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### Problem Summary
- We are given that the proportion of students who earned a bachelor’s degree within six years from two-year public institutions is 0.389.
- A president of a junior college believes the proportion of students in her institution is lower than 0.389.

We need to address the following questions:

#### (a) State the null and alternative hypotheses in words.
- **Null Hypothesis (H₀)**: The proportion of students in the junior college who earn a bachelor's degree within six years is equal to 0.389.
- **Alternative Hypothesis (H₁)**: The proportion of students in the junior college who earn a bachelor's degree within six years is less than 0.389.

#### (b) State the hypotheses symbolically.
- **H₀**: \( p = 0.389 \)
- **H₁**: \( p < 0.389 \)

#### (c) Explain what it would mean to make a Type I error.
- **Type I Error**: A Type I error occurs when we reject the null hypothesis when it is actually true. In this context, it would mean that the president concludes the proportion of students who earn a bachelor’s degree within six years is less than 0.389, when in reality, the proportion is actually 0.389.

#### (d) Explain what it would mean to make a Type II error.
- **Type II Error**: A Type II error occurs when we fail to reject the null hypothesis when the alternative hypothesis is true. In this context, it would mean that the president concludes the proportion of students who earn a bachelor’s degree within six years is not less than 0.389 (i.e., equal to or greater than 0.389), when in reality, the proportion is less than 0.389.

Let's fill in the information accordingly in the image you provided:

1. **(b) Hypotheses Symbolically:**
   - **H₀**: \( p = 0.389 \)
   - **H₁**: \( p < 0.389 \)

2. **(c) Type I Error:**
   - The president **rejects** the hypothesis that the proportion of students who earn a bachelor's degree within six years is **0.389** when, in fact, the proportion is **0.389**.

3. **(d) Type II Error:**
   - The president **fails to reject** the hypothesis that the proportion of students who earn a bachelor's degree within six years is **0.389** when, in fact, the proportion is **less than 0.389**.

Would you like any additional details or have any further questions?

### Relative Questions:
1. What is the significance level and how does it affect Type I and Type II errors?
2. How would you calculate the test statistic for this hypothesis test?
3. What sample size is required to achieve a desired power for detecting a smaller proportion?
4. How do confidence intervals relate to hypothesis testing in this scenario?
5. What is the role of p-value in determining the rejection of the null hypothesis?

### Tip:
In hypothesis testing, always clearly define the hypotheses and understand the implications of Type I and Type II errors for making informed decisions.

For students who first enrolled in two-year public institutions in a recent semester, the proportion who earned a bachelor’s degree within six years was 0.389. The president of a certain junior college believes that the proportion of students who enroll in her institution has a lower completion rate. (a) State the null and alternative hypotheses in words. (b) State the null and alternative hypotheses symbolically. (c) Explain what it would mean to make a Type I error. (d) Explain what it would mean to make a Type II error.

This problem explores hypothesis testing with a focus on proportions and understanding Type I and Type II errors. The scenario involves analyzing whether the completion rate of students in a junior college is lower than a standard proportion. Suitable for college-level students studying statistics.

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