Question

Let's suppose we are concerned about the future environment of Canada. We conduct a survey of a sample of 810 to ask people whether they would agree or disagree to increase taxes in order to reduce pollution to the environment. The result of the survey is shown in the following table. Answer the questions below based on the contingency table provided. Write a 2-page report, including the table with percentages. Use double-space and 12 pitch fonts in your answers. Gender Support Higher Tax To Reduce Pollution Female Male 249 246 167 148 Agree Disagree 495 315 Total Number 3. Explain the level of measurement of each variable. 4. Calculate the percent in each cell and construct a percentage table based on your calculations. 5. Analyze the data in the percentage table in Step 4 and explain the relationship between the two variables.

Answer 1

3. We find that the data is given in the form of Count / frequencies for each category. The given data has 2 variables namely,

a. Gender b.Support Higher tax To Reduce Pollution

with 2 and 3 categories respectively. Since, the variables are categorical, the level of measurement for both the variables would be "Nominal".

4. Computing the percentage in each cell, i.e. dividing the frequency in each cell by the Grand total = 315 + 495 = 810:

Gender Support Hiher tax To Total Reduce Pollution Male Female Agree 167 249 416 Disa 148 246 394 Total Number 315 495 810 Obtaining the percentages for each cell: Support Highertax To Gender Reduce Pollution Total Male Female 167/810-0.206 249/810-0.307 416/810 0.514 Agree Disa 148/810-0.183 246/810-0.304394/810-0.486 Total Number 315/810-0.389 495/810 0.611 1.000 We get the Percentage Table: Gender Support Highertax To Total Reduce Pollution Male Female 0.206 0.307 0.514 Disagree 0.183 0.304 0.486 Total Number 0.359 0.611 1.000

To find the gender wise probabilities, we divide each cell frequency by its corresponding column totals:

For cell 1: Probability = 167 / 315 = 0.53. We get the Probability table as:

Gender Support Higher tax To Female Male Disa 0.470 LAS

Looking at the percentages, little difference in the categories is observed. We might not have sufficient evidence to claim a significant relationship between the two variables.

5. To test whether there exist a significant relationship between the two (nominal) variables, we may go for a Chi square test of association.

To test: H₀: No significant relation exist between Gender and opinion on whether they support Higher tax to Reduce pollution Vs H_a: A significant relation exist between Gender and their opinion on whether they support Higher tax to Reduce pollution.

The appropriate test statistic here, would be Chi Square, given by,

$\chi ^{2}=\frac{\sum (O_{i}-E_{i})^{2}}{E_{i}}$

with critical region given by, $\chi ^{2}>\chi ^{2}_{Critical}$

Where Oi = Observed Frequency and Ei = Expected frequency

Expected frequencies can be computed for each cell by obtaining the product of the corresponding row and column totals and dividing it by the Grand Total:

Gender Support Higher tax To Reduce Pollution Female Mak 315 416/810-161.78 495 416/810 254.22 Disa 315 394/810-153.22 (495 394/810 240.75

Substituting the values in the test statistic formula, we get

Oi-Ei 167 161.78 27.2484 0.17 249 254.22 27.2484 27.24840.11 148 153.22 246 240.75 27.2484 0.57

Comparing the test statistic obtained with its critical value (Obtained from chi square tables for (No. of rows - 1)(No. of Columns - 1) = (2-1)(2-1) = 1 degree of freedom, at say, $\alpha =0.05$ level of significance:

Chi-square Distribution Table d.f. 99599 975 95 1 05 025 01 10.00 0.00 0.00 0.00 0.02 2.71 3.84 5.02 6.63 0.07 0.11 0.22 0.35 0.58 6.25 7.81 9.35 11.34 40.21 0.30 0.48 0. 1.06 7.789.49 11.14 13.28 0.4 0.55 0.83 115 1.61 9.24 11.07 12.83 15.09 6 0.68 0.871.4 64 2.20 10.6412.59 14.45 16.81 7 0.99 1.24 .69 2.17 2.83 12.02 14.07 16.01 18.48 81.34 1.65 2.18 2.73 3.49 13.36 15.51 17.5320.09

Since, does not lie in the critical / rejection region, we fail to reject H₀ at 5% level of significance.

2 = 0.57< 3.84

We may conclude that, the given data does not provide sufficient evidence to claim that there exist a significant relation between Gender and their opinion on whether they support Higher tax to Reduce pollution.