Normal Probability Distributions

  1. Find all values of k so that the following is a probability distribution:

x / p(x)
1 / 0.15
2 / 2k
3 / 0.52
4 / k
  1. Find the mean and standard deviation of the following probability distribution:

x / p(x)
1 / 0.3
2 / 0.5
3 / 0.2
  1. An arsenal contains several identical boxes of ammunition. If the number of defective bullets per box has the following distribution, find the mean and standard deviation for x.

x / 0 / 1 / 2
p(x) / 0.90 / 0.07 / 0.03
  1. Thirty percent of hospital admissions for diabetic patients are related to problems with the kidneys. In a sample of 10 diabetic hospital admissions, what is the probability that none will be for a kidney problem?
  1. A binomial distribution is based on trials and success probability. What is the probability that the binomial random variable equals its mean value?
  1. Find the mean and standard deviation of the binomial distribution when n = 60 and p = 1/6. Note that this would correspond to the number of times a “one” would appear in 60 tosses of a fair die.
  1. Given the probability function P(x) = , for x = 1, 2, 3, 4, or 5. Find the mean and standard deviation.

In questions 8. to 11., the random variable z has a standard normal distribution

  1. Find P(z0.63).
  2. Find P(1.21 < z < 1.37).
  3. Find P(0.31 < z < 1.31).
  4. Find P(z1.61).
  1. A traffic study at one point on an interstate highway shows that vehicle speeds are normally distributed with a mean of 61.3 mph and a standard deviation of 3.3 mph. If a vehicle is randomly checked, find the probability that its speed is between 55.0 mph and 60.0 mph.
  1. Reading comprehension scores for junior high students in a school district are normally distributed with a mean of 80.0 and a standard deviation 5.0.
  1. What percent have scores greater than 87.5?
  2. What percent have scores between 75 and 85?

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Normal Probability Distributions

  1. X has a normal distribution with a mean of 75.0 and a standard deviation of 2.5. Find the following probabilities:
  1. P(X < 70.0)
  2. P(72.5 <X < 80.0)
  3. P(X > 82.5)
  1. Waiting times to see a doctor at a large clinic are normally distributed with a mean of 68.2 minutes and a standard deviation of 14.8 minutes. Find the probability that the waiting time to see a doctor is less than 45.0 minutes.
  1. Consider a binomial distribution with 15 identical trials, and probability of success of 0.5.
  1. Find the probability that x = 2 using the binomial tables.
  2. Use the normal approximation to find the probability that x = 2.
  1. Consider a binomial distribution with 12 identical trials, probability of success of 0.3.
  1. Find the probability that x = 3 using the binomial tables.
  2. Use the normal approximation to find the probability that x = 3.
  1. A machine cuts circular filters from large rolls of material. If 7.3% of the filters fail to meet specifications, use the normal approximation to the binomial to compute the probability that a sample of 100 of the filters will contain 5 or fewer that fail to meet specifications.
  1. If 25% of all students entering a certain university drop out during or at the end of their first year, what is the probability that more than 550 of this year’s entering class of 2000 will drop out during or at the end of their first year?
  1. A test-scoring machine is known to record an incorrect grade on 5% of the exams it grades. Find, by the appropriate method, the probability that the machine records
  1. 2 wrong grades in a set of 10 exams.
  2. no more than 2 wrong grades in a set of 10 exams.
  3. no more than 2 wrong grades in a set of 15 exams.
  4. no more than 2 wrong grades in a set of 150 exams.

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