The time required to finish a test is normally distributed (i.e. in a Gaussian distribution) with a mean of 60 minutes and a standard deviation of 10 minutes. What is the probability that a student will finish the test in less than 70 minutes?
1 Answer
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Consider the Gaussian distribution function:
#f(t) = 1/sqrt(2pisigma^2) e^(-(t-t_"avg")^2//2sigma^2)#
You were given the average time and the standard deviation in the time:
#t_"avg" = "60 minutes"# #sigma = "10 minutes"#
So now we have an explicit expression for
#f(t) = 1/sqrt(200pi) e^(-(t-60)^2//200)#
This function, when integrated over a certain range, will give the area under the curve, and that gives the probability of one student finishing in the allotted time.
In other words,
#P = int_(0)^(t) f(t)dt#
is the probability that one student will finish the test within
Thus, in order to finish the test in less than
#int_(0)^(70) f(t)dt = ???#
This is a non-elementary integral, so you can either use a calculator or Wolfram Alpha. And so:
#1/sqrt(200pi) int_(0)^(70) e^(-(t-60)^2//200)dt#
#= 0.841345#
So, there is about a