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August 21, 2014

August 21, 2014

Posted by **Kayse** on Saturday, August 4, 2007 at 12:21am.

For many years the numbers of accidents per week at a hazardous intersection have been studied. The number of accidents per week varies with mean 2.2 and standard deveiation 1.4.

1. Could the exact distributin of the number of accidents per week be described by a normal distribution?

I've drawn the normal curve showing the mean of 2.2 in the center of the curve and deviating both directions 1.4

2. Looking at only 52 weeks we can calculate x-bar the mean number of accidents per week. What is the approximate distribution of x-bar according to the Central Limit Theorem?

N(2.2, 0.194)

3. What is the approximate probability that x-bar is greater than 2 accidents per week?

P(x-bar > 2)

4. What is the probability that there are fewer than 100 accidents at the intersection in a year?

1. Could the exact distribution of the number of accidents per week be described by a normal distribution?

No, because the number of accidents is always an integer larger or equal than zero.

But the mean number of accidents per week, computed over some time T does tend to the normal distribution for

T ---> infinity. The standard deviation of the mean over a period of T is the original standard deviation divided by sqrt(T).

Thanks, what about probability that there are fewer than 100 accidents at the intersection in a year? The hint says to "restate the event in terms of x-bar minus mean number of accidents per week.

If you have 100 accidents in a given year, then in that year the average per week was x-bar = 100/52. You know what the true average is and you know what the standard deviation of the yearly average is (sigma/sqrt[52]).

You then look up the probability that

Z < (x-bar - average)/[sigma/sqrt[52]]

from the standard normal distribution table, or using statistical software.

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