The average commute time via train from the Chicago O'Hare Airport to downtown is 60 minutes with a standard deviation of 15 minutes. Assume that the commute times are normally distributed. What proportion of commutes would be:
longer than 80 minutes?
less than 50 minutes?
between 45 and 75 minutes?
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 calculated.
a. Longer than 80 minutes?
z(80) = (80-60)/15 = 4/3
P(x> 80) = P(z> 4/3) = normal cdf(4/3,100) = 0.0912
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b. Less than 50 minutes?
z(50) = (50-60)/15 = -2/3
P(x< 50) = P(z< -2/3) = normal cdf(-100,-2/3) = 0.2525
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c. Between 45 and 75 minutes?
z(45) = (45-60)/15 =
z(75) = (75-60)/15 =
P (x=60) = P(z= ) = normal cdf (-100, ) =
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?
To find the proportion of commutes that fall within certain time intervals, we need to calculate the z-scores for the given times and use a standard normal distribution table.
1. To find the proportion of commutes longer than 80 minutes:
First, we need to calculate the z-score using the formula:
z = (x - μ) / σ
where x is the given value, μ is the mean, and σ is the standard deviation.
In this case, x = 80, μ = 60, and σ = 15.
So, z = (80 - 60) / 15 = 1.33
Using the standard normal distribution table, we look up the proportion corresponding to a z-score of 1.33 and find that it is approximately 0.908.
Therefore, approximately 0.908 or 90.8% of commutes would be longer than 80 minutes.
2. To find the proportion of commutes less than 50 minutes:
Using the same z-score formula, and given x = 50, μ = 60, and σ = 15:
z = (50 - 60) / 15 = -0.67
Looking up the proportion corresponding to a z-score of -0.67 in the standard normal distribution table, we find it to be approximately 0.251.
Hence, approximately 0.251 or 25.1% of commutes would take less than 50 minutes.
3. To find the proportion of commutes between 45 and 75 minutes:
We need to find the z-scores for both 45 and 75 using the formula as before.
For 45, z = (45 - 60) / 15 = -1
For 75, z = (75 - 60) / 15 = +1
Next, we need to find the respective proportions for both z-scores from the standard normal distribution table. The proportion for a z-score of -1 is approximately 0.158, and for a z-score of +1, it is approximately 0.841.
To find the proportion between these two z-scores, we subtract the smaller proportion from the larger proportion:
0.841 - 0.158 = 0.683
Therefore, approximately 0.683 or 68.3% of commutes would take between 45 and 75 minutes.
In summary:
- Proportion of commutes longer than 80 minutes: approximately 0.908 or 90.8%
- Proportion of commutes less than 50 minutes: approximately 0.251 or 25.1%
- Proportion of commutes between 45 and 75 minutes: approximately 0.683 or 68.3%