Answer to Question #230451 in Statistics and Probability for Caro

Question

A new manufacturing method is supposed to increase the average life span of electronic com-

ponents, while the variance of the life span is expected to stay the same. Using the previous

manufacturing method, the average life span was 112.5 hours with a variance 12 hours. The man-

ufacturer wishes to establish the new average life span by measuring the life spans of a sample of

components manufactured using the new method.

(a) What sample size should be used, if the manufacturer wishes to establish the new average

life span to within 1 hours, with 90% level of confidence? (8)

(b) How will the required sample size change, if the manufacturer wishes to establish the new

average life span to within 1 hours, with 95% level of confidence? (6)

(c) How will the required sample size change, if the manufacturer wishes to establish the new

average life span to within 1/2 hours, with 90% level of confidence? (

Answer 1

(a) The critical value for $\alpha=0.1$ is $z_c=z_{1-\alpha/2}=1.6449.$

$$z_c\times\dfrac{\sigma}{\sqrt{n}}\leq1$$

$$n\geq(\dfrac{z_c\sigma}{1})^2$$

$$n\geq(\dfrac{1.6449(12)}{1})^2$$

$$n\geq390$$

(b) The critical value for $\alpha=0.05$ is $z_c=z_{1-\alpha/2}=1.96.$

$$z_c\times\dfrac{\sigma}{\sqrt{n}}\leq1$$

$$n\geq(\dfrac{z_c\sigma}{1})^2$$

$$n\geq(\dfrac{1.96(12)}{1})^2$$

$$n\geq554$$

(c) The critical value for $\alpha=0.1$ is $z_c=z_{1-\alpha/2}=1.6449.$

$$z_c\times\dfrac{\sigma}{\sqrt{n}}\leq1/2$$

$$n\geq(\dfrac{z_c\sigma}{1/2})^2$$

$$n\geq(\dfrac{1.6449(12)}{1/2})^2$$

$$n\geq1559$$

The required sample size should be increased by 4 times to establish the new average life span to within 1/2 hours, with 90% level of confidence.