Colas per week	Bone Mineral Density (grams per cubic centimeter)
0	0.907
0	0.896
1	0.886
2	0.852
2	0.862
2	0.845
2	0.844
3	0.829
5	0.783
5	0.788
5	0.784
5	0.783
7	0.735
7	0.744
8	0.718

 Find the​ least-squares regression line treating cola consumption per week as the explanatory variable.

To find the least-squares regression line, we need to perform linear regression using the given data. The regression equation takes the form:

\[
\hat{y} = b_0 + b_1 x
\]

Where:
- \( \hat{y} \) is the predicted bone mineral density (BMD).
- \( x \) is the number of colas consumed per week.
- \( b_0 \) is the intercept (the BMD when cola consumption is 0).
- \( b_1 \) is the slope (the change in BMD for each additional cola consumed).

The slope \( b_1 \) is given by:

\[
b_1 = \frac{n\sum{(xy)} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2}
\]

The intercept \( b_0 \) is given by:

\[
b_0 = \frac{\sum{y} - b_1\sum{x}}{n}
\]

Where:
- \( n \) is the number of data points.
- \( x \) is the cola consumption per week.
- \( y \) is the bone mineral density.

### Step-by-step Process:

1. Calculate \( \sum{x} \), \( \sum{y} \), \( \sum{xy} \), and \( \sum{x^2} \).
2. Use these to compute \( b_1 \) and \( b_0 \).
3. Write the least-squares regression equation.

Let me calculate it.The least-squares regression line is:

\[
\hat{y} = 0.900 - 0.023x
\]

This equation tells us that:
- The intercept \( b_0 = 0.900 \), meaning when no colas are consumed, the predicted bone mineral density is approximately 0.900 grams per cubic centimeter.
- The slope \( b_1 = -0.023 \), meaning for each additional cola consumed per week, the bone mineral density decreases by approximately 0.023 grams per cubic centimeter.

Would you like further clarification or additional details on any step?

Here are 5 related questions for further exploration:
1. What is the interpretation of a negative slope in this context?
2. How would the correlation between cola consumption and BMD be evaluated?
3. How can the regression equation be used to predict BMD for 4 colas per week?
4. What is the coefficient of determination \( R^2 \) for this model?
5. Can we determine if the regression line is statistically significant?

**Tip:** Regression analysis assumes a linear relationship between variables. Always visualize your data first to confirm the relationship is approximately linear!

Find the least-squares regression line treating cola consumption per week as the explanatory variable.

Learn how to calculate the least-squares regression line for cola consumption per week and bone mineral density. This statistical problem involves key concepts in linear regression and the least squares method, with real data provided. Suitable for high school and early college students.

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