Question

Among the data collected for the World Health Organization air quality monitoring project is a measure of suspended particles in \(\mu g / m^{3} .\) Let \(X\) and \(Y\) equal the concentration of suspended particles in \(\mu g / m^{3}\) in the city center (commercial district) for Melbourne and Houston, respectively. Using \(n=13\) observations of \(X\) and \(m=16\) observations of \(Y\), we shall test \(H_{0}: \mu_{X}=\mu_{Y}\) against \(H_{1}: \mu_{X}<\mu_{Y}\). (a) Define the test statistic and critical region, assuming that the unknown variances are equal. Let \(\alpha=0.05\). (b) If \(\bar{x}=72.9, s_{x}=25.6, \bar{y}=81.7\), and \(s_{y}=28.3\), calculate the value of the test statistic and state your conclusion.

Short answer

The exact answer will depend on the calculated critical value and test statistic. You would compare the test statistic to the critical value to decide whether to reject or not reject the null hypothesis, concluding accordingly with a statement about the mean concentrations of suspended particles in the city centers of Melbourne and Houston.

Step-by-step solution

Define the Test Statistic

The test statistic for comparing two means when the variances are assumed to be equal is given by the pooled t statistic. This can be represented as follows: \[t= \frac{\(\bar{X} - \bar{Y}\) - (\mu_x - \mu_y)}{s_p \sqrt{\frac{1}{n} + \frac{1}{m}}}\] where \(s_p\) is the pooled standard deviation, \(\bar{X}\) and \(\bar{Y}\) are the sample means, \(n\) and \(m\) are the sample sizes, and \(\mu_x\) and \(\mu_y\) are the population means. Under the null hypothesis, \(\mu_x = \mu_y\), so the equation simplifies to: \[t= \frac{\(\bar{X} - \bar{Y}\)}{s_p \sqrt{\frac{1}{n} + \frac{1}{m}}}\]

Define the Critical Region

The critical region is defined by values that the test statistic must exceed in order for us to reject the null hypothesis. The critical value of \(t\) for a left-tailed test with \(\alpha = 0.05\) and \((n + m - 2)\) degrees of freedom can be found in a t-table. After looking up these numbers in the table, we can finalize the critical region for this test.

Calculate the Test Statistic

The test statistic can now be calculated by substituting the given values into the formula. This involves calculating the pooled standard deviation \(s_p\) using the given standard deviations \(s_x\) and \(s_y\), and then calculating the test statistic \(t\) using the calculated \(s_p\), the given sample sizes \(m\) and \(n\), and the given sample means \(\bar{x}\) and \(\bar{y}\).

Make a Conclusion

After calculating the test statistic, we compare it with the critical value. If the test statistic is less than the critical value (indicating that the mean concentration of suspended particles in the city center of Melbourne is lower than in Houston), we reject the null hypothesis and accept the alternative hypothesis.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method that is used to make decisions about the properties of a population based on a sample. In this context, it's used to determine whether there is a significant difference between the air quality in Melbourne and Houston. The process starts with the formulation of two hypotheses:
- The null hypothesis \(H_0\) suggests no difference between the means, meaning the concentration of particles in Melbourne equals that in Houston (- The alternative hypothesis \(H_1\) posits that the concentration in Melbourne is less than that in Houston ( \(H_1: \mu_X < \mu_Y\)).
We then use statistical tests to determine the likelihood that the observed data would occur if the null hypothesis were true. A significance level \(\alpha\) of 0.05 is typically chosen, which means there's a 5% risk of incorrectly rejecting the null hypothesis.
This method of hypothesis testing ensures a systematic approach in evaluating the differences in data sets, guiding decision-making, and providing a way to gauge the effectiveness of solutions.

Pooled Variance

When comparing two groups, we often need to assume that their variances are equal—this is where the concept of pooled variance comes in. Pooled variance is an estimate of the combined variance of the two samples. It serves as a weighted average of each sample's variance, providing a singular variance measure across the datasets. Here's how we can find the pooled sample variance, \ where:
- \(s_X^2\) and \(s_Y^2\) are the variances of the two samples
- \(n\) and \(m\) represent the respective sample sizes.
The formula is as follows:\[s_p^2 = \frac{(n-1)s_X^2 + (m-1)s_Y^2}{n+m-2}\]Using this approach, we assume the two groups being compared have approximately the same variance, simplifying our calculations in hypothesis testing. It also helps in accurately estimating the test statistic, leading to precise conclusions.

Critical Region

The critical region is a crucial part of hypothesis testing as it determines whether we reject the null hypothesis. To establish this region, we first find critical values from a statistical table—typically a t-distribution table since we're often dealing with t-tests.
In the scenario provided, with \(\alpha = 0.05\), we're using a left-tailed test, meaning that we're mainly interested in whether the test statistic is significantly less than a specific threshold. The degrees of freedom \(df\) for the test are calculated by \(df = n + m - 2\).If the calculated test statistic falls within the critical region derived from the t-table, we have reason to reject the null hypothesis and accept the alternative hypothesis. This critical region thus helps in making data-driven decisions based on observed evidence.

Test Statistic

The test statistic is a number calculated from a statistical test's data, which is used to determine if the null hypothesis should be rejected. For a two-sample t-test assuming equal variances, we calculate the t-statistic using:
\[t = \frac{(\bar{X} - \bar{Y})}{s_p \sqrt{\frac{1}{n} + \frac{1}{m}}}\]Here:

• \(\bar{X}\) and \(\bar{Y}\) are the sample means of Melbourne and Houston, respectively
• \(s_p\) is the pooled standard deviation
• \(n\) and \(m\) are the sizes of Melbourne and Houston's samples.

By substituting the sample data, we calculate the actual t-value.
This value is then compared to the critical region's threshold. If it falls within the region, it indicates strong enough evidence against the null hypothesis, leading to its rejection. This process confirms whether observed differences in sample data are statistically significant.