Question

Income East and West of the Mississippi For a random sample of households in the US, we record annual household income, whether the location is east or west of the Mississippi River, and number of children. We are interested in determining whether there is a difference in average household income between those east of the Mississippi and those west of the Mississippi. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) What statistic(s) from the sample would we use to estimate the difference?

Short answer

The parameter for households East of the Mississippi River is denoted as \( \mu1 \) and for households West of the Mississippi River as \( \mu2 \). The null hypothesis would be \( H0: \mu1 - \mu2 = 0 \), meaning there's no difference in average household income, and the alternative hypothesis is \( Ha: \mu1 - \mu2 \neq 0 \), meaning there is a difference. The sample means \( \bar{X1} \) and \( \bar{X2} \) are used to estimate the difference.

Step-by-step solution

Defining Parameters

The parameters of interest here would be \( \mu1 \), the population mean income of households located east of the Mississippi River, and \( \mu2 \), the population mean income of households located west of the Mississippi River.

Null and Alternative Hypothesis

The null hypothesis \( H0 \) is that there is no difference in average household income between the two locations. In mathematical terms, \( H0: \mu1 - \mu2 = 0 \). The alternative hypothesis \( Ha \) is that there is a difference in the average income: \( Ha: \mu1 - \mu2 \neq 0 \).

Identify Statistics

To estimate the difference in average household incomes, we would use the sample means from both locations. Thus, the statistics from the sample would be \( \bar{X1} \) and \( \bar{X2} \), where \( \bar{X1} \) represents the sample mean income of households located east of the Mississippi River and \( \bar{X2} \) represents the sample mean income of households located west of the Mississippi River.

Key concepts

Null and Alternative Hypotheses

Understanding null and alternative hypotheses is crucial when embarking on statistical hypothesis testing. In essence, the null hypothesis (\( H_0 \)) is a statement of no effect or no difference. For instance, in comparing household incomes, the null hypothesis posits there is no significant difference in average incomes based on the location relative to the Mississippi River. That is to say, it predicts that the mean income to the east, \( \mu1 \), is equal to the mean income to the west, \( \mu2 \).

Contrarily, the alternative hypothesis (\( H_a \)) is the statement we are seeking to test and potentially validate. It essentially suggests the existence of an effect or a difference. In our example, the alternative hypothesis contends that there is a discrepancy in household incomes between the two regions (\( \mu1 eq \mu2 \)). It leaves open the direction of the difference; the incomes could be higher or lower on either side of the river, as long as they are not equal.

Sample Mean

The sample mean \( \bar{X} \) is a statistic that represents the average value of a sample from a population. It serves as an estimate for the population mean \( \mu \), and calculating it involves summing all the values for a particular variable in the sample and then dividing by the total number of observations.

When comparing two groups, as in the case with household incomes on either side of the Mississippi River, we calculate two sample means \( \bar{X1} \) and \( \bar{X2} \). These averages provide critical insights into the central tendencies of the respective datasets and lay the groundwork for hypothesis testing. They are essential in estimating the difference between the respective population means.

Population Mean

The population mean (\( \mu \) ) represents the average value of a particular variable for an entire population. Unlike the sample mean, which is derived from a subset of the population, the population mean includes every individual instance within the defined group. It's an ideal parameter that usually remains unknown and is estimated using the sample mean.

In the context of our household income example, we are observing two population means: \( \mu1 \) and \( \mu2 \), representing the average incomes east and west of the Mississippi River, respectively. Understanding these theoretical averages is vital as they serve as benchmarks for the true state of affairs in the entire population.

Estimation of Difference

Estimation of difference involves calculating the degree to which two sample statistics vary from each other. This is often used to infer whether there is a likely difference in the population parameters. For example, the estimated difference in average household incomes between the two locations would be the subtraction of the sample mean to the west \( \bar{X2} \) from the sample mean to the east \( \bar{X1} \) (e.g., \( \bar{X1} - \bar{X2} \)).

This estimate provides a snapshot of how incomes compare, while the significance of this difference will be addressed through statistical hypothesis testing. In practice, such estimations are accompanied by a measure of variability or uncertainty, like a confidence interval, that adds more context to the point estimate.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to determine if there is enough statistical evidence in a sample of data to infer that a certain condition holds true for the entire population. The procedure involves several steps, beginning with the formulation of both null and alternative hypotheses. Then, a test statistic that represents the data is calculated, and its value is used to make a decision regarding the hypotheses.

In the scenario of comparing household incomes across different regions, a test statistic derived from the sample means \( \bar{X1} \) and \( \bar{X2} \) would be evaluated against a critical value. If the test statistic falls within a certain range (typically determined by a significance level, commonly 0.05), we would reject the null hypothesis, suggesting that there is a statistically significant difference in household incomes by geography.