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.