Question

Michael, has been asked by “Custom Calculators” to address an issue
the company is having with their warranty system. The company’s email to
Michael. "Dear Michael,
we produce and sell calculators. So, we know from market research that a calculator made by our factory will stop working on average after 3000 days, according to a bell-curve (that’s what the market research person told us, but we don’t understand what he means). Anyway, we decided to set our warranty at 2000 days (so that anyone who bought a calculator from us could have it replaced provided it was bought no later than 2000 days ago), but we’ve found that we’re replacing 10% of calculators we sell. We can’t really afford this, and have only budgeted to replace 3% of calculators we sell.
Can you tell us what we should change our warranty time to so that we only have to replace 3%

What should Michael’s advice be?

Answer 1

Answer on Question #42077 – Math - Statistics and Probability

By the statement of this question, if $P(\xi < 2000) = 0.1$, $E(\xi) = 3000$, then

$P(3000 + b\eta < 2000) = 0.1$, $b$ is standard deviation of $\xi$ (it is unknown, it can be found from assumptions of this assignment). Rewrite the last equality

$$P \left(\eta < \frac {2000 - 3000}{b}\right) = 0.1, \text{ it means}$$

$F_{\eta}\left(-\frac{1000}{b}\right) = 0.1$, where $F_{\eta}$ is the standard normal cumulative distribution function.

Take the argument corresponding to a value of 0.1 of the standard normal cumulative distribution function from corresponding tables. We define

$$- \frac {1000}{b} \approx -1.28 \quad \text{(more exactly} \quad - \frac {1000}{b} = \text{NORMSINV}(0,1) = -1.28155 \text{ via an Excel function)}.$$ $$\begin{array}{l} \text{So, } - \frac {1000}{b} = -1.28155, \text{ from where we conclude} \\ b = \frac {1000}{1.28155} = 780.3051. \end{array}$$

Rewrite

$$P (\xi < 2000 + x) = 0.03 \text{ and obtain } P (3000 + b\eta < 2000 + x) = 0.03 \text{ or } P \left(\eta < \frac {x - 1000}{b}\right) = 0.03.$$

In similar way,

$$\begin{array}{l} \frac {x - 1000}{b} = -1.88079, \text{ whence} \\ x = -1.88079 * b + 1000 = -1.88079 * 780.3051 + 1000 = -467.59. \end{array}$$

Finally, we set a new value $2000 + x = 2000 - 467.59 = 1532.41$

Answer: 1532.41

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