Problem 46 - Basic Statistics for Business and Economics with Student

Question

A United Nations report shows the mean family income for Mexican migrants to the United States is 27,000$ per year. A FLOC (Farm Labor Organizing Committee) evaluation of 25 Mexican family units reveals a mean to be 30,000 with a sample standard deviation of 10,000 . Does this information disagree with the United Nations report? Apply the 0.01 significance level.

Step-by-Step Solution

Answer

The information does not disagree with the UN report at a 0.01 significance level.

Step 1: Define the Hypotheses

The null hypothesis (\(H_0\)) represents the statement to be tested and usually indicates no effect or no difference. In this scenario, it posits that the mean family income for Mexican migrants is \(27,000\). Thus, \(H_0: \mu = 27,000\). The alternative hypothesis (\(H_a\)) suggests that the mean family income is not \(27,000\), so \(H_a: \mu eq 27,000\).

Step 2: Determine the Test Statistic

Since the sample size is 25, which is considered small and the population standard deviation is not known, we use the t-test. The test statistic formula is:\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]where \(\bar{x} = 30,000\) is the sample mean, \(\mu = 27,000\) is the population mean under the null hypothesis, \(s = 10,000\) is the sample standard deviation, and \(n = 25\) is the sample size. Substituting these values gives:\[ t = \frac{30,000 - 27,000}{10,000 / \sqrt{25}} \]

Step 3: Calculate the Test Statistic Value

First, calculate the standard error: \(SE = 10,000/\sqrt{25} = 2,000\). Now substitute into the t-test formula: \[ t = \frac{3,000}{2,000} = 1.5 \] The calculated t-test statistic value is 1.5.

Step 4: Determine the Critical Value and Decision Rule

For a two-tailed test at the 0.01 significance level with 24 degrees of freedom (\(n - 1 = 25 - 1\)), the critical t-value can be found in t-distribution tables or calculated with statistical software. It is approximately \(\pm2.807\). The decision rule is to reject the null hypothesis if the calculated t-value is less than -2.807 or greater than 2.807.

Step 5: Make the Conclusion

The calculated t-value from Step 3 is 1.5. Since 1.5 is within the range of -2.807 to 2.807, we fail to reject the null hypothesis at the 0.01 significance level. Thus, there is not enough evidence to suggest that the mean family income of Mexican migrants differs from \(27,000\) USD.

Key Concepts

t-test

A t-test is a powerful statistical tool used to determine if there is a significant difference between the means of two groups. In our exercise, the t-test helps us decide whether the average family income of Mexican migrants differs from an established benchmark figure provided by the United Nations. The formula for calculating the t-test statistic is:\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]Here, \(\bar{x}\) represents the sample mean, \(\mu\) is the mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size. We use the t-test in this scenario because the population standard deviation is unknown and the sample size is relatively small (n=25). The result of the t-test allows us to see if the observed sample mean deviates significantly from what the null hypothesis predicts based on the statistical evidence. If the test statistic exceeds a specific threshold (critical value), it suggests that such a deviation is unlikely under the assumption that the null hypothesis is true.

significance level

The significance level is a critical concept in hypothesis testing that determines the threshold for determining statistical significance. In our exercise, this threshold is set at 0.01, meaning we are willing to accept a 1% risk of concluding that a difference exists when it actually does not. The significance level, often denoted by \(\alpha\), is selected before conducting the test and represents the probability of making a Type I error. A Type I error occurs when the null hypothesis is incorrectly rejected. Choosing a lower significance level, like 0.01, means that we demand stronger evidence before rejecting the null hypothesis compared to a higher significance level, such as 0.05. In practical terms, a 0.01 significance level makes it harder to declare a result as statistically significant, promoting more confidence in our findings when we do reject the null hypothesis.

null hypothesis

In hypothesis testing, the null hypothesis is a starting assumption or statement we aim to test. For the exercise in question, the null hypothesis (\(H_0\)) states that the mean family income for Mexican migrants remains at 27,000 USD per year, precisely as claimed by the United Nations report. This forms the status quo or baseline from which deviations are measured. We essentially ask whether the observed data (sample mean of 30,000 USD) can naturally occur under this null hypothesis—all statistical tests aim to prove or disprove this claim. Rejecting the null hypothesis usually suggests that there is indeed a significant effect or difference worth investigating further in the data being analyzed. Thus, the null hypothesis serves as a useful benchmark for comparison and plays a foundational role in assessment and decision-making in statistics.

alternative hypothesis

The alternative hypothesis (\(H_a\)) represents the statement we consider if the null hypothesis seems implausible. In this context, the alternative hypothesis posits that the mean family income is different from 27,000 USD—highlighting any deviation we observe from the reported mean. This directionality is essential since it guides us on what differences or changes we should be looking for.A statistical test can support the alternative hypothesis by showing that any observed deviation in our sample is unlikely to occur if the null hypothesis holds true. Evidence favoring \(H_a\) leads to questioning and potentially rejecting the null hypothesis. In our two-tailed test (examining both directions of difference), the exercise looks for any significant discrepancy, whether an increase or decrease, from the mean specified by \(H_0\). Therefore, the alternative hypothesis provides the more interesting claim, driving the analysis to reveal new insights or confirm existing beliefs in statistical investigations.