Using the data below and a 0.05 significant level test the claim that the responses occurred with percentages of 15% 20% 25% 25% 15% respectively determine the test statistic and critical value. Response a, B, C, D, E frequency a 12, 15, 16, 18, 19,
First, we need to calculate the total number of responses in order to determine the expected frequencies for each response category.
Total number of responses = 12 + 15 + 16 + 18 + 19 = 80
Next, we need to calculate the expected frequencies for each response category assuming the percentages claimed.
Expected frequency for response A = 80 * 0.15 = 12
Expected frequency for response B = 80 * 0.20 = 16
Expected frequency for response C = 80 * 0.25 = 20
Expected frequency for response D = 80 * 0.25 = 20
Expected frequency for response E = 80 * 0.15 = 12
Now we can perform a Chi-squared Goodness of Fit test to determine if the observed frequencies match the expected frequencies.
Null Hypothesis: The observed frequencies match the expected frequencies.
Alternative Hypothesis: The observed frequencies do not match the expected frequencies.
Calculating the test statistic:
χ² = ∑( (Observed frequency - Expected frequency)² / Expected frequency )
χ² = ( (12-12)² / 12 ) + ( (15-16)² / 16 ) + ( (16-20)² / 20 ) + ( (18-20)² / 20 ) + ( (19-12)² / 12 )
χ² = (0² / 12) + (1² / 16) + (4² / 20) + (2² / 20) + (49² / 12)
χ² = 0 + 0.0625 + 0.8 + 0.2 + 20.4167
χ² = 21.4792
Degrees of Freedom = Number of categories - 1 = 5 - 1 = 4
Using a Chi-squared distribution table with 4 degrees of freedom and a significance level of 0.05, the critical value is approximately 9.488.
Since the test statistic (21.4792) is greater than the critical value (9.488), we reject the null hypothesis. This means that the observed frequencies do not match the expected frequencies and there is evidence to suggest that the claim is not accurate.