Problem 4 Suppose that 44% of all drivers stop at an intersection having flashin

Question

Problem 4 Suppose that 44% of all drivers stop at an intersection having flashing red lights when no other cars are visible. Of 360 randomly selected drivers coming to an intersection under these conditions, let X be the number of those who stop. Suppose we are interested in finding P(105 sX 3 165) What is the exact distribution of X? Remember to provide the name of the exact distribution and the value(s) of the parameter(s) b) Use the exact distribution of X to compute the exact probability P(105 s X S 165) in R Now let's see how well the Normal approximation to the Binomial works. c) Check the appropriate condition(s) for the approximation. d) Compute the mean and the standard deviation of the approximate distribution of X Now calculate the approximate probability P(105 X 165) using R (remember to use the continuity correction factor of 0.5!) Finally, let's compare the approximate probability to the exact one above Comment on how well/poorly the Normal approximation to the Binomial works here, i.e compare the result of e) to the exact (true) probability value from part b) 1) Let's also see if there is any benefit from using the continuity correction in the approximation process g) Perform the approximation of P(105 X S 165) without the continuity correction step, i.e. apply the normal approximation to "uncorrected" original probability. And compare the result to the exact value from part b). Comment on whether the use of continuity correction improves the Normal approximation to Binomial.

Explanation / Answer

a) X is binomial distribution, the parameters n = 360, p = 0.44
b) Using Binomial distribution

c)
Condition: np = 360*0.44 = 158.4 which is greater than 5
and nq = 360*0.56 = 201.6 which is greater than 5
thus, normal approximation conditions are satiesfied

d) Mean = np = 360*0.44 = 158.4, SD = sqrt(npq) = sqrt(360,0.44*0.56) = 9.4183

e) continuity correction

f) Part (b) and Part(e) results are very close to equal

g) without continuity correction

Part (b) and Part(g) results are different

> pbinom(165,360,0.44)-pbinom(104,360,0.44)
[1] 0.774785

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