According to a bank, the time taken by its customers to use cash dispensing machines is normally distributed with a mean of 18 seconds and a standard deviation of 3 seconds.
I) What is the probability that a customer selected at random takes less than 13 seconds?
II) Find the proportion of customers that take :
a) Between 15 and 21 seconds to use the cash dispensers?
b) Between 12 and 24 seconds to use the cash dispensers?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To solve these probability questions, we can use the standard normal distribution table or calculator. However, we need to first standardize the given values using the formula:
Z = (X - μ) / σ
where Z is the standard score (Z-score), X is the given value, μ is the mean, and σ is the standard deviation.
I) What is the probability that a customer selected at random takes less than 13 seconds?
To find this probability, we need to find the area under the normal curve to the left of 13 seconds. First, we calculate the Z-score:
Z = (13 - 18) / 3 = -5/3 = -1.67
By referring to the standard normal distribution table or using a calculator, we find that the area to the left of -1.67 is approximately 0.0475.
Therefore, the probability that a customer selected at random takes less than 13 seconds is approximately 0.0475, or 4.75%.
II) Find the proportion of customers that take:
a) Between 15 and 21 seconds to use the cash dispensers?
To find this proportion, we need to calculate the Z-scores for the two given values:
Z1 = (15 - 18) / 3 = -1
Z2 = (21 - 18) / 3 = 1
We then find the areas corresponding to these Z-scores under the standard normal distribution curve. Using the Z-table or calculator, we find:
Area to the left of Z1 = 0.1587
Area to the left of Z2 = 0.8413
To find the proportion between 15 and 21 seconds, we subtract the smaller area from the larger area:
Proportion = Area(Z2) - Area(Z1) = 0.8413 - 0.1587 = 0.6826
Therefore, approximately 68.26% of customers take between 15 and 21 seconds to use the cash dispensers.
b) Between 12 and 24 seconds to use the cash dispensers?
Similarly, we calculate the Z-scores for the given values:
Z1 = (12 - 18) / 3 = -2
Z2 = (24 - 18) / 3 = 2
Using the Z-table or calculator, we find:
Area to the left of Z1 = 0.0228
Area to the left of Z2 = 0.9772
To find the proportion between 12 and 24 seconds, we subtract the smaller area from the larger area:
Proportion = Area(Z2) - Area(Z1) = 0.9772 - 0.0228 = 0.9544
Therefore, approximately 95.44% of customers take between 12 and 24 seconds to use the cash dispensers.