A Secchi disk is an 8-inch-diameter weighted disk that is painted black and whit

Question

A Secchi disk is an 8-inch-diameter weighted disk that is painted black and white and attached to a rope. The disk is lowered into water and the depth (in inches) at which it is no longer visible is recorded. The measurement is an indication of water clarity. An environmental biologist is interested in determining whether the water clarity of a lake is improving. She takes measurements at the same location on six dates during the course of a year and repeats the measurements on the same dates 5 years later. Complete parts (a) through (c) below.

(c) Construct a 90% 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 left parenthesis nothing comma nothing right parenthesis.

(Round to three decimal places as needed.)

Observation Date Initial_depth_(Xi) Depth_five_years_later_(Yi) 1 1–25 62.5 67.3 2 3–19 57.4 64.9 3 5–30 40.5 36.6 4 7–3 63.9 67.5 5 9–13 65.8 69.3 6 11–7 47.2 44.9

Explanation / Answer

Confidence Interval
CI = d ± t a/2 * (Sd/ Sqrt(n))
Where,
d = di/n
Sd = Sqrt( di^2 – ( di )^2 / n ] / ( n-1 ) )
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
d = ( di/n ) =-13.2/6=-2.2
Pooled Sd( Sd )= Sqrt [ 125- (-13.2^2/6 ] / 5 = 4.381
Confidence Interval = [ -2.2 ± t a/2 ( 2.529/ Sqrt ( 6) ) ]
= [ -2.2 - 2.015 * (1.788) , -2.2 + 2.015 * (1.788) ]
= [ -5.804 , 1.404 ]

X Y X-Y (X-Y)^2 62.5 67.3 -4.8 23.04 57.4 64.9 -7.5 56.25 40.5 36.6 3.9 15.21 63.9 67.5 -3.6 12.96 65.8 69.3 -3.5 12.25 47.2 44.9 2.3 5.29 -13.2 125

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