The number of accidents per week at a hazardous intersection varies with mean 2.2
and standard deviation 1.4.
a. What is the distribution of Xbar, the mean number of accidents in one year, (52
weeks)? {Note you do not have to use R for this question, write down the
distribution with parameters here and then proceed]
b. What is the probability that Xbar is less than 2?
c. What is the probability that Xbar is between 5 and 20?
d. What is the probability that there are fewer than 100 accidents in a year?
You are dealing here with a distribution of means rather than raw scores.
The standard error of the mean (SEm) = SD/√(n-1)
Would n = number of weeks?
Z = (score-mean)/SEm
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z scores.
100 accidents/year = 1.923/week
a. To determine the distribution of Xbar, the mean number of accidents in one year (52 weeks), we can use the Central Limit Theorem. According to the Central Limit Theorem, if we take a large enough sample size from any distribution, the distribution of the sample means will tend to follow a normal distribution, regardless of the underlying distribution.
In this case, since the number of accidents per week follows a distribution with a mean of 2.2 and a standard deviation of 1.4, we can consider this as the population distribution.
Using the Central Limit Theorem, the distribution of Xbar, the sample mean number of accidents in one year, can be approximated as a normal distribution with a mean equal to the population mean (2.2) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (1.4 / sqrt(52)).
Therefore, the distribution of Xbar is approximately N(2.2, 0.193).
b. To calculate the probability that Xbar is less than 2, we need to standardize the value of 2 using the mean and standard deviation of the distribution of Xbar.
Using the formula for standardization, z = (X - mean) / standard deviation, we can calculate the z-score for X = 2:
z = (2 - 2.2) / 0.193
Once we have the z-score, we can use a standard normal distribution table or a statistical software to find the probability associated with this z-score.
c. To calculate the probability that Xbar is between 5 and 20, we again need to standardize the values of 5 and 20 using the mean and standard deviation of the distribution of Xbar.
Calculate the z-score for X = 5 and X = 20 using the formula: z = (X - mean) / standard deviation.
Then, we can find the probabilities associated with these two z-scores using a standard normal distribution table or statistical software. Finally, subtract the probability associated with the smaller z-score from the probability associated with the larger z-score to get the probability that Xbar is between 5 and 20.
d. To calculate the probability of having fewer than 100 accidents in a year, we need to standardize the value of 100 using the mean and standard deviation of the distribution of Xbar.
Calculate the z-score for X = 100 using the formula: z = (X - mean) / standard deviation.
Then, we can find the probability associated with this z-score using a standard normal distribution table or statistical software. The resulting probability will give us the probability of having fewer than 100 accidents in a year.