Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5. Assume that the groups consist of 17 couples. Complete parts (a) through (c) below.
a. Find the mean and the standard deviation for the numbers of girls in groups of 17 births.
The value of the mean is = 8.5 .
(Type an integer or a decimal. Do not round.)
The value of the standard deviation is o = 2.1 .
(Round to one decimal place as needed.)
b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.
Values of girls or fewer are significantly low.
(Round to one decimal place as needed.)

Let's work through the problem step by step:

### Part (a): 
The number of girls in the group follows a binomial distribution, where:
- \( n = 17 \) (the number of couples or trials),
- \( p = 0.5 \) (the probability of a girl being born).

#### Mean:
The mean (\( \mu \)) of a binomial distribution is given by:
\[
\mu = n \cdot p
\]
Substitute the values:
\[
\mu = 17 \cdot 0.5 = 8.5
\]
So, the mean is \( \mu = 8.5 \), which matches the provided value.

#### Standard Deviation:
The standard deviation (\( \sigma \)) of a binomial distribution is given by:
\[
\sigma = \sqrt{n \cdot p \cdot (1 - p)}
\]
Substitute the values:
\[
\sigma = \sqrt{17 \cdot 0.5 \cdot 0.5} = \sqrt{17 \cdot 0.25} = \sqrt{4.25} \approx 2.1
\]
Thus, the standard deviation is approximately \( \sigma = 2.1 \), which also matches the given value.

### Part (b): Range Rule of Thumb
The range rule of thumb estimates "usual" results as those within 2 standard deviations of the mean. To find the values separating significantly low and significantly high results, use the following formulas:
- Significantly low: \( \mu - 2\sigma \)
- Significantly high: \( \mu + 2\sigma \)

#### Significantly Low:
\[
\mu - 2\sigma = 8.5 - 2 \cdot 2.1 = 8.5 - 4.2 = 4.3
\]
So, any number of girls less than or equal to **4.3** is considered significantly low.

#### Significantly High:
\[
\mu + 2\sigma = 8.5 + 2 \cdot 2.1 = 8.5 + 4.2 = 12.7
\]
So, any number of girls greater than or equal to **12.7** is considered significantly high.

### Final Answer for Part (b):
- **Significantly low** values are 4.3 or fewer girls.
- **Significantly high** values are 12.7 or more girls.

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Would you like more details on any part? 

Here are 5 additional questions related to this topic:
1. How do you calculate the probability of getting exactly 10 girls in a group of 17 births?
2. What is the probability of getting fewer than 5 girls in 17 births?
3. How would the mean and standard deviation change if the probability of having a girl increased to 0.6?
4. What is the cumulative probability of getting between 6 and 10 girls?
5. How would the values for significantly low and high numbers of girls change with a different sample size?

**Tip:** The standard deviation gives you a measure of how spread out the results are from the mean. In a binomial distribution, it depends on both the number of trials and the probabilities.

Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5. Assume that the groups consist of 17 couples. Complete parts (a) through (c) below. a. Find the mean and the standard deviation for the numbers of girls in groups of 17 births. The value of the mean is = 8.5. The value of the standard deviation is 2.1. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

Learn how to calculate the mean and standard deviation for a group of 17 births with a 50% probability of having a girl using binomial distribution. Includes the application of the range rule of thumb to find significantly low and high values. Suitable for Grades 10-12 students.

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