To evaluate the truth of each statement, we need to consider how income and happiness are typically related and the nature of the line of best fit in statistical analysis. Here's an analysis of each statement:
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The slope of the line of best fit could be around m = -1/2000.
- This could be true if there is a slight negative correlation; however, it would be very weak. Depending on the context of the given data, it is possible but not certain.
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As a person's income goes up, their happiness trends down.
- This statement suggests a negative correlation. Generally, research has shown mixed results, but many studies indicate that income is positively correlated with happiness. Hence, this statement is likely false.
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This is a moderate positive correlation.
- If this is referring to the relationship between income and happiness, then in most contexts, this statement is more likely false than true.
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The line of best fit must show the middle score of the data as being on the line.
- This statement is false. The line of best fit (like a regression line) is drawn to minimize the distance of all points from the line; the median or middle score does not have to lie on the line.
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The line of best fit would have a positive slope representing as income goes up, happiness goes up.
- This statement is true if we assume a positive correlation exists between income and happiness.
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The line of best fit should include the lowest and highest scores.
- This statement is false. The line of best fit does not have to pass through all points; it is a general trend line created from statistical methods like least squares.
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The y-intercept of the line of best fit would show a happiness score of around 40 when there is $0 income.
- This can be true if the dataset supports such a y-intercept and if it’s consistent with the data being analyzed. Without actual data, we cannot definitively prove this, but it is plausible in many datasets.
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There is no correlation between happiness and income.
- Generally, research indicates that there is at least some degree of correlation between happiness and income (though the strength and nature of that correlation can vary). Therefore, this statement is likely false.
Summary of true statements:
- The slope of the line of best fit could be around m = -1/2000 (conditional).
- The line of best fit would have a positive slope representing as income goes up, happiness goes up (conditional).
- The y-intercept of the line of best fit would show a happiness score of around 40 when there is $0 income (conditional).
You should decide on whether you view the first and last statements as definitely true based on the context of the data. If you're looking for clear "true" statements without ambiguity, option 5 is definitely true in a common context where income and happiness are positively correlated.