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Chapter 3

Basic Concepts in Statistics

and Probability

3.7 Choice of

Statistical Distributions

• The distributions are simply models of reality (not

reality themselves)

• Always check the assumptions

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3.8 Statistical Inference

• Methods of Statistical Inference used to estimate

the parameters of the statistical distributions: – Central Limit Theorem

– Point Estimation • Maximum Likelihood Estimation

– Confidence Intervals

– Tolerance Intervals

– Hypothesis Tests • Probability Plots

• Likelihood Ratio Tests

– Bonferroni Intervals

• If X ~ N(, 2), then 𝑋 ~𝑁(𝜇,𝜎2

𝑛)

• If X ~ unknown, 𝑋 ~𝑁 when n is large – If X distribution closes to normal, n ~ 15 – 20

– If X distribution differs from normal greatly, n ~ 100

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3.8.1 Central Limit Theorem

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Central Limit Theorem

The Central Limit Theorem

Let X1,…,Xn be a sequence of independent random variables with means 1 2 … n and variances 1

2, 22

…n2, then

𝑍 = 𝑋𝑖 − 𝜇𝑖

𝑛𝑖=1

𝑛𝑖=1

𝜎𝑖2𝑛

𝑖=1

Approaches the standard normal distribution as n approaches infinity.

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Rule of Thumb

For most populations, if the sample size is greater than 30,

the Central Limit Theorem approximation is good.

Normal approximation to the Binomial:

If X ~ Bin(n,p) and if np > 5, and n(1– p) > 5, then

X ~ N(np, np(1-p)) approximately.

Normal Approximation to the Poisson:

If X ~ Poisson(λ), where λ > 10, then X ~ N(λ, λ2).

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• The statistical distributions have 1 or more parameters (usually represented by Greek letters)

• The value of these parameters are generally unknown and must be estimated by sample statistics.

• Point estimator: the form of the estimation

• Point estimates: the numerical value of an estimator

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3.8.2 Point Estimation

Parameters Sample Statistics

𝑋

2 S2

p 𝑝

3.8.2.1 Maximum Likelihood

Estimation

• A method for obtaining point estimates such that

the probability of observing the data that are in a

sample is maximized.

• Likelihood function: 𝑓 𝑥1, … , 𝑥𝑛 = 𝑓 𝑥1 … 𝑓(𝑥𝑛)

• Likelihood function for of Normal distribution

𝐿(𝜇, 𝜎2; 𝑥1, … , 𝑥𝑛) =1

𝜎 2𝜋

𝑛exp[−

1

2𝜎2 (𝑥𝑖 − 𝜇)2]𝑛

𝑖=1

• Setting the derivative = 0, (𝑥𝑖 − 𝜇)2= 0,𝑛𝑖=1 then

𝜇 = 𝑋

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• An interval estimator will contain the unknown parameter with (approximately) a specified probability.

• The desired degree of confidence determines the width of the interval.

• For a fixed sample size, an increase in the width will increase the degree of confidence.

• Increasing the sample size will decrease the width of a confidence interval.

• Narrower confidence intervals are more meaningful (lower uncertainty)

• However, a low level of uncertainty is contingent upon the requisite assumptions being met.

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3.8.3 Confidence Intervals

• Confidence intervals that utilize either t or z are always of the form

𝜃 ± 𝑡 𝑜𝑟𝑧 𝑠𝜃 Where: is the parameter to be estimated 𝜃 is the point estimator 𝑠𝜃 is the estimated standard deviation of the point

estimator

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Confidence Intervals

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Constructing a CI

To see how to construct a confidence interval, let

represent the unknown population mean and let 2

be the unknown population variance. Let

X1,…,X100 be the 100 amperages of the sample

batteries. The observed value of the sample mean

is 185.5. Since is the mean of a large sample,

and the Central Limit Theorem specifies that it

comes from a normal distribution with mean and

whose standard deviation is .

X

100/ X

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Computing a 95% Confidence

Interval The 95% confidence interval (CI) is .

So, a 95% CI for the mean is 185.5 1.96 (5/√100) or 185.5 0.98 or (184.52, 186.48). We can use the sample standard deviation as an estimate for the population standard deviation, since the sample size is large.

We can say that we are 95% confident, or confident at the 95% level, that the population mean amperage lies between 184.52 and 186.48.

Warning: The methods described here require that the data be a random sample from a population. When used for other samples, the results may not be meaningful.

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XX 96.1

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Illustration of Capturing True

Mean

Here is a normal curve, which represents the distribution of .

The middle 95% of the curve, extending a distance of 1.96

on either side of the population mean , is indicated. The

following illustrates what happens if lies within the middle

95% of the distribution:

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X

X

X

Illustration of Not Capturing True

Mean

If the sample mean lies outside the middle 95% of the curve:

Only 5% of all the samples that could have been drawn fall

into this category. For those more unusual samples the 95%

confidence interval fails to cover the true

population mean .

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XX 96.1

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Question?

Does this 95% confidence interval actually cover the population mean ? • It depends on whether this particular sample happened

to be one whose mean came from the middle 95% of the distribution or whether it was a sample whose mean was unusually large or small, in the outer 5% of the population.

• There is no way to know for sure into which category this particular sample falls.

• In the long run, if we repeated these confidence intervals over and over, then 95% of the samples will have means in the middle 95% of the population. Then 95% of the confidence intervals will cover the population mean.

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Pieces of CI

• Recall that the CI was 185.5 0.98.

• 185.5 was the sample mean which is a point estimate for the population mean.

• We call the plus-or-minus number 0.98 the margin of error

• The margin of error is the product of 1.96 and = 0.5.

• We refer to which is the standard deviation of , as the standard error.

• In general, the standard error is the standard deviation of the point estimator.

• The number 1.96 is called the critical value for the confidence interval. The reason that 1.96 is the critical value for a 95% CI is that 95% of the area under the normal curve is within – 1.96 and 1.96 standard errors of the population mean.

X

X

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X

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Other CI Levels

• Suppose we are interested in 68% confidence

intervals, then we know that the middle 68% of the

normal distribution is in an interval that extends

1.0 on either side of the population mean.

• It follows that an interval of the same length around

specifically, will cover the population mean for 68%

of the samples that could possibly be drawn.

• For our example, a 68% CI for the diameter of

pistons is 185.5 1.0(0.5), or (185.0, 186.0).

X

X

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100(1 - )% CI

Let X1,…,Xn be a large (n > 30) random sample from a population with mean and standard deviation , so that is approximately normal. Then a level 100(1 - )% confidence interval for is

where . When the value of is unknown, it can be replaced with the sample standard deviation s.

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X

XzX 2/

nX

/

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100(1 - )% CI

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Specific Intervals for

• is a 68% interval for .

• is a 90% interval for .

• is a 95% interval for .

• is a 99% interval for .

• is a 99.7% interval for .

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n

sX

n

sX 645.1

n

sX 96.1

n

sX 58.2

n

sX 3

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More About CI’s

• The confidence level of an interval measures the reliability of the method used to compute the interval.

• A level 100(1 - )% confidence interval is one computed by a method that in the long run will succeed in covering the population mean a proportion 1 - of all the times that it is used.

• In practice, there is a decision about what level of confidence to use.

• This decision involves a trade-off, because intervals with greater confidence are less precise.

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Probability vs. Confidence

• In computing CI, such as the one of amperage of batteries: (184.52, 186.48), it is tempting to say that the probability that lies in this interval is 95%.

• The term probability refers to random events, which can come out differently when experiments are repeated.

• 184.52 and 186.48 are fixed, not random. The population mean is also fixed. The mean diameter is either in the interval or not.

• There is no randomness involved.

• So, we say that we have 95% confidence that the population mean is in this interval.

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Probability vs. Confidence

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a) 68% CI

b) 95% CI

c) 99.7% CI

Determining Sample Size

• In Example 5.4, a 95% CI was given by 12.68 1.89.

• This interval specifies the mean to within 1.89. Now assume that the interval is too wide to be useful.

• Assume that it is desirable to produce a 95% confidence interval that specifies the mean to within 0.5.

• To do this, the sample size must be increased.

• The width of a CI is specified by

• If the desired with is w then

• Solving this equation for n yields / 2

/ .w z n

2 2 2

/ 2/ .n z w

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/ 2/ .z n

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One-Sided Confidence Intervals

• We are not always interested in CI’s with both an upper

and lower bound.

• For example, we may want a confidence interval on

battery life. We are only interested in a lower bound on

the battery life.

• With the same conditions as with the two-sided CI, the

level 100(1-)% lower confidence bound for is

and the level 100(1-)% upper confidence bound for

is

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.X

zX

.X

zX

Small Sample CIs for a

Population Mean

• The methods that we have discussed for a

population mean previously require that the

sample size be large.

• When the sample size is small, there are no

general methods for finding CI’s.

• If the population is approximately normal, a

probability distribution called the Student’s t

distribution can be used to compute confidence

intervals for a population mean.

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More on CI’s

• What can we do if is the mean of a small sample?

• If the sample size is small, s may not be close to , and may not be approximately normal. If we know nothing about the population from which the small sample was drawn, there are no easy methods for computing CI’s.

• However, if the population is approximately normal, it will be approximately normal even when the sample size is small. It turns out that we can use the quantity , but since s may not be close to , this quantity has a Student’s t distribution.

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X

X

)//()( nsX

Student’s t Distribution

• Let X1,…,Xn be a small (n < 30) random sample from

a normal population with mean . Then the quantity

has a Student’s t distribution with n -1 degrees of

freedom (denoted by tn-1).

• When n is large, the distribution of the above quantity

is very close to normal, so the normal curve can be

used, rather than the Student’s t.

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./

)(

ns

X

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More on Student’s t

• The probability density of the Student’s t

distribution is different for different degrees of

freedom.

• The t curves are more spread out than the

normal.

• Table C, called a t Distribution, provides

probabilities associated with the Student’s t

distribution.

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Student’s t CI

Let X1,…,Xn be a small random sample from a normal population with mean . Then a level 100(1 - )% CI for is

To be able to use the Student’s t distribution for calculation and confidence intervals, you must have a sample that comes from a population that is approximately normal. Samples such as these rarely contain outliers. So if a sample contains outliers, this CI should not be used.

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.2/,1n

stX n

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Other CI’s

Let X1,…,Xn be a small random sample from a normal

population with mean .

• Then a level 100(1 - )% upper confidence bound for is

• Then a level 100(1 - )% lower confidence bound for is

• Occasionally a small sample may be taken from a normal

population whose standard deviation is known. In these cases, we do not use the Student’s t curve, because we are not approximating with s. The CI to use here is the one using the z table, which we discussed in the first section.

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.,1n

stX n

.,1n

stX n

3.8.4 Tolerance Intervals

• A confidence interval for a parameter is an

interval that is likely to contain the true value of

the parameter.

• Prediction and tolerance intervals are concerned

with the population itself and with values that may

be sampled from it in the future.

• These intervals are only useful when the shape of

the population is known, here we assume the

population is known to be normal.

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Prediction Interval

• A prediction interval is an interval that is likely to

contain the value of an item that will be sampled

from the population at a future time.

• We “predict” that a value that is yet to be sampled

from the population will fall within the predication

interval.

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100(1 – α)% Prediction Interval

Let X1,…,Xn be a random sample from a normal population. Let Y be another item to be sampled from this population, whose value has not yet been observed. The 100(1 – α)% prediction interval for Y is

𝑋 ± 𝑡𝑛−1,𝛼/2𝑠 1 +1

𝑛

The probability is 1 – α that the value of Y will be contained in this interval.

One sided intervals may also be constructed.

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Comparing CI and PI

• The formula for the PI is similar to the formula for the CI of a mean of normal population.

• The prediction interval has a small adjustment to the standard error with the additional + 1 under the square root.

• This reflects the random variation in the value of the sampled item that is to be predicted.

• Prediction intervals are sensitive to the assumption that the population is normal.

• If the shape of the population differs much from the normal curve, the prediction interval may be misleading.

• Large samples do not help, if the population is not normal then the prediction interval is invalid.

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Tolerance Intervals

• A tolerance interval is an interval that is likely to contain a specified proportion of the population.

• First assume that we have a normal population whose mean μ and standard deviation σ are known.

• To find an interval that contains 90% of the population, we have μ ± 1.645σ.

• In general, the interval μ ± zγ/2σ will contain 100(1 – γ)% of the population.

• In practice, we do not know μ or σ. Instead we use the sample mean and sample standard deviation.

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Consequences

• Since we are estimating the mean and standard

deviation from the sample,

– We must make the interval wider than it would be if μ

and σ were known.

– We cannot be 100% confident that the interval actually

contains the required proportion of the population.

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Construction of

Tolerance Interval • We must specify the proportion 100(1 – γ)% of the

population that we wish the interval to contain.

• We must also specify the confidence 100(1 – α)% that the interval actually contains the specified proportion.

• It is then possible to find a number kn,α,γ such that the interval

will contain at least 100(1 – γ)% of the population with confidence 100(1 – α)%. Values of kn,α,γ are presented in various statistical books (for example Table A.4 in Principles of Statistics for Engineers and Scientists)

skX n ,,

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Tolerance Interval Summary

Let X1,…,Xn be a random sample from a normal

population. A tolerance interval for containing at least

100(1 – γ)% of the population with confidence

100(1 – α)% is

Of all the tolerance intervals that are computed by this

method, 100(1 – α)% will actually contain at least

100(1 – γ)% of the population.

skX n ,,

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1. Formulate a hypothesis that is to be tested 2. Use the data to test the hypothesis 3. Determine whether or not the hypothesis should be

rejected

• The hypothesis that is being tested is called the null hypothesis (H0), which is tested against an alternative hypothesis (H1).

• Hypothesis tests are used implicitly when control charts are employed.

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3.8.4 Hypothesis Tests

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• Hypothesis tests using t or z are of the form

𝑡 𝑜𝑟𝑧 =𝜃 − 𝜃

𝑠𝜃

Where: is the parameter to be estimated 𝜃 is the point estimator 𝑠𝜃 is the estimated standard deviation of the point

estimator

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Hypothesis Tests

• In general, hypotheses that are tested are hardly ever true.

• A hypothesis that is being tested is more likely to be rejected for a very large sample size than for a very small sample size.

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Hypothesis Tests

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P-Value

• The P-value measures the plausibility of H0.

• The smaller the P-value, the stronger the

evidence is against H0.

• If the P-value is sufficiently small, we may

be willing to abandon our assumption that

H0 is true and believe H1 instead.

• This is referred to as rejecting the null

hypothesis.

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Steps in Performing a

Hypothesis Test 1. Define H0 and H1. 2. Assume H0 to be true. 3. Compute a test statistic. A test statistic is a statistic

that is used to assess the strength of the evidence against H0. A test that uses the z-score as a test statistic is called a z-test.

4. Compute the P-value of the test statistic. The P-value is the probability, assuming H0 to be true, that the test statistic would have a value whose disagreement with H0 is as great as or greater than what was actually observed. The P-value is also called the observed

significance level.

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One and Two-Tailed Tests

• When H0 specifies a single value for , both

tails contribute to the P-value, and the test

is said to be a two-sided or two-tailed

test.

• When H0 specifies only that is greater

than or equal to, or less than or equal to a

value, only one tail contributes to the P-

value, and the test is called a one-sided or

one-tailed test. 45

Drawing Conclusions from the

Results of Hypothesis Tests

• There are two conclusions that we draw when we are finished with a hypothesis test, – We reject H0. In other words, we concluded that H0 is

false.

– We do not reject H0. In other words, H0 is plausible.

• One can never conclude that H0 is true. We can just conclude that H0 might be true.

• We need to know what level of disagreement, measured with the P-value, is great enough to render the null hypothesis implausible.

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More on the P-value

• The smaller the P-value, the more certain we can

be that H0 is false.

• The larger the P-value, the more plausible H0

becomes but we can never be certain that H0 is

true.

• A rule of thumb suggests to reject H0 whenever P

0.05. While this rule is convenient, it has no

scientific basis.

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Comments

• Some people report only that a test significant at a certain level, without giving the P-value. Such as, the result is “statistically significant at the 5% level.”

• This is poor practice.

• First, it provides no way to tell whether the P-value was just barely less than 0.05, or whether it was a lot less.

• Second, reporting that a result was statistically significant at the 5% level implies that there is a big difference between a P-value just under 0.05 and one just above 0.05, when in fact there is little difference.

• Third, a report like this does not allow readers to decide for themselves whether the P-value is small enough to reject the null hypothesis.

• Reporting the P-value gives more information about the strength of the evidence against the null hypothesis and allows each reader to decide for himself or herself whether to reject the null hypothesis.

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Comments on P

Let be any value between 0 and 1. Then, if P ,

The result of the test is said to be significantly significant

at the 100% level.

The null hypothesis is rejected at the 100% level.

When reporting the result of the hypothesis test, report

the P-value, rather than just comparing it to 5% or 1%.

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Significance

• When a result has a small P-value, we say that it

is “statistically significant.”

• In common usage, the word significant means

“important.”

• It is therefore tempting to think that statistically

significant results must always be important.

• Sometimes statistically significant results do not

have any scientific or practical importance.

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Hypothesis Tests and CI’s

• Both confidence intervals and hypothesis tests are concerned with determining plausible values for a quantity such as a population mean .

• In a hypothesis test for a population mean , we specify a particular value of (the null hypothesis) and determine if that value is plausible.

• A confidence interval for a population mean can be thought of as a collection of all values for that meet a certain criterion of plausibility, specified by the confidence level 100(1-)%.

• The values contained within a two-sided level 100(1-)% confidence intervals are precisely those values for which the P-value of a two-tailed hypothesis test will be greater than .

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Small Sample Test for a

Population Mean • When we had a large sample we used the sample

standard deviation s to approximate the population deviation .

• When the sample size is small, s may not be close to , which invalidates this large-sample method.

• However, when the population is approximately normal, the Student’s t distribution can be used.

• The only time that we don’t use the Student’s t distribution for this situation is when the population standard deviation is known. Then we are no longer approximating and we should use the z-test.

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Hypothesis Test

• Let X1,…, Xn be a sample from a normal population

with mean and standard deviation , where is

unknown.

• To test a null hypothesis of the form H0: 0, H0:

≥ 0, or H0: = 0.

• Compute the test statistic

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./

0

ns

Xt

P-value

Compute the P-value. The P-value is an area under the Student’s t curve with n – 1 degrees of freedom, which depends on the alternate hypothesis as follows.

• If the alternative hypothesis is H1: > 0, then the P-value is the area to the right of t.

• If the alternative hypothesis is H1: < 0, then the P-value is the area to the left of t.

• If the alternative hypothesis is H1: 0, then the P-value is the sum of the areas in the tails cut off by t and -t.

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Fixed-Level Testing

• A hypothesis test measures the plausibility of the null hypothesis by producing a P-value.

• The smaller the P-value, the less plausible the null.

• We have pointed out that there is no scientifically valid dividing line between plausibility and implausibility, so it is impossible to specify a “correct” P-value below which we should reject H0.

• If a decision is going to be made on the basis of a hypothesis test, there is no choice but to pick a cut-off point for the P-value.

• When this is done, the test is referred to as a fixed-level test.

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Conducting the Test

To conduct a fixed-level test:

• Choose a number , where 0 < < 1. This is

called the significance level, or the level, of the test.

• Compute the P-value in the usual way.

• If P , reject H0. If P > , do not reject H0.

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Comments

• In a fixed-level test, a critical point is a value of the test statistic that produces a P-value exactly equal to .

• A critical point is a dividing line for the test statistic just as the significance level is a dividing line for the P-value.

• If the test statistic is on one side of the critical point, the P-value will be less than , and H0 will be rejected.

• If the test statistic is on the other side of the critical point, the P-value will be more than , and H0 will not be rejected.

• The region on the side of the critical point that leads to rejection is called the rejection region.

• The critical point itself is also in the rejection region.

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Errors

When conducting a fixed-level test at

significance level , there are two types of

errors that can be made. These are

Type I error: Reject H0 when it is true.

Type II error: Fail to reject H0 when it is false.

The probability of Type I error is never

greater than .

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Power of Tests

• A hypothesis test results in Type II error if H0 is not

rejected when it is false.

• The power of the test is the probability of rejecting

H0 when it is false. Therefore,

Power = 1 – P(Type II error).

• To be useful, a test must have reasonable small

probabilities of both type I and type II errors.

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More on Power

• The type I error is kept small by choosing a small value of as the significance level.

• If the power is large, then the probability of type II error is small as well, and the test is a useful one.

• The purpose of a power calculation is to determine whether or not a hypothesis test, when performed, is likely to reject H0 in the event that H0 is false.

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Computing the Power

This involves two steps:

1. Compute the rejection region.

2. Compute the probability that the test statistic falls in the rejection region if the alternate hypothesis is true. This is power.

When power is not large enough, it can be

increased by increasing the sample size.

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Example

Find the power of the 5% level test of H0: 80 versus

H1: > 80 for the mean yield of the new process under

the alternative = 82, assuming n = 50 and = 5.

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3.8.5.1 Probability Plots

• Scientists and engineers often work with data that can be thought of as a random sample from some population. In many cases, it is important to determine the probability distribution that approximately describes the population.

• More often than not, the only way to determine an appropriate distribution is to examine the sample to find a sample distribution that fits.

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Finding a Distribution

Probability plots are a good way to determine an appropriate distribution.

Here is the idea: Suppose we have a random sample X1,…,Xn. We first arrange the data in ascending order. Then assign evenly spaced values between 0 and 1 to each Xi. There are several acceptable ways to this; the simplest is to assign the value (i – 0.5)/n to Xi.

The distribution that we are comparing the X’s to should have a mean and variance that match the sample mean and variance. We want to plot (Xi, F(Xi)), if this plot resembles the cdf of the distribution that we are interested in, then we conclude that that is the distribution the data came from.

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Probability Plot: Example

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i Xi (i-.5)/n Qi

1 3.01 0.1 2.4369 2 3.35 0.3 3.9512 3 4.79 0.5 5.0000 4 5.96 0.7 6.0488 5 7.89 0.9 7.5631

0.0000

1.0000

2.0000

3.0000

4.0000

5.0000

6.0000

7.0000

8.0000

0 2 4 6 8 10

Qi

Probability Plot: Example

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Software

Many software packages take the (i – 0.5)/n assigned to each Xi, and calculate the quantile (Qi) corresponding to that number from the distribution of interest. Then it plots each (Xi, Qi). If this plot is a reasonably straight line then you may conclude that the sample came from the distribution that we used to find quantiles.

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Normal Probability Plots

The sample plotted on the left comes from a population that is not close to normal.

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Normal Probability Plots

The sample plotted on the left comes from a population that is not close to normal. The sample plotted on the right comes from a population that is close to normal.

3.8.5.2 Likelihood Ratio Tests

• The general idea is to form the ratio of the likelihood function using the hypothesized value and the likelihood function using an alternative value.

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3.8.6 Bonferroni Intervals

• If 𝓀 100(1-)% confidence intervals are constructed for each of 𝓀 parameters, the probability that every interval contains the unknown parameter value that it estimates is at least 1 − 𝓀𝛼.

• Thus, if it is desired to have the probability of coverage for all 𝓀 intervals equal to at least 1 − 𝛼, each interval

should then be a 100 1 −𝛼

𝓀% confidence interval.

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Bonferroni Method in

Hypothesis Tests

• Sometimes a situation occurs in which it is necessary to perform many hypothesis tests.

• The basic rule governing this situation is that as more tests are performed, the confidence that we can place in our results decreases.

• The Bonferroni method provides a way to adjust P-values upward when several hypothesis tests are performed.

• If a P-value remains small after the adjustment, the null hypothesis may be rejected.

• To make the Bonferroni adjustment, simply multiply the P-value by the number of test performed.

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Example

Four different coating formulations are tested to see if they reduce the wear on cam gears to a value below 100 m. The null hypothesis H0: 100 m is tested for each formulation and the results are

Formulation A: P = 0.37

Formulation B: P = 0.41

Formulation C: P = 0.005

Formulation D: P = 0.21

The operator suspects that formulation C may be effective, but he knows that the P-value of 0.005 is unreliable, because several tests have been performed. Use the Bonferroni adjustment to produce a reliable P-value.

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3.9 Enumerative Studies v.s.

Analytic Studies

• An enumerative study is conducted for the purpose of

determining the “current state of affairs” relative to a

fixed frame (population).

• Example of enumerative study: random sampling of

typing errors made by clerical workers.

• Analytic study focuses on determining the cause(s) of

the errors that were made, with an eye toward

reducing the number.

• Making inferential and descriptive statements

regarding a fixed frame <> Determining how to

improve future performance.