To determine which statements are true, let's analyze each one based on general statistical principles regarding lines of best fit (regression lines) and relationships between variables:
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The line of best fit must show the middle score of the data as being on the line.
- False: The line of best fit does not have to pass through the median or mean of the data points; it minimizes the sum of squared distances from each point to the line.
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The slope of the line of best fit could be around m = -1/2000.
- True or Possibly True: It is possible for the slope to be negative and very small, though this depends on the actual data observed. Without specific data, we can't definitively say if there's a correlation of this type.
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The line of best fit should include the lowest and highest scores.
- False: The line of best fit does not have to include the lowest or highest scores; it is determined by the overall trend of the data, and can fall above, below, or between these values.
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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.
- True or Possibly True: This can be true if the regression model indicates a y-intercept of around 40 when income is equal to 0. However, this needs specific data to verify.
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There is no correlation between happiness and income.
- False: Often there is some degree of correlation between happiness and income, but the strength and nature of that correlation can vary.
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The line of best fit would have a positive slope representing as income goes up, happiness goes up.
- True or Possibly True: This could be true if the data shows a positive correlation between income and happiness, which is commonly observed. However, it is not always guaranteed without data.
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This is a moderate positive correlation.
- True or Possibly True: Again, this may be valid; it heavily depends on the actual data and the strength of the relationship.
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As a person's income goes up, their happiness trends down.
- False: This statement describes a negative correlation. Generally, the expectation is that as income increases, happiness tends to increase, though this is not universally applicable.
Summary of Responses
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True or Possibly True:
- The slope of the line of best fit could be around m = -1/2000.
- The y-intercept of the line of best fit would show a happiness score of around 40 when there is $0 income.
- The line of best fit would have a positive slope representing as income goes up, happiness goes up (only if data supports this).
- This is a moderate positive correlation (only if data supports this).
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False:
- The line of best fit must show the middle score of the data as being on the line.
- The line of best fit should include the lowest and highest scores.
- There is no correlation between happiness and income.
- As a person's income goes up, their happiness trends down.
Without specific data, some responses can only be cautiously deemed true or false.