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A financial analyst is examining the relationship between stock prices and earnings per share. She chooses sixteen publicly traded companies at random and records each company's current stock price and the company's earnings per share reported for the past 12 months. Her data are given below, with [tex]\(x\)[/tex] denoting the earnings per share from the previous year, and [tex]\(y\)[/tex] denoting the current stock price (both in dollars). Based on these data, she computes the least-squares regression line to be [tex]\(\hat{y} = -0.236 + 0.046x\)[/tex]. This line, along with a scatter plot of her data, is shown below.

[tex]\[
\begin{array}{|c|c|}
\hline
\text{Earnings per share } x \text{ (in dollars)} & \text{Current stock price } y \text{ (in dollars)} \\
\hline 17.17 & 0.50 \\
\hline 21.90 & 0.71 \\
\hline 36.03 & 1.52 \\
\hline 56.66 & 2.14 \\
\hline 29.56 & 0.96 \\
\hline 41.00 & 1.76 \\
\hline 49.15 & 1.64 \\
\hline 59.16 & 2.70 \\
\hline 30.16 & 1.35 \\
\hline 35.09 & 1.74 \\
\hline 36.38 & 1.23 \\
\hline 26.26 & 1.02 \\
\hline 52.84 & 2.83 \\
\hline 42.31 & 1.49 \\
\hline 42.11 & 0.95 \\
\hline 16.04 & 0.64 \\
\hline
\end{array}
\][/tex]

Based on the analyst's data and regression line, complete the following:

(a) For these data, current stock prices that are greater than the mean of the current stock prices tend to be paired with values for earnings per share that are (Choose one: above/below) the mean of the values for earnings per share.

(b) According to the regression equation, for an increase of one dollar in earnings per share, there is a corresponding increase of how many dollars in current stock price?
Earnings per share (in dollars) [tex]\(\square\)[/tex]

Sagot :

GinnyAnswer
Alright, let's break down the solution based on the given data and regression equation step by step.

### (a) Mean Relationship

First, we need to find the mean values of current stock prices ([tex]\( y \)[/tex]) and earnings per share ([tex]\( x \)[/tex]) based on the given data.

The mean current stock price, [tex]\( \bar{y} \)[/tex], is calculated using the sum of all current stock prices divided by the number of data points.

Similarly, the mean earnings per share, [tex]\( \bar{x} \)[/tex], is calculated using the sum of all earnings per share divided by the number of data points.

The calculated mean values are:
- Mean current stock price, [tex]\( \bar{y} = 1.44875 \)[/tex] dollars
- Mean earnings per share, [tex]\( \bar{x} = 36.98875 \)[/tex] dollars

Based on these values, we can state the relationship:
- Current stock prices that are greater than the mean current stock price ([tex]\(1.44875\)[/tex] dollars) tend to be paired with values for earnings per share that are greater than the mean value for earnings per share ([tex]\(36.98875\)[/tex] dollars).

This conclusion derives from the assumption that if [tex]\(y\)[/tex] is greater than its mean, using the positive trend suggested by the regression line, [tex]\(x\)[/tex] will generally also be above its mean.

### (b) Slope Interpretation

The regression equation given is:
[tex]\[ \hat{y} = -0.236 + 0.046x \][/tex]

In a regression equation of the form [tex]\( \hat{y} = a + bx \)[/tex]:
- [tex]\( a \)[/tex] is the intercept.
- [tex]\( b \)[/tex] is the slope.

The slope [tex]\( b = 0.046 \)[/tex] indicates the amount by which [tex]\( y \)[/tex] (the current stock price) changes for a one-dollar change in [tex]\( x \)[/tex] (earnings per share).

Thus, according to the regression equation:
- For an increase of one dollar in earnings per share, there is a corresponding increase of [tex]\( 0.046 \)[/tex] dollars in the current stock price.

### Conclusion

(a) Current stock prices greater than the mean of the current stock prices tend to be paired with values for earnings per share that are greater than the mean of the values for earnings per share.

(b) According to the regression equation, for an increase of one dollar in earnings per share, there is a corresponding increase of 0.046 dollars in the current stock price.
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