Math Problem Statement

Identify the reasons the following statements are incorrect. Not all reasons will be used.

Solution

Let's go through the problem from the image together.

You are asked to identify why some of the statements regarding correlation might be incorrect. You're given a list of possible reasons from the "Answer Bank," and these are the statements being evaluated:

  1. Researchers found that the correlation between the number of hours students study for a test and their test scores is r=0.56r = 0.56 hours.
  2. The correlation between the advertised selling price of cars and the makes of the cars is r=0.69r = -0.69.
  3. The amount of time spent on an exercise and the number of calories burned are highly correlated r=1.06r = 1.06.
  4. The correlation between total daily snowfall in inches and daily low temperatures in degrees Fahrenheit differs from the correlation between daily snowfall in inches and daily low temperatures in degrees Celsius.

Let's break down each statement:

  1. First statement: r=0.56r = 0.56 hours.

    • Issue: The correlation coefficient rr is unitless. The mention of “hours” here is problematic because correlation does not have units.
    • Reason: "Correlation should not depend on units."
  2. Second statement: r=0.69r = -0.69 for the correlation between selling price and makes of cars.

    • Issue: The "make" of a car is categorical, not numerical. Correlation applies to numerical variables.
    • Reason: "One of the variables should be categorical."
  3. Third statement: r=1.06r = 1.06.

    • Issue: Correlation values must lie between -1 and 1. A correlation of 1.06 is impossible.
    • Reason: "Correlation should be between -1 and 1."
  4. Fourth statement: The comparison between correlations in Fahrenheit and Celsius.

    • Issue: The correlation between two variables remains the same even if the units are changed (from Fahrenheit to Celsius, for example).
    • Reason: "Correlation should not depend on units."

Now you can match these explanations to the options from the "Answer Bank":

  • "Correlation should not depend on units" — for the first and fourth statements.
  • "One of the variables should be categorical" — for the second statement.
  • "Correlation should be between -1 and 1" — for the third statement.

Would you like more details on any of these?


Here are 5 follow-up questions to expand on this topic:

  1. How is the correlation coefficient rr calculated?
  2. What is the significance of a positive versus a negative correlation?
  3. Can categorical data be used in correlation analysis? If so, how?
  4. Why is it important that correlation coefficients remain between -1 and 1?
  5. How would changing units (like Fahrenheit to Celsius) typically affect a dataset?

Tip: When interpreting correlation, always remember that correlation does not imply causation—there might be other variables influencing the relationship.

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Math Problem Analysis

Mathematical Concepts

Correlation
Statistics

Formulas

Correlation coefficient formula: r = Σ[(Xi - X̄)(Yi - Ȳ)] / √[Σ(Xi - X̄)^2 * Σ(Yi - Ȳ)^2]

Theorems

Properties of the correlation coefficient

Suitable Grade Level

Grades 10-12