Homework Handout for Using the Normal Distribution to Approximate the Normal Distribution

  1. The distributions below are binomial with number of trials n and probability of success p.  Determine which of the distributions can be approximated by the normal distribution.  If it can be approximated by the normal distribution, give the mean and standard deviation of the normal approximation.      Solution
    1. n = 75, p = 0.8
    2. n = 30, p = 0.9
    3. n = 50, p = 0.06


  2. The distributions below are binomial with number of trials n and probability of success p.  Determine which of the distributions can be approximated by the normal distribution.  If it can be approximated by the normal distribution, give the mean and standard deviation of the normal approximation.
    1. n = 100, p = 0.3
    2. n = 20, p = 0.1
    3. n = 24, p = 0.95
    4. n = 22, p = 0.48

  3. For each of the following probability statements involving the binomial distribution and variable r number of successes, write down the corresponding probability statement that uses the normal distribution variable X.  Make sure you use the continuity correction.     Solution
    1.  P(r < 18)
    2.  P(r > 59)
    3.  P(r < 22)
    4.  P(r > 37)
    5. P(11 < r < 19)

  4. For each of the following probability statements involving the binomial distribution and variable r number of successes, write down the corresponding probability statement that uses the normal distribution variable X.  Make sure you use the continuity correction.
    1.  P(r < 25)
    2.  P(r > 17)
    3.  P(r < 80)
    4.  P(r > 35)
    5. P(35 < r < 45)

  5. Each of the probability statements involve the binomial distribution.  Use the continuity correction to write a probability statement that uses the normal distribution.
    1. The probability that the more than 29 of the students woke up before 8:00 AM.     Solution
    2. The probability that at least 22 of the cars are hybrid vehicles.
    3. The probability that fewer than 41 of the customers returned the next week.
    4. The probability that at most 17 of the women wear makeup.
    5. The probability that between 14 and 26 of the cats fall on their feet.


  6. Each of the probability statements involve the binomial distribution.  Use the continuity correction to write a probability statement that uses the normal distribution.
    1. The probability that the number of yes votes is more that 18.
    2. The probability that the number of people with blue eyes is at least 22.
    3. The probability that fewer than 50 of the rodents survive.
    4. The probability that at most 32 of the penguins jump into the water.
    5. The probability that between 21 and 28 of the students pass the exam.

For exercises 7 through 9, use the normal distribution to approximate the binomial distribution.

  1. Suppose that 20% of all college students are vegetarians.  If 80 students are randomly selected, what is the probability that
    1. fewer than 13 of them are vegetarians?
    2. more than 14 of them are vegetarians?
    3. at least 20 of them are vegetarians?
    4. at most 17 of them are vegetarians?
    5. Between 13 and 16 of them are vegetarians?

        Explain why it was ok to use the normal approximation to the binomial distribution for these calculations.     Solution

  1. Suppose that 84% of all college students travel during winter break.  If 45 students are randomly selected, what is the probability that
    1. fewer than 40 of them travel during winter break?
    2. more than 35 of them travel during winter break?
    3. at least 38 of them travel during winter break?
    4. at most 32 of them travel during winter break?
    5. Between 37 and 41 of them travel during winter break?

    2.         Explain why it was ok to use the normal approximation to the binomial distribution for these calculations.

  2. Only 7% of all people who receive CPR survive.  If a researcher randomly studied 120 cases of people who received CPR, what is the probability that
    1. Less than 12 of them survived.
    2. more than 7 of them survived.
    3. at least 10 of them survived.
    4. at most 9 of them survived.
    5. Between 8 and 13 of them survived.

        Explain why is was ok to use the normal approximation to the binomial distribution for these calculations.