91Ó°ÊÓ

In a study of bacterial concentration in surface and subsurface water ("Pb and Bacteria in a Surface Microlayer" journal of Marine Research [1982]: \(1200-\) 1206 ), the accompanying data were obtained. Concentration \(\left(\times 10^{6} / \mathrm{mL}\right)\) \(\begin{array}{llllll}\text { Surface } & 48.6 & 24.3 & 15.9 & 8.29 & 5.75 \\\ \text { Subsurface } & 5.46 & 6.89 & 3.38 & 3.72 & 3.12 \\ \text { Surface } & 10.8 & 4.71 & 8.26 & 9.41 & \\ \text { Subsurface } & 3.39 & 4.17 & 4.06 & 5.16 & \end{array}\) Summary quantities are \(\sum x=136.02 \sum y=39.35\) \(\sum x^{2}=3602.65 \sum y^{2}=184.27 \quad \sum x y=673.65\) Using a significance level of \(.05,\) determine whether the data support the hypothesis of a linear relationship between surface and subsurface concentration.

Step by step solution

01

Calculate the mean of both sets of data

Use the formula for calculating mean. The mean of the surface concentration \(x\) can be calculated as \(\bar{x} = \frac{\sum x}{n}\), where \(n\) is the total number of observations for surface concentrations. Similarily, calculate mean of the subsurface concentration \(y\), \(\bar{y} = \frac{\sum y}{n}\).

02

Calculate the correlation coefficient

Now, we calculate the correlation coefficient using the formula: \(r = \frac{n\sum xy - \sum x\sum y}{\sqrt{n\sum x^2 - (\sum x)^2} \sqrt{n\sum y^2 - (\sum y)^2}}\). Here, \(n\) is the total number of observations, \(\sum xy\) is the sum of the product of corresponding \(x\) and \(y\) values, \(\sum x\) and \(\sum y\) are the sums of \(x\) values and \(y\) values respectively and \(\sum x^2\) and \(\sum y^2\) are the sums of the squares of \(x\) and \(y\) values respectively.

03

Determine the critical value for correlation coefficient

To test the hypothesis of a linear relationship, we need to determine the critical value for correlation coefficient. If our calculated correlation coefficient is beyond this critical value, we will reject the null hypothesis (that the two sets of data have no correlation). The critical value can be found on a table of critical values for correlation. We are testing at a significance level of .05, which will be used to find the critical value.

04

Test the hypothesis

Now that we have the calculated correlation value and critical value, we can test our hypothesis. If our calculated correlation coefficient is beyond the critical value, we reject the null hypothesis and conclude that the data supports the hypothesis of a linear relationship between surface and subsurface concentration. If the computed correlation is within the bounds of the critical value, we fail to reject the null hypothesis and conclude that the data does not support a linear relationship.

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Key Concepts

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

Linear Relationship

A linear relationship indicates that as the value of one variable changes, the value of another variable changes in a predictable way. In a linear relationship, data points typically form a pattern that can be roughly approximated by a straight line when plotted on a graph.
Detecting a linear relationship involves looking for consistency in how two variables rise or fall together. This exercise examines whether surface and subsurface bacterial concentrations have such a relationship.
If you can fit a straight line through the plotted observations without substantial deviations, you likely have a linear relationship. Understanding if a linear trend exists is vital as it helps predict one variable's behavior based on another.

Correlation Coefficient

The correlation coefficient, often represented by the symbol \( r \), gauges the strength and direction of a linear relationship between two variables.
It is a numerical representation that ranges from -1 to 1. An \( r \) value of 1 suggests a perfect positive linear relationship, where both variables increase together, while -1 indicates a perfect negative linear relationship, where one variable increases as the other decreases.
A correlation coefficient close to 0 implies no linear relationship.

• Positive \( r \): As one variable increases, the other also increases.
• Negative \( r \): As one variable increases, the other decreases.
• Zero \( r \): No linear relationship exists.

Calculating \( r \) for surface and subsurface concentrations indicates the strength of their linear connection, shedding light on how tightly the variables are interrelated.

Critical Value

The critical value is a threshold against which you compare your calculated statistic to determine statistical significance. For correlation coefficients, finding the critical value involves using a table that considers the degrees of freedom (which is the number of data pairs minus 2) and the desired significance level.
Given a significance level of 0.05, the critical value helps ascertain if the detected correlation is strong enough to conclude a linear relationship. If your \( r \) value exceeds the critical value, the linear relationship is deemed statistically significant.
This step safeguards against making premature conclusions about the strength of the correlation.

Significance Level

The significance level, denoted by \( \alpha \), is a probability threshold set to decide whether to reject the null hypothesis.
Commonly used significance levels include 0.05, 0.01, and 0.10, which correspond to 5%, 1%, and 10% chances, respectively, of incorrectly rejecting the null hypothesis (a false positive). In hypothesis testing, the null hypothesis typically suggests no relationship.
In our case, with a significance level of 0.05, we are willing to accept a 5% risk of concluding that a linear relationship exists between surface and subsurface concentrations when it does not.
Setting a significance level is crucial as it frames the confidence with which we can make conclusions from the data. This measure helps ensure that results from the tests are reliable and valid.