In each of the following questions, use the normal approximation to the binomial distribution.
1. Find the approximate probability that, in the next 50 tosses of a fair coin, the proportion of heads observed will be less than or equal to 0.46.
2. In a large city. 12% of the workforces are unemployed. If 300 people from the workforce are selected at random; find the approximate probability that more than 10% or the people surveyed are unemployed.
3. It is known that on average 50% of the children born at a particular hospital are female. Find the approximate probability that more than 60% of the next 25 children born at that hospital will be female.
4. A car manufacturers expects 10% of cars produced to require minor adjustments before they are certified as ready for sale. What is the approximate probability that more than 15% of the next 200 cars inspected will require minor adjustments?
5. Past records show that on average 30% of the workers at a particular company have had one or more accidents in the workplace. What is the approximate probability that less than 20% of a random sample of SO workers have had one or more accidents?
6. Sacha is shooting at a target which she has a probability of 0.6 of hitting. What is the approximate probability that:
a. The proportion of times she hits the target in her next 100 attempts is less than 0.8
b. The proportion or times she hits the target in her next 1oo attempts is between 0.6 and 0.8
c. The proportion of times she hits the target in her next 100 attempts is between 0.7 and 0.8. Given that it is more than 0.6?
7. Find the approximate probability that, in the next 100 losses or a fair coin, the proportion or heads will be between 0.4 and 0.6.
8. A machine has a probability of 0.1 of producing a defective item.
a. What is the approximate probability that, in the next batch or 1000 items produced? The proportion of defective items will be between 0.08 and 0.12?
b. What is the approximate probability that. In the next batch of 1000 items produced. The proportion of defective items will be between 0.o8 and 0.12. Given that we know that it is greater than 0. 10?
9. The proportion or voters in the population who favour Candidate A is 52%. Or a random sample or 400 voters, 230 indicated that (hey would vote for Candidate A at the next election.
a. what is the value of the sample proportion. p?
b. Find the approximate probability that, in a random sample of 400 voters, the proportion who favour Candidate A is greater than or equal to the value of p observed in (his particular sample.
10. A manufacturer claims that 90% of (heir batteries will last more than 100 hours. Of a random sample of 250 batteries. 212 lasted more than 100 hours.
a. What is the value of the sample proportion. p?
b. Find the approximate probability that. In a random sample of 250 batteries, the proportion lasting more than 100 hours is less than or equal to the value of /observed in this particular sample.
c. Does your answer to b cause you to doubt the manufacturer's claim?