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A recent article concerning bullish and bearish sentiment about the stock market reported that \(41 \%\) of investors responding to an American Institute of Individual Investors (AAII) poll were bullish on the market and \(26 \%\) were bearish (USA Today, January 11,2010 ). The article also reported that the long-term average measure of bullishness is .39 or \(39 \%\) Suppose the AAII poll used a sample size of \(450 .\) Using .39 (the long-term average) as the population proportion of investors who are bullish, conduct a hypothesis test to determine if the current proportion of investors who are bullish is significantly greater than the longterm average proportion. a. State the appropriate hypotheses for your significance test. b. Use the sample results to compute the test statistic and the \(p\) -value. c. Using \(\alpha=.10,\) what is your conclusion?

Step by step solution

01

Define Hypotheses

We start by defining the null and alternative hypotheses for the hypothesis test. The null hypothesis \(H_0\) states that the current proportion of bullish investors is equal to the long-term average, \(p = 0.39\). The alternative hypothesis \(H_a\) suggests that the current proportion is greater than the long-term average, \(p > 0.39\). Thus, we have:\[H_0: p = 0.39 \H_a: p > 0.39\]

02

Calculate Test Statistic

To calculate the test statistic, we use the formula for a one-sample proportion test:\[z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\]where \(\hat{p} = 0.41\) (sample proportion), \(p_0 = 0.39\) (population proportion), and \(n = 450\) (sample size). Substituting these values provides:\[z = \frac{0.41 - 0.39}{\sqrt{\frac{0.39(1-0.39)}{450}}} = \frac{0.02}{\sqrt{\frac{0.2379}{450}}}\]Calculating further:\[z \approx \frac{0.02}{0.0221} \approx 0.905\]

03

Determine p-value

The next step is to find the p-value corresponding to the calculated z-score. Since we have a right-tailed test, we look up the value of \(z = 0.905\) in the standard normal distribution table or use a calculator to find the p-value. For \(z = 0.905\), the p-value \(\approx 0.182\).

04

Draw Conclusion

To draw a conclusion, we compare the p-value with the significance level \(\alpha = 0.10\). Since the p-value \(0.182\) is greater than \(0.10\), we fail to reject the null hypothesis.This means that there is not enough evidence to suggest that the current proportion of bullish investors is significantly greater than the long-term average of \(39\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Test

A proportion test is a statistical method used to determine if there is a significant difference between a sample proportion and a known or assumed population proportion. In practice, it involves comparing the observed proportion from sample data to a theoretical proportion from a larger population.
For example, if a poll reveals that 41% of investors are bullish about the stock market, we can use a proportion test to see if this figure significantly differs from a long-term average of 39%.
The formula for the test statistic in a proportion test is:

• Formula: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\)
• Where \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesis population proportion, and \(n\) is the sample size.

By calculating the z-score and finding the p-value, we can see if our sample's proportion is significantly different from the population proportion.

Significance Level

The significance level in hypothesis testing is a measure that decides the threshold for rejecting the null hypothesis. It is denoted by the Greek letter \(\alpha\).
Typically set at 0.05 or 0.10, the significance level indicates the probability of incorrectly rejecting a true null hypothesis, which is known as making a Type I error. In simpler terms, it's the "risk" we are willing to take in drawing a wrong conclusion.
For instance, in our stock market sentiment example, a significance level of \(\alpha = 0.10\) means we're allowing a 10% risk of claiming there is a significant difference when there isn't actually one.
This level determines the critical value or threshold for the p-value, which we use to decide whether the results are statistically significant.

Null Hypothesis

The null hypothesis, denoted as \(H_0\), is a statement we seek to test in hypothesis testing. It represents the default or established statement that there is no effect or no difference.
In context, the null hypothesis for the stock market example states that the current proportion of bullish investors is equal to the long-term average of 39%. In mathematical terms, it is expressed as:\[H_0: p = 0.39\]
The null hypothesis serves as a reference point against which we measure the sample's statistics. We assume it to be true unless the evidence strongly suggests otherwise.
Failing to reject it means the sample data does not provide sufficient evidence to prove that the observed effect exists.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\) or \(H_1\), posits a new claim that we aim to challenge or provide evidence for against the null hypothesis.
In our exercise on stock market sentiment, the alternative hypothesis suggests that the proportion of bullish investors is greater than the long-term average. This can be expressed as:

• \(H_a: p > 0.39\)

The alternative hypothesis is accepted if the sample data provides strong enough evidence to reject the null hypothesis within the limits of the chosen significance level.
In simpler terms, if our calculated p-value is below the set significance level, we have founded evidence that supports our alternative hypothesis.