Buy-side vs. sell-side analysts' earnings forecasts. Refer to the Financial Analysts Journal (Jul. /Aug. 2008) study of financial analysts' forecast earnings, Exercise 2.86 (p. 112). Recall that data were collected from $\mathbf{3}\mathbf{,}\mathbf{526}$ buy-side analysts and $\mathbf{58}\mathbf{,}\mathbf{562}$forecasts made by sell-side analysts, and the relative absolute forecast error was determined for each. The mean and standard deviation of forecast errors for both types of analysts are given in the table.

a. Construct a $\mathbf{95}\mathbf{\%}$ confidence interval for the difference between the mean forecast error of buy-side analysts and the mean forecast error of sell-side analysts.

b. Based on the interval, part a, which type of analysis has the greater mean forecast error? Explain.

c. What assumptions about the underlying populations of forecast errors (if any) are necessary for the validity of the inference, part b?

After we get the critical value, find the 95% confidence interval with the use of this formula below:

$({\overline{\mathrm{x}}}_1-{\overline{\mathrm{x}}}_2)\pm {\mathrm{z}}_{\mathrm{\alpha }/2}\sqrt{\frac{{\mathrm{s}}_1^2}{{\mathrm{n}}_1}+\frac{{\mathrm{s}}_2^2}{{\mathrm{n}}_2}}$

Where${\overline{x}}_1$ is the mean forecast error of the buy-side analyst${\overline{\mathrm{x}}}_2$ is the mean forecast error of the sell-side analyst$s_1^2$ squared the standard deviation of the buy-side analyst$s_2^2$ squared the standard deviation of the sell-side analyst$n_1$ total number of samples - buy-side analyst total $n_2$number of samples - sell-side analyst$z_{\alpha /2}$ is the critical values of $z$-test.

Given the below:

One survey was done on financial analysts. The information was gathered from 3,526 buy-side analyst projections and 58,562 sell-side analyst estimates. The information is represented in the table.

${\mathrm{n}}_1=3,526,{\mathrm{n}}_2=58,562,{\overline{\mathrm{x}}}_1=0.85,{\overline{\mathrm{x}}}_2=-0.05,{\mathrm{s}}_1=1.93\mathrm{and}{\mathrm{s}}_2=0.85$

Let ${\mathrm{\mu }}_1$be the mean forecast error of buy-side analysts and

Let${\mathrm{\mu }}_2$be the mean forecast error of sell-side analysts.

Critical value:

The level of confidence is $95\%$.

$\mathrm{\alpha }=0.05$

$\frac{\mathrm{\alpha }}{2}=\frac{0.05}{2}$

$=0.025$

Hence, the cumulative area to the left is as follows:

Area to the left - Area to the right

$=1-0.025$

$=0.975$

From Table II of the standard normal distribution in Appendix D, the critical value is $1.96$

Confidence interval:

$$\begin{gathered} \mathrm{CI}=({\overline{\mathrm{x}}}_1-{\overline{\mathrm{x}}}_2)\pm {\mathrm{z}}_{\frac{\mathrm{a}}{2}}\sqrt{\left(\frac{{\mathrm{\sigma }}^{2}1}{{\mathrm{n}}_1}+\frac{{\mathrm{\sigma }}^{2}1}{{\mathrm{n}}_2}\right)} \\ =(0.85-(-0.05\left)\right)\pm 1.96\sqrt{\left(\frac{1.{93}^2}{3,526}+\frac{0.{85}^2}{58,562}\right)} \\ =0.90\pm 0.0640 \\ =(.8359,0.9640) \end{gathered}$$

Thus, the $95\%$confidence interval for the difference between the mean forecast error of buy-side analysts and the mean forecast error of sell-side analysts is$(0.8359,0.9640).$

The$95\%$confidence interval for the difference between the mean forecast error of buy-side analysts and the mean forecast error of sell-side analysts is$(.8359,0.9640)$.

The interval does not contain $0$. So, the mean difference is greater than zero. There is evidence to say that the difference is present between the two groups.

All values in the interval are positive; therefore, it indicates that the mean forecast error of buy-side analysts is more than the mean forecast error of sell-side analysts.

The assumptions for this test are the following:

• Two samples are randomly selected from the two target populations.
• Sample sizes are both large.