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Let's break down the problem:

**a. How many out of 300 would you expect to have an anxiety disorder if the 18% is correct?**

If 18% of adult Americans suffer from an anxiety disorder, then we can expect 18% of 300 random college students to also have an anxiety disorder.

The expected number is:
\[
\text{Expected number} = 0.18 \times 300 = 54
\]
So, we would expect **54 students** out of 300 to have an anxiety disorder if 18% is correct.

---

**b. Hypothesis Testing**

We are testing the hypothesis that the population proportion of college students suffering from an anxiety disorder is not 0.18 at the 0.05 significance level. This is a two-tailed hypothesis test.

- Null Hypothesis (\(H_0\)): \( p = 0.18 \)
- Alternative Hypothesis (\(H_a\)): \( p \neq 0.18 \)

Now, the sample proportion (\( \hat{p} \)) is calculated as:
\[
\hat{p} = \frac{50}{300} = 0.1667
\]
The standard error (SE) of the sampling distribution for proportions is given by:
\[
SE = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.18 \times (1 - 0.18)}{300}} = \sqrt{\frac{0.18 \times 0.82}{300}} \approx 0.022
\]

Next, we compute the test statistic (z-value):
\[
z = \frac{\hat{p} - p}{SE} = \frac{0.1667 - 0.18}{0.022} \approx -0.606
\]
Using a z-table or a standard normal distribution, we find the p-value associated with \( z = -0.606 \).

Since this is a two-tailed test, we double the p-value. The p-value turns out to be **greater than 0.05** (approximately 0.544).

---

**Conclusion**:
- Since the p-value is larger than 0.05, we **fail to reject the null hypothesis**. This means that we do not have sufficient evidence to say that the proportion of college students with anxiety disorders is different from 18%.

---

Would you like further details on hypothesis testing or any other concepts from this problem?

Here are 5 related questions to think about:
1. What is the interpretation of a p-value in hypothesis testing?
2. How does the significance level affect the decision in a hypothesis test?
3. What assumptions must hold for the standard error formula used in proportion tests?
4. How does sample size affect the results of a hypothesis test?
5. What would happen if the sample proportion were exactly 0.18?

**Tip**: The closer your sample proportion is to the hypothesized population proportion, the higher the chance you will fail to reject the null hypothesis, assuming other factors like sample size remain constant.

Anxiety Disorders According to Time magazine (December 5, 2011), 18% of adult Americans suffer from an anxiety disorder. Suppose a test of 300 random college students showed that 50 suffered from an anxiety disorder. a. How many out of 300 would you expect to have an anxiety disorder if the 18% is correct? b. Suppose you are testing the hypothesis that the population proportion of college students suffering from an anxiety disorder is not 0.18 at the 0.05 significance level. Choose the correct figure and interpret the p-value.

Learn how to perform hypothesis testing for the proportion of college students with anxiety disorders. This problem involves testing whether the proportion of students with anxiety is different from 18% using a significance level of 0.05. It includes detailed calculations of expected values, standard error, and p-value interpretation.

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