Question

Keeping water supplies clean requires regular measurement of levels of pollutants. The measurements are indirecta typical analysis involves forming a dye by a chemical reaction with the dissolved pollutant, then passing light through the solution and measuring its "absorbence." To calibrate such measurements, the laboratory measures known standard solutions and uses regression to relate absorbence and pollutant concentration. This is usually done every day. Here is one series of data on the absorbence for different levels of nitrates. Nitrates are measured in milligrams per liter of water.

Nitrates	50	50	100	200	400	800	1200	1600	2000	2000
Absorbence	7.0	7.4	12.9	24.0	47.0	94.0	138.0	183.0	229.0	226.0

(a) Chemical theory says that these data should lie on a straight line and if the correlation is not at least 0.997 then the calibration procedure is repeated. Find the correlation. (Use 4 decimal places.)
r =

(b) Must the calibration be done again?

YesNo    

(c) The calibration process sets nitrate level and measures absorbence. Once established, the linear relationship will be used to estimate the nitrate level in water from a measurement of absorbance. What is the equation of the line used for estimation? (Use 2 decimal places for intercept and 3 decimal places for slope.)
=  +  x

(d) What is the estimated nitrate level in a water specimen with absorbence 39? (Use 1 decimal place.)
mg/l

(e) Do you expect estimates of nitrate level from absorbence to be quite accurate?

This prediction should be very inaccurate because the relationship is too perfectly linear.
This prediction is of a value that is not in the range of the data and therefore cannot be accurate.
This prediction should be very accurate because the relationship is so strong as indicated by r.
Since the calibration is so important, it is inaccurate to use this regression to predict.

Answer 1

	Nitrates (y)	Absorbence(x)	xy	x2	y2
	50	7	350	49	2500
	50	7.4	370	54.76	2500
	100	12.9	1290	166.41	10000
	200	24	4800	576	40000
	400	47	18800	2209	160000
	800	94	75200	8836	640000
	1200	138	165600	19044	1440000
	1600	183	292800	33489	2560000
	2000	229	458000	52441	4000000
	2000	226	452000	51076	4000000
Sum	8400	968.3	1469210	167941.17	12855000

a)

Formula for correlation coefficient is

where sx and sy are the sample standard deviations, and sxy is the sample covariance.

= 72870.889

Now

The correlation coefficient is 0.9999

b)

No, Since the correlation is greater than 0.997, the calibration cannot be done again.

c)

The equation of regression line is

y = b0 + b1x

where

b1 = slope

b0 = intercept

Formula for finding slope (b1) is

So,

Formula for finding y-intercept (b) is

where

So,

The equation for regression line is

d)

Given

Absorbence, X= 39

The equation for regression line is

So,

Therefore,the estimated nitrate level in a water specimen with absorbence 39 is 328.7 mg/l.

e)

This prediction should be very accurate because the relationship is so strong as indicated by r.