*Academic advising:* In 2014, the Community College Survey of Student Engagement reported that 32% of the students surveyed rarely or never use academic advising services. Suppose that in reality, 42% of community college students rarely or never use academic advising services at their college. In a simulation we select random samples from this population. For each sample we calculate the proportion who rarely or never use academic advising services.

If we randomly sample 200 students from this population repeatedly, the standard error is approximately 3.5%. Is it unusual to see 32% who rarely or never use academic advising services in one of these samples?

Group of answer choices

Yes, this is unusual because 32% is 10% lower than 42%

No, this is unusual because the error is only 10%

Yes, this is unusual because 32% is 2.9 standard errors below 42%. It is very rare for a sample to be nearly three standard erors from the mean.

No, this is unusual because 32% is 2.9 standard errors below 42%.

Let's analyze the question step by step using statistical concepts:

1. **Population Proportion (\( p \)):** The true proportion of community college students who rarely or never use academic advising services is 42% (\( p = 0.42 \)).
   
2. **Sample Proportion (\( \hat{p} \)):** The sample proportion given in the problem is 32% (\( \hat{p} = 0.32 \)).

3. **Standard Error (SE):** The standard error measures the expected variability of the sample proportion from the population proportion. In this case, the standard error is given as 3.5% or \( 0.035 \).

4. **How Unusual is 32%?** To determine if the observed 32% is unusual, we calculate how many standard errors it is away from the population proportion (42%).

   - The formula for the **z-score** (how many standard errors a sample proportion is from the population proportion) is:

     \[
     z = \frac{\hat{p} - p}{SE}
     \]
     Substituting the values:
     \[
     z = \frac{0.32 - 0.42}{0.035} = \frac{-0.10}{0.035} \approx -2.86
     \]
     So, the sample proportion of 32% is about 2.86 standard errors below the population proportion of 42%.

5. **Conclusion:** Since a z-score of 2.86 is nearly 3 standard errors below the mean, this is considered unusual. In a normal distribution, a value that is more than 2 standard errors away from the mean is considered rare (typically less than 5% chance).

### Answer:
The correct option is:
**Yes, this is unusual because 32% is 2.9 standard errors below 42%. It is very rare for a sample to be nearly three standard errors from the mean.**

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Would you like further details on this concept, or have any questions?

Here are five related questions you may want to explore:
1. How is the standard error calculated for a proportion?
2. What does a z-score represent in the context of sample proportions?
3. What is considered an "unusual" event in statistics?
4. How can the sample size affect the standard error?
5. What is the difference between population proportion and sample proportion?

**Tip:** A z-score close to ±3 means the event is in the extreme tails of a normal distribution, implying it is very rare.

In 2014, the Community College Survey of Student Engagement reported that 32% of the students surveyed rarely or never use academic advising services. Suppose that in reality, 42% of community college students rarely or never use academic advising services at their college. In a simulation, we select random samples from this population. For each sample, we calculate the proportion who rarely or never use academic advising services. If we randomly sample 200 students from this population repeatedly, the standard error is approximately 3.5%. Is it unusual to see 32% who rarely or never use academic advising services in one of these samples?

Explore a statistical problem involving community college students using academic advising services. Learn how to calculate standard error, z-scores, and determine if a sample proportion of 32% is unusual given a population proportion of 42%. The standard error is 3.5%, and this problem illustrates concepts like the Central Limit Theorem and normal distribution.

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