Question

If the manufacturer will replace bulbs that burn out in less than 620 hours, how many extras should she make?

Short answer

The manufacturer would need to make an extra 8200 bulbs if she were to replace bulbs that burn out in less than 620 hours.

Step-by-step solution

Formula for z

$\mathit{z}\mathbf{=}\frac{\mathbf{x}\mathbf{-}\overline{\mathbf{x}}}{\mathbf{s}}$.

Step 2: The manufacturer plans to sell a million bulbs

The manufacturer gives an offer of replacing any bulb that burns out in less than 620 hours, free of cost.

Use the Gaussian curve from Figure 4-1.

Calculation of the number of bulbs to be replaced:

From figure 4-1, we know that the mean is$\left(\overline{x}\right)=845.2$and the standard deviation is$\left(s\right)=94.2$

Express the desired interval in multiples of the standard deviation and calculate asfollows:

$$\begin{gathered} z=\frac{x-\overline{x}}{s} \\ =\frac{620-845.2}{94.2} \\ =\frac{-225.2}{94.2} \\ =-2.390 \\ =-2.4. \end{gathered}$$

From Table 4-1, the area under the curve between the mean value and $z=-2.4$is$0.4918$.

The entire area from $-\infty$to the mean value is 0.5000. So, the area from $-\infty$to $-2.4$must be$0.500-0.4918=0.0082$.

Determine the number of bulbs

The area to the left of 620 hours in figure $4-1$is only $0.82\%$of the entire curve. Since $0.0082\times 100=0.82\%$,

$0.82\%$of the bulbs are expected to fail in fewer than 620 hours.

Therefore, the number of bulbs required to be replaced from one million bulbs is calculated below.

No. of bulbs required$\frac{0.82}{100}\times 1000000=8200.$

Hence, the number of bulbs needed for the replacement is$8200\mathrm{bulbs}$.