Question at position 29 All but one of the following statements contains an error. Which statement could be correct?We found a correlation of r = -0.63 between gender and political party preference.The correlation between the distance travelled by a hiker and the time spent hiking is r = 0.9 meters per second.We found a high correlation between the height and age of children: r = 1.12.There is a correlation of 0.54 between the position a football player plays and his weight.The correlation between mid-August soil moisture and the per-acre yield of tomatoes is r = 0.53.

Question

BlackTom AI · Accepted Answer

When evaluating these statements, the key is to check whether the correlation coefficient is being applied to appropriate types of variables and whether the numerical value lies within its valid range.
Option 1: 'We found a correlation of r = -0.63 between gender and political party preference.' Here, gender is typically a categorical variable (often binary or nominal), and correlation coefficients like r assume at least ordinal or continuous data. While one could encode categories numerically, the Pearson correlation in this straightforward form is not appropriate for purely categorical data, making this statement questionable or misleading. The underlying issue is using r with a nominal/categorical variable without clear, justifiable encoding.
Option 2: 'The correlation between the distance travelled by a hiker and the time spent hiking is r = 0.9 meters per second.' This mixes a dimensionless statistic (correlation coefficient r) with a physical unit (meters per second). The correlation coefficient is unitless by definition; attaching a unit to r is incorrect. Additionally, even if distance and time are related, the value of r should be between -1 and 1 and not expressed with a physical unit.
Option 3: 'We found a high correlation between the height and age of children: r = 1.12.' A correlation coefficient must lie within the interval [-1, 1]. A value of 1.12 is outside this range, indicating an impossible or miscomputed statistic. This is a clear mathematical error.
Option 4: 'There is a correlation of 0.54 between the position a football player plays and his weight.' Similar to Option 1, 'position' is a categorical variable (the role or position on a team), while 'weight' is continuous. Pearson r is not directly appropriate for a purely categorical variable paired with a continuous one unless the position is encoded in a meaningful, justified numeric way and certain statistical assumptions are met. Without such encoding justifications, this statement is dubious as a straightforward correlation.
Option 5: 'The correlation between mid-August soil moisture and the per-acre yield of tomatoes is r = 0.53.' Both soil moisture and yield are continuous variables, and a correlation of 0.53 falls within the valid range [-1, 1]. This statement is plausible and consistent with how correlation is typically reported in agronomic studies, assuming data collection and analysis were done correctly. It does not involve units attached to r itself, and it does not rely on inapt variable types.
In summary, Options 1, 2, 3, and 4 each contain a conceptual or mathematical flaw relevant to the measurement or interpretation of correlations, while Option 5 adheres to the proper use of the correlation coefficient with appropriate variable types and a valid value. The presence of a mathematically invalid scale (Option 3), unit attachment to a dimensionless statistic (Option 2), inappropriate use of r with categorical data (Options 1 and 4), and a plausible, correctly formed statement (Option 5) collectively point to Option 5 as the only statement that could be correct.

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Of the following, the correlation coefficient that indicates the weakest relationship between the two variables being measured is ______________.

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