Exercises - Binomial Approximations to Normal Distributions

  1. For each, find the missing value using an appropriate continuity correction.

    1. $P_{\textrm{binomial}}(x \ge 3) = P_{\textrm{normal}}(x \gt ?)$

    2. $P_{\textrm{binomial}}(x \le 3) = P_{\textrm{normal}}(x \lt ?)$

    3. $P_{\textrm{binomial}}(x \gt 3) = P_{\textrm{normal}}(x \gt ?)$

    4. $P_{\textrm{binomial}}(x \lt 3) = P_{\textrm{normal}}(x \lt ?)$

  2. Use the normal approximation to the binomial with $n = 30$ and $p = 0.5$ to find the probability $P(X = 18)$.

  3. Use the normal approximation to the binomial with $n = 10$ and $p = 0.5$ to find the probability $P(X \ge 7)$.

  4. Use the normal approximation to the binomial with $n = 50$ and $p = 0.6$ to find the probability $P(X \le 40)$.

  5. According to recent surveys, 53% of households have personal computers. If a random sample of 175 households is selected, what is the probability that more than 75 but fewer than 110 have a personal computer?

  6. According to Mars, 24% of M&M plain candies are blue. In a given sample of 100 M&Ms, 27 are found to be blue. Assuming that the claimed rate of 24% is correct, find the probability of randomly selecting 100 M&Ms and getting 27 or more that are blue. Based on the result, is 27 (out of 100) an unusually high number of blue M&Ms?

  7. A Boeing 767-300 aircraft has 213 seats. When someone buys a ticket for a flight there is a 0.0995 probability that the person will not show up for the flight. A ticket agent accepts 236 reservations for a flight that uses a Boeing 767-300. Find the probability that not enough seats will be available. Is this probability low enough so that overbooking is not a real concern? If not, how many tickets should be sold so that the probability is less than 10% that at least one person will not have a seat?

  8. Let $X$ be the random variable that represents a count of the number of heads showing when a coin is tossed 12 times. Find the following binomial probabilities exactly, and then compute their corresponding normal approximations: (a) P(at least 8 heads), (b) P(exactly 5 heads), (c) P(at most 5 heads)

  9. Toss a coin 16 times. Let X be the number of heads that appear. Find the probability that there will be more than 13 heads. (a) Work as a binomial. (b) Use the standard normal to approximate the probability. (c) Was the approximation reasonable? Explain.

  10. Assume that the probability of a college student having a car on campus is .30. A random sample of 12 students is taken. What is the probability that at least 4 will have a car on campus? (a) Work the problem as a binomial. (b) Approximate the probability using the standard normal. (c) Is the approximation reasonable? Explain clearly.