Chapter PVII, Problem 16RE

(a)

To determine

To explain will you conduct a test of homogeneity or independence and why.

Answer to Problem 16RE

Chi-square test of independence will be conducted.

Explanation of Solution

It is given in the question that the Gallup poll conducted a representative telephone survey during the first quarter. The contingency table for the political parties- Republican, Democratic and Independent with different ages is also given as:

		Observed
Age	Republican	Democratic	Independent	Total
18−29	241	351	409	1001
30−49	299	330	370	999
50−64	282	341	375	998
65+	279	382	343	1004
Total	1101	1404	1497	4002

Since the association between age and political affiliation needs to be examined so the chi-square test of independence will be conducted, as they are different groups not from the same sample.

(b)

To determine

To test an appropriate hypothesis.

Answer to Problem 16RE

They are not independent.

Explanation of Solution

It is given in the question that the Gallup poll conducted a representative telephone survey during the first quarter. Let us first check the condition as:

Counted Data Condition: We have counts of the number of sample in categories.

Independence Assumption: The sample is largely independent of each other.

Randomization Condition: The sample is randomly selected.

Expected Cell Frequency Condition: The expected values shown are all at least $5$ .

So, all the conditions are satisfied. Thus, to test a hypothesis of homogeneity or independence, we will use the calculator $\text{TI}-89$ so, we need to enter the data as a matrix. From the home screen, press APPS and select $6:$ Data/Matrix Editor, then select $3:$ New. Specify type as Matrix and name the matrix in the Variable box. Specify the number of rows and columns. Type the entries, pressing ENTER after each. Press $2\text{nd ESC}$ to leave the editor. To do the test, choose $\text{8:Chi2 2-way}$from the STAT Tests menu.The contingency table for the political parties- Republican, Democratic and Independent with different ages is also given as:

		Observed
Age	Republican	Democratic	Independent	Total
18−29	241	351	409	1001
30−49	299	330	370	999
50−64	282	341	375	998
65+	279	382	343	1004
Total	1101	1404	1497	4002

The hypothesis is defined as:

Null hypothesis: The political parties are independent.

Alternative hypothesis: The political parties are not independent.

And the number of rows and column is:

  $\begin{array}{l}r=4\\ c=3\end{array}$

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}{\chi }^2=16.66\\ df=6\\ P=0.0106\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that the variables are not independent.

(c)

To determine

To state your conclusion including an analysis of differences you find.

Explanation of Solution

It is given in the question that the Gallup poll conducted a representative telephone survey during the first quarter. The contingency table for the political parties- Republican, Democratic and Independent with different ages is also given as:

		Observed
Age	Republican	Democratic	Independent	Total
18−29	241	351	409	1001
30−49	299	330	370	999
50−64	282	341	375	998
65+	279	382	343	1004
Total	1101	1404	1497	4002

The hypothesis is defined as:

Null hypothesis: The political parties are independent.

Alternative hypothesis: The political parties are not independent.

Thus, by using the calculator $\text{TI}-89$ , the test statistics and the P-value is as:

  $\begin{array}{l}{\chi }^2=16.66\\ df=6\\ P=0.0106\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that the political parties are not independent. So, the political parties are dependent on each other which mean that any effect on anyone effect the other as well.