The average expenses of the Dela Cruz
family amount to P12,000 every half of the
month with a standard deviation of
P1,570. 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, we need to use the properties of the normal distribution.

Given:
- Mean (μ) = P12,000
- Standard deviation (σ) = P1,570
- 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. **Convert the value to a z-score**:
   The z-score represents how many standard deviations a value is from the mean. The formula to calculate the z-score \( z \) for a value \( X \) is:

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

   Plugging in the values:

   \[
   z = \frac{11,000 - 12,000}{1,570} = \frac{-1,000}{1,570} \approx -0.637
   \]

2. **Find the probability associated with the z-score**:
   We now need to find the probability that corresponds to \( z = -0.637 \). This can be done using the standard normal distribution table or a calculator.

   Looking up \( z = -0.637 \) in the standard normal distribution table, we find:

   \[
   P(Z < -0.637) \approx 0.261
   \]

   This means there is a 26.1% 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 **26.1%**.

---

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 P13,500 in half a month?
2. What is the probability that their expenses fall between P11,000 and P12,500 in half a month?
3. If the family's expenses are more than P14,000, what is the probability of this happening?
4. How would the probability change if the standard deviation increased to P2,000?
5. What is the probability that the family spends exactly P12,000?

**Tip:** Understanding how to convert a value to a z-score is crucial for working with normally distributed data.

Explore how to calculate the probability of Dela Cruz family spending less than P11,000 in half a month using the properties of normal distribution. Given mean μ = P12,000 and standard deviation σ = P1,570.

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