Question

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=2.13603469
b=0.38350216
r^2=0.03852450403
r=0.62068111

Write the equation of the least-squares regression line. Use the full accuracy shown in the calculator output (do not round your answers).
Answer
What is the correlation between the interest rate and the yield of the stock?
Answer
Predict the price when the prime interest rate is 7%. Round the answer to at least four decimal places.
Answered by GPT-4o mini
The equation of the least-squares regression line can be written in the form:

\[
y = a + bx
\]

Where:
- \( a \) is the y-intercept.
- \( b \) is the slope.

From the given values:
- \( a = 2.13603469 \)
- \( b = 0.38350216 \)

Substituting these values into the equation, we get:

\[
y = 2.13603469 + 0.38350216x
\]

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

\[
y = 2.13603469 + 0.38350216x
\]
Answered by GPT-4o mini
The correlation between the interest rate and the yield of the stock is represented by the value of \( r \), which stands for the correlation coefficient.

From the output provided, we have:

\[
r = 0.62068111
\]

This value indicates a moderately strong positive correlation between the prime interest rate and the price of the stock. In general terms:
- An \( r \) value close to 1 indicates a strong positive correlation.
- An \( r \) value close to -1 indicates a strong negative correlation.
- An \( r \) value close to 0 indicates no correlation.

In this case, since \( r = 0.62068111 \), it suggests that as the prime interest rate increases, the stock price tends to increase as well, but the correlation is not perfect.
Answered by GPT-4o mini
To predict the price of the stock when the prime interest rate (\(x\)) is 7%, we can use the least-squares regression line equation that we previously derived:

\[
y = 2.13603469 + 0.38350216x
\]

Now, substituting \(x = 7\):

\[
y = 2.13603469 + 0.38350216(7)
\]

Calculating it step by step:

1. Calculate \(0.38350216 \times 7\):
\[
0.38350216 \times 7 = 2.68451412
\]

2. Add the result to \(2.13603469\):
\[
y = 2.13603469 + 2.68451412 = 4.82054881
\]

Thus, the predicted price when the prime interest rate is 7% is:

\[
\boxed{4.8205}
\]

This value is rounded to four decimal places.