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9-1 Hypothesis Testing9-1.1 Statistical Hypotheses

Statistical hypothesis testing and confidence intervalestimation of parameters are the fundamental methodsused at the data analysis stage of a comparative

Definition

, ,example, in comparing the mean of a population to aspecified value.

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For example, suppose that we are interested in the burning rate of a solid propellant used to power aircrewescape systems.

Now burning rate is a random variable that can bedescribed by a probability distribution.

Suppose that our interest focuses on the mean burningrate (a parameter of this distribution).

Specifically, we are interested in deciding whether or

not the mean burning rate is 50 centimeters per second.

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null hypothesis

alternative hypothesis

Two-sided Alternative Hypothesis

One-sided Alternative Hypotheses

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9-1 Hypothesis Testing

9-1.1 Statistical Hypotheses

Test of a Hypothesis

A procedure leading to a decision about a particular hypothesis

Hypothesis-testing procedures rely on using the informationin a random sample from the population of interest .

If this information is consistent with the hypothesis, then wewill conclude that the hypothesis is true ; if this information isinconsistent with the hypothesis, we will conclude that thehypothesis is false .

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9-1.2 Tests of Statistical Hypotheses

Definitions

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9-1.2 Tests of Statistical Hypotheses

Sometimes the type I error probability is called thesignificance level , or the -error , or the size of the test.

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Figure 9-3 Theprobability of type IIerror when = 52 andn = 10.

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Figure 9-4 Theprobability of type IIerror when = 50.5and n = 10.

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Figure 9-5 Theprobability of type IIerror when = 2 and n= 16.

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Definition

- ,

the probability of correctly rejecting a false null hypothesis. Weoften compare statistical tests by comparing their power properties.

For example, consider the propellant burning rate problem whenwe are testing H 0 : = 50 centimeters per second against H 1 : notequal 50 centimeters per second . Suppose that the true value of themean is = 52. When n = 10, we found that = 0.2643, so the

power of this test is 1 - = 1 - 0.2643 = 0.7357 when = 52.

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9-1.3 One-Sided and Two-Sided Hypotheses

Two-Sided Test :

One-Sided Tests :

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Example 9-1

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The bottler wants to be sure that the bottles meet the

specification on mean internal pressure or bursting strength,which for 10-ounce bottles is a minimum strength of 200 psi. The bottler has decided to formulate the decision proce ure or a spec c o o o es as a ypo es s es ng problem. There are two possible formulations for this problem, either

or

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9-1.4 P-Values in Hypothesis Tests

Definition

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9-1.5 Connection between Hypothesis Tests andConfidence Intervals

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9-1.6 General Procedure for Hypothesis Tests

1. From the problem context, identify the parameter of interest.

2. State the null hypothesis, H 0 .

3. Specify an appropriate alternative hypothesis, H 1.

4. Choose a significance level, .5. Determine an appropriate tst statistic.

6. State the rejection region for the statistic.

7. Compute any necessary sample quantities, substitute these into theequation for the test statistic, and compute that value.

8. Decide whether or not H 0 should be rejected and report that in the problem context.

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9-2 Tests on the Mean of a Normal

Distribution, Variance Known9-2.1 Hypothesis Tests on the Mean

We wish to test :

The test statistic is:

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9-2 Tests on the Mean of a NormalDistribution, Variance Known

Example 9-2

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Example 9-2

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Example 9-2

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9-2.1 Hypothesis Tests on the Mean

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9-2.1 Hypothesis Tests on the Mean (Continued)

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9-2.1 Hypothesis Tests on the Mean (Continued)

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P -Values in Hypothesis Tests

9 2 T h M f N l

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Distribution, Variance Known9-2.2 Type II Error and Choice of Sample Size

Finding the Probability of Type II Error

9 2 T h M f N l

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Distribution, Variance Known

9-2.2 Type II Error and Choice of Sample Size


9 2 T t th M f N l

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Figure 9-7 The distribution of Z 0 under H 0 and H 1

9 2 T t th M f N l

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Sample Size Formulas

For a two-sided alternative hypothesis:

9 2 Tests on the Mean of a Normal

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Sample Size Formulas

For a one-sided alternative hypothesis:


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Distribution, Variance KnownExample 9-3


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Using Operating Characteristic Curves


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Using Operating Characteristic Curves


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Distribution, Variance KnownExample 9-4

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Distribution, Variance Unknown

9-3.1 Hypothesis Tests on the Mean

Figure 9-9 The reference distribution for H 0: = 0 with criticalregion for (a) H 1: 0 , (b) H 1: > 0, and (c) H 1: < 0.


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Distribution, Variance UnknownExample 9-6


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Figure 9-10Normal probability

lot of the

coefficient of restitution datafrom Example 9-6.

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Distribution, Variance Unknown9-3.2 P -value for a t -Test

The P -value for a t -test is just the smallest level of significanceat which the null hypothesis would be rejected.

Notice that t 0 = 2.72 in Example 9-6, and that this is between twotabulated values, 2.624 and 2.977. Therefore, the P -value must be

between 0.01 and 0.005. These are effectively the upper and lower bounds on the P -value.

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9-4 Hypothesis Tests on the Variance and

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Standard Deviation of a Normal Distribution9-4.1 Hypothesis Test on the Variance


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Standard Deviation of a Normal DistributionExample 9-8

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Standard Deviation of a Normal DistributionExample 9-9

9-5 Tests on a Population Proportion

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9 5 Tests on a Population Proportion

9-5.1 Large-Sample Tests on a Proportion

Many engineering decision problems include hypothesis testing

about p.

An appropriate test statistic is


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p p

Example 9-10


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p p

Example 9-10


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p p

Another form of the test statistic Z 0 is

or


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p p


For a two-sided alternative

t e a ternat ve s p < p0

If the alternative is p > p0


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For a two-sided alternative

For a one-sided alternative


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Example 9-11


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Example 9-11

9-7 Testing for Goodness of Fit

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The test is based on the chi-square distribution.

Assume there is a sample of size n from a population whose probability distribution is unknown.

i .

Let E i be the expected frequency in the ith class interval.

The test statistic is


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Example 9-12


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Example 9-12


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Example 9-12


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Example 9-12


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Example 9-12


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Example 9-12

9-8 Contingency Table Tests

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Many times, the n elements of a sample from a population may be classified according to two different

criteria. It is then of interest to know whether the twomethods of classification are statistically independent;


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