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Transcript of Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1...
![Page 1: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/1.jpg)
Chapter 9 Title and Outline 1
9Tests of Hypotheses for a Single Sample
CHAPTER OUTLINE
![Page 2: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/2.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
Learning Objectives for Chapter 9After careful study of this chapter, you should be able to do the
following:1. Structure engineering decision-making as hypothesis tests.2. Test hypotheses on the mean of a normal distribution using either a Z-test or a
t-test procedure.3. Test hypotheses on the variance or standard deviation of a normal
distribution.4. Test hypotheses on a population proportion.5. Use the P-value approach for making decisions in hypothesis tests.6. Compute power & Type II error probability. Make sample size selection
decisions for tests on means, variances & proportions.7. Explain & use the relationship between confidence intervals & hypothesis
tests.8. Use the chi-square goodness-of-fit test to check distributional assumptions.9. Use contingency table tests.
2Chapter 9 Learning Objectives
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.1 Statistical Hypotheses
Definition
Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental methods used at the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a specified value.
![Page 4: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/4.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.1 Statistical Hypotheses
For example, suppose that we are interested in the burning rate of a solid propellant used to power aircrew escape systems.
• Now burning rate is a random variable that can be described by a probability distribution.
• Suppose that our interest focuses on the mean burning rate (a parameter of this distribution).
• Specifically, we are interested in deciding whether or not the mean burning rate is 50 centimeters per second.
![Page 5: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/5.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.1 Statistical Hypotheses
null hypothesis
alternative hypothesis
One-sided Alternative Hypotheses
Two-sided Alternative Hypothesis
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.1 Statistical HypothesesTest of a Hypothesis
• A procedure leading to a decision about a particular hypothesis
• Hypothesis-testing procedures rely on using the information in a random sample from the population of interest.
• If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.
![Page 7: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/7.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.2 Tests of Statistical Hypotheses
Figure 9-1 Decision criteria for testing H0: = 50 centimeters per second versus H1: 50 centimeters per second.
![Page 8: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/8.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.2 Tests of Statistical Hypotheses
Definitions
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.2 Tests of Statistical Hypotheses
Sometimes the type I error probability is called the significance level, or the -error, or the size of the test.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.2 Tests of Statistical Hypotheses
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
Figure 9-3 The probability of type II error when = 52 and n = 10.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
![Page 14: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/14.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
Figure 9-4 The probability of type II error when = 50.5 and n = 10.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
Figure 9-5 The probability of type II error when = 2 and n = 16.
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
Definition
• The power is computed as 1 - , and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often 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 : not equal 50 centimeters per second . Suppose that the true value of the mean 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.
![Page 20: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/20.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.3 One-Sided and Two-Sided Hypotheses Two-Sided Test:
One-Sided Tests:
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
Example 9-1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
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 procedure for a specific lot of bottles as a hypothesis testing problem. There are two possible formulations for this problem, either
or
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.4 P-Values in Hypothesis Tests
Definition
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.4 P-Values in Hypothesis Tests
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.4 P-Values in Hypothesis Tests
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.5 Connection between Hypothesis Tests and Confidence Intervals
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-1 Hypothesis Testing
9-1.6 General Procedure for Hypothesis Tests
1. From the problem context, identify the parameter of interest.
2. State the null hypothesis, H0 .
3. Specify an appropriate alternative hypothesis, H1.
4. Choose a significance level, .
5. Determine an appropriate test statistic.
6. State the rejection region for the statistic.
7. Compute any necessary sample quantities, substitute these into the equation for the test statistic, and compute that value.
8. Decide whether or not H0 should be rejected and report that in the problem context.
![Page 28: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/28.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.1 Hypothesis Tests on the Mean
We wish to test:
The test statistic is:
![Page 29: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/29.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.1 Hypothesis Tests on the Mean
Reject H0 if the observed value of the test statistic z0 is either:
z0 > z/2 or z0 < -z/2
Fail to reject H0 if -z/2 < z0 < z/2
![Page 30: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/30.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
![Page 31: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/31.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-2
![Page 32: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/32.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-2
![Page 33: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/33.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-2
![Page 34: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/34.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.1 Hypothesis Tests on the Mean (Eq. 9-10 & 11)
![Page 35: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/35.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.1 Hypothesis Tests on the Mean (Continued) (Eq. 9-12 & 18)
![Page 36: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/36.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.1 Hypothesis Tests on the Mean (Continued)
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9-2 Tests on the Mean of a Normal Distribution, Variance Known
P-Values in Hypothesis Tests
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeFinding the Probability of Type II Error (Eq. 9-19)
![Page 39: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/39.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeFinding the Probability of Type II Error (Eq. 9-20)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeFinding the Probability of Type II Error (Figure 9-9)
Figure 9-9 The distribution of Z0 under H0 and H1
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeSample Size Formulas
For a two-sided alternative hypothesis: (Eq. 9-22)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeSample Size Formulas
For a one-sided alternative hypothesis: (Eq. 9-23)
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-3
![Page 44: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/44.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-3
![Page 45: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/45.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeUsing Operating Characteristic Curves (Eq. 9-24)
![Page 46: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/46.jpg)
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.2 Type II Error and Choice of Sample SizeUsing Operating Characteristic Curves
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-2 Tests on the Mean of a Normal Distribution, Variance Known
Example 9-4
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9-2 Tests on the Mean of a Normal Distribution, Variance Known
9-2.3 Large Sample Test
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
9-3.1 Hypothesis Tests on the MeanOne-Sample t-Test
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© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
9-3.1 Hypothesis Tests on the Mean (Figures 9-10, 11, 12)
Figure 9-10 The reference distribution for H0: = 0 with critical region for (a) H1: 0 , (b) H1: > 0, and (c) H1: < 0.
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
Example 9-6
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
Example 9-6
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
Example 9-13
Figure 9-13 Normal probability plot of the coefficient of restitution data from Example 9-6.
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
Example 9-6
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
9-3.2 P-value for a t-Test
The P-value for a t-test is just the smallest level of significance at which the null hypothesis would be rejected.
Notice that t0 = 2.72 in Example 9-6, and that this is between two tabulated 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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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
9-3.3 Type II Error and Choice of Sample Size
The type II error of the two-sided alternative (for example) would be
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9-3 Tests on the Mean of a Normal Distribution, Variance Unknown
Example 9-7
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
9-4.1 Hypothesis Test on the Variance (Eq. 9-34, 35)
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
9-4.1 Hypothesis Test on the Variance
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
9-4.1 Hypothesis Test on the Variance (Eq. 9-33, 34)
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
9-4.1 Hypothesis Test on the Variance (Figure 9-14)
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
Example 9-8
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
Example 9-8
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
9-4.2 Type II Error and Choice of Sample Size
For the two-sided alternative hypothesis:
Operating characteristic curves are provided in Charts VIi and VIj:
![Page 65: Chapter 9 Title and Outline 1 9 Tests of Hypotheses for a Single Sample 9-1 Hypothesis Testing 9-1.1 Statistical Hypotheses 9-1.2 Tests of Statistical.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649dc85503460f94abea4e/html5/thumbnails/65.jpg)
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9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution
Example 9-9
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9-5 Tests on a Population Proportion
9-5.1 Large-Sample Tests on a Proportion (Eq. 9-40)
Many engineering decision problems include hypothesis testing about p.
An appropriate test statistic is
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9-5 Tests on a Population Proportion
Example 9-10
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9-5 Tests on a Population Proportion
Example 9-10
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9-5 Tests on a Population Proportion
Another form of the test statistic Z0 is
or
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9-5 Tests on a Population Proportion
9-5.2 Type II Error and Choice of Sample Size
For a two-sided alternative (Eq. 9-42)
If the alternative is p < p0 (Eq. 9-43)
If the alternative is p > p0 (Eq. 9-44)
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9-5 Tests on a Population Proportion
9-5.3 Type II Error and Choice of Sample Size
For a two-sided alternative (Eq. 9-45)
For a one-sided alternative (Eq. 9-46)
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9-5 Tests on a Population Proportion
Example 9-11
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9-5 Tests on a Population Proportion
Example 9-11
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9-7 Testing for Goodness of Fit
• 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.
• Let Oi be the observed frequency in the ith class interval.
• Let Ei be the expected frequency in the ith class interval.
The test statistic is (Eq.9-47)
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Example 9-12
9-7 Testing for Goodness of Fit
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9-7 Testing for Goodness of Fit
Example 9-12
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9-7 Testing for Goodness of Fit
Example 9-12
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9-7 Testing for Goodness of Fit
Example 9-12
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9-7 Testing for Goodness of Fit
Example 9-12
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9-7 Testing for Goodness of Fit
Example 9-12
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9-8 Contingency Table Tests
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 two methods of classification are statistically independent;
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9-8 Contingency Table Tests
(Eq. 9-48)
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9-8 Contingency Table Tests
(Eq.9-49)
(Eq. 9-50)
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9-8 Contingency Table Tests
Example 9-14
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9-8 Contingency Table Tests
Example 9-14
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9-8 Contingency Table Tests Example 9-14
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9-8 Contingency Table Tests Example 9-14
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Important Terms & Concepts of Chapter 9
Chapter 9 Summary 88