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To answer this question, let's analyze the given information and perform a statistical calculation to determine if it would be unusual for less than 21% of a sample of 120 freshmen to major in STEM.

### Steps to Solve:
1. **Identify Key Data:**
   - Population proportion (\( p \)) = 21% = 0.21
   - Sample size (\( n \)) = 120
   - Threshold to find: probability of observing less than 21% in the sample.

2. **Assume the sampling distribution of the sample proportion \( \hat{p} \) is approximately normal:**
   - The mean of \( \hat{p} \) (sampling distribution) is equal to \( p = 0.21 \).
   - The standard deviation of \( \hat{p} \) is given by:
     \[
     \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.21 \times 0.79}{120}}
     \]

3. **Calculate the Standard Deviation:**
   \[
   \sigma_{\hat{p}} \approx \sqrt{\frac{0.1659}{120}} \approx \sqrt{0.0013825} \approx 0.0372
   \]

4. **Find the Probability:**
   - The question asks if it would be unusual to observe less than 21% (or \( p = 0.21 \)), which is exactly the mean of our distribution. This means we need to find the probability of observing a proportion less than 0.21.
   - Since 0.21 is the mean of the distribution, the probability of observing less than 0.21 is exactly 50%, which is a typical value in a normal distribution.
   
Thus, **it would not be unusual** to observe less than 21%, as the probability is about 50%. This indicates that the observation falls at the mean of the normal distribution, so it is not unusual. 

If you want to calculate the exact probability of observing significantly lower values (such as a specific proportion less than 0.21), a Z-score calculation would be needed. However, based on this, it's reasonable to conclude it would not be unusual.

Would you like a more detailed breakdown or further explanation? Here are some follow-up questions:

1. How do we calculate the probability of observing a proportion significantly lower than 21%?
2. What Z-score would represent a value significantly lower than 21% in this context?
3. How does sample size affect the standard deviation of the sample proportion?
4. What does "unusual" typically mean in statistical terms regarding probability?
5. How would the answer change if the population proportion was different from 21%?

**Tip:** For small sample sizes or when conditions for normal approximation aren't met, consider using exact binomial probabilities instead of normal approximation.

Would it be unusual if less than 21% of the freshmen in the sample of 120 were planning to major in a STEM discipline?

Determine whether it would be unusual for less than 21% of freshmen in a sample of 120 to plan for STEM majors using normal distribution and probability theory. Includes calculations for population proportion and sampling distribution with a standard deviation of 0.0372.

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