Question

One company produces movie trailers with mean 130 seconds and standard deviation 30 seconds, while a second company produces trailers with mean 110 seconds and standard deviation 20 seconds. Assume these trailers vary normally. What is the probability that two randomly selected trailers, one from each company, will combine to less than 250 seconds?

Question 6 options:

	.46
	.54
	.31
	.61

Answer 1

Given that,

$X \sim N(\mu_1 = 130, \sigma_1 = 30)$

$Y \sim N(\mu_2 = 110, \sigma_2 = 20)$

Therefore,

$X+Y \sim N(\mu_1 + \mu_2 , \sqrt {\sigma_1^2+ \sigma_2^2 })$

$\mu_1+ \mu_2 = 130 +110 = 240$

$\sqrt { \sigma_1^2 + \sigma_2^2} = \sqrt { (30)^2 +(20)^2} = \sqrt { 900 + 400}= \sqrt {1300}= 36.0555$

$X+Y \sim N(\mu_1 + \mu_2 = 240 , \sqrt {\sigma_1^2+ \sigma_2^2 } = 36.0555)$

We want to find, the probability that two randomly selected trailers, one from each company, will combine to less than 250 seconds. That is to find, P(X + Y < 250)

$P(X &plus; Y < 250)$

$P(\frac {(X &plus; Y) -(\mu_1 &plus; \mu_2) }{(\sqrt {\sigma_1^2 &plus; \sigma_2^2)}} < \frac {250-240}{36.0555}) = P(Z < 0.28)= 0.6103$

$P(X &plus;Y < 250) = 0.6103 \approx 0.61$

Answer: 0.61