To calculate the correlation coefficient ($r$), we use the formula:

[ r = \frac{n \sum(xy) - \sum(x)\sum(y)}{\sqrt{\left[n \sum(x^2) - (\sum(x))^2\right]\left[n \sum(y^2) - (\sum(y))^2\right]}} ]

Where:

• $x$: Hours spent studying
• $y$: Midterm grades
• $n$: Number of data points

Step 1: Organize the data and compute the necessary sums

We need to calculate:

1. $\sum(x)$ (sum of hours)
2. $\sum(y)$ (sum of grades)
3. $\sum(x^2)$ (sum of squared hours)
4. $\sum(y^2)$ (sum of squared grades)
5. $\sum(xy)$ (sum of products of hours and grades)

The data is:

$x$ (Hours)	$y$ (Grades)	$x^2$	$y^2$	$xy$
1.5	60	2.25	3600	90
2	63	4	3969	126
2.5	66	6.25	4356	165
3	75	9	5625	225
3.5	81	12.25	6561	283.5
4	84	16	7056	336
4.5	90	20.25	8100	405
5	93	25	8649	465
5.5	96	30.25	9216	528
6	99	36	9801	594

Now, calculate the sums: $\sum(x) = 37.5, \quad \sum(y) = 807, \quad \sum(x^2) = 161.25, \quad \sum(y^2) = 67333, \quad \sum(xy) = 3217.5$

Step 2: Plug the sums into the formula for $r$

1. $n = 10$
2. Numerator: $n \sum(xy) - \sum(x)\sum(y) = 10(3217.5) - (37.5)(807) = 32175 - 30262.5 = 1912.5$
3. Denominator: [ \sqrt{\left[n \sum(x^2) - (\sum(x))^2\right] \left[n \sum(y^2) - (\sum(y))^2\right]} ]

• First term: $n \sum(x^2) - (\sum(x))^2 = 10(161.25) - (37.5)^2 = 1612.5 - 1406.25 = 206.25$
• Second term: $n \sum(y^2) - (\sum(y))^2 = 10(67333) - (807)^2 = 673330 - 651249 = 22081$
• Denominator: $\sqrt{206.25 \times 22081} = \sqrt{4551686.25} = 2134.32$

4. Calculate $r$: $r = \frac{1912.5}{2134.32} \approx 0.896$

Final Answer:

The correlation coefficient $r$ is approximately 0.896.

Would you like details on interpreting this result or any related calculations?

Related Questions:

1. What does the correlation coefficient indicate about the relationship between study hours and grades?
2. How can we calculate the coefficient of determination ($r^2$)?
3. What are the limitations of using $r$ for this dataset?
4. How would the correlation change if the grades were scaled differently?
5. Can we predict a grade if a student studies for 7 hours using linear regression?

Tip:

Always check the scatterplot of your data to ensure the relationship is linear before interpreting