91Ó°ÊÓ

The article "Pine Needles as Sensors of Atmospheric Pollution" (Environ. Monitoring, 1982: 273-286) reported on the use of neutron-activity analysis to determine pollutant concentration in pine needles. According to the article's authors, "These observations strongly indicated that for those elements which are determined well by the analytical procedures, the distribution of concentration is lognormal. Accordingly, in tests of significance the logarithms of concentrations will be used." The given data refers to bromine concentration in needles taken from a site near an oil-fired steam plant and from a relatively clean site. The summary values are means and standard deviations of the log-transformed observations. \begin{tabular}{lccc} Site & Sample Size & Mean Log Concentration & SD of Log Concentration \\ \hline Steam plant & 8 & \(18.0\) & \(4.9\) \\ Clean & 9 & \(11.0\) & \(4.6\) \\ \hline \end{tabular} Let \(\mu_{1}^{*}\) be the true average log concentration at the first site, and define \(\mu_{2}^{*}\) analogously for the second site. a. Use the pooled \(t\) test (based on assuming normality and equal standard deviations) to decide at significance level .05 whether the two concentration distribution means are equal. b. If \(\sigma_{1}^{*}\) and \(\sigma_{2}^{*}\) (the standard deviations of the two log concentration distributions) are not equal, would \(\mu_{1}\) and \(\mu_{2}\) (the means of the concentration distributions) be the same if \(\mu_{1}^{*}=\mu_{2}^{*}\) ? Explain your reasoning.

Step by step solution

01

State the Hypotheses

We are tasked with comparing the mean log concentrations at the two sites, therefore we set up the hypotheses for the pooled t-test as:\[ H_0: \mu_1^* = \mu_2^* \] (the means are equal) versus \[ H_a: \mu_1^* eq \mu_2^* \] (the means are different).

02

Calculate the Pooled Standard Deviation

The pooled standard deviation (SD) is calculated using the formula:\[ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} \]where \( s_1 = 4.9 \), \( s_2 = 4.6 \), \( n_1 = 8 \), and \( n_2 = 9 \).\[ s_p = \sqrt{\frac{(8-1)(4.9)^2 + (9-1)(4.6)^2}{8+9-2}} = 4.73 \]

03

Compute the Test Statistic

The test statistic for the pooled t-test is given by:\[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]Substituting the given values:\[ t = \frac{18.0 - 11.0}{4.73 \sqrt{\frac{1}{8} + \frac{1}{9}}} = 2.96 \]

04

Determine the Critical Value and Decision

With degrees of freedom \( df = n_1 + n_2 - 2 = 15 \), we look up the critical value for a two-tailed test at the \( \alpha = 0.05 \) level in the t-distribution table. The critical value is approximately 2.131. Since the calculated \( t = 2.96 \) exceeds this critical value, we reject the null hypothesis.

05

Consideration of Unequal Standard Deviations

If \( \sigma_1^* eq \sigma_2^* \), the means \( \mu_1 \) and \( \mu_2 \) of the original concentration distributions are unlikely to be equal even if \( \mu_1^* = \mu_2^* \). This is due to the fact that lognormal distributions are not symmetric, and variation in spread can significantly affect the mean of the distributions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lognormal Distribution

A lognormal distribution is a statistical distribution of a random variable whose logarithm is normally distributed. This means that if you take the natural logarithm of data following a lognormal distribution, the resulting dataset will have a normal distribution.
This type of distribution is common in environmental studies and other fields because it models data that is positive-sided and multiplicative.

• For example, the distribution of pollutant concentrations, as found in the pine needles study, often follows a lognormal pattern.
• Lognormal distributions are right-skewed, meaning they have a long right tail, and they are not symmetrical like the normal distribution.

When data is lognormally distributed, statistical techniques like taking the logarithm of the data can improve the accuracy of significance testing by reducing skewness. By transforming data into its logarithmic form, researchers can utilize statistical tests designed for normally distributed data, making analysis more reliable.

Significance Testing

Significance testing is a method of using statistical analysis to determine whether the observations in an experiment reflect real effects or are likely due to random chance. In the context of the pine needles study, significance testing is used to decide whether the differences in mean log concentrations of pollutants at two different sites are statistically significant.
During the testing process:

• We start by setting up a null hypothesis ( H_0 ), which is usually a statement of no effect or no difference. In this case, it is that the log concentrations are equal between the two sites.
• The alternative hypothesis ( H_a ) would be the opposite, suggesting that there is a difference in the concentrations between the sites.

The result of the significance test is a p-value, which indicates the probability of observing the data if the null hypothesis is true. A low p-value, often less than a predefined threshold such as 0.05, suggests the data is inconsistent with the null hypothesis, leading to its rejection in favor of the alternative. In the exercise, this is accomplished through a pooled t-test.

Statistical Hypothesis Testing

Statistical hypothesis testing is a core method in statistics for testing assumptions or claims about a dataset. It forms the basis of decision-making in statistical analyses by allowing researchers to draw conclusions from sample data.
In hypothesis testing:

• You begin by establishing two hypotheses: the null hypothesis ( H_0 ) and the alternative hypothesis ( H_a ).
• Data from samples are used to calculate a test statistic, which is then compared to a critical value derived from a statistical distribution.

This comparison will tell you whether to reject the null hypothesis. A test such as the pooled t-test, which assumes equal variances, compares the means of two groups to determine if they are likely to have come from the same population.
Understanding the impact of unequal variances, like those mentioned in the exercise, is critical. It can challenge the assumption of a pooled t-test and indicate that results may differ if variability is not accounted for. Thus, hypothesis testing requires careful consideration of the correct methods and assumptions to make valid inferences from data.