Suppose you want to eat lunch at a popular restaurant the...

Suppose you want to eat lunch at a popular restaurant. The restaurant does not take reservations, so there is usually a waiting time before you can be seated. Let x represent the length of time waiting to be seated. From past experience, you know that the mean waiting time is μ = 18.4 minutes with σ = 4.4 minutes. You assume that the x distribution is approximately normal. (Round your answers to four decimal places.) (a) What is the probability that the waiting time will exceed 20 minutes, given that it has exceeded 15 minutes? Hint: Compute P(x > 20|x > 15). (b) What is the probability that the waiting time will exceed 25 minutes, given that it has exceeded 18 minutes? Hint: Compute P(x > 25|x > 18).

Transcript

00:01 So for this problem, i'm first just going to note that i'll be using the notation, phi of x, which is equal to the cumulative probability distribution function for a standard normal distribution.

00:12 So that is the probability that z is less than or equal to x, where you can calculate using excel or using a table of values, something along those lines.

00:23 Now, for part a, we are looking for the probability that x is greater than 20, given that x is greater than 15.

00:34 We can calculate that by taking the probability that x is greater than 20 and dividing it by the probability that x is greater than 15.

00:44 Using the typical basian conditional probability formula, x greater than 20 means that x would also be greater than 15, but typically when we're doing a given b, we do this as probability of a and b, divided by probability of b that being said we can find probability that x is greater than 20 by taking one minus phi evaluated at the z score corresponding to 20 which would be 20 minus the mean value 18 .4 divided by the standard deviation of 4 .4 divide that by 1 minus phi evaluated at 15 minus 18 .4 divided by 4 .4, which in turn would be 1 minus phi at, let me double check my calculations off screen here.

01:47 Okay.

01:47 So for 20, that would give us about 0 point, or the z score, corresponding would be 0 .364.

01:54 Then we're dividing by 1 minus phi evaluated at negative 0 .773, which will give us a result of for the upper value that's going to be 0 .358 for the lower value oh pardon me uh double checking okay yes that does make sense sorry for the lower value we should have 0 .78 so we get that our probability comes out to about 0 .5 4 5 9 when rounded to 3 decimal places or actually you're asked for 4 so rounded to 4 decimal places it would actually round to 0 .5 then for part b we're doing a very similar process probability of x greater than 25 given x greater than 18 i'll go directly to the statement form where we've calculated out the corresponding z scores but we would just do z score for 25 in the top and z score for 18 in the bottom so we'd have one minus oh, one second here...

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