The following display from the TI-84 Plus calculator presents the least-squares regression line for predicting the price of a certain stock (y) from the prime interest rate in percent (x).

LinReg
y= a+bx
a=2.17306922
b=0.32743551
r^2=0.3704382065
r=0.60863635
Write the equation of the least-squares regression line. Use the full accuracy shown in the calculator output (do not round your answers).

5 answers

The least-squares regression line is given by the equation \( y = a + bx \), where \( a \) is the y-intercept and \( b \) is the slope of the line.

From the calculator output:

  • \( a = 2.17306922 \)
  • \( b = 0.32743551 \)

Substituting these values into the equation, we get:

\[ y = 2.17306922 + 0.32743551x \]

So, the equation of the least-squares regression line is:

\[ y = 2.17306922 + 0.32743551x \]

What is the correlation between the interest rate and the yield of the stock?

The correlation between the interest rate and the yield of the stock is represented by the correlation coefficient \( r \). From the calculator output, the value of \( r \) is:

\[ r = 0.60863635 \]

This indicates a positive correlation between the prime interest rate and the yield of the stock. The value of \( r \) can range from -1 to 1, where:

  • \( r = 1 \) indicates a perfect positive correlation,
  • \( r = -1 \) indicates a perfect negative correlation,
  • \( r = 0 \) indicates no correlation.

In this case, an \( r \) value of approximately \( 0.6086 \) suggests a moderate positive correlation between the interest rate and the yield of the stock. This implies that, as the interest rate increases, the yield of the stock tends to increase as well.

Predict the price when the prime interest rate is 6%. Round the answer to at least four decimal places.

To predict the price of the stock when the prime interest rate (\( x \)) is 6%, you can substitute \( x = 6 \) into the least-squares regression line equation:

\[ y = 2.17306922 + 0.32743551x \]

Substituting \( x = 6 \):

\[ y = 2.17306922 + 0.32743551(6) \]

Calculating the product:

\[ 0.32743551 \times 6 = 1.96461306 \]

Now, adding it to the y-intercept:

\[ y = 2.17306922 + 1.96461306 = 4.13768228 \]

Therefore, the predicted price when the prime interest rate is 6% is:

\[ \boxed{4.1377} \]

(Rounded to four decimal places.)