Chapter 10, Problem 47SE

a.

To determine

Find a 95% confidence interval for the difference between the bullish sentiment measures for the most recent two weeks.

Answer to Problem 47SE

The 95% confidence interval for the difference between the bullish sentiment measures for the most recent two weeks is $\left(-0.297,-0.127\right)$.

Explanation of Solution

Calculation:

It is given that the bullish sentiment of individual investors for the week end was 27.6%, 48.7% for one week earlier and 39.7% for one month earlier. A sample of 240 is considered.

Confidence interval:

The interval estimate of the difference between two population proportions is,

$\text{Confidence interval}={\overline{p}}_1-{\overline{p}}_2\pm z_{\frac{\alpha }{2}}\sqrt{\frac{{\overline{p}}_1\left(1-{\overline{p}}_1\right)}{n_1}+\frac{{\overline{p}}_2\left(1-{\overline{p}}_2\right)}{n_2}}$

where $1-\alpha$ is the confidence coefficient, ${\overline{p}}_1\text{and }{\overline{p}}_2$ the point estimates for population proportion, $n_1\text{and }{n}_2$ are sample sizes for the two group..

Here, $n_1=n_2=n_3=240,{\overline{p}}_1=0.276,{\overline{p}}_2=0.487,{\overline{p}}_3=0.397$,  and

$\begin{array}{c}{x}_1=\left(240\right)\left(0.276\right)\\ =66.24\\ \approx 66\end{array}$

$\begin{array}{c}{x}_2=\left(240\right)\left(0.487\right)\\ =116.88\\ \approx 117\end{array}$

$\begin{array}{c}{x}_3=\left(240\right)\left(0.397\right)\\ =95.28\\ \approx 95\end{array}$

Software procedure:

Step-by-step software procedure to obtain the p-value using MINITAB software is as follows,

• Choose Stat > Basic Statistics > 2 Proportions.
• Choose Summarized data.
• In First sample, enter Trials (n) as 240 and Events as (x) 66.
• In Second sample, enter Trials (n) as 240 and Events as 117.
• Click Options.
• In Confidence level, Enter 95.
• In Alternative select not equal.
• Click OK in all dialogue boxes.

Output using MINITAB software is as follows,

Thus, the 95% confidence interval is $\left(-0.297,-0.127\right)$.

Interpretation:

There is 95% confidence that the difference between the bullish sentiment measures for the most recent two weeks lies between –0.297 and –0.127.

b.

To determine

State the null and alternative hypothesis so that rejection of the null hypothesis will allow to conclude that the most recent bullish sentiment is weaker than that of one month ago.

Answer to Problem 47SE

The null and alternative hypotheses are given below:

Null hypothesis: $H_0:p_1-p_3\ge 0$

Alternative hypothesis: $H_a:p_1-p_3<0$

Explanation of Solution

Calculation:

Here $p_1$ represents the proportion of the bullish sentiment of individual investors for the most recent week and $p_3$ represents the proportion of the bullish sentiment of individual investors for one month ago.

State the hypothesis:

The test hypotheses are as follows,

Null hypothesis:

$H_0:p_1-p_3\ge 0$

That is, the proportion of the bullish sentiment of individual investors for the most recent week is greater than the proportion of the bullish sentiment of individual investors for one month ago.

Alternative hypothesis:

$H_a:p_1-p_3<0$

That is, the proportion of the bullish sentiment of individual investors for the most recent week is less than the proportion of the bullish sentiment of individual investors for one month ago.

c.

To determine

Test the hypothesis in part (b) at $\alpha =0.01$ and make a conclusion.

Answer to Problem 47SE

There is sufficient evidence to conclude that, there is decrease in bullish sentiment over the past month.

Explanation of Solution

Calculation:

Test statistic:

The test statistic for hypothesis tests about $p_1-p_3$ is,

$z=\frac{\left({\overline{p}}_1-{\overline{p}}_3\right)}{\sqrt{\overline{p}\left(1-\overline{p}\right)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$

Where ${\overline{p}}_1\text{and }{\overline{p}}_2$ the point estimates for population proportion, $n_1\text{and }{n}_2$ are sample sizes for the two groups, and $\overline{p}$ is the pooled proportion

Software procedure:

Step-by-step software procedure to obtain the p-value using MINITAB software is as follows,

• Choose Stat > Basic Statistics > 2 Proportions.
• Choose Summarized data.
• In First sample, enter Trials (n) as 240 and Events as (x) 66.
• In Second sample, enter Trials (n) as 240 and Events as 95.
• Click Options.
• In Confidence level, Enter 99.
• In Alternative select less than.
• Click OK in all dialogue boxes.

Output using MINITAB software is as follows,

Thus, the p-value is 0.002.

Rejection rule:

If the $p-value\le \alpha$, then reject the null hypothesis.

If the $p-value>\alpha$, then do not eject the null hypothesis.

Conclusion:

Here the p-value 0.002 is less than the significance level 0.01.

That is, $p\text{-value}\left(0.002\right)<\alpha \left(0.01\right)$

Thus, the null hypothesis is rejected.

Therefore, there is sufficient evidence to conclude that, the proportion of the bullish sentiment of individual investors for the most recent week is less than the proportion of the bullish sentiment of individual investors for one month ago.

Interpretation:

Here the null hypothesis $H_0$ is rejected. That is, there is decrease in bullish sentiment over the past month.