Question

Victoria is doing a chi-square test to see if people in California have the same color preferences as people in Arizona. A sample of 100 Californians were polled. Thirty-five people prefer yellow, fifty people prefer green, and fifteen people prefer red. A sample of 50 Arizonans were polled. Ten people prefer yellow, ten people prefer green, and thirty people prefer red. Calculate the test statistic. State the critical value. Come to a conclusion about the null hypothesis. And state what Victoria should conclude about Californian’s and Arizonan’s color preferences? Let α = .05.

Answer 1

california

sample size=100

color	no of people
yellow	35
green	50
red	15

Arizona

sample size=50

color	no of people
yellow	10
green	10
red	30

Victoria is doing a chi-square test to see if people in California have the same color preferences as people in Arizona, thus the

HYPOTHESIS STATEMENTS be:

H₀: people in California have the same color preferences as people in Arizona

H_a: people in California have different color preferences as people in Arizona( α = .05.)

Step 1.LAY OUT THE DATA IN A TABLE

state\color	yellow	green	red
california	35	50	15
arizona	10	10	30

Step 2.  Select the appropriate test statistic.

The formula for the test statistic is:

In the test statistic, O = observed frequency and E=expected frequency in each of the response categories

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.

Step3: Add up rows and columns:

state\color	yellow	green	red
california	35	50	15	100
arizona	10	10	30	50
	45	60	45	150

Step4: Calculate "Expected Value" for each entry:

Multiply each row total by each column total and divide by the overall total:

state\color	yellow	green	red
california	(45*100)/150	(60*100)/150	(45*100)/150	100
arizona	(45*50)/150	(60*50)/150	(45*50)/150	50
	45	60	45	150

Which gives us:

state\color	yellow	green	red
california	30	40	30	100
arizona	15	20	15	50
	45	60	45	150

Subtract expected from observed, square it, then divide by expected:

In other words, use formula (O−E)² /E where

• O = Observed (actual) value
• E = Expected value

state\color	yellow	green	red
california				100
arizona				50
	45	60	45	150

Which gets us:

state\color	yellow	green	red
california	0.833	2.5	7.5	100
arizona	1.66	5	15	50
	45	60	45	150

Now add up those calculated values:

0.833+2.5+7.5+1.66+5+15 = 32.493

Chi-Square is 32.493

Step 5. 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.

The row variable is the residential state which are 2 , thus r=2. The column variable is color preference and 3 responses are considered, thus c=3. For this test, df=(2-1)(3-1)=2.

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=2 and a 5% level of significance, the appropriate critical value is 5.99 and the decision rule is as follows: Reject H₀ if chi ²> 5.99.

Step 6. Conclusion

We reject H₀ because 32.493 > 5.99.

Thus Victoria has statistically significant evidence at a =0.05 to show that H₀ is false or people in California have different color preferences as people in Arizona..