91影视

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

$({\stackrel{炉}{\mathrm{x}}}_1-{\stackrel{炉}{\mathrm{x}}}_2)卤{\mathrm{z}}_{\mathrm{伪}/2}\sqrt{\frac{{\mathrm{s}}_1^2}{{\mathrm{n}}_1}+\frac{{\mathrm{s}}_2^2}{{\mathrm{n}}_2}}$

Where${\stackrel{炉}{x}}_1$ is the mean forecast error of the buy-side analyst${\stackrel{炉}{\mathrm{x}}}_2$ 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$z_{伪/2}$ 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.

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

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

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

Critical value:

The level of confidence is $95\%$.

$\mathrm{伪}=0.05$

$\frac{\mathrm{伪}}{2}=\frac{0.05}{2}$

$=0.025$

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:

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

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.