QMB 3200 – Homework #2

Instructions:

1. Solve all the problems. Each problem carries 10 points. Maximum score possible for this Homework is 100 points.

• Presenting only the final answer is not sufficient to get complete credit. Show the steps in solution approach. That way partial credit can be earned to various steps in final solution.  It is your responsibility to demonstrate mastery of the subject matter through your answers.

• Submit your report as an Excel file. Solve each problem on a separate tab (worksheet). Your Excel file must show Excel functions and formulas in cells. Word document submissions and Pasting images into Excel file will not be acceptable submissions.  Organize your solutions on the Excel worksheet properly. Show where your answers are for each problem and the sections of the problem. Use Proper formatting. Name Your File to show your Full Name and the HW Number. Upload your report file on Canvas and verify if everything is fine by opening up the uploaded file. It is your responsibility to ensure your report is uploaded properly.

• Do not wait until the last minute. The deadline is strictly enforced by Canvas. No hardcopy submissions are accepted. No e-mail submissions are accepted. If your file does not appear on Canvas by the deadline, zero points will be recorded for you for that HW. No exceptions are entertained for any reason under any circumstance in this regard.

HW Problems:

1. The number of students taking the Scholastic Aptitude Test (SAT) has risen to an all-time high of more than 1.5 million. Students are allowed to repeat the test in hopes of improving the score that is sent to college and university admission offices. The number of times the SAT was taken and the number of students are as follows.

Number of Times SAT is taken            Number of Students

1                                                  721,769

2                                                  601,325

3                                                  166,736

4                                                  22,299

5                                                  6,730

1. Let “X” be the random variable indicating the number of times a student takes the SAT. Prepare the probability distribution table for this random variable.
   1. What is the probability that a student takes the SAT more than one time?
   1. What is the probability that a student takes the SAT three or more times?
   1. What is the expected value of the number of times the SAT is taken? What is your interpretation of the expected value?
   1. What is the variance and standard deviation for the number of times the SAT is taken?

• Use Excel functions and prepare the Binomial Distribution (with sample size = 15 and probability of success = 0.30) and the Cumulative Probability Distribution in a table and plot the distributions.

• In San Francisco, 30% of workers take public transportation daily. In a sample of 10 workers,
  • Clearly state what the random variable in this problem is?
  • What is an appropriate distribution to be used for this problem and why?
  • What is the probability that exactly three workers take public transportation daily?
  • What is the probability that NONE of workers take public transportation daily?
  • What is the probability that more than five workers take public transportation daily?
  • What is the probability that less than seven workers take public transportation daily?
  • What is the probability at least two but no more than eight workers take public transportation daily?

• In a typical month, an insurance agent presents life insurance plans to 40 potential customers.  Historically, one in four such customers choose to buy life insurance from this agent.  You may treat this as a binomial experiment.
  • What is the probability of success for this problem?
  • What is the total number of trials in this problem?
  • Create a probability distribution table which includes the value for the random variable and the probability of each possible outcome of the random variable.  Also create the cumulative probability column.
  • What is the probability that exactly five customers will buy life insurance from this agent in the coming month?
  • What is the probability that no more than 10 customers will buy life insurance from this agent in the coming month?
  • What is the probability that at least 20 customers will buy life insurance from this agent in the coming month?
  • Determine the mean and standard deviation of the number of customers who will buy life insurance from this agent in the coming month?
  • What is the probability that the number of customers who buy life insurance from this agent in the coming month will lie within two standard deviations of the mean?
• XYZ University finds that 25% of its students withdraw without completing the calculus course. Assuming 20 students have registered for the course
• Clearly state what the random variable in this problem is?
• What is an appropriate distribution to be used for this problem and why?
• Compute the expected number of withdrawals
• Compute the probability that exactly four will withdraw
• Compute the probability that more than three will withdraw
• Compute the probability that two or fewer will withdraw
• Use Excel functions and prepare the Poisson Distribution (with mean = 5.5) and the Cumulative Probability Distribution in a Table and plot the distributions.

• During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes.
  • Clearly state what the random variable in this problem is?
  • What is an appropriate distribution to be used for this problem and why?
  • What is the expected number of calls in one hour?
  • What is the probability of receiving three calls in five minutes?
  • What is the probability of receiving NO calls in a 10-minute period?
  • What is the probability of receiving more than five calls in a 10-minute period?
  • What is the probability of receiving less than seven calls in 15-minutes?
  • What is the probability of receiving at least three but no more than 10 calls in 12 minutes?

• The annual number of industrial accidents occurring in a particular manufacturing plant is known to follow a Poisson distribution with mean 12.
  • What is the probability of observing of observing exactly 12 accidents during the coming year?
  • What is the probability of observing no more than 12 accidents during the coming year?
  • What is the probability of observing at least 15 accidents during the coming year?
  • What is the probability of observing between 10 and 15 accidents (including 10 and 15) during the coming year?
  • Find the smallest integer k such that we can be at least 99% sure that the annual number of accidents occurring will be less than k.
• Suppose the number of customers arriving each hour at the only checkout counter in a local pharmacy is approximately Poisson distributed with an expected arrival rate of 20 customers per hour. 
  • Find the probability that exactly 10 customers arrive in a given hour.
  • Find the probability that at least five customers arrive in a given hour.
  • Find the probability that no more than 25 customers arrive in a given hour.
  • Find the probability that between 10 and 30 customers (inclusive) arrive in a given hour.
• The Safety Council estimates that off-the-job accidents cost businesses almost $500 billion annually in lost productivity. Based on their estimates, companies with hundred employees are expected to have six off-the-job accidents per year.
• Clearly state what the random variable in this problem is?
• What is an appropriate distribution to be used for this problem and why?
• What is the probability of no off-the-job accidents during the next six months?
• What is the probability of at least four off-the-job accidents during a one-year period?
• What is the probability that the number of off-the-job accidents is more than two but less than six during a four-month period?
• What is the probability of at the most three off-the-job accidents during the next eight months?