The mean one-way commute to work is 25.8 minutes. The longest one-way travel time is in New York City, where the mean time is 38.9 minutes. Assume the distribution of travel times in New York City follow the normal probability distribution and the standard deviation is 7.9 minutes.
A: What percent of New York City commutes for less than 30 minutes?
B: What percent are between 30 and 34 minutes?
C: What percent are between 30 and 41 minutes?
A: 30 minutes is ... [(38.9 - 30) / 7.9] s.d. below the mean
... use a z-score table to find the percentage
B: 34 minutes is ... [(38.9 - 34) / 7.9] s.d. below the mean
... use a z-score table to find the percentage
... find the difference in the percentage from A:
A: 41 minutes is ... [(41 - 38.9) / 7.9] s.d. above the mean
... use a z-score table to find the percentage
... find the difference in the percentage from A:
To find the percent of New York City commutes for less than 30 minutes, we will use the z-score formula:
z = (x - μ) / σ
where
x = threshold value (30 minutes)
μ = mean (38.9 minutes)
σ = standard deviation (7.9 minutes)
Step 1: Calculate the z-score for 30 minutes:
z = (30 - 38.9) / 7.9
Step 2: Look up the z-score in the standard normal distribution table or use a calculator. A z-score of -1.11392 corresponds to the cumulative probability of 0.1304.
Step 3: Convert the cumulative probability to a percentage by multiplying it by 100:
Percentage = 0.1304 * 100 = 13.04%
So, approximately 13.04% of New York City commutes for less than 30 minutes.
To find the percent of New York City commutes between 30 and 34 minutes, we will repeat the same process:
Step 1: Calculate the z-scores for 30 minutes and 34 minutes:
z1 = (30 - 38.9) / 7.9
z2 = (34 - 38.9) / 7.9
Step 2: Look up the z-scores in the standard normal distribution table or use a calculator.
For z1 = -1.11392, the cumulative probability is 0.1304.
For z2 = -0.62025, the cumulative probability is 0.2679.
Step 3: Find the difference between the two cumulative probabilities:
Difference = 0.2679 - 0.1304 = 0.1375
Step 4: Convert the difference to a percentage by multiplying it by 100:
Percentage = 0.1375 * 100 = 13.75%
So, approximately 13.75% of New York City commutes are between 30 and 34 minutes.
To find the percent of New York City commutes between 30 and 41 minutes, we will repeat the same process:
Step 1: Calculate the z-scores for 30 minutes and 41 minutes:
z1 = (30 - 38.9) / 7.9
z2 = (41 - 38.9) / 7.9
Step 2: Look up the z-scores in the standard normal distribution table or use a calculator.
For z1 = -1.11392, the cumulative probability is 0.1304.
For z2 = 0.26582, the cumulative probability is 0.6056.
Step 3: Find the difference between the two cumulative probabilities:
Difference = 0.6056 - 0.1304 = 0.4752
Step 4: Convert the difference to a percentage by multiplying it by 100:
Percentage = 0.4752 * 100 = 47.52%
So, approximately 47.52% of New York City commutes are between 30 and 41 minutes.
To answer these questions, we will use the z-score formula and the standard normal distribution table.
The z-score formula is given by:
z = (x - μ) / σ
where:
z is the z-score
x is the value we want to find the probability for
μ is the mean
σ is the standard deviation
Step 1: Calculate the z-scores for the given values.
For question A: x = 30
z = (30 - 38.9) / 7.9 = -1.12
For question B: x = 30 and y = 34
z₁ = (30 - 38.9) / 7.9 = -1.12
z₂ = (34 - 38.9) / 7.9 = -0.62
For question C: x = 30 and y = 41
z₁ = (30 - 38.9) / 7.9 = -1.12
z₂ = (41 - 38.9) / 7.9 = 0.27
Step 2: Use the z-score to find the probability using the standard normal distribution table.
For question A, we want to find the probability of getting a z-score less than -1.12. Looking at the table, the probability for a z-score of -1.12 is 0.1314. So approximately 13.14% of New York City commutes for less than 30 minutes.
For question B, we want to find the probability of getting a z-score between -1.12 and -0.62. From the table, the probability for -1.12 is 0.1314, and the probability for -0.62 is 0.2660. Subtracting the two probabilities, we get 0.2660 - 0.1314 = 0.1346. So approximately 13.46% of New York City commutes are between 30 and 34 minutes.
For question C, we want to find the probability of getting a z-score between -1.12 and 0.27. From the table, the probability for -1.12 is 0.1314, and the probability for 0.27 is 0.6051. Subtracting the two probabilities, we get 0.6051 - 0.1314 = 0.4737. So approximately 47.37% of New York City commutes are between 30 and 41 minutes.