3.	If you look at the middle 26% of the distribution of survey responses given by all inner suburb residents, what responses will you find at the lowest and highest ends of this portion of the distribution? (1 mark)

To determine the lowest and highest responses within the middle 26% of the distribution, we need to understand the concept of **percentiles**.

We assume that the distribution of survey responses follows a **normal distribution** (since that’s common for survey data analysis), and we are asked to focus on the middle 26%. Here's how to approach this:

### 1. Middle 26% of the Distribution
The middle 26% refers to the central part of the distribution. To find the boundaries at the lowest and highest ends of this range, we will calculate the corresponding **percentiles**. 

- 100% - 26% = 74%, so the middle 26% leaves **37%** on each tail of the distribution. 
- To find the percentiles that cut off the lowest and highest ends of the middle 26%, we split the remaining 74% into two equal parts, each 37%:

  - The **lower percentile** will be at \( \frac{100\% - 26\%}{2} = 37\% \).
  - The **upper percentile** will be at \( 100\% - 37\% = 63\% \).

Therefore, the lowest value of this middle 26% is at the **37th percentile**, and the highest value is at the **63rd percentile**.

### 2. Interpretation
The responses that lie between the **37th percentile** and the **63rd percentile** of the distribution represent the range of responses in the middle 26%. These values can be found using statistical tables or software if specific data points are provided.

### Final Answer:
- The responses at the **37th percentile** and the **63rd percentile** will define the lowest and highest ends of the middle 26% of the distribution. 

Let me know if you'd like help with further calculations or if you have specific data! 

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**Relative Questions:**
1. How do you calculate percentiles for non-normal distributions?
2. What is the importance of using the middle portion of a distribution?
3. How would you calculate the z-scores for the 37th and 63rd percentiles?
4. How do extreme values (outliers) affect the percentiles of a dataset?
5. What methods can be used to visualize the middle percentage of a dataset?

**Tip:** Understanding percentiles helps in comparing individual data points to a larger population in terms of position, not actual values!

If you look at the middle 26% of the distribution of survey responses given by all inner suburb residents, what responses will you find at the lowest and highest ends of this portion of the distribution?

Understand how to calculate the lowest and highest responses within the middle 26% of a normal distribution. This problem covers percentiles, the empirical rule, and how to find boundaries of survey data. Perfect for advanced statistics students in Grades 11-12.

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