Page 32 - Statistics for Environmental Engineers

L1592_Frame_C02 Page 23 Tuesday, December 18, 2001 1:40 PM

Taylor, J. K. (1987). Quality Assurance of Chemical Measurements, Chelsea, MI: Lewis Publishers, Inc.
Watts, D. G. (1991). “Why Is Introductory Statistics Difficult to Learn? And What Can We Do to Make It
Easier?” Am. Statistician, 45, 4, 290–291.

Exercises
2.1 Concepts I. Define (a) population, (b) sample, and (c) random variable.
2.2 Concepts II. Define (a) random error, (b) noise, and (c) experimental error.
2.3 Randomization. A laboratory receives 200 water specimens from a city water supply each
day. This exceeds their capacity so they randomly select 20 per day for analysis. Explain how
you would select the sample of n = 20 water specimens.
2.4 Experimental Errors. The measured concentration of phosphorus (P) for n = 20 identical
specimens of wastewater with known concentration of 2 mg/L are:
1.8 2.2 2.1 2.3 2.1 2.2 2.1 2.1 1.8 1.9
2.4 2.0 1.9 1.9 2.2 2.3 2.2 2.3 2.1 2.2

Calculate the experimental errors. Are the errors random? Plot the errors to show their
distribution.
2.5 Summary Statistics. For the phosphorus data in Exercise 2.4, calculate the average, variance,
and standard deviation. The average and standard deviation are estimated with how many
degrees of freedom?
2.6 Bias and Precision. What are the precision and bias of the phosphorus data in Exercise 2.4?
2.7 Concepts III. Define reproducibility and repeatability. Give an example to explain each. Which
of these properties is more important to the user of data from a laboratory?
2.8 Concepts IV. Define normality, randomness, and independence in sampled data. Sketch plots
of “data” to illustrate the presence and lack of each characteristic.
2.9 Normal Distribution. Sketch the normal distribution for a population that has a mean of 20
and standard deviation of 2.
2.10 Normal Probabilities. What is the probability that the standard normal deviate z is less than 3;
that is, P(z ≤ 3.0)? What is the probability that the absolute value of z is less than 2; that is,
P(|z | ≤ 2)? What is the probability that z ≥ 2.2?
2.11 t Probabilities. What is the probability that t ≤ 3 for ν = 4 degrees of freedom; that is, P(t ≤ 3.0)?
What is the probability that the absolute value t is less than 2 for ν = 30; that is, P(|t | ≤ 2)?
What is the probability that t > 6.2 for ν = 2?
y
2.12 t Statistic I. Calculate the value of t for sample size n = 12 that has a mean of = 10 and a
standard deviation of 2.2, for (a) η = 12.4 and (b) η = 8.7.
2.13 Sampling Distributions I. Below are eight groups of five random samples drawn from a normal
distribution which has mean η = 10 and standard deviation σ = 1. For each sample of five
(i.e., each column), calculate the average, variance, and t statistic and plot them in the form
of Figure 2.11.

1 2 3 4 5 6 7 8
9.1 9.1 8.9 12.1 11.7 11.7 8.4 10.4
9.5 9.0 9.2 7.8 11.1 9.0 10.9 9.7
10.1 10.4 11.2 10.4 10.4 10.6 12.1 9.3
11.9 9.7 10.3 8.6 11.3 9.2 11.2 8.7
9.6 9.4 10.6 11.6 10.6 10.4 10.0 9.1

© 2002 By CRC Press LLC