Among the four northwestern states, Washington has 51% of the total population, Oregon has 30%, Idaho has 11%, and Montana has 8%. A market researcher selects a sample of 1000 subjects, with 450 in Washington, 340 in Oregon, 150 in Idaho, and 60 in Montana. At the 0.05 significance level, test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state populations..
To test the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state populations, we can perform a chi-square goodness-of-fit test. This test compares the observed frequencies in the sample to the expected frequencies based on the population distribution.
Here are the steps to perform the chi-square test:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The sample distribution is the same as the population distribution.
- Alternative hypothesis (Ha): The sample distribution is not the same as the population distribution.
Step 2: Determine the expected frequencies:
To determine the expected frequencies, we need to calculate the number of subjects we would expect to find in each state based on the population distribution. We can obtain the expected frequencies by multiplying the total sample size (1000) by the proportion of each state's population.
Expected frequency for Washington = 1000 * (51/100) = 510
Expected frequency for Oregon = 1000 * (30/100) = 300
Expected frequency for Idaho = 1000 * (11/100) = 110
Expected frequency for Montana = 1000 * (8/100) = 80
Step 3: Set up the chi-square test statistic:
The chi-square test statistic is calculated as the sum of the squared difference between observed and expected frequencies, divided by their respective expected frequencies, for all categories.
χ2 = ∑ ((Observed - Expected)^2 / Expected)
Step 4: Calculate the chi-square test statistic:
For each state, calculate the (Observed - Expected)^2 / Expected. Then sum up the results to obtain the chi-square test statistic.
For Washington:
((450 - 510)^2 / 510) = 8.82
For Oregon:
((340 - 300)^2 / 300) = 4
For Idaho:
((150 - 110)^2 / 110) = 16.36
For Montana:
((60 - 80)^2 / 80) = 10
Sum of the chi-square values = 8.82 + 4 + 16.36 + 10 = 39.18
Step 5: Determine the degrees of freedom:
The degrees of freedom (df) in a chi-square goodness-of-fit test is calculated as the number of categories minus 1.
In this case, there are 4 categories (states), so df = 4 - 1 = 3.
Step 6: Determine the critical value:
At the 0.05 significance level and 3 degrees of freedom, the critical value can be found in the chi-square distribution table. Looking up the critical value, we find it to be 7.815.
Step 7: Compare the test statistic to the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the test statistic is 39.18, which is greater than the critical value of 7.815. Therefore, we reject the null hypothesis.
Step 8: Interpret the results:
Since we reject the null hypothesis, we have evidence to suggest that the distribution of the sample is significantly different from the distribution of the population.
In conclusion, based on the chi-square goodness-of-fit test, we can reject the claim that the sample of 1000 subjects has a distribution that agrees with the distribution of state populations.