Problem 59 - Basic Statistics for Business and Economics with Student

Question

The Gallup Organization in Princeton, New Jersey, is one of the best-known polling organizations in the United States. It often combines with USA Today or CNN to conduct polls of current interest. It also maintains a website at: http://www.gallup.com/. Consult this website to find the most recent polling results on presidential approval ratings. You may need to click on Gallup Poll. Test whether the majority (more than 50 percent) approve of the president's performance. If the article does not report the number of respondents included in the survey, assume that it is \(1,000,\) a number that is typically used.

Step-by-Step Solution

Answer

The majority approve of the president's performance, as the test statistic exceeds the critical value.

Step 1: Obtain Approval Percentage

Visit the Gallup website or its polling results section and find the latest presidential approval rating percentage. Let's assume it states that 55% approve of the president's performance.

Step 2: Hypothesis Formulation

Formulate the null and alternative hypothesis. The null hypothesis will be that the proportion of approval, \( p \), is equal to 0.5, and the alternative hypothesis is that \( p \) is greater than 0.5. \( H_0: p = 0.5 \); \( H_a: p > 0.5 \).

Step 3: Calculate Test Statistic

Use the formula for the test statistic for a proportion: \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \), where \( \hat{p} \) is the sample proportion (0.55), \( p_0 \) is the hypothesized proportion (0.5), and \( n \) is the sample size (1,000). Plug in the values: \( z = \frac{0.55 - 0.5}{\sqrt{\frac{0.5 \times 0.5}{1000}}} \approx 3.1622 \).

Step 4: Determine Threshold for Significance

Choose a significance level, commonly \( \alpha = 0.05 \). For a one-tailed test, find the critical value of \( z \) which is approximately 1.645.

Step 5: Make a Decision

Compare the calculated test statistic \( z \approx 3.1622 \) with the critical value 1.645. Since 3.1622 > 1.645, you reject the null hypothesis.

Key Concepts

Proportion Test

A proportion test is a type of statistical hypothesis test used to determine whether the proportion of a certain outcome in a sample is significantly different from a specified value.
It is commonly used in situations where you want to compare the observed proportion of a feature in a population to a theoretical or expected proportion.

In our example about presidential approval rating, we want to know if the sample proportion (55%) truly reflects a majority of approval, or more than 50%.

• The test involves determining whether the observed sample proportion differs significantly from the hypothesized proportion.
• This involves calculating a test statistic, which in this case is a z-score, using a mathematical formula.
• Once the test statistic is calculated, it is compared against critical values to determine if the observed difference is statistically significant.

By following these steps, the proportion test gives us a measure of confidence that the observed proportion is different from what we assumed in the null hypothesis.

Statistical Significance

Statistical significance is a crucial concept that helps us decide whether to reject or accept a hypothesis.
It shows if an effect or difference observed in a study is strong enough to be considered beyond random chance.

In any hypothesis test, we choose a significance level, typically denoted as \( \alpha \), which is a threshold for making this decision.

• Common values for \( \alpha \) are 0.05 or 0.01, meaning there's a 5% or 1% chance to incorrectly reject a true null hypothesis.
• We then compare our calculated test statistic to the critical value that corresponds to our significance level.
• If the test statistic falls beyond the critical value, the results are considered statistically significant, allowing us to reject the null hypothesis.

For example, in the presidential approval test, the z-score calculation provided a value beyond the critical limit, indicating that the proportion is significantly more than 50% with our chosen significance level.

Null and Alternative Hypothesis

Before any hypothesis test begins, we first need to define our hypotheses.
This involves setting up the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \).

The null hypothesis represents the statement we are trying to test against. It usually reflects the idea that there is no effect or no difference, serving as a default or baseline assumption. In our example:

• The null hypothesis \( H_0: p = 0.5 \) suggests that exactly 50% of the population approve of the president's performance.

The alternative hypothesis, on the other hand, is what we're testing for. It represents the possibility of an effect or difference being present. In our scenario:

• The alternative hypothesis \( H_a: p > 0.5 \) suggests that more than 50% of the population approve of the president's performance.

Hypothesis testing revolves around these two hypotheses, providing a framework for statistical analysis to evaluate if the initial assumption (null hypothesis) stands or should be rejected in favor of the alternative hypothesis.