Approximate each of these binomial distributions with the normal distribution.
A continuity correction will be needed with each question.
1. A seed researcher finds that 7% of a certain type of seed type fails to germinate. If 600 seeds are planted, find the probability that at least 40 will fail.
2. It has been shown that 70% of students in the whole of New South Wales in Australia pass a test. If a random sample of 90 students sitting the test is chosen, what is the probability that more than 71 will pass the test?
3. A manufacturer of batteries knows that 4% of the batteries made will be faulty before being checked. If 900 batteries are tested calculate the probability that more than 40 will be faulty and need replacing.
4. A factory produces a large number of appliances, which are supposed to meet given specifications. A proportion, π, of the appliances are faulty because they do not meet these specifications.
A random sample of n appliances is taken, and each appliance is inspected. Let the random variable X be the number of faulty appliances in the sample.
Suppose that n = 25 and π = 0.3
a. What distribution will approximate the distribution of X? (Give the name of the distribution and its parameter(s)). Justify your choice of distribution.
b. Use this approximation to find the probability that the sample contains fewer than two faulty appliances.
