To determine the expected frequencies, we need to check if the two variables ("result of picking the Crush" and "bet placed the following week") are independent. We can use the formula for calculating expected frequency in a contingency table:

$\text{Expected Frequency} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$

Let's calculate each missing value one by one:

Given Data

Bet Placed the Following Week	Picked Crush to Win	Picked Crush to Lose	Total
Result of Picking the Crush
Crush Won	36	19	55
Crush Lost	37	8	45
Total	73	27	100

Missing Values Calculations

1. Cell for Crush Won / Picked Crush to Lose:

   Using the row total for "Crush Won" (55) and the column total for "Picked Crush to Lose" (27): $\frac{55 \times 27}{100} = 14.85$

2. Cell for Crush Lost / Picked Crush to Win:

   Using the row total for "Crush Lost" (45) and the column total for "Picked Crush to Win" (73): $\frac{45 \times 73}{100} = 32.85$

3. Cell for Crush Lost / Picked Crush to Lose:

   Using the row total for "Crush Lost" (45) and the column total for "Picked Crush to Lose" (27): $\frac{45 \times 27}{100} = 12.15$

Filled Table with Expected Frequencies

Bet Placed the Following Week	Picked Crush to Win	Picked Crush to Lose	Total
Result of Picking the Crush
Crush Won	36	14.85	55
Crush Lost	32.85	12.15	45
Total	73	27	100

Let me know if you’d like further clarification or have any questions!

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Here are five questions to deepen your understanding:

1. What is the meaning of “independence” in the context of a contingency table?
2. Why do we use expected frequencies to test for independence?
3. How would a Chi-square test confirm whether the variables are independent?
4. What would happen to expected values if the total number of bets increased?
5. How would these results be interpreted in a practical context, like betting behavior?

Tip: Expected frequencies are essential in hypothesis testing, especially when comparing observed data with a hypothesized distribution.