Question

Let X be the number of accidents per week at the Hillsborough Street roundabout by the NCSU Bell Tower. Assume X varies with mean 2.2 and standard deviation 1.4. Note that X has only small whole-number values so the distribution of X cannot be reasonably approximated with a normal model. Let x be the mean number of accidents per week at the roundabout during a year (52 weeks). What is the probability that x is less than 2?

Answer 1

The probability that the mean number of accidents per week is less than 2 at the Hillsborough Street roundabout is approximately 0.6515 or 65.15%.

Step-by-step explanation:

To find the probability that the mean number of accidents per week is less than 2, we need to use the Poisson distribution. The average number of accidents per week is given as 2.2, which means the rate parameter (λ) is also 2.2. Using the Poisson distribution formula, we can calculate the probability.

P(x < 2) = e^(-λ) * ((λ^0)/0!) + e^(-λ) * ((λ^1)/1!)

= e^(-2.2) * ((2.2^0)/0!) + e^(-2.2) * ((2.2^1)/1!)

= (0.1108057 * 1) + (0.2457727 * 2.2)

= 0.1108057 + 0.5407001

= 0.6515058

Therefore, the probability that x is less than 2 is approximately 0.6515 or 65.15%.

Answer 2

To find the probability that x is less than 2, calculate the cumulative probability for x = 0 and x = 1, then subtract from 1.

Step-by-step explanation:

To find the probability that x is less than 2, we need to calculate the cumulative probability for x = 0 and x = 1, and then subtract it from 1.

First, let's find the cumulative probability for x = 0:

P(X ≤ 0) = P(X = 0) = e-2.2 * (2.2)0 / 0! = e-2.2 ∼ 0.111

Next, let's find the cumulative probability for x = 1:

P(X ≤ 1) = P(X = 0) + P(X = 1) = e-2.2 * (2.2)0 / 0! + e-2.2 * (2.2)1 / 1! = e-2.2 + (2.2)e-2.2 ∼ 0.303

Finally, we can calculate the probability that x is less than 2:

P(x < 2) = 1 - P(X ≤ 1) = 1 - 0.303 = 0.697