Construct A 95% Confidence Interval About The Population Mea ✓ Solved

Construct a 95% confidence interval about the population mean

A Researcher takes measurements of water clarity at the same location in a lake on the same dates during the course of a year and repeats the measurements on the same dates 5 years later. The collected data is given in the table below:

Observation Dates: 1/25, 3/19, 5/30, 7/3, 9/13, 11/7

Initial Clarity: 37.4, 35.9, 37.6, 47.7, 63.3, 55.5

Clarity After Five Years: 37.8, 33.6, 37.4, 54.0, 67.6, 54.6

Construct a 95% confidence interval about the population mean difference. Compute the difference as (initial depth of clarity) minus (depth of clarity after five years). Interpret your results. The confidence interval is (……..),(……..). (Round to three decimal places as needed)

Paper For Above Instructions

In this study, we analyze the changes in water clarity over a period of five years, utilizing data collected at six specific observation dates. The initial measurements of water clarity and the measurements taken five years later provide a basis for determining the impact of time on water quality in the studied lake. The main objective is to construct a 95% confidence interval for the mean difference in water clarity.

Data Analysis

The initial measurements of water clarity in the lake are: 37.4, 35.9, 37.6, 47.7, 63.3, and 55.5. The measurements taken after five years are: 37.8, 33.6, 37.4, 54.0, 67.6, and 54.6. We can calculate the differences for each observation date as follows:

  • 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

Calculating these differences gives us the following values: -0.4, 2.3, 0.2, -6.3, -4.3, and 0.9.

Mean Difference Calculation

Next, we compute the mean of these differences:

Mean Difference = (−0.4 + 2.3 + 0.2 − 6.3 − 4.3 + 0.9) / 6 = −1.4 / 6 = −0.2333

We will also need the standard deviation of the differences to construct the confidence interval. First, we need to calculate the variance:

Variance = [Σ(difference - mean difference)²] / (n - 1)

Calculating the squared deviations:

  • (−0.4 + 0.2333)² = (−0.1667)² = 0.0278
  • (2.3 + 0.2333)² = (2.5333)² = 6.4211
  • (0.2 + 0.2333)² = (0.4333)² = 0.1878
  • (−6.3 + 0.2333)² = (−6.0667)² = 36.81
  • (−4.3 + 0.2333)² = (−4.0667)² = 16.4939
  • (0.9 + 0.2333)² = (1.1333)² = 1.2854

Summing these squared deviations gives us:

Σ = 0.0278 + 6.4211 + 0.1878 + 36.81 + 16.4939 + 1.2854 = 60.225

The variance then becomes:

Variance = 60.225 / (6 - 1) = 60.225 / 5 = 12.045

Standard deviation (SD) = √12.045 = 3.4697

Constructing the Confidence Interval

To construct a 95% confidence interval around the mean difference, we use the following formula:

CI = mean difference ± (t-value) * (SD / √n)

Where:

  • mean difference = -0.2333
  • t-value for 95% CI with (n - 1) = 5 degrees of freedom ≈ 2.571
  • SD = 3.4697
  • n = 6

Substituting these values into the formula gives:

CI = -0.2333 ± (2.571 * (3.4697 / √6))

Calculating the standard error (SE): SE = 3.4697 / √6 ≈ 1.419

Calculating the margin of error:

Margin of error = 2.571 * 1.419 ≈ 3.6444

The confidence interval is:

CI = -0.2333 ± 3.6444

This results in:

  • Lower limit = -0.2333 - 3.6444 = -3.8777
  • Upper limit = -0.2333 + 3.6444 = 3.4111

Rounding to three decimal places, the 95% confidence interval for the mean difference in water clarity is:

(-3.878, 3.411)

Interpretation of Results

The 95% confidence interval for the mean difference in water clarity between the initial measurements and those taken after five years is (-3.878, 3.411). This interval suggests that there may not be a statistically significant change in water clarity over the five-year period since our confidence interval includes zero. The fact that the interval encompasses both positive and negative values indicates that the clarity levels could have either improved or worsened without showing a definitive trend based on the data collected.

Conclusion

In conclusion, the analysis of water clarity over a five-year period reveals no statistically significant change in clarity levels at the observed lake. Further studies with larger sample sizes and additional observation dates may provide deeper insights into the trends in water quality.

References

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