Let's work through these calculations:

Finding the value of $A$ (Observed Count Table)

In the observed count table, we know that the sum for the sassafras row should equal 64. Therefore, we can solve for $A$ by subtracting the known values in the sassafras row from the total:

$A = 64 - (24 + 16 + 16) = 64 - 56 = 8$

So, $A = 8$.

Finding the value of $B$ (Expected Count Table)

The expected count $B$ for blackwood in column 2 (16-21) can be calculated using the formula for expected frequency in contingency tables:

$\text{Expected Count}_{\text{(blackwood, 16-21)}} = \frac{(\text{Row Total for blackwood}) \times (\text{Column Total for 16-21})}{\text{Grand Total}}$

Plugging in the numbers:

$B = \frac{73 \times 36}{137} = \frac{2628}{137} \approx 19.175$

So, $B \approx 19.175$.

Final Answers

1. $A = 8$
2. $B \approx 19.175$

Would you like a breakdown of these calculations, or do you have further questions?

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Here are 5 related questions to deepen your understanding:

1. How does one calculate expected counts in a contingency table for any given cell?
2. What does the chi-squared value signify in each cell of the table?
3. Why are expected values and chi-squared tests useful in studying associations between categorical variables?
4. How would this analysis change if there were more categories for age?
5. What additional information can be derived by examining residuals in the chi-squared test?

Tip: In contingency tables, double-check sums and totals after calculating any missing values to ensure accuracy.