SOLUTION: (10 marks) One-way commutes to a workplace have a normal distribution with mean 25.5 minutes and standard deviation 6.1 minutes. The workplace has less than a hundred employees.

Question 1191659: (10 marks) One-way commutes to a workplace have a normal distribution with mean 25.5 minutes and standard deviation 6.1 minutes. The workplace has less than a hundred employees.
a What proportion of commutes are longer than 30 minutes?
b What proportion of commutes are between 15 minutes and 30 minutes?

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem:
**a) Proportion of commutes longer than 30 minutes:**
1. **Calculate the Z-score:** The Z-score tells us how many standard deviations a particular value is away from the mean.
Z = (X - μ) / σ
Where:
* X = the value we're interested in (30 minutes)
* μ = the mean (25.5 minutes)
* σ = the standard deviation (6.1 minutes)
Z = (30 - 25.5) / 6.1
Z ≈ 0.74
2. **Look up the probability:** Use a Z-table (or a calculator with normal distribution functions) to find the probability associated with this Z-score. We want the proportion *longer* than 30 minutes, so we're looking for the area to the *right* of the Z-score.
P(Z > 0.74) ≈ 0.2296 (This is the area to the *right* of 0.74 on the standard normal curve.)
Therefore, approximately 22.96% of commutes are longer than 30 minutes.
**b) Proportion of commutes between 15 minutes and 30 minutes:**
1. **Calculate the Z-scores for both values:**
* For 15 minutes:
Z1 = (15 - 25.5) / 6.1
Z1 ≈ -1.72
* For 30 minutes (we already did this in part a):
Z2 = (30 - 25.5) / 6.1
Z2 ≈ 0.74
2. **Look up the probabilities:** Use a Z-table or calculator.
* Find the probability for Z1 = -1.72. This gives you the area to the *left* of -1.72. P(Z < -1.72) ≈ 0.0427
* Find the probability for Z2 = 0.74. This gives you the area to the *left* of 0.74. P(Z < 0.74) ≈ 0.7704
3. **Find the probability between the two values:** Subtract the smaller probability from the larger probability.
P(-1.72 < Z < 0.74) = P(Z < 0.74) - P(Z < -1.72)
P(-1.72 < Z < 0.74) ≈ 0.7704 - 0.0427
P(-1.72 < Z < 0.74) ≈ 0.7277
Therefore, approximately 72.77% of commutes are between 15 and 30 minutes.

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