91Ó°ÊÓ

In Chapter 6 , Exercise 25 , we looked at collected samples of water from streams in the Adirondack Mountains to investigate the effects of acid rain. Researchers measured the pH (acidity) of the water and classified the streams with respect to the kind of substrate (type of rock over which they flow). A lower pH means the water is more acidic. Here is a boxplot of the \(\mathrm{pH}\) of the streams by substrate (limestone, mixed, or shale): Here are selected parts of a software analysis comparing the pH of streams with limestone and shale substrates: 2 -Sample \(t\) -Test of \(\mu_{1}-\mu_{2}\) Difference Between Means \(=0.735\) \(t\) -Statistic \(=16.30 \mathrm{w} / 133 \mathrm{df}\) \(\mathrm{p} \leq 0.0001\) a. State the null and alternative hypotheses for this test. b. From the information you have, do the assumptions and conditions appear to be met? c. What conclusion would you draw?

Step by step solution

01

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that there is no significant difference between the means of pH values of streams flowing over limestone and shale substrates. That is, the mean difference \(\mu_{1}-\mu_{2}\) is 0. The alternative hypothesis (\(H_a\)) is that there is a significant difference between the means, i.e., \(\mu_{1}-\mu_{2} \neq 0\).

02

Check for the Assumptions and Conditions

To perform a t-test, the assumptions are: 1) The samples are independent. 2) Each population is normally distributed. 3) The populations have the same variance. Since the problem does not provide enough information to confirm or reject these assumptions, they need to be checked with the original data or with the researcher.

03

Draw the Conclusion

Given that the t-Statistic is 16.30 with 133 degrees of freedom and the p-value is less than 0.0001. This p-value is less than any common significance level \(\alpha\) (such as 0.05 or 0.01), we reject the null hypothesis. Hence, there is a statistically significant difference between the mean pH levels of streams flowing over limestone and shale substrates.

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Key Concepts

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

t-test

A t-test is a statistical test used to compare the means of two groups, which helps to determine if there is a significant difference between them. In the case of the water pH levels in streams with limestone and shale substrates, a t-test was used. This test is particularly beneficial when dealing with small sample sizes, and it assumes that the data is normally distributed and that the two groups have similar variances.

• The t-test calculates a t-statistic, which, when compared to a critical value from the t-distribution, helps decide whether to reject or fail to reject the null hypothesis.
• A higher t-statistic suggests a greater difference between group means.
• In the given example, a t-statistic of 16.30 was obtained, indicating a large difference between the pH levels of the streams over different substrates.

null hypothesis

The null hypothesis (\(H_0\)) is a statement asserting there is no effect or no difference. It's the hypothesis that researchers attempt to test, and either validate or refute. For the stream pH comparison in the Adirondack Mountains, the null hypothesis was that there is no significant difference between the mean pH levels for streams running over limestone and shale substrates.

• The null hypothesis serves as a baseline or starting point for statistical testing.
• If the data provides sufficient evidence against\(H_0\), it is rejected in favor of the alternative hypothesis.
• In this scenario, researchers set\(H_0: \mu_1 - \mu_2 = 0\)

alternative hypothesis

The alternative hypothesis (\(H_a\)) suggests that there is a significant effect or difference, contrary to the null hypothesis. It is what a researcher aims to prove. In the example of pH levels, the alternative hypothesis proposed that there is indeed a difference between the mean pH of streams over limestone compared to those over shale.

• The alternative hypothesis can be two-sided, indicating a difference in either direction, or one-sided, indicating a specific direction.
• In our case,\(H_a: \mu_1 - \mu_2 eq 0\), which is a two-sided hypothesis suggesting differences in any direction.
• The purpose of the t-test is to provide evidence to either reject or not reject this hypothesis.

The conclusion from hypothesis testing is drawn based upon whether the evidence strongly supports the alternative hypothesis.

p-value

The p-value is a metric that helps determine the strength of the results from a hypothesis test. It shows the probability that the observed data would occur under the null hypothesis.

• A small p-value (<0.05) indicates strong evidence against the null hypothesis, suggesting it should be rejected.
• In our example, the p-value was reported as less than 0.0001, strongly indicating a significant difference in pH levels between the stream substrates.
• A p-value helps decide if the results are not likely due to random chance.

The smaller the p-value, the stronger the evidence supporting the alternative hypothesis.

statistical significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. It's an essential part of hypothesis testing to assess if the test results are meaningful.

• When a p-value is less than the chosen significance level (\(\alpha\)), typically 0.05 or 0.01, the result is termed statistically significant.
• In this exercise, because the p-value was less than 0.0001, the difference in the mean pH levels between limestone and shale streams was considered statistically significant.
• This means researchers are confident in concluding that a real difference exists rather than attributing the result to chance alone.

Recognizing statistical significance aids in making informed decisions in research.