A researcher studies 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 researcher immerses a weighted disk painted black and white and measures the depth (in inches) at which it is no longer visible. The collected data is given in the table below. Complete parts (a) through (c) below.

Observation | Date | Initial Depth, [tex]X_i[/tex] | Depth Five Years Later, [tex]Y_i[/tex]
--- | --- | --- | ---
1 | 1/25 | 47.7 | 56.0
2 | 3/19 | 38.3 | 37.4
3 | 5/30 | 43.9 | 49.7
4 | 7/3 | 41.2 | 44.5
5 | 9/13 | 49.5 | 54.6
6 | 11/7 | 51.7 | 53.8

(a) Why is it important to take the measurements on the same date?

A. Those are the same dates that all biologists use to take water clarity samples.
B. Using the same dates makes it easier to remember to take samples.
C. Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date. (Correct answer)
D. Using the same dates maximizes the difference in water clarity.

(b) Does the evidence suggest that the clarity of the lake is improving at the [tex]\alpha = 0.05[/tex] level of significance? Note that the normal probability plot and boxplot of the data indicate that the differences are approximately normally distributed with no outliers. Let [tex]d = X_i - Y_i[/tex].

Identify the null and alternative hypotheses.
- Null hypothesis ([tex]H_0[/tex]): [tex]\mu_d = 0.050[/tex]
- Alternative hypothesis ([tex]H_1[/tex]): [tex]\mu_d < 0.050[/tex]

Determine the test statistic for this hypothesis test. (Round to two decimal places as needed.)

a) The correct answer is C. Using the same dates makes the second sample dependent on the first and reduces date-related variability in water clarity.

b) Null Hypothesis is 0.050 and Alternative Hypothesis <0.050.

a) The correct answer is C. Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date. By taking measurements on the same dates, the researcher ensures that the measurements are not influenced by seasonal variations or other time-dependent factors, thus reducing the variability in the data.

b) To determine if the clarity of the lake is improving, the null and alternative hypotheses are:

Null hypothesis (H0): The mean difference in water clarity (μd) is 0.

Alternative hypothesis (H1): The mean difference in water clarity (μd) is less than 0.

The test statistic for this hypothesis test can be obtained by performing a paired t-test using the given data. The P-value can then be compared to the significance level (α = 0.05) to make a conclusion about the evidence for the improvement in water clarity.