Simon, a statistician, has been asked by “Cameras Are Us” to address an issue th

Question

Simon, a statistician, has been asked by “Cameras Are Us” to address an issue the company is having with their warranty system. Following is the company’s email.

Dear Simon

Your friend Jimmy works for us, and said that you might be able to help.

We sell low-priced cameras. As you might know, cameras all eventually brake down, and the last manager said their market research team reported that cameras broke down according to a “bell curve” (I have no idea what this means).

Anyway, I first decided to set a warranty of 650 days and found that we were replacing 20% of cameras. We can’t afford this, so I set the warranty to 550 days and it reduced to 10%. Does that mean
MXB101 Semester 1 2017 Worksheet 6 Page 2 out of 5
that if I set it to 450 days it will be at 0%? That doesn’t sound right, as I know that some of cameras definitely break in fewer than 450 days.

Anyway, I’ve worked out that I can afford to replace 5% of cameras, and while I think this means I need to set the warranty to 500 days, Jimmy said that I should email you as clearly I have no idea what I’m talking about!

So, in your expert opinion, what should I set our warranty to? Thanks. :)

- Cam E. Rah

Simon decides to use the Normal distribution to model the breakage time of these cameras. What should be Simon’s advice to Cameras Are Us?

Explanation / Answer

Let's is the mean time to failure for camera and is the standard deviation of that failure time. As failure time is normally distributed we will use probability values from normal distribution.

so as it is given set a warranty of 650 days and found that we were replacing 20% of cameras

Pr (x < = 650 ; ; ) = 0.20

from Z - table the value of Z for the given probability = -0.84

z = (x - )/

- 0.84 = (650 -  )/

650 = - 0.84 ......(i)

similarly, for  550 days and it reduced to 10%

Pr (x < = 650 ; ; ) = 0.10

from Z - table the value of Z for the given probability = -1.28

z = (x - )/

- 1.28 = (650 -  )/

550 = - 1.28 ....(ii)

so (i) - (ii)

100 = (1.28 - 0.84) = 0.44

= 100/0.44 = 227.27 days

= 550 + 1.28 * 227.27 = 841 days

so we can afford 5% failure rate of cameras

Pr( x< X ; 841; 227.27) = 0.05

so Z - value = - 1.645

(x - 841)/ 227.27 = -1.645

x = - 1.645 * 227.27 + 841 = 467.14 days = 467 days

so he should set his warrenty to 467 days

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