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Binomial Distributions, Geometric Distributions, and Sampling Distributions Free Response Practice Problems for AP Statistics

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By — McGraw-Hill Professional
Updated on Feb 4, 2011

Review the following concepts if necessary:

Problems

  1. A factory manufacturing tennis balls determines that the probability that a single can of three balls will contain at least one defective ball is 0.025. What is the probability that a case of 48 cans will contain at least two cans with a defective ball?
  2. A population is highly skewed to the left. Describe the shape of the sampling distribution of drawn from this population if the sample size is (a) 3 or (b) 30.
  3. Suppose you had gobs of time on your hands and decided to flip a fair coin 1,000,000 times and note whether each flip was a head or a tail. Let X be the count of heads. What is the probability that there are at least 1000 more heads than tails? (Note: this is a binomial but your calculator will not be able to do the binomial computation because the numbers are too large for it).
  4. In Chapter 9, we had an example in which we asked if it would change the proportion of girls in the population (assumed to be 0.5) if families continued to have children until they had a girl and then they stopped. That problem was to be done by simulation. How could you use what you know about the geometric distribution to answer this same question?
  5. At a school better known for football than academics (a school its football team can be proud of ), it is known that only 20% of the scholarship athletes graduate within 5 years. The school is able to give 55 scholarships for football. What are the expected mean and standard deviation of the number of graduates for a group of 55 scholarship athletes?
  6. Consider a population consisting of the numbers 2, 4, 5, and 7. List all possible samples of size two from this population and compute the mean and standard deviation of the sampling distribution of . Compare this with the values obtained by relevant formulas for the sampling distribution of . Note that the sample size is large relative to the population—this may affect how you compute σ by formula.
  7. Approximately 10% of the population of the United States is known to have blood type B. If this is correct, what is the probability that between 11% and 15%, inclusive, of a random sample of 500 adults will have type B blood?
  8. Which of the following is/are true of the central limit theorem? (More than one answer might be true.)
    1. μ = μ.
    2. σ = (if N ≥10n).
    3. The sampling distribution of a sample mean will be approximately normally distributed for sufficiently large samples, regardless of the shape of the original population.
    4. The sampling distribution of a sample mean will be normally distributed if the population from which the samples are drawn is normal.
  9. A brake inspection station reports that 15% of all cars tested have brakes in need of replacement pads. For a sample of 20 cars that come to the inspection station,
    1. what is the probability that exactly 3 have defective breaks?
    2. what is the mean and standard deviation of the number of cars that need replacement pads?
  10. A tire manufacturer claims that his tires will last 40,000 miles with a standard deviation of 5000 miles.
    1. Assuming that the claim is true, describe the sampling distribution of the mean lifetime of a random sample of 160 tires. Remember that "describe" means discuss center, spread, and shape.
    2. What is the probability that the mean life time of the sample of 160 tires will be less than 39,000 miles? Interpret the probability in terms of the truth of the manufacturer's claim.
  11. The probability of winning a bet on red in roulette is 0.474. The binomial probability of winning money if you play 10 games is 0.31 and drops to 0.27 if you play 100 games. Use a normal approximation to the binomial to estimate your probability of coming out ahead (that is, winning more than 1/2 of your bets) if you play 1000 times. Justify being able to use a normal approximation for this situation.
  12. Crabs off the coast of Northern California have a mean weight of 2 lbs with a standard deviation of 5 oz. A large trap captures 35 crabs.
    1. Describe the sampling distribution for the average weight of a random sample of 35 crabs taken from this population.
    2. What would the mean weight of a sample of 35 crabs have to be in order to be in the top 10% of all such samples?
  13. The probability that a person recovers from a particular type of cancer operation is 0.7. Suppose 8 people have the operation. What is the probability that
    1. exactly 5 recover?
    2. they all recover?
    3. at least one of them recovers?
  14. A certain type of light bulb is advertised to have an average life of 1200 hours. If, in fact, light bulbs of this type only average 1185 hours with a standard deviation of 80 hours, what is the probability that a sample of 100 bulbs will have an average life of at least 1200 hours?
  15. Your task is to explain to your friend Gretchen, who knows virtually nothing (and cares even less) about statistics, just what the sampling distribution of the mean is. Explain the idea of a sampling distribution in such a way that even Gretchen, if she pays attention, will understand.
  16. Consider the distribution shown at the right. Describe the shape of the sampling distribution of for samples of size n if
    1. n = 3.
    2. n = 40.
  17. After the Challenger disaster of 1986, it was discovered that the explosion was caused by defective O-rings. The probability that a single O-ring was defective and would fail (with catastrophic consequences) was 0.003 and there were 12 of them (6 outer and 6 inner). What was the probability that at least one of the O-rings would fail (as it actually did)?
  18. Your favorite cereal has a little prize in each box. There are 5 such prizes. Each box is equally likely to contain any one of the prizes. So far, you have been able to collect 2 of the prizes. What is:
    1. the probability that you will get the third different prize on the next box you buy?
    2. the probability that it will take three more boxes to get the next prize?
    3. the average number of boxes you will have to buy before getting the third prize?
  19. We wish to approximate the binomial distribution B(40, 0.8) with a normal curve N(μ, σ). Is this an appropriate approximation and, if so, what are μ and σ for the approximating normal curve?
  20. Opinion polls in 2002 showed that about 70% of the population had a favorable opinion of President Bush. That same year, a simple random sample of 600 adults living in the San Francisco Bay Area showed only 65% had a favorable opinion of President Bush. What is the probability of getting a rating of 65% or less in a random sample of this size if the true proportion in the population was 0.70?
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