Interpreting a Correlation Between Study Time and Test Scores
A secondary school mathematics teacher is analysing the relationship between the number of hours eight students spent studying for a test and their scores. The data are: hours (x): 2, 3, 4, 5, 6, 7, 8, 9; scores (y): 50, 55, 62, 66, 72, 79, 84, 90. The teacher wants to know whether study time is a good predictor of score, and whether the school should recommend a minimum number of study hours. She calculates the correlation coefficient and the least-squares regression line. A student who studied 10 hours wants to know what score to expect. Another student, who studied 12 hours, asks the same question, but the teacher is concerned about extrapolating beyond the observed range. The teacher also notes that some students who studied longer also attended a revision workshop, so she is unsure whether study time alone is responsible for the higher scores. She must present her findings to the year head.
1. Calculate the correlation coefficient and interpret it in context.
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For the eight pairs,
[
\bar x=5.5,\qquad \bar y=69.75.
]
Compute the sums of squares and products:
[
S_{xx}=\sum (x-\bar x)^2=42,
]
[
S_{yy}=\sum (y-\bar y)^2=1385.5,
]
[
S_{xy}=\sum (x-\bar x)(y-\bar y)=241.
]
Then
[
r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}
=\frac{241}{\sqrt{42(1385.5)}}
=\frac{241}{\sqrt{58191}}
\approx\frac{241}{241.23}\approx0.999.
]
The correlation coefficient is about 0.999, indicating a very strong positive linear relationship. In context, students who spent more hours studying tended to score higher on the test. However, correlation does not prove causation, and the sample is small.
2. Find the equation of the least-squares regression line and use it to predict the score of a student who studies 10 hours. Comment on the reliability of this prediction.
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The slope is
[
b=\frac{S_{xy}}{S_{xx}}=\frac{241}{42}\approx5.738.
]
The intercept is
[
a=\bar y-b\bar x=69.75-5.738(5.5)\approx69.75-31.56=38.19.
]
So the least-squares regression line is approximately
[
\hat y=5.74x+38.19.
]
For (x=10) hours:
[
\hat y=5.738(10)+38.19\approx57.38+38.19=95.57.
]
The predicted score is about 95.6.
This prediction is a mild extrapolation because the observed study times run only from 2 to 9 hours. The prediction is close to the maximum possible score, so it should be treated with caution: the linear trend may not continue, and there may be diminishing returns. It is reasonable as an approximate guide but not as a precise forecast.
3. The teacher predicts a score of 107 for a student who studies 12 hours. Explain why this prediction is problematic, and suggest two other factors that could affect the relationship.
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Using the regression line:
[
\hat y=5.738(12)+38.19\approx68.86+38.19=107.05.
]
A predicted score of about 107 is impossible if the test is marked out of 100. The problem is extrapolation: 12 hours is outside the observed range of 2 to 9 hours, and the linear relationship may not continue beyond that range. Scores are capped at 100, and additional study time may produce smaller gains (diminishing returns).
Two other factors that could affect the relationship are:
- Confounding variables such as attendance at a revision workshop. If students who studied longer also attended the workshop, the higher scores may be partly due to the workshop rather than study time alone.
- Prior ability or motivation, which could influence both how much a student studies and how well they score.
Other acceptable factors include sleep, attendance, teaching quality, or test anxiety. The teacher should present the correlation as an association, not as proof that increasing study hours alone will raise scores.