For three independent samples, each with n= 100, the respective sample proportions are 0.30, 0.35, and 0.25. Use the 0.05 level in testing whether the three population proportions could be the same.
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To test whether the three population proportions could be the same, we can use the chi-square test for independence. Here's how to calculate it step by step:
Step 1: State the hypothesis:
H0 (null hypothesis): The three population proportions are equal.
HA (alternative hypothesis): At least one population proportion is different.
Step 2: Calculate the expected frequencies:
To do this, we need to assume that the null hypothesis is true and calculate the expected frequencies for each category. In this case, we have three categories (the three sample proportions), so we need to calculate the expected frequency for each category.
Expected frequency for each category = (total sample size) * (proportion under H0)
In this case, the total sample size is 100 for each sample. Under H0, since the three proportions are equal, the proportion under H0 is the average of the three proportions (0.30 + 0.35 + 0.25) / 3 = 0.30.
So, the expected frequency for each category is: 100 * 0.30 = 30.
Step 3: Calculate the chi-square statistic:
To calculate the chi-square statistic, we use the formula:
χ^2 = Σ((observed frequency - expected frequency)^2 / expected frequency)
In this case, we have three categories. Let's calculate the chi-square statistic for each category and sum them up.
Category 1: (0.30 * 100 - 30)^2 / 30 = 10
Category 2: (0.35 * 100 - 30)^2 / 30 = 6.67
Category 3: (0.25 * 100 - 30)^2 / 30 = 6.67
Summing up the three chi-square statistics, we get: χ^2 = 10 + 6.67 + 6.67 = 23.34.
Step 4: Determine the critical value:
To determine the critical value for a chi-square test with three categories and a significance level of α = 0.05, we need to consult a chi-square distribution table or use statistical software. For this example, the critical value is 7.815.
Step 5: Make a decision:
Compare the calculated chi-square statistic (23.34) with the critical value (7.815). If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, 23.34 > 7.815, so we reject the null hypothesis.
Step 6: State the conclusion:
We conclude that there is sufficient evidence to suggest that at least one of the population proportions is different.