ARG/PDW: MCEN4027F00 III: 15
APPLIED STATISTICAL ANALYSIS
Introduction
Variability in the Measurement versus Variability in the Population Under Study
In carrying out experiments we utilize measurement systems that interact with the physical system being studied.
As a result of this interaction, we are able to estimate the value of various physical parameters.
We then use the measured results to evaluate hypotheses concerning the system under study.
To test our hypotheses we must be able to assess the quality of the measurements, i.e., we must be able to estimate whether the measurements are in error, and if so, by how much.
Otherwise, we are not able to make a legitimate decision regarding the hypotheses.
Uncertainty analysis results in a statement regarding variability introduced by the measurement system (due to noise or resolution limits), NOT variability in the sample or population being studied.
Uncertainty analysis can be used to:
· Control systematic errors (through calibration);
· Guide the investigator as to which of a number of possible measurement techniques may be best;
· Obtain improvement in the overall accuracy of a measurement by reducing variability in certain critical parameters.
Our goal is make the uncertainty in the measurement (system) as small as possible.
Basic Concepts
Continuous Random Variables and Distributions
A continuous random variable X is the set of all possible numbers over a given interval, e.g. A £ X £ B.
The probability distribution (density function) f(x) of continuous random variable X is
where
The cumulative distribution function F(x) for continuous random variable X is
This integral represents the area under the curve, f(x), from -¥ to x so that
Mean and Variance
The expected value of x, E(x), is the mean value of x, designated by the symbol m
The variance of x, V(x) is defined as
We can show that
Normal Distribution
The normal or Gaussian distribution is defined by the continuous density function:
for -¥<x<¥ where -¥m¥ and s >0.
The distribution is symmetric about x = m so m is the mean value of x.
In addition, we can show that V(x) is equal to s2, and s is defined as the standard deviation of the normal distribution function.
Various members of the normal family are shown in the figures.
In the special case where m=0 and s=1 we transform f(x) to the standard normal distribution fz(z) for -¥<z<¥:
A figure showing the standard normal density function with some corresponding important probabilities is shown below:
68% of the population lies within ± 1s
95% of the population lies within ± 2s
99.7% of the population lies within ± 3s
Nonstandard Normal Distributions
Probabilities involving X of a nonstandard normal distribution can be obtained by relating X to the normal distribution variable Z using
In this way tables for standard normal distributions can be used to determine probabilities for non-standard normal distributions.
Example
The reaction time of drivers to a break light from a vehicle in front of theirs is modeled as a normal distribution with a mean and standard deviation of 1.25 an 0.46 s, respectively. What is the probability of a reaction time between 1 and 2 s?
Standardize by writing:
For Xmin =1 and Xmax =2, we obtain -0.543 < Z < 1.630
Then P(1 < X < 2) = P(-0.543 < Z < 1.630)
= F(1.630) - F(-0.543)
= 0.948 – 0.294 = 0.654
Hypothesis Testing
A statistical hypothesis is a conjecture regarding a random variable or the probability distribution of a random variable.
Testing a statistical hypothesis involves:
· Determination of a test statistic
· Utilization of a sample value of the test statistic to choose between a given hypothesis, termed the null hypothesis (Ho), and a competing hypothesis, termed the alternative hypothesis (Ha).
Implicit in the formulation of a testing procedure is the recognition that a statistic is a random variable defined as a function of several random variables X1, X2, …. Xn comprising a random sample, so that
The testing procedure is to observe V and, based upon the observation, decide which hypothesis, Ho or Ha, is to be adopted.
In terms of the sampling distribution of the test statistic V, two regions are defined:
· The acceptance region, comprised of those values of V resulting in the adoption of Ho;
· The rejection (critical) region, comprised of those values of V resulting in the adoption of Ho.
Since there are competing hypotheses regarding the population random variable X, the distribution of the test statistic V is not fully known.
Indeed, it is the observation of V that is to lead to a decision regarding which hypothesis is to be adopted.
For instance, let X be given as a f(x;q) where the unknown parameter q possesses one of two values, qo or qa.
The choice as to which value of q to accept is determined by an observation of the test statistic.
The choices can be represented by
Ho: q = qo
Ha: q = qa
The possible outcomes of the chosen test statistic are divided into two mutually exclusive classes: those in the acceptance region A and those in the rejection region C.
Upon observation, if V falls in A, then Ho is accepted; if V falls in C, then Ho is rejected.
Example
Having purchased a shipment of synthetic rubber seals for disk-brake calipers, a truck manufacturer suspects that substandard seals have been substituted for the ones ordered. From experience, the manufacturer knows that, when subjected to rigorous testing conditions, only 10% of the ordered seals will fail but 30% of the substandard seals will fail. To detect whether or not there has been a bogus shipment, the manufacturer plans to test 20 seals and make a determination based upon the number of brake failures due to seal rupture.
From the perspective of hypothesis testing, there is a population described by a random variable X, with p, the probability of failure being unknown. Assuming the shipment to be large in number, the sampling procedure can be treated as random. If the test statistic V counts the number of seals that fail the testing procedure, then
V = X1 + X2 + ….. X20
where X1 + X2 + ….. X20 comprise a random sample, and the sampling distribution of V has parameters n =20 and p. The two competing hypotheses can be described by
Ho: p = 0.3
Ha: p = 0.1
The null hypothesis represents the conjecture that there has been a bogus shipment.
Having chosen a test statistic, the next problem is to determine a critical region. The greater the number of failures, the more reason there is to accept the null hypothesis; the fewer the failures, the more reason there is to reject the null hypothesis and accept the alternative hypothesis that the seals shipped are the genuine ones ordered.
A reasonable choice for the critical region might be
C = {0,1,2,3,4}
where up to 4/20 or 0.2 of the test seals fail. The corresponding acceptance region would be
A = {5,6, …..20}
If 4 or fewer seals fail the test, then the manufacturer rejects the null hypothesis and accepts the alternative hypothesis. On the other hand, if 5 or more seals fail, then the manufacturer accepts the null hypothesis that there has been a bogus shipment.
Now suppose 20 seals are subjected to testing and only 3 fail. Then according to the hypothesis test just designed, the manufacturer rejects the null hypothesis and concludes the shipment does not consist of substandard seals.
Is the manufacturer certain of this conclusion? No!
The conclusion is an inference based upon the outcome of an experiment. A different outcome might yield a different inference.
Rejection Regions for the Normal Distribution
Two Types of Error
Type I Error: An error of type I occurs if the null hypothesis Ho reflects the true state of nature but the test statistic falls into the critical region, thereby leading to a rejection of Ho.
Type II Error: An error of type II occurs if the null hypothesis Ho does not reflect the true state of nature but the test statistic falls into the acceptance region, thereby leading to an acceptance of Ho.
True State of Nature
Ho / H1Decision / Reject Ho / Type I error / Correct decision
Accept Ho / Correct decision / Type II error
There are always errors in testing: Our goal is to minimize both type I and type II errors.
The probability of a type I error is denoted by a.
The probability of a type II error is denoted by b.
In an experiment in which statistics will be employed, one always sets the desired values of a and b in the design protocol before any experimentation has been initiated.
Selection of the Null and Alternative Hypotheses
A common issue that arises in practical applications is whether to conduct a one- or two-tailed test.
The decision depends on what one wants to detect.
Example
Suppose you operate a chemical plant that produces a variable amount y of product per day and if m, the mean value of y, s less than 100 tons/day, you will eventually be bankrupt.
If m exceeds 100 tons/day, you are financially safe.
In order to determine whether your process is leading to financial disaster, you will want to detect whether m < 100 tons, and you will conduct a one-tailed test of Ho: m ³ 100 versus H1: m < 100.
If you were to conduct a two-tailed test for this situation, you would reduce your chance of detecting values of m less than 100 tons, i.e., you would increase the values of b for alternative values of m < 100 tons.
Example
Suppose you have designed a new drug so that its mean potency is some specific level, say 10%.
As the mean potency tends to exceed 10%, you lose money.
If it is less than 10% by some specified amount, the drug becomes ineffective as a pharmaceutical (you lose money).
To conduct a test of the mean potency m for this situation, you would want to detect values of m either larger or smaller than m = 10.
Consequently, you would select H1: m ¹ 10 and conduct a two-tailed statistical test.
These examples demonstrate that a statistical test is an attempt to detect departure from the null hypothesis; the key to the test is to define the specific alternatives that you wish to detect.
It should be stressed, however, that Ho and H1 should be constructed prior to obtaining and observing the sample data.
If you use information in the sample to aid in selecting the hypotheses, the prior information gained from the sample biases the test results; specifically, the true probability of a Type I error will be larger than the preselected value of a.