(c) Would it be unusual to observe 170 smokers who started smoking before turning 18 years old in a random sample of 200 adult smokers? A. Yes, because 170 is greater than $$ \mu+2 \sigma $$. B. Yes, because 170 is between $$ \mu-20 $$ and $$ \mu+2 \alpha $$ C. No, because 170 is less than $$ \mu-2 \alpha $$ D. Yes, because 170 is between $$ \mu-20 $$ and $$ \mu+2 \sigma $$. E. No, because 170 is between $$ \mu-2 \sigma $$ and $$ \mu+2 \sigma $$.

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To determine whether observing 170 smokers who started smoking before turning 18 years old in a random sample of 200 adult smokers is unusual, we need to consider the context of the problem, specifically the mean ($$ \mu $$) and standard deviation ($$ \sigma $$) of the distribution of adult smokers who started smoking before 18.

However, since the values of $$ \mu $$ and $$ \sigma $$ are not provided in the question, we can analyze the options based on the general statistical principles regarding unusual observations.

1. **Unusual Observations**: Typically, in statistics, an observation is considered unusual if it lies more than 2 standard deviations away from the mean (i.e., $$ \mu \pm 2\sigma $$).

2. **Sample Size**: In this case, we have a sample size of 200. If we denote the proportion of smokers who started before 18 as $$ p $$, then the expected number of smokers who started before 18 in a sample of 200 would be $$ 200p $$.

3. **Calculating the Thresholds**: If we assume that $$ p $$ is the proportion of smokers who started before 18, then:
   - The mean number of smokers who started before 18 would be $$ \mu = 200p $$.
   - The standard deviation for a binomial distribution can be calculated as $$ \sigma = \sqrt{200p(1-p)} $$.

4. **Evaluating 170**: To determine if 170 is unusual, we would check if it lies outside the range $$ \mu - 2\sigma $$ and $$ \mu + 2\sigma $$.

Given the options:

- **A**: Yes, because 170 is greater than $$ \mu + 2\sigma $$. (This could be true if $$ \mu $$ and $$ \sigma $$ are such that 170 exceeds the upper limit.)
- **B**: Yes, because 170 is between $$ \mu - 20 $$ and $$ \mu + 2\alpha $$. (This is not a standard way to evaluate unusualness.)
- **C**: No, because 170 is less than $$ \mu - 2\alpha $$. (This is not a standard way to evaluate unusualness.)
- **D**: Yes, because 170 is between $$ \mu - 20 $$ and $$ \mu + 2\sigma $$. (This is not a standard way to evaluate unusualness.)
- **E**: No, because 170 is between $$ \mu - 2\sigma $$ and $$ \mu + 2\sigma $$. (This is the correct interpretation if 170 falls within the usual range.)

Without specific values for $$ \mu $$ and $$ \sigma $$, we cannot definitively conclude. However, if we assume that 170 is indeed within the range of $$ \mu \pm 2\sigma $$, then the correct answer would be:

**E. No, because 170 is between $$ \mu - 2\sigma $$ and $$ \mu + 2\sigma $$.**

This indicates that observing 170 smokers who started smoking before 18 years old is not unusual.
170 smokers who started before 18 is not unusual if it falls within $$ \mu - 2\sigma $$ and $$ \mu + 2\sigma $$.

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