Questions: A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is 69%. During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 153 passed the course. The instructor carried out a hypothesis test and found that the observed value of the test statistic was 2.29. The p-value associated with this test statistic is 0.0109. Explain the meaning of the p-value in this context. Based on this result, should the instructor believe the success rate has improved? State the hypotheses that were used for the test. Choose the correct answer below. A. H0: p<0.69 and Ha: p>0.69 B. H0: p=0.69 and H3: p<0.69 C. H0: p>0.69 and H3: p<0.69 D. H0: p=0.69 and Ha: p>0.69 E. H0: p=0.69 and Ha: p ≠ 0.69

A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is 69%. During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 153 passed the course. The instructor carried out a hypothesis test and found that the observed value of the test statistic was 2.29. The p-value associated with this test statistic is 0.0109. Explain the meaning of the p-value in this context. Based on this result, should the instructor believe the success rate has improved?

State the hypotheses that were used for the test.
Choose the correct answer below.
A. H0: p<0.69 and Ha: p>0.69
B. H0: p=0.69 and H3: p<0.69
C. H0: p>0.69 and H3: p<0.69
D. H0: p=0.69 and Ha: p>0.69
E. H0: p=0.69 and Ha: p ≠ 0.69

Transcript text: A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is $69 \%$. During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 153 passed the course. The instructor carried out a hypothesis test and found that the observed value of the test statistic was 2.29. The p-value associated with this test statistic is 0.0109. Explain the meaning of the $p$-value in this context. Based on this result, should the instructor believe the success rate has improved? State the hypotheses that were used for the test. Choose the correct answer below. A. $H_{0}: p<0.69$ and $H_{a}: p>0.69$ B. $H_{0}: p=0.69$ and $H_{3}: p<0.69$ C. $H_{0}: p>0.69$ and $H_{3}: p<0.69$ D. $H_{0}: p=0.69$ and $H_{a}: p>0.69$ E. $H_{0}: p=0.69$ and $H_{a}: p \neq 0.69$

Solution

Solution Steps

Step 1: Hypotheses Formulation

We set up the hypotheses for the test as follows:

Null hypothesis ($H_0$): $p = 0.69$ (the passing rate is 69%)
Alternative hypothesis ($H_a$): $p > 0.69$ (the passing rate is greater than 69%)

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:

$\hat{p} = \frac{153}{200} = 0.765$
$p_0 = 0.69$
$n = 200$

Calculating the test statistic: \[ Z = \frac{0.765 - 0.69}{\sqrt{\frac{0.69(1 - 0.69)}{200}}} = 2.2934 \]

Step 3: P-value Calculation

The p-value associated with the calculated test statistic $Z = 2.2934$ is found to be: \[ \text{P-value} = 0.0109 \]

Step 4: Critical Region Determination

For a significance level of $\alpha = 0.05$ in a one-tailed test, the critical value is: \[ Z > 1.6449 \] This means that if the calculated $Z$ exceeds $1.6449$, we reject the null hypothesis.

Step 5: Conclusion

Since the calculated test statistic $Z = 2.2934$ is greater than the critical value $1.6449$ and the p-value $0.0109$ is less than $\alpha = 0.05$, we reject the null hypothesis. This suggests that the use of embedded tutors has likely improved the success rate in introductory chemistry courses.