Q:The table below shows a distribution and the observed frequencies of the values of a variable from a simple random sample of a population. Perform a chi-square goodness-of-fit test, at the specified significance level, to decide whether the distribution of the variable differs from the given distribution.
Distribution: 0.2 0.2 0.1 0.2 0.3
Observed freq: 37 15 12 23 43
significance level=0.05
To perform a chi-square goodness-of-fit test in this scenario, we need to follow several steps.
Step 1: Formulate hypotheses
- Null Hypothesis (H0): The observed distribution of the variable is the same as the given distribution.
- Alternative Hypothesis (Ha): The observed distribution of the variable differs from the given distribution.
Step 2: Determine the critical value
- The critical value is based on the significance level and the degrees of freedom.
Step 3: Calculate the test statistic
- The test statistic in this case is the chi-square statistic.
Step 4: Compare the test statistic with the critical value
- If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is evidence to suggest that the observed distribution differs from the given distribution.
Now, let's calculate the test statistic and determine the critical value.
Step 1: Formulate hypotheses
- H0: The observed distribution of the variable is the same as the given distribution.
- Ha: The observed distribution of the variable differs from the given distribution.
Step 2: Determine the critical value
- The significance level is given as 0.05.
- The degrees of freedom for this test are the number of categories (k) minus 1.
- In this case, there are 5 categories, so the degrees of freedom (df) = 5 - 1 = 4.
- We can use a chi-square table or a statistical software to find the critical value for the given significance level and degrees of freedom.
- For a significance level of 0.05 and 4 degrees of freedom, the critical value is approximately 9.488.
Step 3: Calculate the test statistic
- The formula for calculating the chi-square statistic is:
X2 = Σ((Observed frequency - Expected frequency)2 / Expected frequency)
First, we need to calculate the expected frequencies for each category. The expected frequency for each category is calculated by multiplying the given distribution by the total number of observations.
Expected freq = given distribution * sum of observed frequencies
Expected freq = (0.2, 0.2, 0.1, 0.2, 0.3) * (37 + 15 + 12 + 23 + 43)
Expected freq = (7.4, 7.4, 3.7, 14.8, 22.2)
Next, we calculate the chi-square statistic:
X2 = [(37 - 7.4)2 / 7.4] + [(15 - 7.4)2 / 7.4] + [(12 - 3.7)2 / 3.7] + [(23 - 14.8)2 / 14.8] + [(43 - 22.2)2 / 22.2]
X2 = 7.297 + 1.973 + 6.703 + 1.562 + 10.811
X2 = 28.346
Step 4: Compare the test statistic with the critical value
- The test statistic (28.346) is greater than the critical value (9.488).
- Therefore, we reject the null hypothesis and conclude that there is evidence to suggest that the observed distribution differs from the given distribution.
In summary, based on the chi-square goodness-of-fit test, the observed distribution of the variable differs significantly from the given distribution at a significance level of 0.05.