From the image you provided, we have the following data points relating minutes walked (x) and calories burned (y):

• $(1, 6)$
• $(6, 27)$
• $(11, 50)$
• $(13, 56)$
• $(16, 70)$

To determine whether the data show a linear relationship, we can calculate the slope between each pair of points and see if it is consistent. If it is, we'll then use the linear equation form $y = mx + b$ to write an equation of a line of best fit. Then we can estimate $y$ when $x = 15$ and explain its meaning.

Let's first check the consistency of the slope between consecutive data points.The calculated slopes between consecutive points are as follows:

• Between $1$ minute and $6$ minutes: $4.2$ calories per minute
• Between $6$ minutes and $11$ minutes: $4.6$ calories per minute
• Between $11$ minutes and $13$ minutes: $3.0$ calories per minute
• Between $13$ minutes and $16$ minutes: $4.67$ calories per minute

The slopes are not perfectly consistent but are somewhat close to each other. This suggests that while not perfectly linear, a linear model might still be a good approximation for this data. Let's fit a linear regression model to find the best line of fit and then use it to estimate $y$ when $x = 15$ minutes.The equation of the line of best fit, based on the given data, is: $y = 4.26x + 1.76$

Using this model, the estimated number of calories burned when walking for 15 minutes is approximately $65.65$ calories.

This means, based on the trend observed in the given data, if you walk for 15 minutes, you can expect to burn around 66 calories. This value is derived from the average rate of calorie burn per minute as calculated from the input data points.

Would you like further details on this topic or any other questions? Here are a few related queries you might consider:

1. How does the calculation of the slope and intercept help in understanding the relationship between variables?
2. What are other methods to check the goodness of fit for a linear regression model?
3. Can we calculate the calorie burn for different durations using the same model?
4. What assumptions do we make when using linear regression on this kind of data?
5. How would outliers in the data affect the model and predictions?

Tip: When modeling real-world data, it’s useful to visualize the data points and the regression line to assess the fit and identify any outliers that might affect the model.