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 4 Initial Depth, X Depth Five Years Later,Y 1/25 3/19 5/30 7/3 9/13 11/7 38.1 72.3 40.4 73.5 54.1 71.7 46.8 69.2 44.1 72.6 62.3 71.0 Click the icon to view the table of critical t-values Determine the test statistic for this hypothesis test. -1.30 (Round to two decimal places as needed.) Find the critical value(s) for this hypothesis test. (Use a comma to separate answers as needed. Round to two decimal places as needed)

Answer 1

Here paired t test will be used. Hypotheses are:

$H_{0}:\mu_{d}=0,H_{a}:\mu_{d}\neq 0$

Following is the output of paired t test:

Hypothesis Test: Paired Observations
	0.0000	hypothesized value
	58.3500	mean Intital Depth, X
	61.0000	mean Depth Five years later, Y
	-2.6500	mean difference (Intital Depth, X - Depth Five years later, Y)
	5.0079	std. dev.
	2.0445	std. error
	6	n
	5	df
	-1.30	t
	.2515	p-value (two-tailed)

The degree of freedom: df=5

The critical values for $\alpha=0.05$ are +/- 2.57.

Excel function used: "=TINV(0.05,5)"