Expected Counts in Two-Way Tables - AP Statistics

1. What is the expected count if Row Total = 55, Column Total = 25, and Grand Total = 250?

   Expected Count = 5.5. Calculate: $\frac{55 \times 25}{250} = \frac{1375}{250} = 5.5$

2. Identify the formula to compute the expected count for a cell in a contingency table.

   Expected Count = $\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$. Standard formula for expected frequencies in contingency tables under independence assumption.

3. How do observed counts relate to expected counts in hypothesis testing?

   Observed counts are compared to expected counts to test independence. Large differences between observed and expected suggest dependence between variables.

4. What conclusion is drawn if observed counts significantly differ from expected counts?

   There may be an association between the variables. Significant differences suggest the variables are not independent of each other.

5. If a table's row total is 25 and column total is 10 with grand total 100, what is the expected count?

   Expected Count = 2.5. Calculate: $\frac{25 \times 10}{100} = \frac{250}{100} = 2.5$

6. What is the expected count for a cell with Row Total = 100, Column Total = 25, Grand Total = 1000?

   Expected Count = 2.5. Calculate: $\frac{100 \times 25}{1000} = \frac{2500}{1000} = 2.5$

7. What is the expected count if Row Total = 60, Column Total = 20, and Grand Total = 240?

   Expected Count = 5. Calculate: $\frac{60 \times 20}{240} = \frac{1200}{240} = 5$

8. Which condition must be met for expected counts in a chi-square test?

   Each expected count should be at least 5. This ensures the chi-square distribution approximation is valid for the test.

9. How do you calculate the expected count for a cell in a two-way table?

   Use Expected Count = $\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$. Apply this formula to find theoretical frequency for any cell in a two-way table.

10. What expected count results from Row Total = 25, Column Total = 40, Grand Total = 200?

    Expected Count = 5. Calculate: $\frac{25 \times 40}{200} = \frac{1000}{200} = 5$

11. Calculate the expected count for a cell: Row Total = 75, Column Total = 50, Grand Total = 300.

    Expected Count = 12.5. Calculate: $\frac{75 \times 50}{300} = \frac{3750}{300} = 12.5$

12. Calculate the expected count for a cell with Row Total = 50 and Column Total = 30 in a 150 total table.

    Expected Count = 10. Calculate: $\frac{50 \times 30}{150} = \frac{1500}{150} = 10$

13. What is the expected count if Row Total is 40, Column Total is 50, and Grand Total is 200?

    Expected Count = 10. Calculate: $\frac{40 \times 50}{200} = \frac{2000}{200} = 10$

14. What is the expected count for a cell with Row Total = 50, Column Total = 100, Grand Total = 1000?

    Expected Count = 5. Calculate: $\frac{50 \times 100}{1000} = \frac{5000}{1000} = 5$

15. What expected count results from Row Total = 80, Column Total = 20, Grand Total = 1600?

    Expected Count = 1. Calculate: $\frac{80 \times 20}{1600} = \frac{1600}{1600} = 1$

16. Calculate the expected count for a cell: Row Total = 45, Column Total = 15, Grand Total = 135.

    Expected Count = 5. Calculate: $\frac{45 \times 15}{135} = \frac{675}{135} = 5$

17. Determine the expected count for a cell with Row Total = 70, Column Total = 45, Grand Total = 315.

    Expected Count = 10. Calculate: $\frac{70 \times 45}{315} = \frac{3150}{315} = 10$

18. How are expected counts used in determining statistical significance?

    They are compared with observed counts to compute the chi-square statistic. The chi-square statistic measures how much observed counts deviate from expected counts.

19. Find the expected count: Row Total = 50, Column Total = 50, Grand Total = 500.

    Expected Count = 5. Calculate: $\frac{50 \times 50}{500} = \frac{2500}{500} = 5$

20. What is the interpretation of an expected count in a two-way table?

    The expected count is the theoretical frequency of a cell if there's no association. Represents what the cell frequency would be if the variables were independent.

21. Find the expected count: Row Total = 60, Column Total = 30, Grand Total = 600.

    Expected Count = 3. Calculate: $\frac{60 \times 30}{600} = \frac{1800}{600} = 3$

22. In which situation would you use expected counts in a two-way table?

    To test the independence of two categorical variables. Expected counts help determine if categorical variables are independent or associated.

23. What is the expected count for a cell with Row Total = 80, Column Total = 20, Grand Total = 200?

    Expected Count = 8. Calculate: $\frac{80 \times 20}{200} = \frac{1600}{200} = 8$

24. What expected count results from Row Total = 36, Column Total = 12, Grand Total = 300?

    Expected Count = 1.44. Calculate: $\frac{36 \times 12}{300} = \frac{432}{300} = 1.44$

25. Find the expected count: Row Total = 120, Column Total = 40, Grand Total = 240.

    Expected Count = 20. Calculate: $\frac{120 \times 40}{240} = \frac{4800}{240} = 20$

26. Determine the expected count for a cell with Row Total = 35, Column Total = 35, Grand Total = 245.

    Expected Count = 5. Calculate: $\frac{35 \times 35}{245} = \frac{1225}{245} = 5$

27. What does a high chi-square statistic indicate about expected vs. observed counts?

    A high chi-square suggests a significant difference between observed and expected counts. Large chi-square values indicate substantial deviation from independence assumption.

28. If a table's row total is 90 and column total is 40 with grand total 360, what is the expected count?

    Expected Count = 10. Calculate: $\frac{90 \times 40}{360} = \frac{3600}{360} = 10$

29. If a table's row total is 100 and column total is 25 with grand total 500, what is the expected count?

    Expected Count = 5. Calculate: $\frac{100 \times 25}{500} = \frac{2500}{500} = 5$

30. What result would indicate no association between variables in a two-way table?

    Observed counts are close to expected counts. Similar observed and expected counts suggest the variables are independent.

31. Which statistical test utilizes expected counts in a two-way table?

    Chi-square test of independence. This test compares observed frequencies to expected frequencies to assess independence.

32. Which values are needed to calculate expected counts in a two-way table?

    Row Total, Column Total, Grand Total. These three marginal totals are required components for the expected count formula.

33. What does it mean if expected counts match observed counts?

    It suggests that the variables are likely independent. Close agreement indicates no significant association between the categorical variables.

34. Find the expected count: Row Total = 30, Column Total = 60, Grand Total = 180.

    Expected Count = 10. Calculate: $\frac{30 \times 60}{180} = \frac{1800}{180} = 10$

35. What is the formula for expected count in a two-way table cell?

    Expected Count = $\frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}$. This formula calculates theoretical frequency assuming independence between row and column variables.

36. Find the expected count: Row Total = 90, Column Total = 10, Grand Total = 300.

    Expected Count = 3. Calculate: $\frac{90 \times 10}{300} = \frac{900}{300} = 3$

37. Identify the effect of increasing sample size on expected counts.

    Larger sample sizes generally increase expected counts. More data points increase the marginal totals, which typically increases expected frequencies.

38. Find the expected count: Row Total = 45, Column Total = 15, Grand Total = 90.

    Expected Count = 7.5. Calculate: $\frac{45 \times 15}{90} = \frac{675}{90} = 7.5$

Study these flashcards