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Among the 30 largest U.S. cities, the mean one-way commute time to work is 25.8 minutes.
The longest one-way travel time is in New York City, where the mean time is 38.3 minutes.
Assume the distribution of travel times in New York City follows the normal probability distribution
and the standard deviation is 7.5 minutes.
(a) What percent of the New York City commutes are for less than 30 minutes?
(b) What percent are between 30 and 35 minutes?
(c) What percent are between 30 and 40 minutes?
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This problem is specially presented in a way to confuse you.
It first tells you about the mean one-way commute time among the 30 largest US cities,
but you do NOT need this info to solve the problem - this info is IRRELEVANT.
Since the problem is then focuced on the New York city ONLY, you only need the info related to New York.
The major starting point in solving this and many similar problems is that
the wording description only DISTRACT your attention.
All you need to know are the parameters of the appropriate/relevant normal curve:
its mean value and its standard deviation.
In your case, the mean is 38.3 minutes; the standard deviation is 7.5 minutes.
The probabilities they want from you, are the areas under this normal curve
in the designed diapasons.
Now see how it works.
(a) In part (a), they want you determine the area under the normal curve
on the left of the raw mark of 30 minutes.
In part (a), same as in other parts (b) and (c), the mean is 38.3 minutes; the standard deviation is 7.5 minutes.
You may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(-9999, 30, 38.3, 7.5)
The calculator gives the answer P = 0.1342, or 13.42%.
(b) In part (b), they want you determine the area under the normal curve
between the raw marks of 30 and 35 minutes.
In part (b), same as in other parts (a) and (c), the mean is 38.3 minutes; the standard deviation is 7.5 minutes.
Again, you may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(30, 35, 38.3, 7.5)
The calculator gives the answer P = 0.1957, or 19.57%.
(c) In part (c), they want you determine the area under the normal curve
between the raw marks of 30 and 40 minutes (very similar to part (b)).
In part (c), same as in other parts (a) and (b), the mean is 38.3 minutes; the standard deviation is 7.5 minutes.
Again, you may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(30, 40, 38.3, 7.5)
The calculator gives the answer P = 0.4554, or 45.54%.
At this point, the problem is just solved.
But I want present you an alternative way to solve this and similar problems.
Go to website https://onlinestatbook.com/2/calculators/normal_dist.html
and use free of charge calculator there.
It allows to do all these calculations, and every time it shows you
the area of interest under the normal curve.
Its interface is very friendly, very simple and very convenient.
It will help you to learn the subject in 5 - 10 minutes.
You may play with this calculator as long as you need, until everything in the subject
becomes clear to you.
After that, you may return to your regular hand calculator TI-83 / TI-84
enriched with full understanding of the subject.