Automobile repair costs continue to rise with the average...

Automobile repair costs continue to rise with the average cost now at $367 per repair. Assume that the cost for an automobile repair is normally distributed with a standard deviation of $88. Answer the following questions about the cost of automobile repairs. (a) What is the probability that the cost will be more than $490? (Round your answer to four decimal places.) (b) What is the probability that the cost will be less than $250? (Round your answer to four decimal places.) (c) What is the probability that the cost will be between $250 and $490? (Round your answer to four decimal places.) (d) If the cost for your car repair is in the lower 5% of automobile repair charges, what is your maximum possible cost in dollars? (Round your answer to the nearest cent.)

Transcript

00:01 For this problem, we're told that the repair cost for automobiles is normally distributed.

00:07 We're told that the mean, which i'll call mu, is given by $367, and the standard deviation, which i'll call sigma, is given by $88.

00:18 So let me say that x is the cost of a randomly selected repair from this population, and we know it's distributed normally with mean 367 and standard deviation 88.

00:31 So i can draw a normal distribution for x as follows, with the mean in the middle here, 367.

00:38 And given this information, we want to answer some questions.

00:42 First, we want the probability that our repair cost will be more than $490, so that's the probability that x is greater than 490.

00:52 In terms of our normal curve, that means we want the area under the curve to the right of 490.

00:57 So if i say that 490 is about here, then we're looking for all the area to the right of that, all the way up to plus infinity.

01:06 There are many ways you could do this using tables, programs, calculators.

01:14 I'm going to use a ti -84 with the normal cdf function.

01:21 This function takes in four inputs and gives you this probability as an output.

01:28 The first input is the lower limit of the probability where the area under the curve starts, that's 490.

01:36 Then the second input is the upper limit of the probability where the area under the curve ends, and for us that's going to be all the way up to infinity since we want all the, everything to the right of 490.

01:51 So we don't have an infinity button on the calculator, so we just have to use a big positive number like 10 to the power 99, which we can write in the calculator as e99.

02:02 Next we need the mean, 367, and the standard deviation, 88.

02:09 So i will now go to my calculator and plug all this in and see what i get.

02:22 Okay, and it looks like this probability is 0 .081124 decimal places.

02:29 So there's our answer for part a.

02:34 Let's look at part b now.

02:38 This time we want the probability that x is going to be less than $250.

02:45 So again, going to our normal curve, that's all the area to the left of 250, off to minus infinity.

02:53 Doesn't make much sense to have negative numbers for a repair cost, but we're just approximating this data using a normal distribution, so it doesn't matter too much.

03:05 Anything in the negatives is going to have a vanishingly low probability anyways, so we can calculate this the same way we did for part a.

03:16 We take our normal cdf function, put in our lower limit, which this time will be minus infinity, so minus e99 on the calculator.

03:26 Our upper limit is 250, and our mean and standard deviation are unchanged, 367 and 88.

03:36 So let me see what this works out to be.

03:42 Okay, i'm getting an answer of 0 .0913...

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