using minitab to do the assignment

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Instructions:

  • In this assignment we will explore using Minitab for probability calculations and distributions. You may discuss this assignment in general terms with other students, but the work you hand in should be your own.
  • Presentation of solutions is important. 10% of the marks for the assignment will be rewarded for the way your solution and thought process are presented in the file you submit.
  • For your convenience, some relevant Minitab commands are listed at the end of the assignment. I welcome questions about certain commands on the discussion board labeled Minitab.

Question 1 – Binomial Distribution (20 points)

To protect its cash-low Southwest Airlines overbook for flights. It particular, it sells 300 tickets for the Kansas City – Denver flight when there are only 270 seats available. On average, we expect 15% of those with tickets to this flight to not show up. Based on this information answer the following, use Minitab for probability calculations.

    1. a) Let X = the number of people with a ticked who show up for the Kansas-City – Denver flight. What is the distribution of X? Explain and specify values of the parameters of the distribution.
    1. b) What is the probability that there will be enough seats for everyone who show up for the Kansas- City – Denver flight? Explain.
    1. c) What should be the capacity of the Kansas-City – Denver flight to guarantee that there is at least 0.999 chance that everyone who show up for the flight will have a seat?

Question 2 – Geometric Distribution (20 points)

In class we explored 2 random variables (and distributions) that are based on Bernoulli trials. In this question you will learn about one more important distribution – the Geometric Distribution. Suppose we toss a fair die until we roll a ‘6’ and stop. Let X = the number of times we tossed the die until we rolled a ‘6’ for the first time.

  1. a) What is the Bernoulli trial in the above scenario? What outcome is considered ‘success’? What is the probability of success p?
  2. b) The distribution of X is known to be Geometric(p) where p is the probability of success you found in part (a). Use Minitab to find the probability that we will need to toss the die exactly 10 times until we roll a ‘6’ for the first time, that is, find P(X = 10).
  3. c) Use Minitab to find the probability that we would need to toss the die at least 5 times until we roll the first ‘6’.
  4. d) Use Minitab to find the probability that you will need to toss the die at least 4 times but no more than 10 times until you roll the first 6.

Question 3 – Poisson Distribution (20 points)

Another important random variable (and distribution) that is used to model the occurrence of ‘special event’ is the Poisson distribution. The parameter of the Poisson distribution, λ, is the average rate of occurrence of this ‘special event’ and is also the mean and variance of this distribution.
Suppose that the number of snow storm per year in Colombia MO is known to follow a Poisson distribution with a rate of λ = 2 per year.

  1. a) Use Minitab to calculate the probability that there will be no snow storm in Colombia next year.
  2. b) Use Minitab to calculate the probability that there will be at least 1 snow storm next year.
  3. c) How are your answers to part (a) and (b) above are related to each other.
  4. d) Use Minitab to calculate the probability that there will be no more than 4 snow storm next year.

Some useful Minitab Commands

  • To calculate exact probabilities such as P(X = 3) we would use: Calc > Probability Distribution then choose the distribution from the menu (e.g., Binomial or Geometric or Poisson), select Probability (at the top) and in the input constant tab enter 3 (the value whose probability you are calculating).
  • For calculations involve Binomial distribution we would need to specify the ‘Number of Trials’ = n and ‘Event Probability’ = p. For calculation involves Geometric distribution we would need to specify and ‘Event Probability’ = p and for calculating related to Poisson distribution we would need to specify the ‘Mean’ = λ.
  • To calculate cumulative probabilities such as P(X ≤ 3) we would use: Calc > Probability Distribution then choose the distribution from the menu, select Cumulative Probability (at the top) and in the input constant tab enter 3 (the value whose cumulative probability you are calculating).
  • To find the value k such that P(X ≤ k) ≥ 0.9 we use: Calc > Probability Distribution then choose the distribution from the menu, select Inverse Cumulative Probability (at the top) and in the input constant tab enter 0.9 (the desired probability).