wo researchers adopted different sampling techniques while investigating the same group of students to find the number of students falling in different intelligence levels. The results are given in Table Q8.1.G:
Researcher	No. of students in each level
Below Avg.	Avg.	Above Avg.
X	77	67	53
Y	50	19	26
Table Q8.1.G: Intelligence Level


To analyze the given data from the two researchers investigating the number of students falling into different intelligence levels, we can organize and compare the results in a table. Here's the data presented in a clear format:

| Researcher | Below Avg. | Avg. | Above Avg. |
|------------|------------|------|------------|
| X          | 77         | 67   | 53         |
| Y          | 50         | 19   | 26         |

Now, let's analyze and compare the results of the two researchers.

1. **Total Number of Students Surveyed by Each Researcher:**
   - Researcher X: \( 77 + 67 + 53 = 197 \)
   - Researcher Y: \( 50 + 19 + 26 = 95 \)

2. **Proportion of Students in Each Category for Each Researcher:**
   - **Researcher X:**
     - Below Avg.: \( \frac{77}{197} \approx 0.39 \)
     - Avg.: \( \frac{67}{197} \approx 0.34 \)
     - Above Avg.: \( \frac{53}{197} \approx 0.27 \)
   - **Researcher Y:**
     - Below Avg.: \( \frac{50}{95} \approx 0.53 \)
     - Avg.: \( \frac{19}{95} \approx 0.20 \)
     - Above Avg.: \( \frac{26}{95} \approx 0.27 \)

3. **Comparison of Findings:**
   - **Below Average:**
     - Researcher X: 39%
     - Researcher Y: 53%
   - **Average:**
     - Researcher X: 34%
     - Researcher Y: 20%
   - **Above Average:**
     - Researcher X: 27%
     - Researcher Y: 27%

### Observations:
- Researcher Y found a higher proportion of students in the "Below Average" category compared to Researcher X.
- Researcher X found a higher proportion of students in the "Average" category compared to Researcher Y.
- Both researchers found the same proportion of students in the "Above Average" category.

### Possible Explanations for Differences:
1. **Sampling Techniques:**
   - The researchers used different sampling techniques which may have led to different subsets of the population being surveyed.
2. **Sample Size:**
   - Researcher X surveyed more students (197) compared to Researcher Y (95). A larger sample size typically provides a more accurate representation of the population.
3. **Random Variation:**
   - Differences can also be due to random variation, especially if the sample sizes are not very large.

### Further Analysis:
To determine if the differences are statistically significant, we could perform a chi-square test of independence. This test would help us see if the observed differences in the distribution of intelligence levels are likely due to the different sampling techniques or just random chance.

Would you like to see the calculations for the chi-square test, or do you have any other specific questions about the data?

### Related Questions:
1. What are the types of sampling techniques commonly used in research?
2. How can sample size affect the results of a survey?
3. What is a chi-square test of independence, and how is it performed?
4. Why is it important to compare proportions when analyzing categorical data?
5. How can we ensure that our sample is representative of the population?
6. What are the potential biases in sampling, and how can they be mitigated?
7. How can random variation impact survey results?
8. What are the benefits of using larger sample sizes in research?

**Tip:** When comparing survey results from different samples, always consider the sampling techniques and sample sizes. These factors can significantly impact the results and their interpretation.

Explore the comparison of intelligence levels between two researchers, X and Y, using different sampling techniques. Analyze the data presented in Table Q8.1.G and understand the differences in the distribution of students across below average, average, and above average intelligence levels.

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