Answer to Question #97590 in Statistics and Probability for mani

Question

19. The manager of Galaxy electric bulb manufacturing company wants to analyze the company’s performance with the help of normal distribution. He has the following data collected from the records: Mean life of electric bulbs manufactured by a firm is 1200 hrs. The standard deviation is 200 hrs. He wants to analyse:
(i) In a lot of 10,000 bulbs, how many bulbs are expected have life 1050 hrs. or more?
(ii) What is the percentage of bulbs which are expected to fuse before 1500 hrs. of service?
OR

Answer 1

Let $X=$ life of electric bulb in hours: $X\sim N(\mu, \sigma^2).$

Then

$$Z={X-\mu \over\sigma/\sqrt{n}}\sim N(0,1)$$

(i)

Given that $\mu=1200\ hrs, \sigma=200\ hrs, n=10000$

$$P(X\geq1050)=1-P(X<1050)=$$

$$=1-P(Z<{1050-1200 \over200/\sqrt{10000}})=1-P(Z<-75)\approx1$$

Approximately all 10,000 bulbs are expected have life 1050 hrs. or more.

(ii)

$$P(X<1500)=P(Z<{1500-1200 \over200/\sqrt{10000}})=$$

$$=P(Z<150)\approx1$$

Approximately 100 percentage of bulbs are expected to fuse before 1500 hrs. of service.