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Figure 4.25 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 71 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 \(\mathrm{pH}\) causes the average mercury level in fish to increase? Why or why not?

Step by step solution

01

Define the null and alternative hypotheses.

The null hypothesis \(H_0\) is that there is no association between pH and mercury level. On the other hand, the alternative hypothesis \(H_1\) is that there is a negative association between pH and mercury level. In mathematical terms, for \(H_0\), the correlation is \(0\) and for \(H_1\), the correlation is less than \(0\).

02

Interpret the p-value.

The p-value is \(0.000017\), it's exceedingly small, and typically, if the p-value is less than \(0.05\), this indicates strong evidence against the null hypothesis, and we reject the null hypothesis.

03

Formulate a conclusion based on the p-value.

Based on the very small p-value, there is strong evidence against the null hypothesis. Therefore, \(H_0\) is rejected in favor of \(H_1\), suggesting there is indeed a negative association between pH and mercury level.

04

Discuss causation.

Although a significant correlation (negative in this case) is observed, it cannot be concluded that low pH causes an increase in average mercury level. This is because correlation does not imply causation. Statistical results only provide evidence of an association, and do not establish a causative relationship, which would require a different type of study design.

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Key Concepts

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

Scatterplot Analysis

Scatterplot analysis is a visual representation used in statistics to examine the relationship between two quantitative variables. In our scenario, this involves pH levels in Florida lakes and the associated mercury levels in fish. Each point on the scatterplot depicts a pair of values, demonstrating how these two variables might be connected.

A scatterplot is beneficial because it quickly reveals trends and patterns, such as positive or negative correlations, clusters, or outliers. In this exercise, the scatterplot presented shows a negative trend, meaning as the pH decreases, mercury levels tend to increase.

To interpret this plot effectively, we should ask questions such as:

• Are most of the points close to a trend line, indicating a strong relationship?
• Does the plot exhibit any significant outliers that do not fit the pattern?
• Is the trend upward or downward, pointing to a positive or negative correlation?

Such insights drive hypotheses and statistical testing to further explore these relationships.

Correlation vs Causation

Understanding the distinction between correlation and causation is crucial in data analysis. Correlation indicates a statistical relationship or association between two variables. However, just because two variables change together doesn't mean one causes the other to change.

In the given exercise, the correlation coefficient ( ") was -0.575, suggesting a moderate negative correlation between pH level and mercury in fish. This implies that as pH decreases, mercury content generally increases. However, this does not infer causation.

Some important points to consider about correlation vs causation:

• Correlation: Quantified by the correlation coefficient (r), ranging from -1 to 1. Negative values indicate inverse relationships, while positive ones point to direct relationships.
• Causation: Requires more than just correlation, often through controlled experiments that rule out outside variables.
• Confounding Variables: Other factors can cause both variables to shift, giving the impression of causality.

Remember, observing a correlation is just the first step. Establishing causation demands thorough investigation and careful experimental design.

P-Value Interpretation

The p-value in hypothesis testing measures the strength of evidence against the null hypothesis. It's the probability of obtaining test results at least as extreme as the observed data, assuming the null hypothesis is true.

In our exercise, the p-value was reported as 0.000017, which is very small. This significantly low value suggests strong evidence against the null hypothesis. Generally, if the p-value is less than the significance level (often set at 0.05), we reject the null hypothesis.

Key points about p-value interpretation:

• When the p-value is low, it indicates our observed data is unlikely under the null hypothesis, leading to its rejection.
• If the p-value exceeds 0.05, we do not have sufficient evidence to reject the null hypothesis.
• The p-value does not tell us about the size or importance of the effect, just the significance of the results.

For our scenario, the low p-value reinforced the conclusion that there is indeed a negative association between pH levels and mercury in fish, supporting the alternative hypothesis.