Question

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 5%. suppose that a batch of 250 boards has been received and that the condition of any particular board is independent of that of any other board.

a. what is the approximate probability that at least 10% of the boards in the batch are defective?
b. what is the approximate probability that there are exactly 10 defectives in the batch?

Answer 1

The probability that at least 10% of the boards are defective can be approximated using a normal distribution, while the exact probability of having 10 defectives in the batch can be calculated using the binomial formula. For large samples, normal approximation can be used for convenience.

Step-by-step explanation:

To find the probability that at least 10% of the boards are defective, we can use the binomial distribution since each board's condition is independent of the other boards. The formula for binomial probability is P(X = k) = (n choose k) * pk * (1-p)(n-k), where n is the number of trials, p is the probability of success on each trial, and k is the number of successes. However, for large sample sizes and when the sample proportion is close to the population proportion, we can approximate the binomial distribution with a normal distribution.

To use the normal approximation, we calculate the mean and standard deviation with the formulas μ = n * p and σ = √(n * p * (1-p)). For the batch of 250 boards with 5% defective rate, μ = 250 * 0.05 = 12.5 and σ = √(250 * 0.05 * 0.95) ≈ 3.4641. We then convert the problem into a z-score and use standard normal distribution tables or software to find the probability that Z > (25 - 12.5)/3.4641.

To find the exact probability of having exactly 10 defectives in the batch, we use the binomial formula since the normal approximation is less accurate for exact probabilities. The calculation would be P(X = 10) = (250 choose 10) * 0.0510 * 0.95240.

Answer 2

The probability that at least 10% of the boards are defective is: 7.35% The probability of exactly 10 units being defective is: 9.634% With 250 samples the distribution of percent of boards failed can be approximated as a normal distibution. The standard deviation of binomial random variable is sqrt(n*p*(1-p)) where p is the probability of failure. This gives a standard deviation of 3.44601 and 10% deviates from the 5% mean by 1.45 standard deviations. Looking this up on a standard deviation chart shows that 7.35% of 250 unit samples will exceed or meet a 10% failure rate The formula for the probabilty of a certain number of sucesses occuring is (n choose k)*(p^n)*((1-p)^(n-k)) where p is the probability of success n is the number of trials n choose k is the best way i have of writing the binomial coefficient and k is the number of successes the answer is found by letting p=.95 n=250 k=240