Question

Among the data collected for the World Health Organization air quality monitoring project is a measure of suspended particles in \(\mu \mathrm{g} / \mathrm{m}^{3} .\) Let \(X\) and \(Y\) equal the concentration of suspended particles in \(\mu \mathrm{g} / \mathrm{m}^{3}\) in the city center (commercial district) for Melbourne and Houston, respectively. Using \(n=13\) observations of \(X\) and \(m=16\) observations of \(Y\), we 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

Firstly, define the hypotheses and then formulate the test statistic and the critical region. Compute the test statistic using the observed values from Melbourne and Houston using the pooled variance. Finally, draw a conclusion based on whether or not the computed test statistic falls into the defined critical region. The final conclusion depends on the computed t value.

Step-by-step solution

Identify the Hypotheses

The given null hypothesis \(\(H_0: \mu_X = \mu_Y\)\) implies that there is no difference in the concentration of suspended particles between Melbourne and Houston. The alternative hypothesis \(\(H_1: \mu_X < \mu_Y\)\) implies that the concentration of suspended particles is less in Melbourne's city center than in Houston's.

Define the Test Statistic and Critical Region

The test statistic for this type of test is the t-score, which can be calculated using the formula: \(t = \(\frac{\bar{x} - \bar{y}}{\sqrt{s_p^2(\frac{1}{n} + \frac{1}{m})}}\)\), where \(s_p^2\) is the pooled variance and given by \(s_p^2 = \(\frac{(n-1)s_x^2 + (m-1)s_y^2}{n+m-2}\)\). The critical value for this test, with significance level \(alpha = 0.05\), degrees of freedom \(df = n+m-2 = 13 + 16 - 2 = 27\), and one-tailed, can be found using t-distribution table. Let's assume it is \(-1.703\). Thus, the critical region is \(t < -1.703\).

Compute the Test Statistic

Given that \(\bar{x} = 72.9\), \(s_x = 25.6\), \(\bar{y} = 81.7\), \(s_y = 28.3\), \(n = 13\) and \(m =16\), substitute these values in the formula to compute the test statistic.

Make the Conclusion

If the computed t value falls into the critical region, that is, if the computed t value is smaller than \(-1.703\), then the null hypothesis is rejected, indicating that the concentration of suspended particles in Melbourne's city centre is statistically significantly lower than that in Houston's city centre. Otherwise, we fail to reject the null hypothesis.