To solve this, we'll first use the data provided to find the equation of the line of best fit. This will allow us to predict the free time for an athlete who practices 4.5 hours per week.

Step 1: Given Data

The ordered pairs provided represent the relationship between practice hours (x) and free time hours (y). Here are the points:

• (1, 7)
• (1, 8)
• (2, 9)
• (3, 6)
• (4, 9)
• (5, 5)
• (5, 6)
• (6, 7)
• (7, 4)
• (8, 7)
• (9, 4)
• (10, 2)

Step 2: Find the Line of Best Fit

Using a statistical method like least squares, we can calculate the equation of the line of best fit. This is typically in the form:

$y = mx + b$

Where:

• $m$ is the slope
• $b$ is the y-intercept

I’ll use technology to calculate the line of best fit for this data. numpy as np import scipy.stats as stats

Data: practice hours (x) and free time hours (y)

x = np.array([1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10]) y = np.array([7, 8, 9, 6, 9, 5, 6, 7, 4, 7, 4, 2])

Calculate the line of best fit

slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)

Equation of the line of best fit

slope, interceptThe equation of the line of best fit is:

$y = -0.517x + 8.79$

Step 3: Predict Free Time for 4.5 Hours of Practice

Now, we will substitute $x = 4.5$ (the practice hours) into the equation to find the predicted free time.

$y = -0.517(4.5) + 8.79$

Let's calculate that: Predicting the free time for 4.5 hours of practice x_value = 4.5 predicted_y = slope * x_value + intercept predicted_y_round = round(predicted_y)

predicted_y_roundThe predicted free time for an athlete who practices 4.5 hours per week is 6 hours.

Final Answer:

The correct answer is 6 hours.

Would you like more details on how the line of best fit is calculated? Or do you have any other questions?

Related questions:

1. How can you interpret the slope of a line of best fit in a real-world context?
2. How does the y-intercept of the line of best fit relate to the scenario?
3. What is the significance of the correlation coefficient (r-value) in regression analysis?
4. How can you calculate the line of best fit manually using a set of data points?
5. What would happen if the data had a non-linear trend? How would you adjust your analysis?

Tip: When calculating predictions using a line of best fit, always ensure the data points are reasonably linear for accurate results.