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The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls.


To determine whether 1 girl in 10 births is a significantly low number of girls, we will apply the **range rule of thumb**. This rule states that most values should lie within two standard deviations from the mean. Values outside this range can be considered significantly low or high.

### Steps:
1. **Mean (μ):** The mean of a probability distribution is calculated as:
   \[
   \mu = \sum (x \cdot P(x))
   \]
   where \(x\) is the number of girls and \(P(x)\) is the probability of getting that number of girls.

2. **Standard deviation (σ):** The standard deviation for a probability distribution is calculated using:
   \[
   \sigma = \sqrt{\sum \left[ (x - \mu)^2 \cdot P(x) \right]}
   \]

3. **Range Rule of Thumb:** We calculate the significantly low value as:
   \[
   \text{Low threshold} = \mu - 2\sigma
   \]
   If the number of girls (1 in this case) is below this threshold, it is considered significantly low.

### Calculation:
Let me calculate these values for you based on the given probabilities.

#### 1. Calculate Mean (μ):
\[
\mu = (0 \times 0.005) + (1 \times 0.014) + (2 \times 0.037) + (3 \times 0.114) + (4 \times 0.203) + (5 \times 0.236) + (6 \times 0.196) + (7 \times 0.113) + (8 \times 0.045) + (9 \times 0.019) + (10 \times 0.018)
\]

#### 2. Calculate Standard Deviation (σ):
\[
\sigma = \sqrt{ \sum \left[ (x - \mu)^2 \cdot P(x) \right] }
\]

Let me calculate these steps for you.Here are the results from the calculations:

- **Mean (μ):** 5.1
- **Standard deviation (σ):** 1.77
- **Low threshold (μ - 2σ):** 1.56

Based on the **range rule of thumb**, a significantly low number of girls would be any number below 1.56. Since 1 girl is below this threshold, we can conclude that **1 girl in 10 births is a significantly low number**.

Would you like more details, or do you have any questions about this? Here are some related questions you might want to explore:

1. How would you interpret the significance of more than 8 girls in 10 births?
2. What is the probability of getting exactly 5 girls in this distribution?
3. How does the standard deviation affect the interpretation of "significant" results?
4. Can you use this method for any number of births, or are there limitations?
5. What are other ways to determine if a value is significant?

**Tip:** The range rule of thumb is a quick method, but a more precise evaluation would involve directly comparing probabilities using cumulative distribution or binomial tests.

The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls.

Learn how to determine whether 1 girl in 10 births is significantly low using the range rule of thumb. This problem covers key probability concepts, including calculating the mean, standard deviation, and using the range rule of thumb for statistical significance. Suitable for high school students in Grades 10-12.

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