The problem asks us to compute the expected frequencies for the given contingency table. The contingency table is as follows: | | Category A | Category B | |-----------|------------|------------| | Category C | 28 | 37 | | Category D | 16 | 14 |

1. Problem Description

The problem asks us to compute the expected frequencies for the given contingency table. The contingency table is as follows:

| | Category A | Category B |

|-----------|------------|------------|

| Category C | 28 | 37 |

| Category D | 16 | 14 |

2. Solution Steps

To find the expected frequencies for each cell in the contingency table, we use the following formula:

$E_{ij} = \frac{(\text{Row Total}) \times (\text{Column Total})}{\text{Grand Total}}$

First, calculate the row totals, column totals, and grand total.

Row Totals:

Category C: $28 + 37 = 65$

Category D: $16 + 14 = 30$

Column Totals:

Category A: $28 + 16 = 44$

Category B: $37 + 14 = 51$

Grand Total: $65 + 30 = 95$ (Also $44 + 51 = 95$)

Now, we calculate the expected frequencies for each cell:

Category C, Category A:

$E_{CA} = \frac{65 \times 44}{95} = \frac{2860}{95} = 30.1052631579 \approx 30.11$

Category C, Category B:

$E_{CB} = \frac{65 \times 51}{95} = \frac{3315}{95} = 34.8947368421 \approx 34.89$

Category D, Category A:

$E_{DA} = \frac{30 \times 44}{95} = \frac{1320}{95} = 13.8947368421 \approx 13.89$

Category D, Category B:

$E_{DB} = \frac{30 \times 51}{95} = \frac{1530}{95} = 16.1052631579 \approx 16.11$

3. Final Answer

The expected frequencies are:

Category C, Category A: 30.11

Category C, Category B: 34.89

Category D, Category A: 13.89

Category D, Category B: 16.11