Question

The polling organization Ipsos conducted telephone surveys in March of 2004, 2005 and 2006. In each year, 1001 people age 18 or older were asked about whether they planned to use a credit card to pay federal income taxes that year. The data are given in the accompanying table. Is there evidence that the proportion falling in the three credit card response categories is not the same for all three years? Test the relevant hypotheses using a .05 significance level. (Use 2 decimal places.)

Intent to Pay Taxes with a Credit Card
	2004	2005	2006
Definitely/Probably Will Might/Might Not/Probably Not Definitely Not	42 163 782	45 180 777	42 190 780

χ² =  
P-value interval

p < 0.0010.001 ≤ p < 0.01    0.01 ≤ p < 0.050.05 ≤ p < 0.10p ≥ 0.10

Answer 1

• Step 1. Set up hypotheses and determine level of significance.

H₀: The three credit card response categories is not the same for all three years i.e they both are independent.

H₁: H₀ is false. α=0.05

• Step 2.  Select the appropriate test statistic.

The formula for the test statistic is:

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The condition for appropriate use of the above test statistic is that each expected frequency is at least 5. In Step 4 we will compute the expected frequencies and we will ensure that the condition is met.

• Step 3. Set up decision rule.

The decision rule depends on the level of significance and the degrees of freedom, defined as df = (r-1)(c-1), where r and c are the numbers of rows and columns in the two-way data table. thus r=3 ,c=3. For this test, df=(3-1)(3-1)=2(2)=4. Again, with χ² tests there are no upper, lower or two-tailed tests. If the null hypothesis is true, the observed and expected frequencies will be close in value and the χ² statistic will be close to zero. If the null hypothesis is false, then the χ² statistic will be large. The rejection region for the χ² test of independence is always in the upper (right-hand) tail of the distribution. For df=4 and a 5% level of significance, the appropriate critical value is 9.488 and the decision rule is as follows: Reject H₀ if c ²> 9.488.

Table of totals=

	2004	2005	2006	Total
Definitely/probablt will	42	45	42	129
Might/Might Not/Probably Not	163	180	190	533
Definitely Not	782	777	780	2339
Total	987	1002	1012	3001

• Step 4. Compute the test statistic.

We now compute the expected frequencies using the formula,

Expected Frequency = (Row Total * Column Total)/N.

The computations can be organized in a two-way table. The top number in each cell of the table is the observed frequency and the bottom number is the expected frequency.

	2004	2005	2006	Total
Definitely/probablt will	42 (42.42)	45 (43.07)	42 (43.50)	129
Might/Might Not/Probably Not	163 (175.29)	180 (177.96)	190 (179.73)	533
Definitely Not	782 (769.27)	777 (780.96)	780 (788.75)	2339
Total	987	1002	1012	3001

Notice that the expected frequencies are taken to one decimal place and that the sums of the observed frequencies are equal to the sums of the expected frequencies in each row and column of the table.

The test statistic is computed as follows:

  

• Step 5. Conclusion.

We accept H₀ because at 5% l.o.s.

Hence The three credit card response categories is not the same for all three years.

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