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Probability, Statistics and Queueing

Which of the following correctly describes the correlation coefficient of two quantities?

  1. AEven if the correlation coefficient is close to 0, there may be a curved relationship between the two quantities
  2. BWhen the correlation coefficient is −0.9, there is almost no relationship between the two quantities
  3. CThe larger the correlation coefficient, the larger the slope of the regression line predicting one quantity from the other, without exception
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Answer: A. Even if the correlation coefficient is close to 0, there may be a curved relationship between the two quantities

The correlation coefficient expresses the strength and direction of the linear relationship between two quantities as a value from −1 to 1. Close to 1, the points lie along a line rising to the right; close to −1, along a line falling to the right; close to 0, the linear relationship is weak. For example, for y = x² with x from −1 to 1, there is a strong valley-shaped relationship, yet the correlation coefficient can be close to 0. The slope of the regression line depends not only on the correlation coefficient but also on the ratio of the standard deviations of the two quantities.

  • A. Correct. The correlation coefficient expresses only linear relationships, so it can miss curved ones.
  • B. −0.9 is a strong negative correlation, indicating a strong linear relationship in which one quantity decreases as the other increases.
  • C. The slope of the regression line is the correlation coefficient multiplied by the ratio of the standard deviations of the two quantities, so it is not determined by the correlation coefficient alone.
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  1. Which of the following is closest to the correlation coefficient of the five data pairs (x, y) = (1, 2), (2, 4), (3, 5), (4, 4), (5, 5)?
  2. Two quantities x and y have a correlation coefficient of 0.8, the standard deviation of x is 5, and the standard deviation of y is 10. Which of the following is the slope of the least-squares regression line predicting y from x?
  3. A sample of size 36 is drawn at random from a population with a standard deviation of 12. Which of the following is the standard error of the sample mean (the standard deviation that describes the spread of the sample mean)?
  4. Which of the following correctly describes the meaning of a 95% confidence interval for the population mean obtained from a sample?
  5. The standard deviation of a population is known to be 10. A sample of size 100 drawn from it has a sample mean of 50.0. Taking the two-sided 95% value of the standard normal distribution to be 1.96, which of the following is the 95% confidence interval for the population mean?