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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.

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Part 1

Use the range rule of thumb to identify a range of values that are not significant.

The maximum value in this range is

enter your response here

girls.

​(Round to one decimal place as​ needed.)

To determine whether 1 girl in 10 births is a significantly low number, we will apply the "range rule of thumb." According to this rule, values are considered significantly low or high if they fall more than 2 standard deviations from the mean.

### Steps:

1. **Calculate the mean (expected value)**:
   The mean, \( \mu \), of a probability distribution is calculated using:
   \[
   \mu = \sum \left( x \times P(x) \right)
   \]
   where \( x \) is the number of girls and \( P(x) \) is the probability associated with \( x \).

2. **Calculate the standard deviation**:
   The standard deviation, \( \sigma \), is given by:
   \[
   \sigma = \sqrt{\sum \left( (x - \mu)^2 \times P(x) \right)}
   \]
   
3. **Determine the range of non-significant values**:
   Using the range rule of thumb:
   \[
   \text{Lower bound} = \mu - 2\sigma
   \]
   \[
   \text{Upper bound} = \mu + 2\sigma
   \]

Let's start by calculating the mean and standard deviation from the table data.

### Step 1: Mean Calculation
We calculate \( \mu \) using the following formula:
\[
\mu = (0 \times 0.005) + (1 \times 0.016) + (2 \times 0.042) + \dots + (10 \times 0.016)
\]

### Step 2: Standard Deviation Calculation
Next, we'll calculate the variance:
\[
\sigma^2 = \sum (x - \mu)^2 \times P(x)
\]
And then, take the square root to get \( \sigma \).

I'll now proceed with these calculations to get the mean, standard deviation, and the range of values that are not considered significant.

Let me calculate that for you.### Results:

1. **Mean (Expected value)**: \( \mu = 5.06 \) (rounded to two decimal places)
2. **Standard deviation**: \( \sigma = 1.74 \) (rounded to two decimal places)
3. **Range of non-significant values**: 
   \[
   \text{Lower bound} = 1.57
   \]
   \[
   \text{Upper bound} = 8.54
   \]

### Conclusion:
According to the range rule of thumb, the number of girls that would not be considered significant lies between approximately **1.57** and **8.54**. Since 1 girl falls below the lower bound of 1.57, it can be considered **significantly low**.

Would you like more details on any of these steps or any clarifications?

### Related Questions:
1. How is the range rule of thumb applied in other distributions?
2. What does a significantly high number of girls in 10 births look like?
3. How do mean and standard deviation affect the significance of a value?
4. What other methods can be used to determine significant deviations in probability?
5. How does the binomial distribution relate to this problem?

**Tip:** Always double-check that the sum of probabilities adds up to 1 in a probability distribution to ensure accuracy!

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.

Use the range rule of thumb to determine if 1 girl out of 10 births is a significantly low number. This problem involves calculating the mean and standard deviation of a probability distribution. The solution covers key probability concepts, including the expected value and the range of non-significant values.

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