Question

The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. The distribution takes only whole-number values, so it is certainly not normal.

A) Let x-bar be the mean number of accidents at the intersection during a year (52 weeks). What is the approximate probability that x-bar is less than 2?

B) what is the approximate probability that there are fewer than 100 accidents at the intersection in a year? Hint: re-state the events in terms of x-bar

Answer 1

SOLUTION:

From given data,

The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.4. The distribution takes only whole-number values, so it is certainly not normal.

Given,

mean=2.2,

s=1.4

A) Let x-bar be the mean number of accidents at the intersection during a year (52 weeks). What is the approximate probability that x-bar is less than 2?

Let us consider

the approximate distribution of (x-bar) according to the central limit theorem

x bar ~Normal(mean=2.2, s=1.4/√52=0.19)
Now,calculating

P(x bar < 2) = P((x bar - mean) / s < (2-2.2) /0.19)

=P(Z<-1.05)

= 0.1468 (check standard normal table)

B) what is the approximate probability that there are fewer than 100 accidents at the intersection in a year? Hint: re-state the events in terms of x-bar

P(X<100) = P(x bar< 100/52)

=P(x bar<1.92)

=P(Z<(1.92-2.2)/0.19)

=P(Z< -1.47)

=0.0708 (check standard normal table)