For the week ended January 15,2009, the bullish sentiment of...

For the week ended January $15,2009,$ the bullish sentiment of individual investors was $27.6 \%$ (AAII Journal). The bullish sentiment was reported to be $48.7 \%$ one week earlier and $39.7 \%$ one month earlier. The sentiment measures are based on a poll conducted by the American Association of Individual Investors. Assume that each of the bullish sentiment measures was based on a sample size of 240 . a. Develop a $95 \%$ confidence interval for the difference between the bullish sentiment measures for the most recent two weeks. b. Develop hypotheses so that rejection of the null hypothesis will allow us to conclude that the most recent bullish sentiment is weaker than that of one month ago. c. Conduct a hypotheses test of part (b) using $\alpha=.01$. What is your conclusion?

For the week ended January $15,2009,$ the bullish sentiment of individual investors was $27.6 \%$ (AAII Journal). The bullish sentiment was reported to be $48.7 \%$ one week earlier and $39.7 \%$ one month earlier. The sentiment measures are based on a poll conducted by the American Association of Individual Investors. Assume that each of the bullish sentiment measures was based on a sample size of 240 .
a. Develop a $95 \%$ confidence interval for the difference between the bullish sentiment measures for the most recent two weeks.
b. Develop hypotheses so that rejection of the null hypothesis will allow us to conclude that the most recent bullish sentiment is weaker than that of one month ago.
c. Conduct a hypotheses test of part (b) using $\alpha=.01$. What is your conclusion?

Key Concepts

Standard Error for Proportions

The standard error is a measure of the variability of a sample statistic. For two proportions, the combined standard error quantifies the uncertainty associated with their difference. It is calculated using the sample proportions and their sizes, being central to both confidence interval construction and the computation of test statistics in hypothesis testing.

Significance Level

This is the probability threshold (commonly denoted as alpha, such as 0.01 or 0.05) used in hypothesis testing to determine whether an observed effect is statistically significant. It represents the risk of rejecting the null hypothesis when it is actually true (Type I error), guiding the decision-making process in tests.

Null and Alternative Hypotheses

In the context of comparing two population proportions, the null hypothesis typically posits that there is no difference (or that a difference is in a specified direction) between the two proportions. The alternative hypothesis represents the claim to be tested – for instance, that one proportion is significantly lower or higher than the other. This formulation is crucial in guiding the direction and interpretation of the statistical test.

Hypothesis Testing for Difference in Proportions

This involves setting up a statistical test to evaluate whether the observed difference between two sample proportions reflects a true difference in the population. The process includes formulating null and alternative hypotheses, calculating a test statistic (often using a Z-test), and comparing it to a critical value or using a p-value to decide on rejecting or not rejecting the null hypothesis.

Two-Sample Proportion Confidence Interval

This concept involves constructing an interval estimate for the difference between two population proportions based on two independent samples. It uses the sample proportions and their estimated standard error, assuming a normal distribution of the sampling difference, to capture the value of the true difference within a specified level of confidence (e.g., 95%).

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