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Part 1
A pediatrician wants to determine the relation that may exist between a​ child's  height and head circumference. She randomly selects 8​ children, measures their height and head​ circumference, and obtains the data shown in the table. The pediatrician wants to use height to predict head circumference.

The pediatrician wants to determine if there is a relationship between a child's height and head circumference, and then use the height to predict head circumference. This type of analysis is typically done using **linear regression**, which models the relationship between two variables. Here’s how you would approach this:

### Part 1: Understanding the Problem
1. **Objective**: Use height (independent variable \(x\)) to predict head circumference (dependent variable \(y\)).
2. **Data**: We have a sample of 8 children with their respective heights and head circumferences.

### Steps to Follow:
1. **Plot the Data**: First, we can visualize the data by plotting height against head circumference to check for any linear pattern.
   
2. **Calculate the Correlation Coefficient**: This will show us the strength and direction of the linear relationship between height and head circumference. The correlation coefficient \(r\) ranges from -1 to 1:
   - \(r = 1\): Perfect positive linear relationship.
   - \(r = -1\): Perfect negative linear relationship.
   - \(r = 0\): No linear relationship.

3. **Perform Linear Regression**: Use linear regression to fit a line to the data. The equation of the line will be of the form:
   \[
   \hat{y} = b_0 + b_1 x
   \]
   where:
   - \( \hat{y} \) is the predicted head circumference,
   - \( b_0 \) is the intercept (head circumference when height is 0),
   - \( b_1 \) is the slope (the change in head circumference for a unit change in height),
   - \(x\) is the height.

4. **Interpret the Results**: From the linear regression equation, you can interpret the relationship between height and head circumference. You can also use the equation to predict head circumference for a given height.

5. **Check Goodness of Fit**: Use \( R^2 \), the coefficient of determination, to assess how well the model fits the data. \( R^2 \) ranges from 0 to 1, with values closer to 1 indicating a better fit.

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Would you like to upload the data or proceed with the calculations manually? 

### 5 Follow-Up Questions:
1. Do you have the data available for the 8 children’s heights and head circumferences?
2. Have you studied linear regression before, or would you like a quick explanation?
3. Are you familiar with how the correlation coefficient is calculated and interpreted?
4. Do you expect a strong or weak correlation between height and head circumference based on your intuition?
5. Would you like to see an example of how predictions are made using the regression model?

### Tip:
Always plot the data before running a regression analysis to visually inspect the potential relationship between variables.

A pediatrician wants to determine the relation that may exist between a​ child's height and head circumference. She randomly selects 8​ children, measures their height and head​ circumference, and obtains the data shown in the table. The pediatrician wants to use height to predict head circumference.

Explore how to use linear regression to determine the relationship between a child's height and head circumference. Learn how to calculate the correlation coefficient, perform regression analysis, and interpret the results using data from 8 children. Suitable for Grades 11-12 and college students studying statistics.

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