To assess the strength of the relationship between the hours spent studying and the exam scores, we can calculate the correlation coefficient (Pearson's r). A correlation coefficient will help us understand how closely the two variables are related.
Step 1: Organize the data
| Hours Studying | Exam Score | |----------------|------------| | 11 | 77 | | 10 | 70 | | 5 | 50 | | 15 | 91 | | 12 | 78 | | 7 | 70 | | 8 | 62 | | 14 | 86 | | 6 | 55 | | 12 | 80 |
Step 2: Calculate means and standard deviations
Next, we can find the mean (average) of the hours studying and the exam scores, as well as their standard deviations.
-
Mean of Hours Studying (\( \bar{x} \)): \[ \bar{x} = \frac{11 + 10 + 5 + 15 + 12 + 7 + 8 + 14 + 6 + 12}{10} = \frac{ 11 + 10 + 5 + 15 + 12 + 7 + 8 + 14 + 6 + 12 }{10} = \frac{ 11 + 10 + 5 + 15 + 12 + 7 + 8 + 14 + 6 + 12 }{10} = 10.5 \]
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Mean of Exam Scores (\( \bar{y} \)): \[ \bar{y} = \frac{77 + 70 + 50 + 91 + 78 + 70 + 62 + 86 + 55 + 80}{10} = \frac{ 77 + 70 + 50 + 91 + 78 + 70 + 62 + 86 + 55 + 80 }{10} = 70.6\]
Next, calculate the standard deviations for both datasets.
Step 3: Pearson's Correlation Coefficient Formula
The formula for Pearson's correlation coefficient \( r \) is: \[ r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} \]
Where:
- \( n \) is the number of data points
- \( x \) is the hours studying
- \( y \) is the exam score
Step 4: Calculate necessary sums
- \( \sum x = 105 \)
- \( \sum y = 706 \)
- \( \sum xy = 1177 + 1070 + 550 + 1591 + 1278 + 770 + 862 + 1486 + 655 + 1280 \)
- \( \sum x^2 = 11^2 + 10^2 + 5^2 + 15^2 + 12^2 + 7^2 + 8^2 + 14^2 + 6^2 + 12^2 \)
- \( \sum y^2 = 77^2 + 70^2 + 50^2 + 91^2 + 78^2 + 70^2 + 62^2 + 86^2 + 55^2 + 80^2 \)
Calculating those:
\[ \sum xy = 847 + 700 + 250 + 1365 + 936 + 490 + 496 + 1204 + 330 + 960 = 7558 \]
\[ \sum x^2 = 121 + 100 + 25 + 225 + 144 + 49 + 64 + 196 + 36 + 144 = 1074 \]
\[ \sum y^2 = 5929 + 4900 + 2500 + 8281 + 6084 + 4900 + 3844 + 7396 + 3025 + 6400 = 39739 \]
Step 5: Plug values into formula
Now, using \( n = 10 \):
\[ r = \frac{10(7558) - (105)(706)}{\sqrt{[10(1074) - (105)^2][10(39739) - (706)^2]}} \]
Calculating this step by step:
\[ = \frac{75580 - 74130}{\sqrt{[10740 - 11025][397390 - 499636]}} \] \[ = \frac{1450}{\sqrt{[10740 - 11025][-10246]}} \] \[ = \frac{1450}{\sqrt{[-285][-10246]}} = \frac{1450}{\sqrt{2918910}} \approx \frac{1450}{1707.89} \] \[ \approx 0.85 \]
Conclusion
Since the correlation coefficient \( r \approx 0.85 \), we can conclude that there is a very strong positive relationship between the hours spent studying and the exam scores. Thus, the answer to your question is:
Very strong.