Question

Problem 3. A quality-control engineer wants to check whether (in accordance with specifications) 95% of the concrete beams shipped by his company pass the strength test (i.e., the strength is greater or equal to 32). To this end, he randomly selects a sample of 20 beams from each large lot ready to be shipped and passes the lot if at most one of the 20 selected beams fails the test; otherwise, each of the beams in the lot is checked. Let the random variable X be the number of selected beams that pass the test.
1. Find the probability that all 20 selected beams pass the test.
2. Find the probability that 2 beams in the sample fail the test.
3. Find the probability that between 17 to 19 beams in the sample pass the test.
4. Find the probabilities that the quality-control engineer will commit the error of holding a lot for further inspection even though 95% of the beams strength is greater or equal to 32 (in accordance with specifications).

Hint: The quality-control engineer hold a lot if 2 or more beams in the sample fail the test.

Answer 1

We will be using binomial distribution to find the probabilities

Answer 1)

Here, we have to find P(X = 20) that is probability all 20 selected beams pass the test.

The probability that all 20 selected beams pass the test is 0.3585

Answer 2)

Exactly two beam fail the test. So 18 beam will past the test.

Here, we have to find P(X = 18) that is probability that all 20 selected beams pass the test.

The probability that 2 beams in the sample fail the test 0.1887

Answer 3)

Here, we have to find P(17 < X < 19) the probability that between 17 to 19 beams in the sample pass the test.

The probability that between 17 to 19 beams in the sample pass the test is 0.6256

Answer 4)

The quality-control engineer hold a lot if 2 or more beams in the sample fail the test. So we have to find P(X > 2)

P(X > 2) = 1 - P(X < 1)

P(X>2) = 1 - 0.7358

P(X > 2) = 0.2642

The probabilities that the quality-control engineer will commit the error of holding a lot for further inspection even though 95% of the beams strength is greater or equal to 32 is 0.2642