Math 114Q, Sections 9, 11
Second Exam
October 30, 2008

General guidelines Here are the questions:

  1. (20 points) Your school's PTO is conducting a raffle. They want to clear $1000 after paying the winners (that is, their profit after paying all of the winners should be $1000). The first prize is $500. There are two second prizes worth $100 each and ten third prizes worth $10 each. Tickets will cost a dollar.

    1. How many tickets should they sell?

    2. What is the "house advantage", expressed as a percentage?

    3. If you buy one ticket, what is the probability that you win some prize?

    4. What is the fair price for a ticket?

    5. (Optional). If you buy three tickets, what is the probability that you win at least one prize? (An estimate is OK.)

  2. (20 points) The Office of Institutional Research at UMass Boston collected data about class size in 2005. Here's what they found. 

    	Class size  number of sections
    	 0 - 9          253
    	 10 - 19        288
    	 20 - 29        425
    	 30 - 39        222
    	 40 - 49        19
    	 50 - 99        45
    	100 - 199       15
    You can find the Excel spreadsheet near the link to this exam, or you can try to download it by clicking here:  www.cs.umb.edu/~eb/114/exam2/ClassSize.xls

    Use Excel to answer the following questions. Write the answers to all but the first question in your blue book (you need not show your work, since that will be visible in the spreadsheet).

    1. Construct a properly labelled histogram displaying this data.
    2. Estimate the mean, median and modal class size.

    3. What percent of the classes are smaller than the median class size?

    4. What percent of the classes are smaller than the mean class size?

    5. What percent of the classes have 50 or more students?  

    6. What percent of the time does a student find herself in a class with 50 or more students?

    7. When you are sure you are done (is everything properly labelled?) print out your spreadsheet and turn in the printout with your exam. Also email the spreadsheet to your instructor ( maura.mast@umb.edu or eb@cs.umb.edu) as an attachment. Put 
      	 Math 114 second exam spreadsheet
      in the subject line.

  3. (20 points) Many states offer a prepaid tuition savings plan that allows parents to invest in a saving account that then contributes a set amount toward tuition and fees at a public institution. The Massachusetts plan's amount is the average tuition paid by students at the four state universities.

    Use the information in the table below to answer the following questions. You may do the arithmetic in Excel, or with a calculator. If you choose to work with Excel you can find a link to the data file near the link to this exam, or download the file by clicking on  www.cs.umb.edu/~eb/114/exam2/TuitionPlan.xls 

     school      undergraduate      in-state tuition & fees
              full-time enrollment  
     Boston          6644                  $4273
     Amherst        19047                 $10232
     Dartmouth       6647                  $8592
     Lowell          6063                  $8731

    1. What is the Massachusetts plan's contribution to tuition? Why can't you simply average the four numbers in the last column to answer this question?

    2. At which schools will the plan pay more than the undergraduate in-state tuition and fees?

  4. (20 points) Typhoid fever is a life-threatening illness caused by the bacterium Salmonella Typhi. In the United States about 400 cases occur each year, and 70% of these are acquired while traveling internationally. Typhoid fever is still common in the developing world, where it affects about 12.5 million persons each year. (source: www.wrongdiagnosis.com/artic/typhoid_fever_general_dbmd.htm)

    1. What is the probability that a person in the United States who has not traveled internationally will have typhoid fever this year?

      Warning. The probability is very small. Be sure you get the decimal point in the right place.

    2. What is the probability that a person in the developing world will have typhoid fever this year?

      Hint: To answer this question you will have to estimate the population in the "developing world."

  5. (20 points) Typhoid fever can be cured if it is treated promptly, so a rapid inexpensive accurate test would be very useful in the developing world. In 2006 The Indian Journal of Medical Research reported on a study of such a test: Prospective evaluation of a rapid diagnostic test Typhidot® for typhoid fever

    In the study, the Typhidot test was given to 545 patients suspected of having the disease. Those patients were also tested using a blood culture known to be very reliable, which showed that 36 patients actually had typhoid fever. For the Typhidot test the study found 6 false positives and 3 false negatives.

    1. Finish filling in the following table summarizing the data in this study: 
                           had typhoid      did not have      total
                              fever         typhoid fever

       test positive                                           

       test negative

       total                   36                              545
    2. Calculate the false positive rate and the false negative rate. Write a sentence explaining what these rates mean.

    3. What percentage of the people tested actually had typhoid fever?

    4. What percentage of the people who had typhoid fever had a positive test?