The time taken by a man to deliver milk is normally distributed with, mean 12 minutes and standard deviation 2 minutes. He delivers milk everyday estimate the probability that he takes.
(a) longer than 17 minutes
(b) between 9 and 13 minutes
0.75
Calculate Z=(x-μ)/σ
and look up the normal distribution table for the probabilities.
To answer these questions, we'll use the standard normal distribution. We need to calculate the z-scores for the given times and then find the corresponding probabilities.
The formula to calculate the z-score is:
z = (x - μ) / σ
where:
x = the value we want to find the probability for,
μ = mean of the distribution, and
σ = standard deviation of the distribution.
(a) To find the probability that the man takes longer than 17 minutes, we'll calculate the z-score for 17 minutes and find the area to the right of this z-score.
z = (17 - 12) / 2
= 5 / 2
= 2.5
Using a standard normal distribution table or calculator, we find that the area to the right of z = 2.5 is approximately 0.0062. This means there is a 0.0062 probability that the man takes longer than 17 minutes to deliver milk.
(b) To find the probability that the man takes between 9 and 13 minutes, we'll calculate the z-scores for both times and find the area between these two z-scores.
For 9 minutes:
z1 = (9 - 12) / 2
= -3 / 2
= -1.5
For 13 minutes:
z2 = (13 - 12) / 2
= 1 / 2
= 0.5
Using a standard normal distribution table or calculator, we find that the area to the left of z = -1.5 is approximately 0.0668 and the area to the left of z = 0.5 is approximately 0.6915.
To find the area between these two z-scores, we subtract the smaller area from the larger area:
0.6915 - 0.0668 = 0.6247
So, there is a 0.6247 probability that the man takes between 9 and 13 minutes to deliver milk.
To estimate the probability in this scenario, we can refer to the normal distribution curve and use the z-score formula. The z-score measures the standard deviations from the mean.
(a) To estimate the probability that he takes longer than 17 minutes:
Step 1: Calculate the z-score for 17 minutes using the formula:
z = (x - μ) / σ
where x is the value (17 minutes), μ is the mean (12 minutes), and σ is the standard deviation (2 minutes).
z = (17 - 12) / 2
z = 5 / 2
z = 2.5
Step 2: Use a z-table (or a statistical calculator) to find the probability corresponding to the z-score of 2.5. The z-table provides the value of the standard normal distribution up to a given z-score. Looking up 2.5 in the table, we find that the probability is approximately 0.9938.
However, we need to find the probability for the right side since we want the probability of taking longer than 17 minutes. We subtract the obtained probability from 1 to get the final result:
P(X > 17) = 1 - 0.9938
P(X > 17) ≈ 0.0062
Therefore, there is approximately a 0.62% probability that he takes longer than 17 minutes to deliver milk.
(b) To estimate the probability that he takes between 9 and 13 minutes:
Step 1: Calculate the z-scores for both 9 and 13 minutes using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 9 minutes:
z1 = (9 - 12) / 2
z1 = -3 / 2
z1 = -1.5
For 13 minutes:
z2 = (13 - 12) / 2
z2 = 1 / 2
z2 = 0.5
Step 2: Use the z-table to find the probabilities corresponding to the z-scores of -1.5 and 0.5. Looking up these values in the table, we find that the probability corresponding to -1.5 is approximately 0.0668, and the probability corresponding to 0.5 is approximately 0.6915.
Step 3: To find the probability between 9 and 13 minutes, we subtract the probability corresponding to the left z-score from the probability corresponding to the right z-score:
P(9 < X < 13) = P(X < 13) - P(X < 9)
P(9 < X < 13) = 0.6915 - 0.0668
P(9 < X < 13) ≈ 0.6247
Therefore, there is approximately a 62.47% probability that he takes between 9 and 13 minutes to deliver milk.