Chapter 12, Problem 1CP

In a study conducted by Zogby International for the Democrat and Chronicle, more than 700 New Yorkers were polled to determine whether the New York state government works. Respondents surveyed were asked questions involving pay cuts for state legislators, restrictions on lobbyists, term limits for legislators, and whether state citizens should be able to put matters directly on the state ballot for a vote. The results regarding several proposed reforms had broad support, crossing all demographic and political lines.

Suppose that a follow-up survey of 100 individuals who live in the western region of New York was conducted. The party affiliation (Democrat, Independent, Republican) of each individual surveyed was recorded, as well as their responses to the following three questions.

1. 1. Should legislative pay be cut for every day the state budget is late?

   Yes ____ No ____

2. 2. Should there be more restrictions on lobbyists?

   Yes ____ No ____

3. 3. Should there be term limits requiring that legislators serve a fixed number of years?

   Yes ____ No ____

The responses were coded using 1 for a Yes response and 2 for a No response. The complete data set is available in the file NYReform.

Managerial Report

1. 1. Use descriptive statistics to summarize the data from this study. What are your preliminary conclusions about the independence of the response (Yes or No) and party affiliation for each of the three questions in the survey?
2. 2. With regard to question 1, test for the independence of the response (Yes and No) and party affiliation. Use α = .05.
3. 3. With regard to question 2, test for the independence of the response (Yes and No) and party affiliation. Use α = .05.
4. 4. With regard to question 3, test for the independence of the response (Yes and No) and party affiliation. Use α = .05.
5. 5. Does it appear that there is broad support for change across all political lines? Explain.

To determine

Find the descriptive statistics to summarize the data from the study.

Explain about the preliminary conclusions about the independence of the response and party affiliation for each of the three questions in the survey.

Explanation of Solution

Calculation:

The given data are related to the survey of 100 individuals, who were asked three questions. The response was coded using “1” for “Yes” response and “2” for “No” response.

The variable ‘Pay Cut?’ represents the response of the question “Should legislative pay be cut for every day the state budget is late?”.

Contingency table:

Software procedure:

Step-by-step software procedure to obtain contingency table using EXCEL software is as follows:

• Open an EXCEL sheet.
• Enter the data as Observation in column A, Party in column B, Pay Cut? in column C, Lobbyists? in column D and Term Limits? in column E.
• Select the all data.
• Choose Insert > PivotTable.
• Under Select a table or range in Table/Range enter Data!$A$1:$E$101.
• Choose New Worksheet.
• Click OK.
• In new sheet enter Party as Row Labels.
• In new sheet enter Pay Cut? as Column Labels.
• In new sheet enter Observation as Values.
• Choose Values > Sum of Observation > Value Field Settings.
• Select Summarize value field by > Count.
• Click OK.
• Output using EXCEL software is given below:

Now, the percentages of responses for the question “Should legislative pay be cut for every day the state budget is late?” are obtained as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{22}{36}\times 100=61.11\%$	$\frac{14}{36}\times 100=38.89\%$
Independent	$\frac{10}{19}\times 100=52.63\%$	$\frac{9}{19}\times 100=47.36\%$
Republican	$\frac{39}{45}\times 100=86.67\%$	$\frac{6}{45}\times 100=13.33\%$

Thus, it can be concluded that since the percentage of Republicans that answered yes is much greater than the percentage of Democrats or Independents that answered yes, the classifications do not appear to be independent.

The variable ‘Lobbyists?’ represents the response of the question “Should there be more restriction on lobbyists?”.

Software procedure:

Step-by-step software procedure to obtain contingency table using EXCEL software is as follows:

• Open an EXCEL sheet.
• Enter the data as Observation in column A, Party in column B, Pay Cut? in column C, Lobbyists? in column D and Term Limits? in column E.
• Select the all data.
• Choose Insert > PivotTable.
• Under Select a table or range in Table/Range enter Data!$A$1:$E$101.
• Choose New Worksheet.
• Click OK.
• In new sheet enter Party as Row Labels.
• In new sheet enter Lobbyists? as Column Labels.
• In new sheet enter Observation as Values.
• Choose Values > Sum of Observation > Value Field Settings.
• Select Summarize value field by > Count.
• Click OK.
• Output using EXCEL software is given below:

Now, the percentages of responses for the question “Should there be more restriction on lobbyists?” are obtained as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{21}{36}\times 100=58.33\%$	$\frac{15}{36}\times 100=41.66\%$
Independent	$\frac{15}{19}\times 100=78.94\%$	$\frac{4}{19}\times 100=21.05\%$
Republican	$\frac{34}{45}\times 100=75.55\%$	$\frac{11}{45}\times 100=24.44\%$

Thus, it can be concluded that since the percentage of Democrats that answered yes is much less than the percentage of Independents or Republicans that answered yes, the classifications do not appear to be independent.

The variable ‘Term Limits?’ represents the response of the question “Should there be term limits requiring that legislators serve a fixed number of years?”.

Contingency table:

Software procedure:

Step-by-step software procedure to obtain contingency table using EXCEL software is as follows:

• Open an EXCEL sheet.
• Enter the data as Observation in column A, Party in column B, Pay Cut? in column C, Lobbyists? in column D and Term Limits? in column E.
• Select the all data.
• Choose Insert > PivotTable.
• Under Select a table or range in Table/Range enter Data!$A$1:$E$101.
• Choose New Worksheet.
• Click OK.
• In new sheet enter Party as Row Labels.
• In new sheet enter Term Limits?  as Column Labels.
• In new sheet enter Observation as Values.
• Choose Values > Sum of Observation > Value Field Settings.
• Select Summarize value field by > Count.
• Click OK.
• Output using EXCEL software is given below:

Now, the percentages of responses for the question “Should there be term limits requiring that legislators serve a fixed number of years?” are obtained as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{17}{36}\times 100=47.22\%$	$\frac{19}{36}\times 100=52.77\%$
Independent	$\frac{10}{19}\times 100=52.63\%$	$\frac{9}{19}\times 100=47.37\%$
Republican	$\frac{32}{45}\times 100=71.11\%$	$\frac{13}{45}\times 100=28.89\%$

Thus, it can be concluded that since the percentage of Republicans that answered yes is much higher than the percentage of Democrats or Independents that answered yes, the classifications do not appear to be independent.

To determine

Perform a test for the independence of the response (Yes or No) and party affiliation with regard to question “Should legislative pay be cut for every day the state budget is late?” at $\alpha =0.05$ level of significance.

Answer to Problem 1CP

There is a statistically significant relation between the response (Yes or No) and party affiliation.

Explanation of Solution

Calculation:

State the test hypotheses.

Null hypothesis:

$H_0:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are independent}\text{.}$

That is, there is no statistically significant relation between the response (Yes or No) and party affiliation.

Alternative hypothesis:

$H_a:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are not independent}\text{.}$

That is, there is a statistically significant relation between the response (Yes or No) and party affiliation.

The row and column total is tabulated below:

Party	Yes (%)	No (%)	Total
Democrat	22	14	36
Independent	10	9	19
Republican	39	6	45
Total	71	29	100

The formula for expected frequency is given below:

$e_{ij}=\frac{\left(\text{Row }i\text{ Total}\right)\left(\text{Coloumn }j\text{ Total}\right)}{\text{Total Sample Size}}$.

The expected frequency for each category is calculated as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{\left(36\right)\left(71\right)}{100}=25.56$	$\frac{\left(36\right)\left(29\right)}{100}=10.44$
Independent	$\frac{\left(19\right)\left(71\right)}{100}=13.49$	$\frac{\left(19\right)\left(29\right)}{100}=5.51$
Republican	$\frac{\left(45\right)\left(71\right)}{100}=31.95$	$\frac{\left(45\right)\left(29\right)}{100}=13.05$

The formula for chi-square test statistic is given as,

${\chi }^2={\displaystyle \sum _{i=1}^r{\displaystyle \sum _{j=1}^c\frac{{\left(f_{ij}-e_{ij}\right)}^2}{e_{ij}}}}$ where $\left(i,j\right)$^th element denotes the frequency corresponding to $\left(i,j\right)$^th cell and r represent the r^th row and c represents the c^th column.

Therefore, the value of chi-square test statistic is,

$\begin{array}{c}{\chi }^2=\frac{{\left(22-25.56\right)}^2}{25.56}+\frac{{\left(14-10.44\right)}^2}{10.44}+\frac{{\left(10-13.49\right)}^2}{13.49}+\frac{{\left(9-5.51\right)}^2}{5.51}+\frac{{\left(39-31.95\right)}^2}{31.95}+\\ \frac{{\left(6-13.05\right)}^2}{13.05}\\ \approx \frac{12.67}{25.56}+\frac{12.67}{10.44}+\frac{12.18}{13.49}+\frac{12.18}{5.51}+\frac{49.70}{31.95}+\frac{49.70}{13.05}\\ \approx 0.49+1.21+0.90+2.21+1.56+3.81\\ =10.18\end{array}$

Thus, the chi-square test statistic is 10.18.

Degrees of freedom:

The degrees of freedom are $\left(r-1\right)\left(c-1\right)$ for r row and c columns.

Thus, for 3 rows and 2 columns the degrees of freedom are,

$\begin{array}{c}df=\left(3-1\right)\left(2-1\right)\\ =\left(2-1\right)\left(1-1\right)\\ =2\end{array}$

Thus, the degree of freedom is 2.

Level of significance:

The given level of significance is $\alpha =0.05$.

p-value:

Software procedure:

Step -by-step software procedure to obtain p-value using EXCEL software is as follows:

• Open an EXCEL sheet and select cell A1.
• In cell A1 enter the formula =CHISQ.DIST.RT(10.18,2).
• Press Enter.
• Output using EXCEL software is given below:

From the EXCEL output, the p-value is 0.0062.

Rejection rule:

• If the $p\text{-value}\le \alpha$, then reject the null hypothesis.
• Otherwise, failed to reject the null hypothesis.

Conclusion:

Here, the p-value is less than the level of significance.

That is, $p\text{-value}\left(=0.0062\right)<\alpha \left(=0.05\right)$

Thus, the decision is “reject the null hypothesis”.

Therefore, the data provide sufficient evidence to conclude that the responses and party affiliation are not independent.

Thus, there is a statistically significant relation between the response (Yes or No) and party affiliation.

To determine

Perform a test for the independence of the response (Yes or No) and party affiliation with regard to question “Should there be more restriction on lobbyists?” at $\alpha =0.05$ level of significance.

Answer to Problem 1CP

There is no statistically significant relation between the response (Yes or No) and party affiliation.

Explanation of Solution

Calculation:

State the test hypotheses.

Null hypothesis:

$H_0:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are independent}$.

That is, there is no statistically significant relation between the response (Yes or No) and party affiliation.

Alternative hypothesis:

$H_a:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are not independent}$.

That is, there is no statistically significant relation between the response (Yes or No) and party affiliation.

The row and column total is tabulated below:

Party	Yes (%)	No (%)	Total
Democrat	21	15	36
Independent	15	4	19
Republican	34	11	45
Total	70	30	100

The formula for expected frequency is given below:

$e_{ij}=\frac{\left(\text{Row }i\text{ Total}\right)\left(\text{Coloumn }j\text{ Total}\right)}{\text{Total Sample Size}}$.

The expected frequency for each category is calculated as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{\left(36\right)\left(70\right)}{100}=25.20$	$\frac{\left(36\right)\left(30\right)}{100}=10.80$
Independent	$\frac{\left(19\right)\left(70\right)}{100}=13.30$	$\frac{\left(19\right)\left(30\right)}{100}=5.70$
Republican	$\frac{\left(45\right)\left(70\right)}{100}=31.50$	$\frac{\left(45\right)\left(30\right)}{100}=13.50$

Therefore, the value of chi-square test statistic is,

$\begin{array}{c}{\chi }^2=\frac{{\left(21-25.20\right)}^2}{25.20}+\frac{{\left(15-10.80\right)}^2}{10.80}+\frac{{\left(15-13.30\right)}^2}{13.30}+\frac{{\left(4-5.70\right)}^2}{5.70}+\frac{{\left(34-31.50\right)}^2}{31.50}+\\ \frac{{\left(11-13.50\right)}^2}{13.50}\\ \approx \frac{17.64}{25.20}+\frac{17.64}{10.80}+\frac{2.89}{13.30}+\frac{2.89}{5.70}+\frac{6.25}{31.50}+\frac{6.25}{13.50}\\ \approx 0.7+1.63+0.22+0.51+0.20+0.46\\ =3.72\end{array}$

Thus, the chi-square test statistic is 3.72.

Degrees of freedom:

The degrees of freedom are $\left(r-1\right)\left(c-1\right)$ for r row and c columns.

Thus, for 3 rows and 2 columns the degrees of freedom are,

$\begin{array}{c}df=\left(3-1\right)\left(2-1\right)\\ =\left(2-1\right)\left(1-1\right)\\ =2\end{array}$

Thus, the degree of freedom is 2.

Level of significance:

The given level of significance is $\alpha =0.05$.

p-value:

Software procedure:

Step -by-step software procedure to obtain p-value using EXCEL software is as follows:

• Open an EXCEL sheet and select cell A1.
• In cell A1 enter the formula =CHISQ.DIST.RT(3.72,2).
• Press Enter.
• Output using EXCEL software is given below:

From the EXCEL output, the p-value is 0.1557.

Conclusion:

Here, the p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.1557\right)>\alpha \left(=0.05\right)$

Thus, the decision is “Do not reject the null hypothesis”.

Therefore, the data do not provide sufficient evidence to conclude that the responses and party affiliation are not independent.

Thus, there is no statistical significant relation between the response (Yes or No) and party affiliation.

To determine

Perform a test for the independence of the response (Yes or No) and party affiliation with regard to question “Should there be term limits requiring that legislators serve a fixed number of years?” at $\alpha =0.05$ level of significance.

Answer to Problem 1CP

There is no statistically significant relation between the response (Yes or No) and party affiliation.

Explanation of Solution

Calculation:

State the test hypotheses.

Null hypothesis:

$H_0:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are independent}$.

That is, there is no statistical significant relation between the response (Yes or No) and party affiliation.

Alternative hypothesis:

$H_a:\text{The response }\left(\text{Yes or No}\right)\text{ and party affiliation are not independent}$.

That is, there is a statistically significant relation between the response (Yes or No) and party affiliation.

The row and column total is tabulated below:

Party	Yes (%)	No (%)	Total
Democrat	17	19	36
Independent	10	9	19
Republican	32	13	45
Total	59	41	100

The formula for expected frequency is given below:

$e_{ij}=\frac{\left(\text{Row }i\text{ Total}\right)\left(\text{Coloumn }j\text{ Total}\right)}{\text{Total Sample Size}}$.

The expected frequency for each category is calculated as follows:

Party	Yes (%)	No (%)
Democrat	$\frac{\left(36\right)\left(59\right)}{100}=21.24$	$\frac{\left(36\right)\left(41\right)}{100}=14.76$
Independent	$\frac{\left(19\right)\left(59\right)}{100}=11.21$	$\frac{\left(19\right)\left(41\right)}{100}=7.79$
Republican	$\frac{\left(45\right)\left(59\right)}{100}=26.55$	$\frac{\left(45\right)\left(41\right)}{100}=18.45$

Therefore, the value of chi-square test statistic is,

$\begin{array}{c}{\chi }^2=\frac{{\left(17-21.24\right)}^2}{21.24}+\frac{{\left(19-14.76\right)}^2}{14.76}+\frac{{\left(10-11.21\right)}^2}{11.21}+\frac{{\left(9-7.79\right)}^2}{7.79}+\frac{{\left(32-26.55\right)}^2}{26.55}+\\ \frac{{\left(13-18.45\right)}^2}{18.45}\\ \approx \frac{19.98}{21.24}+\frac{17.98}{14.76}+\frac{1.46}{11.21}+\frac{1.46}{7.79}+\frac{29.70}{26.55}+\frac{29.70}{18.45}\\ \approx 0.94+1.22+0.13+0.19+1.12+1.61\\ =5.21\end{array}$

Thus, the chi-square test statistic is 5.21.

Degrees of freedom:

The degrees of freedom are $\left(r-1\right)\left(c-1\right)$ for r row and c columns.

Thus, for 3 rows and 2 columns the degrees of freedom are,

$\begin{array}{c}df=\left(3-1\right)\left(2-1\right)\\ =\left(2-1\right)\left(1-1\right)\\ =2\end{array}$

Thus, the degree of freedom is 2.

Level of significance:

The given level of significance is $\alpha =0.05$.

p-value:

Software procedure:

Step -by-step software procedure to obtain p-value using EXCEL software is as follows:

• Open an EXCEL sheet and select cell A1.
• In cell A1 enter the formula =CHISQ.DIST.RT(5.21,2).
• Press Enter.
• Output using EXCEL software is given below:

From the EXCEL output, the p-value is 0.0739.

Conclusion:

Here, the p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.0739\right)>\alpha \left(=0.05\right)$

Thus, the decision is “Fail to reject the null hypothesis”.

Therefore, the data do not provide sufficient evidence to conclude that the responses and party affiliation are not independent.

Thus, there is no statistically significant relation between the response (Yes or No) and party affiliation.

To determine

Explain whether it appears that there is broad support for change across all political lines.

Explanation of Solution

The tests conducted in Parts 3 and 4 do not produce significant p-values and imply that there is no dependence between the party affiliation and opinions regarding the restriction for lobbyists and term limits of legislators.

However, the test conducted in Part 2 suggests that there is a significant relationship between one’s party affiliation and their opinion regarding legislative pay cut. The Republicans appear to be much more in favor of legislative pay cut for everyday state budget than the Democrats or the Independents.

Thus, it appears that there is no broad support for change across all political lines, but only across some political lines.