A machine in a factory is responsible for filling boxes of cereal. The weight of cereal in each box is has a mean of 500g and a variance of 20g. A box is picked at random. Calculate the probability that it contains less than 490g of cereal.

To calculate the probability that a randomly picked box contains less than 490g of cereal, we'll assume that the weights of the cereal boxes are normally distributed. We can use the properties of the normal distribution to find this probability.

### Step 1: Identify the Given Information
- Mean weight (\( \mu \)): 500g
- Variance (\( \sigma^2 \)): 20g
- Standard deviation (\( \sigma \)): \( \sqrt{20} \approx 4.47g \)
- The value we are interested in (\( X \)): 490g

### Step 2: Standardize the Value
To find the probability, we first need to standardize the value of 490g to a z-score using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
Substituting the values:
\[
z = \frac{490 - 500}{4.47} \approx \frac{-10}{4.47} \approx -2.24
\]

### Step 3: Find the Probability
Now, we need to find the probability corresponding to the z-score of -2.24. This is the probability that a standard normal variable is less than -2.24.

Using standard normal distribution tables or a calculator, we find:
\[
P(Z < -2.24) \approx 0.0125
\]

### Step 4: Conclusion
The probability that a randomly picked box contains less than 490g of cereal is approximately **0.0125** or **1.25%**.

---

If you have any further questions or need clarification, let me know! Here are some related questions you might be interested in:

1. What is the probability that a box contains more than 510g of cereal?
2. How would the probability change if the variance increased to 30g?
3. What is the probability that a box contains between 495g and 505g of cereal?
4. How do you find the z-score for a different weight, such as 495g?
5. What assumptions are made when using the normal distribution in this context?

**Tip:** Remember that the z-score tells you how many standard deviations away a value is from the mean. It helps in finding the probability for any value in a normal distribution.

Learn how to calculate the probability that a randomly picked box contains less than 490g of cereal using the properties of normal distribution. Understand the mean, variance, and standard deviation concepts. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation