Question

The life span of 100 W light bulbs manufactured by a company are tested. It is found that (8 %) of the light bulbs are rejected. A random sample of 15 bulbs is taken from stock and tested. The random variable X is the number of bulbs that is rejected.

a. Give four reasons why ? will have a binomial distribution.

b. Use a formula to find the probability that 2 light bulbs in the sample are rejected.

c. If the true probability of a rejected light bulb is 0.5340. Among the next 6 randomly selected bulbs, what is the probability that at least one of them is accepted?

d. If life span of light bulbs is adjusted so that the mean now is ?.

Find the value of ?:

i. Given that ?(?=0)=0.25.

ii. Given that the variance of ? is 2.4.

Answer 1

Answer:-

Given that:-

Let X: No.of bulbs that is rejecetd .

P=prob .of rejecting bulb=0.06, q=1-p= 0.94

i) Here, there are only two possible outcomes i.e bulb is either rejected or accepted.

ii)p=prob of success.

iii) Each trial is independent trials.

Here

Its pmf is

otherwise

b) prob that 2 light bulbs in the sample are rejected is

c) If and ,

otherwise

prob that atleast one is accepeted =prob that zero bulbs are rejected

d) &  

we know that

Now,  

ii) &  

Again  

can not be determined.

Also , for binomial distribution mean > variance

Here mean is gives to be p which lies between 0 and 1 & variance is given to be 2.4 i.e mean < variance which is not true hence we can not determine p.