Question

A researcher is interested in voting preferences on change of the governing constitution in a certain country controlled by two main parties A and B. A questionnaire was developed and sent to a random sample of voters. The cross tabs are as follows Favour Neutral Oppose Membership: Party A 70 90 85 Party B 50 50 155 Test at α = 0.05 whether party membership and voting preference are associated and state the conditions required for chi-square test results to be valid.

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Clarabelle

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To test whether party membership and voting preference are associated, we can use a chi-square test of independence. The null hypothesis (H0) is that there is no association between party membership and voting preference, while the alternative hypothesis (H1) is that there is an association.
The observed values for the cross tabs are as follows:
Favour Neutral Oppose
Party A 70 90 85
Party B 50 50 155
To perform the chi-square test, we need to calculate the expected values under the assumption of independence. The expected values can be obtained by assuming that party membership and voting preference are independent and calculating the expected counts based on the row and column totals.
The expected values are as follows:
Favour Neutral Oppose
Party A 58.33 58.33 128.34
Party B 61.67 61.67 135.66
Now, we can set up the chi-square test:
1. Set the significance level (α) to 0.05.
2. Calculate the degrees of freedom (df) as (number of rows - 1) * (number of columns - 1). In this case, df = (2-1) * (3-1) = 2.
3. Calculate the chi-square test statistic using the formula:
χ^2 = Σ [(O - E)^2 / E],
where Σ represents the sum of the cells, O is the observed value, and E is the expected value.
4. Compare the calculated chi-square test statistic with the critical chi-square value from the chi-square distribution table with the given degrees of freedom and significance level.
5. If the calculated chi-square test statistic is greater than the critical chi-square value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
Performing the calculations, we obtain a chi-square test statistic of approximately 22.88.
Looking up the critical chi-square value with 2 degrees of freedom and a significance level of 0.05, we find a critical chi-square value of approximately 5.99.
Since the calculated chi-square test statistic (22.88) is higher than the critical chi-square value (5.99), we reject the null hypothesis.
Therefore, we can conclude that there is evidence to suggest an association between party membership and voting preference in this country.
Conditions required for the chi-square test results to be valid:
1. Random sampling: The sample of voters should be randomly selected to ensure representativeness.
2. Independence: The observations in the cross tabs should be independent of each other.
3. Sufficient sample size: The expected count for each cell in the cross tabs should be at least 5. If not, the chi-square test may not be valid, and alternative methods should be considered.

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