Questions: Question 6, 10.1.13 HW Score: 19.05%, 4 of 21 points Part 4 of 5 Points: 0 of 1 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. A certain junior college believes that the proportion of students who enroll in her institution have 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. Among students who enroll at the certain junior college, the completion rate is greater than 0.389 Among students who first enroll in two-year public institutions, the completion rate is 0.389 Among students who enroll at the certain junior college, the completion rate is less than 0.389 (b) State the hypotheses symbolically. H0: p=0.389 H1: p<0.389 (Type integers or decimals. Do not round) (c) What would it mean to make a Type I error? The president fails to reject the hypothesis that the proportion of students who earn a bachelor's degree within six years is

Question 6, 10.1.13 HW Score: 19.05%, 4 of 21 points Part 4 of 5 Points: 0 of 1 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. A certain junior college believes that the proportion of students who enroll in her institution have 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. Among students who enroll at the certain junior college, the completion rate is greater than 0.389 Among students who first enroll in two-year public institutions, the completion rate is 0.389 Among students who enroll at the certain junior college, the completion rate is less than 0.389 (b) State the hypotheses symbolically. H0: p=0.389 H1: p<0.389 (Type integers or decimals. Do not round) (c) What would it mean to make a Type I error? The president fails to reject the hypothesis that the proportion of students who earn a bachelor's degree within six years is
Transcript text: Question 6, 10.1.13 HW Score: 19.05%, 4 of 21 points Part 4 of 5 Points: 0 of 1 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. A certain junior college believes that the proportion of students who enroll in her institution have 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. Among students who enroll at the certain junior college, the completion rate is greater than 0.389 Among students who first enroll in two-year public institutions, the completion rate is 0.389 Among students who enroll at the certain junior college, the completion rate is less than 0.389 (b) State the hypotheses symbolically. $H_{0}: p=0.389$ $H_{1}: p<0.389$ (Type integers or decimals. Do not round) (c) What would it mean to make a Type I error? The president fails to reject the hypothesis that the proportion of students who earn a bachelor's degree within six years is
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Solution

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Solution Steps

Step 1: Hypothesis Formulation

We are testing the completion rate of students who enroll in a certain junior college. The hypotheses are formulated as follows:

  • Null Hypothesis (\(H_0\)): The proportion of students who earn a bachelor's degree within six years is \(p = 0.389\).
  • Alternative Hypothesis (\(H_1\)): The proportion of students who earn a bachelor's degree within six years is less than \(p = 0.389\).
Step 2: Test Statistic Calculation

The test statistic is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values, we find:

\[ Z = -0.8 \]

Step 3: P-value and Critical Region

The calculated P-value for the test statistic is:

\[ \text{P-value} = 0.2119 \]

The critical region for a significance level of \(\alpha = 0.05\) in a left-tailed test is defined as:

\[ Z < -1.6449 \]

Step 4: Type I Error Explanation

A Type I error occurs when we reject the null hypothesis when it is actually true. In this context, it means concluding that the completion rate is less than \(0.389\) when it is, in fact, \(0.389\).

Final Answer

  • (a) Null Hypothesis: \(H_0: p = 0.389\)
  • (b) Alternative Hypothesis: \(H_1: p < 0.389\)
  • (c) Type I Error: Rejecting \(H_0\) when it is true.

Thus, the final boxed answers are:

\[ \boxed{H_0: p = 0.389} \] \[ \boxed{H_1: p < 0.389} \] \[ \boxed{\text{Type I Error: Rejecting } H_0 \text{ when it is true.}} \]

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