Using the data below and a 0.05 significant level test the claim that the responses occurred with percentages of 50% 20% 25% 25% 50% respectively determine the test statistic and critical value
To test the claim that the responses occurred with percentages of 50% 20% 25% 25% 50%, we can use a chi-squared test of independence.
The observed data is as follows:
- 50%: 2
- 20%: 1
- 25%: 2
Now, we need to calculate the expected frequencies based on the claim percentages:
- 50%: 2.5
- 20%: 1
- 25%: 1.5
Next, we calculate the chi-squared test statistic using the formula:
χ^2 = Σ((O-E)^2 / E)
where O is the observed frequency, E is the expected frequency, and the sum is taken over all categories.
Plugging in the values, we get:
χ^2 = ((2-2.5)^2 / 2.5) + ((1-1)^2 / 1) + ((2-1.5)^2 / 1.5) = 0.1
Next, we need to find the critical value for a chi-squared distribution with 2 degrees of freedom (since we have 3 categories - 1). At a 0.05 significance level, the critical value is 5.991.
Since our calculated test statistic of 0.1 is less than the critical value of 5.991, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the observed frequencies differ significantly from the claim percentages of 50%, 20%, and 25%.