A Researcher Takes Measurements Of Water Clarity At The Same

A Researcher Takes Measurements Of Water Clarity At The Same Location

A researcher collected water clarity measurements at the same location in a lake on specific dates during a year and repeated the process five years later on the same dates. The data includes measurements from initial and subsequent years, and the goal is to analyze the difference in water clarity over time. Specifically, the researcher wants to construct a 95% confidence interval for the mean difference in water clarity, calculated as the initial measurement minus the measurement five years later.

The data provided are as follows:

• Observation Dates: 1/25, 3/19, 5/30, 7/3, 9/13, 11/7
• Initial measurements (in units of water clarity): 37.4, 35.9, 37.6, 47.7, 63.3, 55.5
• Measurements after five years: 37.8, 33.6, 37.4, 54.0, 67.6, 54.6

Paper For Above instruction

The assessment of water quality, particularly water clarity, is crucial for understanding ecological health and environmental changes over time. In this study, measurements of water clarity taken at the same lake location on the same dates across two different periods—initially and five years later—serve as data for evaluating temporal changes. The core objective is to determine whether water clarity has significantly changed over five years by constructing a 95% confidence interval for the mean difference between the initial and subsequent measurements. This analysis involves statistical methods suited for paired data, acknowledging that the measurements are matched observations from the same location and dates.

The dataset provided reveals a set of six paired observations. The differences between the initial measurements and those taken five years later are calculated as (initial measurement) minus (measurement after five years). These differences indicate the magnitude and direction of change in water clarity at each observation point. Calculating these differences is essential to perform statistical inference, including estimating the mean difference and constructing the confidence interval.

Methodology

First, compute the differences for each paired observation:

• 1/25: 37.4 - 37.8 = -0.4
• 3/19: 35.9 - 33.6 = 2.3
• 5/30: 37.6 - 37.4 = 0.2
• 7/3: 47.7 - 54.0 = -6.3
• 9/13: 63.3 - 67.6 = -4.3
• 11/7: 55.5 - 54.6 = 0.9

The next step involves calculating the sample mean difference (\(\bar{d}\)) and the sample standard deviation (s_d) of these differences:

\(\bar{d} = \frac{-0.4 + 2.3 + 0.2 - 6.3 - 4.3 + 0.9}{6} = \frac{-7.6}{6} \approx -1.267\)

Standard deviation of differences involves calculating each squared deviation from the mean, summing, dividing by degrees of freedom (n-1 = 5), and taking the square root. Doing so yields a standard deviation of approximately 4.419.

This data allows us to construct the confidence interval using the t-distribution, given the small sample size. For a 95% confidence level and degrees of freedom (df) = 5, the t-critical value (t_{0.975, 5}) is approximately 2.571.

Calculations

The standard error (SE) of the mean difference is:

SE = s_d / √n = 4.419 / √6 ≈ 1.801

The margin of error (ME) is:

ME = t SE = 2.571 1.801 ≈ 4.636

Therefore, the confidence interval for the mean difference is:

\(\bar{d} \pm ME = -1.267 \pm 4.636\)

Converting to interval bounds:

• Lower bound: -1.267 - 4.636 ≈ -5.903
• Upper bound: -1.267 + 4.636 ≈ 3.369

Rounding to three decimal places, the 95% confidence interval for the mean difference is approximately (-5.903, 3.369).

Interpretation

This confidence interval suggests that, on average, the water clarity difference over five years could range from a decrease of about 5.903 units to an increase of about 3.369 units. Since the interval includes zero, there is no statistically significant evidence at the 95% confidence level to conclude that the mean water clarity has changed over this period. The results highlight that local changes in water clarity might be variable and not consistently trending in a particular direction. Further, larger sample sizes could provide more precise estimates and aid in detecting subtle changes over time.

Conclusion

In summary, the constructed 95% confidence interval for the mean difference in water clarity at this lake location spans from approximately -5.903 to 3.369 units. The inclusion of zero within this interval indicates an absence of conclusive evidence for a true change—either improvement or decline—in water clarity over the five-year span studied. Continued monitoring and expanded datasets are recommended for more definitive insights into temporal trends in water quality.

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