The average expenses of the Dela Cruz
family amount to P12,000 every half of the
month with a standard deviation of
P570. If their expenses are normally
distributed, what is the probability of the
family consuming less than P11,000 in half
a month?

To find the probability that the Dela Cruz family spends less than P11,000 in half a month, given the updated standard deviation, we will follow the same approach as before using the normal distribution.

### Given Information:

- Mean (\(\mu\)) = P12,000
- Standard deviation (\(\sigma\)) = P570
- We want to find \(P(X < 11,000)\), where \(X\) is the amount spent by the family in half a month.

### Step-by-Step Solution

1. **Calculate the z-score**:
   The z-score is calculated using the formula:

   \[
   z = \frac{X - \mu}{\sigma}
   \]

   For \(X = 11,000\):

   \[
   z = \frac{11,000 - 12,000}{570} = \frac{-1,000}{570} \approx -1.754
   \]

2. **Find the probability associated with the z-score**:
   Next, we need to find the probability that corresponds to \(z = -1.754\). This can be done using the standard normal distribution table or a calculator.

   Looking up \(z = -1.754\) in the standard normal distribution table, we find:

   \[
   P(Z < -1.754) \approx 0.0396
   \]

   This means there is a 3.96% probability that the Dela Cruz family will spend less than P11,000 in half a month.

### Conclusion

The probability of the Dela Cruz family consuming less than P11,000 in half a month is approximately **3.96%**.

---

Would you like to know more details about this calculation or have any questions?

Here are some related questions for further practice:

1. What is the probability that the Dela Cruz family spends more than P13,000 in half a month?
2. What is the probability that their expenses fall between P11,500 and P12,500 in half a month?
3. If the family’s expenses are more than P12,700, what is the probability of this happening?
4. How would the probability change if the standard deviation decreased to P400?
5. What is the probability that the family spends exactly P12,000?

**Tip:** To quickly determine probabilities for normally distributed data, learn to use the z-score and standard normal distribution tables or tools like statistical calculators or software.

Calculate the probability of the Dela Cruz family spending less than P11,000 in half a month using normal distribution and standard deviation concepts. Understand how to apply the z-score formula and interpret results in statistical analysis.

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