Question

4 from a data set relating height (cm) and weight (kg) for a group of students it was determined that the correlation coefficient was \\(r = 0.75\\). it was also found that the mean height for the group was \\(174.5\text{ cm}\\), with a standard deviation of \\(9.3\text{ cm}\\), and that the mean weight was \\(65.9\text{ kg}\\), with a standard deviation of \\(10.8\text{ kg}\\).
a find the slope of the least squares regression line which would enable weight to be predicted from height. round your answer to three significant figures.
b find the intercept for this line. round your answer to three significant figures.
c hence, write down the equation of the least squares regression line in terms of weight and height.

Explanation:

⚡ Using what you learned: regression

Step 1: Identify given values and variables

Let height be \( x \) (independent variable) and weight be \( y \) (dependent variable, to be predicted from height).

• Mean of \( x \) (\(\bar{x}\)) = \( 174.5\text{ cm} \)
• Standard deviation of \( x \) (\(s_x\)) = \( 9.3\text{ cm} \)
• Mean of \( y \) (\(\bar{y}\)) = \( 65.9\text{ kg} \)
• Standard deviation of \( y \) (\(s_y\)) = \( 10.8\text{ kg} \)
• Correlation coefficient (\(r\)) = \( 0.75 \)

Step 2: Find the slope (a)

The slope \( m \) of the least squares regression line is:
\[ m = r \frac{s_y}{s_x} \]

\[ m = 0.75 \times \frac{10.8}{9.3} \approx 0.870967 \]

Rounding to three significant figures:
\[ m \approx 0.871 \]

Step 3: Find the intercept (b)

The \(y\)-intercept \( c \) of the least squares regression line is:
\[ c = \bar{y} - m\bar{x} \]

Using the unrounded value of \( m \):
\[ c = 65.9 - (0.870967 \times 174.5) \approx 65.9 - 151.9837 \approx -86.0837 \]

Rounding to three significant figures:
\[ c \approx -86.1 \]

Step 4: Write down the regression equation (c)

The equation of the least squares regression line is:
\[ y = mx + c \]

Substituting the rounded values:
\[ y = 0.871x - 86.1 \]

In terms of weight and height:
\[ \text{weight} = 0.871 \times \text{height} - 86.1 \]

Answer:

a The slope of the regression line is \( 0.871 \).

b The intercept of the regression line is \( -86.1 \).

c The equation of the regression line is \( \text{weight} = 0.871 \times \text{height} - 86.1 \) (or \( y = 0.871x - 86.1 \)).