91Ó°ÊÓ

A Secchi disk is an 8 -inch-diameter weighted disk that is painted black and white and attached to a rope. The disk is lowered into water and the depth (in inches) at which it is no longer visible is recorded. The measurement is an indication of water clarity. An environmental biologist interested in determining whether the water clarity of the lake at Joliet Junior College is improving takes measurements at the same location on eight dates during the course of a year and repeats the measurements on the same dates five years later. She obtains the following results: $$ \begin{array}{lcccccccc} \text { Observation } & \mathbf{1} & \mathbf{2} & \mathbf{3} & \mathbf{4} & \mathbf{5} & \mathbf{6} & \mathbf{7} & \mathbf{8} \\ \text { Date } & \mathbf{5 / 1 1} & \mathbf{6 / 7} & \mathbf{6 / 2 4} & \mathbf{7 / 8} & \mathbf{7 / 2 7} & \mathbf{8 / 3 1} & 9 / 30 & \mathbf{1 0 / 1 2} \\ \hline \begin{array}{l} \text { Initial } \\ \text { depth, } X_{i} \end{array} & 38 & 58 & 65 & 74 & 56 & 36 & 56 & 52 \\ \hline \begin{array}{l} \text { Depth five } \\ \text { years later, } Y_{i} \end{array} & 52 & 60 & 72 & 72 & 54 & 48 & 58 & 60 \\ \hline \end{array} $$ (a) Why is it important to take the measurements on the same date? (b) Does the evidence suggest that the clarity of the lake is improving at the \(\alpha=0.05\) level of significance? Note: A normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. (c) Draw a boxplot of the differenced data. Does this visual evidence support the results obtained in part (b)?

Step by step solution

01

Understand the Importance of Same Date Measurements

It’s important to take the measurements on the same date to control for seasonal and weather-related variations. Environmental factors such as temperature, rainfall, and sunlight can affect water clarity, so using the same dates each year ensures that any changes observed are more likely due to long-term trends rather than short-term variations.

02

State Hypotheses for Testing Improvement

Let \(d_i = Y_i - X_i\) be the difference in water clarity measurements on the same date five years apart. The null hypothesis (H_0) is that the mean difference \( \mu_d \) is 0, indicating no improvement. The alternative hypothesis (H_a) is that \( \mu_d \) > 0, indicating an improvement in water clarity.\[H_0: \mu_d = 0\]\[H_a: \mu_d > 0\]

03

Calculate the Differences and Their Statistics

Calculate each \(d_i\):\(d_1 = 52-38 = 14\)\(d_2 = 60-58 = 2\)\(d_3 = 72-65 = 7\)\(d_4 = 72-74 = -2\)\(d_5 = 54-56 = -2\)\(d_6 = 48-36 = 12\)\(d_7 = 58-56 = 2\)\(d_8 = 60-52 = 8\)Calculate the sample mean (\overline{d}) and sample standard deviation (\s_d) of these differences.

04

Perform a t-Test

Use the t-statistic formula: \[t = \frac{\overline{d} - \mu_0}{s_d / \sqrt{n}} \]. Substituting the values, where \(n = 8\), \(\mu_0 = 0\), \(\overline{d} = 5.125\), and \(s_d = 5.27\):\[t = \frac{5.125 - 0}{5.27 / \sqrt{8}} \approx 2.76\].

05

Determine the Critical Value and Conclusion

At \(\alpha = 0.05\) and \(df = n-1 = 7\), the critical t-value for a one-tailed test is \( t_{0.05, 7} = 1.895 \). Since our calculated t-value (2.76) is greater than the critical value (1.895), we reject the null hypothesis, suggesting evidence that water clarity has improved.

06

Draw a Boxplot of the Differences

Create a boxplot for the differences \(d_i\). Since \(\overline{d}=5.125\), looking for the median, quartiles, and any potential outliers.

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Key Concepts

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

paired t-test

The paired t-test is a statistical method used to compare two related samples. In our exercise, we want to see if water clarity has improved over five years. Since we have measurements taken at the same location on the same dates five years apart, we use the paired t-test. This test considers the differences for each pair of observations, allowing us to account for any variability that might affect both sets of measurements in a similar way. By focusing on the differences, we can more precisely determine if there has been a significant change over time.

hypothesis testing

Hypothesis testing is a method to decide if there is enough evidence to support a particular claim about a population. For our exercise, the claim is that water clarity has improved. We start with a null hypothesis (\text{H_0}) that suggests there is no improvement (\text{{\boldmath{\text{\(\text\text{\textbackslash (D_\text{on}}}}}}}} = 0\text) and an alternative hypothesis (\text{H_a}) which indicates improvement (\text{{\boldmath{\text{\)\text\text{\textbackslash (A_\text{NITY}}}}}}}} > 0\text).\text\text\text. We cannot prove the alternative hypothesis directly; instead, we seek evidence that would lead us to reject the null hypothesis. By performing a paired t-test, we compute a t-value and compare it to a critical value determined by our significance level (\text <\( \)\text {alpha}>\text = 0.05\text). If the t-value is greater than the critical value, we reject H_0 and accept H_a, concluding that there is evidence that water clarity has improved.

environmental statistics

Environmental statistics involve the application of statistical analysis to environmental science. Our exercise uses these principles to track changes in water clarity over time. By taking regular measurements of the same lake and using statistical tests, we can interpret whether observed changes are due to natural variability or actual environmental improvements. Key factors in environmental statistics include ensuring measurements are taken consistently (such as on the same date to control for seasonal variations) and using appropriate statistical methods (like the paired t-test) to analyze the data.

data interpretation

Data interpretation is the process of making sense of numerical data collected during an experiment. In our exercise, we calculate differences between measurements taken five years apart then use statistical tools to understand these differences. We start by computing the mean and standard deviation of the differences. A positive mean suggests an overall improvement in water clarity. We then perform a paired t-test to see if this observed improvement is statistically significant. Finally, we use visual tools like a boxplot to help us understand the spread and any potential outliers in our data. By combining these methods, we can confidently interpret whether the lake's water clarity has genuinely improved.