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caculate the mean y and tell me what is this 1000 there and when we multipy by it and why not

To calculate the mean of \( Y \) (the expenses in \( DA \times 10^3 \)) from the given contingency table, we will follow these steps:

### Step 1: Understanding the table and the notation

- The number of people per family is represented by \( X \) (2, 3, 4, 5).
- The different ranges of expenses \( Y \) are grouped into intervals: [25-35[, [35-45[, [45-55[, [55-65[ (in \( DA \times 10^3 \)).
- The numbers in the table represent the frequency of families that fall into each category based on their family size and the interval of their expenses.
- The total number of families surveyed is 30 (sum of all the frequencies in the table).

### Step 2: Filling in missing totals and calculating the marginal totals

We first need to calculate the marginal totals \( n_j \) for each column (representing the sum of frequencies for each interval of \( Y \)) and \( n_i \) for each row (representing the sum of frequencies for each family size \( X \)).

From the table, the observed frequencies are:
- For \( X = 2 \): [5, 2, 0, 0]
- For \( X = 3 \): [3, 7, 0, 0]
- For \( X = 4 \): [0, 1, 5, 3]
- For \( X = 5 \): [0, 0, 2, 2]

### Step 3: Calculating the Mean of \( Y \)

1. **Midpoints for each interval** of \( Y \) (in \( DA \times 10^3 \)) are calculated as follows:
   - For [25-35[: midpoint = \( 30 \)
   - For [35-45[: midpoint = \( 40 \)
   - For [45-55[: midpoint = \( 50 \)
   - For [55-65[: midpoint = \( 60 \)

2. **Sum of weighted frequencies**: We multiply the midpoint of each interval by the total number of families that fall into that interval.

Let \( n_j \) represent the total number of families in each interval:

- \( n_1 = 5 + 3 + 0 + 0 = 8 \)
- \( n_2 = 2 + 7 + 1 + 0 = 10 \)
- \( n_3 = 0 + 0 + 5 + 2 = 7 \)
- \( n_4 = 0 + 0 + 3 + 2 = 5 \)

Next, calculate the total expenses by summing the products of the midpoints and the total number of families in each interval:

\[
\text{Total expenses} = (30 \times 8) + (40 \times 10) + (50 \times 7) + (60 \times 5)
\]
\[
\text{Total expenses} = 240 + 400 + 350 + 300 = 1290
\]

3. **Mean of \( Y \)**:
\[
\text{Mean of } Y = \frac{\text{Total expenses}}{\text{Total number of families}} = \frac{1290}{30} = 43
\]

### Step 4: Interpreting the 1000 factor

The expenses \( Y \) are measured in \( DA \times 10^3 \), meaning that each value of \( Y \) is in thousands of dinars. The reason for multiplying by 1000 is to convert the units back to standard dinars (DA) rather than \( DA \times 10^3 \). 

If the mean \( Y \) calculated is 43, this means the mean expenses are \( 43 \times 10^3 = 43,000 \) DA.

### Conclusion:

- The mean of \( Y \) (expenses) is 43,000 DA (when scaled back to normal dinars by multiplying by 1000).
- The factor of 1000 is there because the expenses are grouped in thousands of dinars, so we need to multiply by 1000 when interpreting the actual dinar value.

Do you have

Caculate the mean y and tell me what is this 1000 there and when we multiply by it and why not

This exercise covers calculating the mean of family expenses in DA using frequency distribution tables. Understand how to interpret data in the table, calculate midpoints, and why we multiply by 1000 in the final step. Perfect for students in Grades 10-12.

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