Math 210
Laboratory 8

Chi-Square Tests: Heart disease, baldness, and Yahtzee

On January 24, 2000 the Holland Sentinel reported on a study that found the greater the hair loss on top of a man's head, the greater the risk for heart disease.  This study, co-authored by Dr. JoAnn Manson from Harvard's Brigham and Women's Hospital in Boston, analyzed baldness patterns of 22,000 male doctors who were 40 to 84 years old when enrolled in the Physician's Health Study.  Eleven years into the study, the researcher asked the doctors to describe their patterns of baldness at age 45.
 

  1. In another study researchers selected a sample of 663 heart disease patients (male) and a control group of 772 males not suffering from heart disease. Each was asked to classify their degree of baldness on a 5-point scale. The results are given in the following table.  Use these data to answer the following questions.
Baldness:
None
Little
Some
Much
Extreme
Heart_Disease
251
165
195
50
2
Control
331
221
185
34
1
    1. Of those in the control group, what percent claimed to have:
      1. Little or no baldness?
      2. Some, much or extreme baldness?
    1. Of those with heart disease, what percent claimed to have:
      1. Little or no baldness?
      2. Some, much or extreme baldness?
    2. Lets now make a bar chart of our data.  To do this, put the data in Minitab in the same order as shown in the table above.  Select Graph > Bar Chart  Under Bars Represent select Values from a table and choose Two-Way Table and Cluster.  Under graph variable put the none to extreme columns.  Under Row labels put the column that has heart disease and baldness.  Click on Columns are outermost categories and rows are innermost. Click on Chart Options and then select Show Y as a Percent and under Take Percent and/or Accumulate click on Within categories at 1 level..  Click OK and OK.  Edit your axis labels appropriately.  If there is no relationship between heart disease and bladness, each part of the graph should look the same.
    3. At this stage in the investigation, state whether you think there is a relationship between heart disease and baldness. Explain your answer.
    4. We are now going to run a test of significance on this.
      1. State the null and alternative hypotheses you would use to test whether there is a relationship between heart disease and baldness.
      2. Complete a chi-square test to determine if there is a relationship between heart disease and baldness.  Use Stat > Tables > Chi-square Test.)  Make sure you give the chi-square statistic, the P-value, and a conclusion.
    1. Briefly discuss whether your results necessarily mean that heart disease is caused by baldness or baldness is caused by heart disease.

In a traditional Yahtzee game, five dice are tossed and the player tries to complete certain categories some of which are similar to poker like three of a kind, full house, or a straight.  If the player gets all five dice to be all the same number, it is called a Yahtzee.  There are hand held electronic versions of this game.  In this game, dice are not actually tossed - the built in computer simulates this.  This might make one wonder if these simulated dice are really random.  To answer this question, a hand held electronic Yahtzee was used to simulate drawing five dice 100 times.  The results are found below.  We are going to conduct a chi-square goodness of fit test on these data to see if the simulated die does not give an even distribution of outcomes. 


number on the dice
 1
 2
 3
 4
 5
 6
observed frequency
 72
 79
 82
 80
 95
 92

  1. Another type of chi-square test is a goodness-of-fit test.  Instead of testing the entire two-way table for independence as done in question 1, this test can determine whether each die is gives an even distribution or not. 
  1. We want to perform a chi-square goodness of fit test to determine if the numbers on the simulated dice are not distributed evenly.  Write down the null and alternative hypotheses for this kind of test.
  2. If the numbers on the dice are distributed evenly, what should each excepted frequency be?
  3. Make a bar graph of the data.  To do this, number on the dice in column 1 and the observed frequency in column 2.  Select Graph > Bar Chart > Simple.  Under Bars Represent select Values from a table.  Your graph variable should be C2 and the Categorical variable should be C1.  Edit your axis labels appropriately.  Do not leave them as C1 and C2.  Do you think there is enough difference in your sample proportions to say that the numbers are not distributed evenly for the population?
  4. To complete the chi-square test put the expected frequencies in column 1.  Now select Stat > Tables > Chi-Square Goodness-of-Fit Test (One Variable).  In Observed Counts, enter C2.  Under Test, choose Equal proportions. Click OK.   Report your test statistic and P-value.
  5. What is your conclusion for your goodness-of-fit test?