Inner suburbs - M=5, SD=6 . 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?

To find the responses at the lowest and highest ends of the middle 26% of a normal distribution (i.e., between the 37th percentile and the 63rd percentile), we need to work through the following steps:

### Step-by-step Solution:

1. **Given Information**:
   - Mean (\(M\)) = 5
   - Standard Deviation (\(SD\)) = 6
   - Middle 26% of the distribution is between the 37th percentile and the 63rd percentile of a normal distribution.

2. **Finding Z-scores**:
   We need to find the Z-scores corresponding to the 37th percentile and the 63rd percentile in a standard normal distribution. Using Z-tables or a calculator:
   - Z-score for the 37th percentile: \(-0.33\)
   - Z-score for the 63rd percentile: \(+0.33\)

3. **Converting Z-scores to Actual Values**:
   To convert the Z-scores back to actual survey responses, use the formula for a normal distribution:
   
   \[
   X = M + (Z \times SD)
   \]
   
   - **For the lowest end (37th percentile, \(Z = -0.33\))**:
   
     \[
     X_{low} = 5 + (-0.33 \times 6) = 5 - 1.98 = 3.02
     \]
   
   - **For the highest end (63rd percentile, \(Z = 0.33\))**:
   
     \[
     X_{high} = 5 + (0.33 \times 6) = 5 + 1.98 = 6.98
     \]

### Conclusion:
The survey responses corresponding to the middle 26% of the distribution are between approximately **3.02** and **6.98**.

Would you like further details on this process or have any questions?

### Further Exploration:
1. What is the interpretation of Z-scores in a normal distribution?
2. How would the results change if the standard deviation was different?
3. What is the relationship between percentiles and standard deviations?
4. How do we calculate the area under the normal curve for different portions?
5. What happens if we look for the middle 50% instead of the middle 26%?

### Tip:
When working with normal distributions, Z-scores allow you to compare values from different distributions, regardless of their mean or standard deviation.

Inner suburbs - M=5, SD=6. 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?

Learn how to find the lowest and highest responses for the middle 26% of a normal distribution with a mean of 5 and a standard deviation of 6. This problem covers key statistical concepts, including Z-scores, percentiles, and the formula X = M + (Z × SD). Suitable for Grades 11-12 and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation