Question

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, X_i   Depth Five Years Later, Y_i
1   1–25 61.7 60.2
2 3–19 40.3    40.9
3 5–30 73.5 71.4
4 7–3 69.7 71.6
5 9–13 74.9 69.2
6 11–7    48.8   50.9

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 the second sample dependent on the first and reduces variability in water clarity attributable to date.

C. Using the same dates makes it easier to remember to take samples.

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 a=0.05 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 di=Xi-Yi. Identify the null and alternative hypotheses.

H0: (p, od, μd) (<,>,=, ≠) BLANK

H1: (p, od, μd) (<,>,=, ≠) BLANK

Determine the test statistic for this hypothesis test.

Find the​ P-value for this hypothesis test.

P-value=

What is your conclusion regarding H0​?

A.Do not reject H0. There is not sufficient evidence at the a= 0.05 level of significance to conclude that the clarity of the lake is improving.

B. Do not reject H0.There is sufficient evidence at the a=0.05 level of significance to conclude that the clarity of the lake is improving.

C. Reject H0. There is sufficient evidence at the a=0.05 level of significance to conclude that the clarity of the lake is improving.

D. Reject H0. There is not sufficient evidence at the a=0.05 level of significance to conclude that the clarity of the lake is improving.

c) Draw a boxplot of the differenced data. Does this visual evidence support the results obtained in part​ b)?

Does this visual evidence support the results obtained in part​ b)?

A.Yes because the boxplot supports that the lake is becoming more​ clear, since most differences are positive or near 0.

B. Yes because the boxplot supports that the lake is becoming more​ clear, since most differences are negative.

C. No because the boxplot supports that the lake is not becoming more​ clear, since most differences are negative.

D. No because the boxplot supports that the lake is not becoming more​ clear, since most differences are positive or near 0.

Answer 1

Answer(a): The correct option is B. Using the same dates makes the second sample dependent on the first and reduces variability in water clarity attributable to date.

Answer(b):

Observation	Date	Initial Depth, xi	Five year later depth,yi	d	d2
1	1–25	61.7	60.2	-1.5	2.25
2	3–19	40.3	40.9	0.6	0.36
3	5–30	73.5	71.4	-2.1	4.41
4	7–3	69.7	71.6	1.9	3.61
5	9–13	74.9	69.2	-5.7	32.49
6	11–7	48.8	50.9	2.1	4.41
total				-4.7	47.53

We have got

We have to test

H0: µd =0

HA: µd >0

we have

n=6

Σd=-4.7

Σd2=47.53

The test statistic to test this null hypothesis is paired t-test given as below:

The test statistic is -0.65

The p-value for above test is 0.7272

Conclusion:

The correct option is A. Do not reject H0.There is not sufficient evidence at the a= 0.05 level of significance to conclude that the clarity of the lake is improving.

Answer(c): The boxplot of the differenced data is as below:

The correct option is C. No because the boxplot supports that the lake is not becoming more ​clear, since most differences are negative.