Determine the estimated score in GECMAT if the number of hours spent in studying is 5 hours 20 minutes. Show your complete solution and make analysis of your answer.

Steps to Solve

1. Convert hours to hours in decimal First, we need to convert 5 hours and 20 minutes into a decimal format.

20 minutes is equal to $\frac{20}{60}$ hours, which simplifies to $\frac{1}{3}$ hours or approximately $0.33$ hours.

So, $5 + 0.33 = 5.33$ hours.

2. Data Setup Next, gather the data provided in the table:

• Scores: 27, 24, 23, 29, 29, 29, 24, 24, 27, 25, 28, 16, 22, 30, 22
• Hours spent studying: 2.6, 3.2, 3, 5, 0.8, 0.4, 5.2, 4.1, 3.8, 1.5, 4, 3.3, 1.2, 2.6, 4.6

3. Create a Scatter Plot Plot the scores against the hours spent studying. This visual representation will help identify any correlation.

4. Calculate the Correlation Coefficient Using the formula for the Pearson correlation coefficient:

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

where $n$ is the number of pairs, $x$ is hours spent studying, and $y$ is scores.

5. Estimate the Score Assuming a linear relationship, you can use the line of best fit (linear regression) to estimate the GECMAT score for 5.33 hours.

The equation of the line can be written as:

$$ y = mx + c $$

Here, $m$ is the slope and $c$ is the y-intercept.

6. Substitute to Find Estimated Score After determining the values of $m$ and $c$, plug in $x = 5.33$ to find the estimated score $y$.