is the difference between these two cumulative probabilities:
P(2 < X < 3.5) = P(z = 0.5) - P(z = -1) ≈ 0.6915 - 0.1587 = 0.5328
This means approximately 53.28% of customers wait between 2 and 3.5 minutes.
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