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Chapter 4: Problem 84

Mercury Levels in Fish Figure 4.26 shows a scatterplot of the acidity (pH) for a sample of \(n=53\) Florida lakes vs the average mercury level (ppm) found in fish taken from each lake. The full dataset is introduced in Data 2.4 on page 68 and is available in FloridaLakes. There appears to be a negative trend in the scatterplot, and we wish to test whether there is significant evidence of a negative association between \(\mathrm{pH}\) and mercury levels. (a) What are the null and alternative hypotheses? (b) For these data, a statistical software package produces the following output: $$ r=-0.575 \quad p \text { -value }=0.000017 $$ Use the p-value to give the conclusion of the test. Include an assessment of the strength of the evidence and state your result in terms of rejecting or failing to reject \(H_{0}\) and in terms of \(\mathrm{pH}\) and mercury. (c) Is this convincing evidence that low pH causes the average mercury level in fish to increase? Why or why not?

Short Answer

Expert verified
In response to (a), the formulated hypotheses are \(H_0: 蟻 = 0\) and \(H_A: 蟻 < 0\). For (b), the extremely low p-value of 0.000017 leads to a rejection of the null hypothesis, pointing towards a significant negative correlation between pH and mercury levels. And for (c), even with robust evidence of a negative correlation, causation can't be established. To claim that low pH causes the average mercury levels in fish to increase necessitates further, more rigorous investigation.

Step by step solution

01

Formulate Null and Alternative Hypotheses

In simple terms, the null hypothesis (\(H_0\)) is the status quo or generally accepted state of things, while the alternative hypothesis (\(H_A\)) is the proposition that challenges it. As per the given problem, the aim is to check whether there is a negative association between pH and mercury levels. Hence, the null hypothesis would state that there is no correlation between pH and mercury levels, and the alternative hypothesis would express that there is a significant negative correlation. So, \(H_0: 蟻 = 0\) and \(H_A: 蟻 < 0\), where 蟻 represents the correlation coefficient.
02

Analyze the p-value result and conclude the test

The p-value is a crucial metric in hypothesis testing 鈥� it signifies the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. A smaller p-value offers stronger evidence against \(H_0\). Here, the p-value is 0.000017, which is extremely low. Therefore, this provides strong evidence against the null hypothesis. Given this result, it can be concluded that the null hypothesis is rejected, meaning there is significant evidence to suggest a negative correlation between pH and mercury level.
03

Evaluate the Evidence for Causation

A common misconception is that correlation implies causation, but that's not necessarily the case. In this scenario, while there is strong evidence of a negative association between pH and mercury levels, correlation alone does not establish causation. Additional research would be needed, potentially controlling other factors, experimental design or longitudinal studies, to confirm if low pH is the cause of increased mercury levels.

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Key Concepts

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

Understanding p-value
The p-value is one of the most important and often misunderstood concepts in statistics. It's a number between 0 and 1 and is used in hypothesis testing to measure the strength of the evidence against the null hypothesis, denoted as ( H_0 ). The p-value tells us how extreme the observed data is, assuming that the null hypothesis is true.

Specifically, the p-value represents the probability of obtaining test results at least as extreme as the ones observed during the test, under the assumption that the null hypothesis is correct. If this value is very low, it suggests that such an extreme result is unlikely to occur if the null hypothesis were true. Consequently, a low p-value (typically less than 0.05) indicates strong evidence against the null hypothesis, leading analysts to reject ( H_0 ).

In the given exercise, the p-value is exceedingly small (0.000017), suggesting a very low probability that the observed negative association between pH levels and mercury levels in fish is due to random chance. Hence, this provides compelling evidence to reject the null hypothesis, supporting the alternative hypothesis that there is, in fact, a significant negative correlation.
Correlation Coefficient Basics
The correlation coefficient, denoted by the Greek letter rho ( 蟻 ), is a statistical measure that calculates the strength and direction of a linear relationship between two variables. On a scale of -1 to 1, this coefficient indicates the degree to which two variables are linearly related.

A value of 1 implies a perfect positive linear correlation, and a value of -1 implies a perfect negative linear correlation. A value of 0, on the other hand, means there is no linear correlation between the variables. The negative correlation coefficient in the exercise ( r=-0.575 ) reflects a moderate negative association between acidity (pH) and mercury levels in the fish. This indicates that as the pH of the lakes decreases, the mercury levels in fish tend to increase.
Defining Null and Alternative Hypotheses
Hypotheses are foundational to the practice of statistical testing. The null hypothesis, ( H_0 ), represents a default position that there is no effect or no association. In the context of the exercise, it posits that there is zero correlation between the pH levels in Florida lakes and mercury levels in fish.

The alternative hypothesis, on the other hand, is what researchers are trying to provide evidence for. It is denoted as ( H_A ) or ( H_1 ) and asserts that there is an effect or a relationship, which in this case is a negative correlation between pH and mercury levels. In hypothesis testing, we never actually 'prove' the alternative hypothesis; instead, we seek evidence that would lead us to reject the null hypothesis in favor of considering the alternative hypothesis as the more plausible explanation.
  • ( H_0: 蟻 = 0 ) 鈥� No correlation between pH and mercury levels.
  • ( H_A: 蟻 < 0 ) 鈥� Negative correlation between pH and mercury levels.
Given the small p-value and the negative correlation coefficient, the evidence leans strongly towards rejecting the null hypothesis in this exercise, suggesting a negative association between pH and mercury levels.

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