The time taken by a milk man to deliver milk to GOR estate is normally distributed with mean 12 minutes and standard deviation of 2 minutes.
Estimate the number of days during a year when he takes between 9 to 13 minutes.
Estimate the number of days during a year when he takes greater than 13 minutes.
Also find P25
To estimate the number of days during a year when the milkman takes between 9 to 13 minutes, we can use the concept of the normal distribution.
Step 1: Find the Z-scores for the lower and upper limits.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
X is the value we want to find the Z-score for (in this case, 9 and 13 minutes)
μ is the mean (given as 12 minutes)
σ is the standard deviation (given as 2 minutes)
For the lower limit Z-score:
Z_lower = (9 - 12) / 2 = -1.5
For the upper limit Z-score:
Z_upper = (13 - 12) / 2 = 0.5
Step 2: Look up the corresponding probabilities for the Z-scores.
Using a standard normal distribution table or a calculator, we can find the probabilities for the Z-scores.
Probability for Z_lower: P(Z < -1.5) = 0.0668 (approximately)
Probability for Z_upper: P(Z < 0.5) = 0.6915 (approximately)
Step 3: Calculate the estimated number of days.
The difference between the probabilities gives us an estimate for the number of days when he takes between 9 to 13 minutes.
Estimated number of days = (0.6915 - 0.0668) * 365 = 232.665 days (approximately)
Therefore, the estimated number of days during a year when the milkman takes between 9 to 13 minutes is approximately 233 days.
To estimate the number of days during a year when the milkman takes greater than 13 minutes, we use the same process.
Step 1: Find the Z-score for the upper limit (13 minutes).
Z_upper = (13 - 12) / 2 = 0.5
Step 2: Find the probability for the Z-score (greater than 0.5).
P(Z > 0.5) = 1 - P(Z < 0.5)
P(Z > 0.5) = 1 - 0.6915 = 0.3085 (approximately)
Step 3: Calculate the estimated number of days.
Estimated number of days = 0.3085 * 365 = 112.4025 days (approximately)
Therefore, the estimated number of days during a year when the milkman takes greater than 13 minutes is approximately 112 days.
To find P25 (the 25th percentile), we need to find the Z-score corresponding to this percentile.
Step 1: Convert the percentile to a Z-score.
P(Z < Z_score) = 0.25
Looking up the Z-score for a cumulative probability of 0.25 in the standard normal distribution table, we find Z_score ≈ -0.674.
Step 2: Find the actual value using the Z-score formula.
Z_score = (X - μ) / σ
-0.674 = (X - 12) / 2
Rearranging the equation, we get:
X - 12 = -0.674 * 2
X - 12 ≈ -1.348
X ≈ 12 - 1.348
X ≈ 10.652
Therefore, the P25 (25th percentile) for the milkman's delivery time is approximately 10.652 minutes.