I have a problem I need some help on. I answered some.
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?
To answer these questions, you can follow these steps:
1. Could the exact distribution of the number of accidents per week be described by a normal distribution?
To determine if the distribution of the number of accidents per week can be approximated by a normal distribution, you can check if the sample is large enough (typically, greater than 30) and also assess if there are any significant departures from normality, such as extreme skewness or outliers. In this case, since the number of accidents per week has been studied for many years, it is reasonable to assume that the sample size is sufficiently large and the distribution can be approximated by a normal distribution.
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?
According to the Central Limit Theorem (CLT), when random sampling is done from a population with any distribution, the distribution of the sample means approaches a normal distribution, regardless of the shape of the original population. In this case, with 52 weeks, the sample mean x-bar will follow a normal distribution with the same mean as the original distribution (2.2 in this case) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (1.4 / sqrt(52) ≈ 0.194). Thus, the approximate distribution of x-bar is N(2.2, 0.194).
3. What is the approximate probability that x-bar is greater than 2 accidents per week?
To calculate the probability that x-bar is greater than 2 accidents per week, you can use the standard normal distribution. First, you need to standardize the value of 2 (subtract the mean and divide by the standard deviation). Then, you can look up the z-score in the standard normal distribution table or use a calculator to find the area under the curve to the right of the z-score. This gives you the probability that x-bar is greater than 2 accidents per week.
4. What is the probability that there are fewer than 100 accidents at the intersection in a year?
To calculate the probability that there are fewer than 100 accidents at the intersection in a year, you can use the normal distribution. However, for this, you need to convert the annual number of accidents to a weekly average. Since there are 52 weeks in a year, divide 100 by 52 to get the average number of accidents per week. Once you have the weekly average, you can use the normal distribution with the mean (2.2) and standard deviation (1.4) to find the probability that there are fewer than that average number of accidents per week.