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To calculate the mean of \( Y \) for \( X = 4 \), we will:

1. Extract the conditional distribution of \( Y \) given \( X = 4 \).
2. Calculate the weighted average based on the number of occurrences.

From the table:

For \( X = 4 \) (row 4), the values of \( Y \) are:
- \( 1 \) family spends in the range \( [35, 45[ \),
- \( 5 \) families spend in the range \( [45, 55[ \),
- \( 3 \) families spend in the range \( [55, 65[ \).

The midpoints of each range for \( Y \) in thousands are:
- Midpoint of \( [35, 45[ \) is \( 40 \).
- Midpoint of \( [45, 55[ \) is \( 50 \).
- Midpoint of \( [55, 65[ \) is \( 60 \).

Now we calculate the mean:

\[
\text{Mean of } Y = \frac{1 \times 40 + 5 \times 50 + 3 \times 60}{1 + 5 + 3}
\]

Let's calculate that now.The mean value of \( Y \) (the expenses in thousands) for families where \( X = 4 \) (i.e., 4 members) is approximately 52.22 (in thousands).

Would you like to go over any specific details or have further questions? Here are five related questions to expand on this:

1. How is the mean value of a variable typically interpreted in statistics?
2. What is the importance of calculating conditional distributions in statistics?
3. Can we assume independence between \( X \) and \( Y \) based on this data?
4. What methods can be used to verify the independence of two variables in contingency tables?
5. How would the mean change if we included a different family size?

**Tip**: When working with contingency tables, always ensure you are considering the correct conditional distributions for accurate interpretations.

Les variables X et Y sont-elles indépendantes ? Donner la distribution conditionnelle de Y sachant X = 4. Et déduire les dépenses moyennes des familles de 4 personnes.

Learn how to determine the independence of variables X (number of persons per family) and Y (food expenses during Ramadan) using a contingency table. Calculate the conditional distribution of Y given X = 4 and find the average food expenses for families with 4 members. This detailed guide covers key statistical concepts, including contingency tables, conditional probability, and mean calculation, suitable for Grades 10-12.

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