Recycling in Two Communities I

  1. α = 0.01.
  2. H0: “Distribution of residents into the methods of disposal is the same for both cities” and HA: “Distribution of residents into the methods of disposal is NOT the same for both cities”.
  • The hypotheses for a chi-square test will usually be “wordy” like this.
  • The H0 will always have the format of the “distribution of INDIVIDUALS into the LEVELS OF THE RESPONSE VARIABLE is the same for ALL POPULATIONS/GROUPS.” Thus, the H0 above simply has the capitalized items replaced with their specific meanings in this example. That is, the individuals are residents, the levels of the response variable are the methods of disposal, and the populations are the two cities.
  • The HA will be exactly like H0 except that it will read “NOT the same.”
  1. A Chi-square test will be used because
    1. two groups/populations are considered (Des Moines and Minneapolis) and
    2. a categorical variable (method for disposing of plastic containers) was recorded.
  • These are the two characteristics that separate a Chi-square test from other tests we will see this semester.
  • Please include some sort of description that shows that the characteristics are met and that you are not just listing the characteristics – in other words, include the parts in parentheses.
  1. This study is observational because the researcher simply asked residents what they did. Who got the survey was random, but they were returned (or not) voluntarily so this is a voluntary response survey rather than a random survey.
  • Recall from the “Data Production” module that you should not be tricked into calling a survey random if the randomly selected respondents could freely choose to respond or not.
  1. The assumptions are met because the expected table has more than five individuals in all six cells
City Recycle Reuse Throw-away Total
Des Moines 49.0 21.5 11.5 82
Minneapolis 58.0 25.5 13.5 97
TOTAL 107 47 25 179


  • The expected table has the same format as the observed table.
  • Each expected value is found by multiplying the respective row and column totals and then dividing by n (or the “total total”). For example, the expected value for Des Moines and Recycle is \(\frac{82\times107}{179}\). Similarly, the expected value for Des Moines and Reuse is computed with \(\frac{82\times25}{179}\) and the expected value for Minneapolis and Throw-away is \(\frac{97\times47}{179}\). The other three values are computed similarly.
  • Expected values are expected counts of individuals, but we traditionally keep at least one decimal place so that our calculations below will be more precise.
  1. The statistic for a Chi-square test is the observed frequency table (which was given in the background in this case).
City Recycle Reuse Throw-away Total
Des Moines 26 37 19 82
Minneapolis 81 10 6 97
TOTAL 107 47 25 179


  • I usually add the marginal totals to my observed tables in preparation for making the expected tables.
  1. The χ2 test statistic is 49.634, as computed from the observed and expected tables as shown below. This test statistic has 2 df.

\[ \chi^{2} = \sum\frac{(obs-exp)^{2}}{exp} = \] \[ \frac{(26-49.0)^{2}}{49.0} + \frac{(37-21.5)^{2}}{21.5} + \frac{(19-11.5)^{2}}{11.5} + \] \[ \frac{(81-58.0)^{2}}{58.0} + \frac{(10-25.5)^{2}}{25.5} + \frac{(6-13.5)^{2}}{13.5} = \] \[ 10.808 + 11.114 + 4.974 + 9.137 + 9.396 + 4.205 = \] \[ 49.634 \]

  • I urge you to write out each of the parts of the test statistic and the intermediate computation for each part. This will help you solve problems if something goes sideways.
  • The df is the number of rows (2) minus 1 times the number of columns (3) minus 1.
  1. p-value<0.00005 (more specifically 1.668e-11) as computed with distrib(49.634,distrib="chisq",df=2,lower.tail=FALSE).
  • When computing the p-value on a chi-square distribution it will always be a “right-of” and, thus, will always include lower.tail=FALSE.
  1. Reject H0 because the p-value < α.
  2. It appears that there is a difference between cities in the distribution of residents into the disposal categories. A comparison of the observed and expected tables suggests that a higher percentage of residents in Minneapolis recycled than did residents in Des Moines.
  • I like to add on the second sentence above as the first sentence is not very satisfying.
  • There is not usually a Step 11 for a Chi-square test. We will not ever do a Step 11 in this class.