To solve this problem, we need to find the probability that less than 30% of the freshmen in a sample are planning to major in a STEM discipline. This is typically done using the normal approximation to the binomial distribution, provided we have sufficient information about the population proportion (let’s denote it as $p$), the sample size ($n$), and the criteria for the approximation to be valid.

Here’s how we would proceed generally:

1. Identify the parameters:

   • $p$: the population proportion of freshmen planning to major in a STEM discipline.
   • $n$: the sample size.
   • We are interested in finding $P(\hat{p} < 0.30)$, where $\hat{p}$ is the sample proportion.

2. Calculate the mean and standard deviation:

   • The mean of the sampling distribution of $\hat{p}$ is $\mu_{\hat{p}} = p$.
   • The standard deviation of $\hat{p}$ is $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$.

3. Use the normal approximation:

   • If the sample size is large enough, we approximate the binomial distribution using the normal distribution. The condition for this is that both $np$ and $n(1-p)$ should be greater than 5.
   • We convert $0.30$ into a z-score: $z = \frac{0.30 - p}{\sigma_{\hat{p}}}$
   • Then, use the z-score to find the probability from the standard normal distribution.

If you can provide the values for the population proportion $p$ and the sample size $n$, I can walk you through the calculation in detail.

Would you like to provide these values or any other information about the problem?

Related Questions

1. What is the z-score and how is it used to find probabilities in a normal distribution?
2. How can the Central Limit Theorem help in approximating probabilities for sample proportions?
3. What conditions must be met for the normal approximation to the binomial distribution to be valid?
4. How do you interpret probabilities in the context of a sampling distribution?
5. What are other methods to calculate probabilities when the normal approximation isn't appropriate?

Tip

Always check whether the conditions for the normal approximation (like $np > 5$ and $n(1-p) > 5$) are met before applying it.