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To determine which correlation coefficient is the most realistic based on the given data for home sizes and their corresponding sale prices, let's take a detailed and step-by-step approach.
### Step 1: Understand the Data
We are given data for the sizes of six homes (in square feet) and their corresponding sale prices (in thousands of dollars).
- Sizes of homes ([tex]$x$[/tex]): 1200, 1350, 1720, 1510, 1440, 1675 square feet
- Sale prices ([tex]$y$[/tex]): 223, 298, 427, 375, 310, 402 thousand dollars
### Step 2: Concept of Correlation Coefficient
The correlation coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:
- [tex]\( r = 1 \)[/tex]: Perfect positive linear relationship
- [tex]\( r = -1 \)[/tex]: Perfect negative linear relationship
- [tex]\( r = 0 \)[/tex]: No linear relationship
### Step 3: Intuitive Analysis
We need to decide which of the provided correlation values ([tex]$r = 0.981$[/tex], [tex]$r = -0.018$[/tex], [tex]$r = 0.318$[/tex], [tex]$r = -0.901$[/tex]) best represents the relationship between the given sizes and prices.
When we plot the sizes and prices on a scatter plot and visualize a potential line of best fit:
1. Strong positive relationship: If sizes increase, the sale prices tend to increase noticeably as well. This suggests a high positive correlation.
2. Nearly no relationship: If an increase in sizes does not consistently align with the increase or decrease in prices, it suggests the correlation is close to zero.
3. Moderate positive relationship: If there is a clear but not strong trend that prices increase with sizes, it implies a moderate positive correlation.
4. Strong negative relationship: If larger home sizes correlate strongly with lower prices, it suggests a high negative correlation.
### Step 4: Interpret the Provided Correlation Values
Given the values:
1. [tex]\( r = 0.981 \)[/tex]: Very close to 1, suggesting a very strong positive relationship.
2. [tex]\( r = -0.018 \)[/tex]: Close to 0, indicating almost no linear relationship.
3. [tex]\( r = 0.318 \)[/tex]: Indicates a moderate positive relationship.
4. [tex]\( r = -0.901 \)[/tex]: Indicates a strong negative relationship.
### Step 5: Determining the Most Realistic Correlation
With the provided data, we consider:
- Visual Alignment: Looking at the data set, the sizes and prices seem to increase together.
- Strong Positive Relationship: Given the trends in real-estate, larger homes tend to cost more. Thus, a very strong positive linear relationship is anticipated.
Therefore, the correlation coefficient [tex]\( r = 0.981 \)[/tex], which suggests a very strong positive relationship, is the most realistic value for the correlation between the sizes of these homes and their sale prices.
### Step 1: Understand the Data
We are given data for the sizes of six homes (in square feet) and their corresponding sale prices (in thousands of dollars).
- Sizes of homes ([tex]$x$[/tex]): 1200, 1350, 1720, 1510, 1440, 1675 square feet
- Sale prices ([tex]$y$[/tex]): 223, 298, 427, 375, 310, 402 thousand dollars
### Step 2: Concept of Correlation Coefficient
The correlation coefficient, denoted as [tex]\( r \)[/tex], measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1:
- [tex]\( r = 1 \)[/tex]: Perfect positive linear relationship
- [tex]\( r = -1 \)[/tex]: Perfect negative linear relationship
- [tex]\( r = 0 \)[/tex]: No linear relationship
### Step 3: Intuitive Analysis
We need to decide which of the provided correlation values ([tex]$r = 0.981$[/tex], [tex]$r = -0.018$[/tex], [tex]$r = 0.318$[/tex], [tex]$r = -0.901$[/tex]) best represents the relationship between the given sizes and prices.
When we plot the sizes and prices on a scatter plot and visualize a potential line of best fit:
1. Strong positive relationship: If sizes increase, the sale prices tend to increase noticeably as well. This suggests a high positive correlation.
2. Nearly no relationship: If an increase in sizes does not consistently align with the increase or decrease in prices, it suggests the correlation is close to zero.
3. Moderate positive relationship: If there is a clear but not strong trend that prices increase with sizes, it implies a moderate positive correlation.
4. Strong negative relationship: If larger home sizes correlate strongly with lower prices, it suggests a high negative correlation.
### Step 4: Interpret the Provided Correlation Values
Given the values:
1. [tex]\( r = 0.981 \)[/tex]: Very close to 1, suggesting a very strong positive relationship.
2. [tex]\( r = -0.018 \)[/tex]: Close to 0, indicating almost no linear relationship.
3. [tex]\( r = 0.318 \)[/tex]: Indicates a moderate positive relationship.
4. [tex]\( r = -0.901 \)[/tex]: Indicates a strong negative relationship.
### Step 5: Determining the Most Realistic Correlation
With the provided data, we consider:
- Visual Alignment: Looking at the data set, the sizes and prices seem to increase together.
- Strong Positive Relationship: Given the trends in real-estate, larger homes tend to cost more. Thus, a very strong positive linear relationship is anticipated.
Therefore, the correlation coefficient [tex]\( r = 0.981 \)[/tex], which suggests a very strong positive relationship, is the most realistic value for the correlation between the sizes of these homes and their sale prices.
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