Use the table to answer each question. Note: Round z-scores to the nearest hundredth and then find the required A values using the table.
The amount of time customers spend waiting in line at a bank is normally distributed, with a mean of 2.0 minutes and a standard deviation of 0.75 minute. Find the probability that the time a customer spends waiting is as follows. (Round your answers to three decimal places.)
(a) less than 2.5 minutes
(b) less than 0.5 minute
To find the probability in each case, we need to convert the given values into z-scores and then use the normal distribution table to find the corresponding probabilities.
For part (a) - less than 2.5 minutes:
Step 1: Convert the value 2.5 minutes into a z-score.
z = (x - μ) / σ
where x = 2.5 minutes, μ = mean = 2.0 minutes, and σ = standard deviation = 0.75 minutes.
z = (2.5 - 2.0) / 0.75
z = 0.5 / 0.75
z ≈ 0.67 (rounded to the nearest hundredth)
Step 2: Use the normal distribution table to find the probability corresponding to the z-score. We want the probability of being less than 2.5 minutes, so we find the area to the left of the z-score on the table.
The closest value in the table to 0.67 is 0.670 (rounded to the nearest thousandth). The corresponding probability is 0.7486 (from the table).
Therefore, the probability that a customer spends less than 2.5 minutes waiting in line at the bank is approximately 0.749.
For part (b) - less than 0.5 minutes:
Step 1: Convert the value 0.5 minutes into a z-score.
z = (x - μ) / σ
where x = 0.5 minutes, μ = mean = 2.0 minutes, and σ = standard deviation = 0.75 minutes.
z = (0.5 - 2.0) / 0.75
z = -1.5 / 0.75
z = -2.0 (rounded to the nearest hundredth)
Step 2: Use the normal distribution table to find the probability corresponding to the z-score. We want the probability of being less than 0.5 minutes, so we find the area to the left of the z-score on the table.
The closest value in the table to -2.0 is 0.022 (rounded to the nearest thousandth). The corresponding probability is 0.0228 (from the table).
Therefore, the probability that a customer spends less than 0.5 minutes waiting in line at the bank is approximately 0.023.