A data set consist of the following 6 values:
The range is 6 .
The median is 5 .
The lower quartile is 4 .
The interquartile range is 3 .
Given that , find the values of and .

Question

A data set consist of the following 6 values: The range is 6 . The median is 5 . The lower quartile is 4 . The interquartile range is 3 . Given that $$ x

UpStudy AI · Accepted Answer

To determine the values of $$ x $$, $$ y $$, and $$ z $$ in the data set, let's analyze the given information step by step.

### Given:
- **Number of values in the data set:** 6
- **Range:** 6
- **Median:** 5
- **Lower Quartile (Q1):** 4
- **Interquartile Range (IQR):** 3
- **Relation:** $$ x < y < z $$

### Step 1: Arrange the Data Set
Let’s denote the ordered data set as:
$$ v_1, v_2, v_3, v_4, v_5, v_6 $$
where $$ v_1 \leq v_2 \leq v_3 \leq v_4 \leq v_5 \leq v_6 $$.

### Step 2: Apply the Given Conditions
1. **Median:**
   - For 6 values, the median is the average of the 3rd and 4th values.
   $$
   \frac{v_3 + v_4}{2} = 5 \implies v_3 + v_4 = 10
   $$
   
2. **Lower Quartile (Q1):**
   - Q1 is the median of the first three values: $$ v_1, v_2, v_3 $$.
   $$
   Q1 = v_2 = 4
   $$
   
3. **Interquartile Range (IQR):**
   - IQR is the difference between the upper quartile (Q3) and Q1.
   $$
   Q3 - Q1 = 3 \implies Q3 = 7
   $$
   - Q3 is the median of the last three values: $$ v_4, v_5, v_6 $$.
   $$
   Q3 = v_5 = 7
   $$
   
4. **Range:**
   $$
   \text{Range} = v_6 - v_1 = 6 \implies v_6 = v_1 + 6
   $$

### Step 3: Determine Possible Values
From the above conditions:
- $$ v_2 = 4 $$
- $$ v_5 = 7 $$
- $$ v_3 + v_4 = 10 $$

Possible integer pairs for $$ (v_3, v_4) $$ that satisfy $$ v_3 \leq v_4 \leq 7 $$:
- $$ (4, 6) $$

Using this pair:
- $$ v_1 \leq 4 $$ (since $$ v_2 = 4 $$)
- $$ v_6 = v_1 + 6 $$
- $$ v_4 = 6 $$ and $$ v_5 = 7 $$

Possible values of $$ v_1 $$ and $$ v_6 $$:
1. If $$ v_1 = 2 $$, then $$ v_6 = 8 $$
   
   The data set becomes: $$ 2, 4, 4, 6, 7, 8 $$

### Step 4: Identify $$ x $$, $$ y $$, and $$ z $$
From the condition $$ x < y < z $$, and based on the data set $$ 2, 4, 4, 6, 7, 8 $$:
- $$ x = 2 $$
- $$ y = 7 $$
- $$ z = 8 $$

### Conclusion
The values satisfying all the given conditions are:
$$
x = 2, \quad y = 7, \quad z = 8
$$

**Final Answer:**
$$
x = 2,\ y = 7,\ \text{and}\ z = 8
$$
$$ x = 2 $$, $$ y = 7 $$, and $$ z = 8 $$.

A data set consist of the following 6 values:
The range is 6 .
The median is 5 .
The lower quartile is 4 .
The interquartile range is 3 .
Given that , find the values of and .

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A data set consist of the following 6 values: The range is 6 . The median is 5 . The lower quartile is 4 . The interquartile range is 3 . Given that , find the values of and .

Upstudy AI Solution

Answer

Solution

The Deep Dive

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Latest Statistics Questions

A data set consist of the following 6 values:
The range is 6 .
The median is 5 .
The lower quartile is 4 .
The interquartile range is 3 .
Given that , find the values of and .