The amount of time a bank teller spends in each customer has a population mean of 3.10 minutes and standard deviation of 0.40 minute. if random number of 16 customers is selected.what is the probablity that the average time spend per customer will be at least 3 minutes
Using the central limit theorem, we can approximate the distribution of the sample mean as a normal distribution with mean μ = 3.10 and standard deviation σ/√n = 0.40/√16 = 0.10.
To find the probability that the average time spend per customer will be at least 3 minutes, we need to find the probability that the sample mean is greater than or equal to 3.
Z = (x - μ) / (σ/√n) = (3 - 3.10) / 0.10 = -1
Using a standard normal distribution table or calculator, we find that the probability of a Z-score being less than -1 is 0.1587. Therefore, the probability of the sample mean being less than 3 is 0.1587.
To find the probability of the sample mean being greater than or equal to 3, we subtract the above probability from 1:
P(X̄ ≥ 3) = 1 - P(X̄ < 3) = 1 - 0.1587 = 0.8413
Therefore, the probability that the average time spend per customer will be at least 3 minutes is 0.8413 or 84.13%.