A university department chair believes that the afternoon section of a large survey class has a consistently lower proportion of students that pass the class compared to the morning section. The chair checks a random sample of student records from each section. The results of the data collected are shown below. Morning section Afternoon section 346 Successes Successes Observations 426 500 400 Observations p-hat_1 0.865 p-hat_2 0.852 Confidence level 95% Z-score -0.5548 p-value 0.2912 -2 -1 0 1 2 3 P = Ex: 1.234 2= Morning section samples: n1 = Ex: 9 Afternoon section samples: n₂ = Sample proportion for morning section samples: p₁ = Ex: 1.234 Sample proportion for afternoon section samples: p2 =

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**Hypothesis Testing: Morning Section vs. Afternoon Section Pass Rate**

**Introduction:**
A university department chair hypothesizes that the afternoon section of a large survey class has a consistently lower proportion of students passing compared to the morning section. To test this, a random sample of student records from each section was analyzed. The results are as follows:

**Data Collected:**
| Section           | Successes | Observations | Proportion (p-hat) |
|-------------------|-----------|--------------|--------------------|
| Morning Section   | 346       | 400          | 0.865              |
| Afternoon Section | 426       | 500          | 0.852              |

*Note*: The table shows the number of successes (students passing) and total observations for each section, along with the calculated sample proportion of passes.

**Statistical Analysis:**
Additional statistical measures were used to test the hypothesis:

- **Confidence Level:** 95%
- **z-Score:** -0.5548
- **p-Value:** 0.2912

**Graphical Representation:**
The provided bell curve graph represents the distribution of the data with:

- The x-axis ranging from -3 to 3 (standard deviations).
- A highlighted region indicating the area under the curve corresponding to the calculated z-score of -0.5548, which falls within the confidence interval.

**Further Analysis:**

1. **Sample Sizes:**
   - Morning section samples: \( n_1 = 400 \)
   - Afternoon section samples: \( n_2 = 500 \)

2. **Sample Proportions:**
   - Proportion for morning section samples: \( \hat{p}_1 = 0.865 \)
   - Proportion for afternoon section samples: \( \hat{p}_2 = 0.852 \)

**Conclusion:**
Given the p-value of 0.2912, which is greater than the significance level of 0.05, there is insufficient evidence to reject the null hypothesis. Therefore, the data do not support the chair's belief that the afternoon section has a consistently lower proportion of students passing compared to the morning section.

To further explore these results, check out the accompanying interactive elements and exercises available on this educational platform!
Transcribed Image Text:**Hypothesis Testing: Morning Section vs. Afternoon Section Pass Rate** **Introduction:** A university department chair hypothesizes that the afternoon section of a large survey class has a consistently lower proportion of students passing compared to the morning section. To test this, a random sample of student records from each section was analyzed. The results are as follows: **Data Collected:** | Section | Successes | Observations | Proportion (p-hat) | |-------------------|-----------|--------------|--------------------| | Morning Section | 346 | 400 | 0.865 | | Afternoon Section | 426 | 500 | 0.852 | *Note*: The table shows the number of successes (students passing) and total observations for each section, along with the calculated sample proportion of passes. **Statistical Analysis:** Additional statistical measures were used to test the hypothesis: - **Confidence Level:** 95% - **z-Score:** -0.5548 - **p-Value:** 0.2912 **Graphical Representation:** The provided bell curve graph represents the distribution of the data with: - The x-axis ranging from -3 to 3 (standard deviations). - A highlighted region indicating the area under the curve corresponding to the calculated z-score of -0.5548, which falls within the confidence interval. **Further Analysis:** 1. **Sample Sizes:** - Morning section samples: \( n_1 = 400 \) - Afternoon section samples: \( n_2 = 500 \) 2. **Sample Proportions:** - Proportion for morning section samples: \( \hat{p}_1 = 0.865 \) - Proportion for afternoon section samples: \( \hat{p}_2 = 0.852 \) **Conclusion:** Given the p-value of 0.2912, which is greater than the significance level of 0.05, there is insufficient evidence to reject the null hypothesis. Therefore, the data do not support the chair's belief that the afternoon section has a consistently lower proportion of students passing compared to the morning section. To further explore these results, check out the accompanying interactive elements and exercises available on this educational platform!
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