https://file.aiquickdraw.com/mathbot/file/1727271898835a9t6ould.jpg

To solve the hypothesis testing problem for cases **a**, **b**, and **c** from the provided image, let's break down each question.

### Hypotheses Setup:
We need to perform hypothesis tests based on the provided data. The hypotheses for each case will be:

1. **Case a**: The leadership style is **higher** than 75% of the expected level.
   - Null Hypothesis (H0): \( \mu \leq 75\% \)
   - Alternative Hypothesis (H1): \( \mu > 75\% \)

2. **Case b**: The leadership style is **lower** than 75% of the expected level.
   - Null Hypothesis (H0): \( \mu \geq 75\% \)
   - Alternative Hypothesis (H1): \( \mu < 75\% \)

3. **Case c**: The leadership style is **different** from 75% of the expected level.
   - Null Hypothesis (H0): \( \mu = 75\% \)
   - Alternative Hypothesis (H1): \( \mu \neq 75\% \)

### Data Information:
The data is in table form, and we have different scores from respondents on a scale of 1 to 4:
- **4** = Sangat Baik (Very Good)
- **3** = Baik (Good)
- **2** = Cukup Baik (Fair)
- **1** = Kurang Baik (Poor)

### Steps:
1. **Convert the Scores to Percentages**: Since we are looking for leadership that exceeds or is lower than **75%**, we need to convert the score ranges into corresponding percentages.
2. **Find the Mean** and **Standard Deviation** of the data.
3. **Conduct a Z-test** or **T-test** (depending on the sample size and distribution of data) to test the hypotheses.
4. Compare the result with the **confidence level (95%)** or **significance level (α = 0.05)** to determine if the null hypothesis can be rejected.

Let me extract the table data and perform the calculations.### Results of the Hypothesis Testing:

1. **Mean of the Data**: The average score is **0.46** (or 46%).
2. **Standard Deviation**: The standard deviation of the data is **0.0667**.
3. **Sample Size (n)**: The number of data points is **40**.
4. **Z-Statistics**:
   - For **case a** (H1: leadership > 75%): \( Z = -27.51 \)
   - For **case b** (H1: leadership < 75%): \( Z = -27.51 \)
   - For **case c** (H1: leadership ≠ 75%): \( Z = -27.51 \)

### Interpretation:
Given that the test statistics for all cases are very negative (around -27.51), it indicates that the mean leadership score (46%) is significantly below the hypothesized 75%. 

- **Case a**: Since the Z-value is extremely negative, we **fail to reject the null hypothesis**. The leadership style is **not higher** than 75%.
- **Case b**: The Z-value supports the alternative hypothesis. The leadership style is indeed **lower** than 75%, so we **reject the null hypothesis**.
- **Case c**: Since the Z-value is far from 0, we **reject the null hypothesis**, indicating that the leadership style is **different** from 75%.

### Conclusion:
- **a**: The leadership style is **not higher** than 75%.
- **b**: The leadership style is **lower** than 75%.
- **c**: The leadership style is **different** from 75%.

Would you like more details on the calculations or any specific part of the results?

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Here are 5 related questions to expand understanding:
1. How is the Z-statistic calculated in hypothesis testing?
2. What would the result be if the mean score was closer to 75%?
3. How would changing the sample size affect the hypothesis test results?
4. Why is a confidence level of 95% commonly used in hypothesis testing?
5. What is the difference between one-tailed and two-tailed tests?

**Tip**: When conducting hypothesis testing, always visualize the data to better understand its distribution before performing any calculations.

Learn how to conduct hypothesis testing for leadership style based on survey data from the Merawang District. This problem involves three cases: testing if leadership is higher, lower, or different from 75%. It covers hypothesis setup, Z-test calculations, and results interpretation.

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