6. The manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips. Approximate the probability that more than 25 chips are defective. [Hint: use normal approximation to binomial

Probability that more than 25 chips are defective is 0.1075.

Step-by-step explanation:

We are given that the manufacturing of semiconductor chips produces 2% defective chips. Assume the chips are independent and that a lot contains 1000 chips.

The above situation can be represented through Binomial distribution;

where, n = number of trials (samples) taken = 1000 chips

            r = number of success = more than 25

           p = probability of success which in our question is % of

                 defective chips, i.e; 2%

LET X = Number of chips that are defective

SO, X ~ Binom(n = 1000, p = 0.02)

Now, here we can't find the probability that more than 25 chips are defective using binomial distribution because the sample size is very large here (n > 30), so we will use Normal approximation to find the respective probability.

So, mean of binomial distribution = E(X) =  

                                                        = = 20

Standard deviation of binomial distribution = S.D.(X) =

                                                                        =  

                                                                        = 4.43

Let Y = Number of chips that are defective for normal approximation;

So, Y ~ Normal()

The z-score probability distribution for normal distribution is given by;

                  Z =  ~ N(0,1)

Now, probability that more than 25 chips are defective is given by = P(Y > 25) = P(Y > 25.5)      --------------{using continuity correction}

        P(Y > 25.5) = P( > ) = P(Z > 1.24) = 1 - P(Z 1.24)

                                                          = 1 - 0.89255 = 0.1075

The above probability is calculated using z table by looking value of  x = 1.24 in z-table.

Therefore, the probability that more than 25 chips are defective is 0.1075.