Problem 47 - Basic Statistics for Business and Economics with Student

Question

Traditionally, two percent of the citizens of the United States live in a foreign country because they are disenchanted with U.S. politics or social attitudes. In order to test if this proportion has increased since the September 11, 2001, terror attacks, U.S. consulates contacted a random sample of 400 of these expatriates. The sample yields 12 people who report they are living overseas because of political or social attitudes. Can you conclude this data shows the proportion of politically motivated expatriates has increased? Use the 0.05 significance level.

Step-by-Step Solution

Answer

No, there is not enough evidence to conclude an increase in the proportion of politically motivated expatriates.

Step 1: Define the Hypotheses

First, we need to define the null and alternative hypotheses for this problem. The null hypothesis (H0) is that the proportion of expatriates with political or social motivations remains at 2% (p = 0.02). The alternative hypothesis (H1) is that this proportion has increased (p > 0.02).

Step 2: Calculate the Sample Proportion

Next, we calculate the sample proportion of politically or socially motivated expatriates. In the sample of 400 expatriates, 12 report political or social motivation. Thus, the sample proportion \( \hat{p} \) is given by: \[ \hat{p} = \frac{12}{400} = 0.03 \]

Step 3: Determine the Test Statistic

We use a one-sample z-test for proportions to determine the test statistic. The formula for the z-test statistic is:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]where \( p_0 = 0.02 \) (the hypothesized proportion), \( \hat{p} = 0.03 \) (the sample proportion), and \( n = 400 \) (the sample size). Let's calculate this value: \[ z = \frac{0.03 - 0.02}{\sqrt{\frac{0.02 (1 - 0.02)}{400}}} \] \[ z = \frac{0.01}{\sqrt{0.000049}} \] \[ z = \frac{0.01}{0.007} \] \[ z \approx 1.43 \]

Step 4: Determine the Critical Value and Compare

Since we are using a significance level \( \alpha = 0.05 \) for a one-tailed test, we check standard normal distribution tables for the critical value of z. The critical z-value for \( 0.05 \) significance level is approximately 1.645. We compare this to our calculated z-value of 1.43.

Step 5: Make a Decision

The calculated test statistic (z = 1.43) is less than the critical value (z = 1.645). Since it does not fall in the rejection region, we fail to reject the null hypothesis. This means we do not have enough evidence to conclude that the proportion of expatriates motivated by political or social reasons has increased beyond 2%.

Key Concepts

Understanding the Null Hypothesis

In hypothesis testing, the null hypothesis plays an essential role. It's a statement of no effect or no difference that we seek to test. For instance, it posits that any observed change in data is purely due to chance.
In our exercise, the null hypothesis ( H0 ) asserts that the proportion of citizens living abroad due to political or social reasons remains unchanged at 2% ( 0.02 ).
This serves as a baseline to compare our sample data against.

• Think of the null hypothesis as an initial stance or default position that we attempt to challenge through data analysis.
• By using a statistical test, we assess whether there is enough evidence to refute this default position.
• If the evidence is insufficient, we continue to assume the null hypothesis is true.

Exploring the Alternative Hypothesis

The alternative hypothesis is a statement that contradicts the null hypothesis. It suggests that there is a noticeable effect or difference in our data.
For the given exercise, the alternative hypothesis ( H1 ) suggests that the proportion of citizens motivated by political or social reasons to live abroad has increased, now exceeding 2% ( p > 0.02 ).

• The alternative hypothesis represents what you aim to prove through hypothesis testing.
• It allows for the possibility of finding a significant effect or difference in your analysis.
• If we find substantial evidence against the null hypothesis, we embrace the alternative hypothesis.

What is a One-Sample Z-Test?

The one-sample z-test is a statistical method used to determine whether there is a significant difference between the mean or proportion of a sample and a known population mean or proportion.
In this scenario, we use it to compare the sample proportion of citizens living abroad ( 0.03 ) to the hypothesized population proportion ( 0.02 ).
This test is suitable when the sample size is large enough (n > 30).

• The one-sample z-test helps to understand if the data from a random sample deviates significantly from the population.
• It's commonly used when the population standard deviation is known, or the sample size is sufficiently large.
• Through the z-test, we calculate a test statistic ( z ) indicating how far the sample proportion is from the hypothesized proportion in terms of standard deviations.

The Importance of Significance Level

The significance level in hypothesis testing is a threshold used to decide whether to reject the null hypothesis. It indicates the probability of rejecting the null hypothesis when it is, in fact, true.
For the exercise, the significance level chosen is 0.05 (or 5%). This means there is a 5% risk of concluding the proportion has increased when it hasn't.

• A lower significance level (e.g., 0.01 ) indicates stronger evidence is required to reject the null hypothesis.
• Conversely, a higher significance level (e.g., 0.10 ) allows for a greater probability of making a Type I error, where the null hypothesis is incorrectly rejected.
• This concept helps assess the strength of the evidence needed to consider the outcome statistically significant.