CH.9 Tests of Hypotheses for a Single Sample

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1 CH.9 Tests of Hypotheses for a Single Sample Hypotheses testing Tests on the mean of a normal distributionvariance known Tests on the mean of a normal distributionvariance unknown Tests on the variance and standard deviation of a normal distribution Tests on a population proportion Testing for goodness of fit 1

2 9-1 Hypothesis Testing Statistical Hypotheses Statistical inference may be divided into two major areas: Parameter estimation Hypothesis testing 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. 2

3 9-1 Hypothesis Testing 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. 3

4 9-1 Hypothesis Testing Statistical Hypotheses Two-sided Alternative Hypothesis null hypothesis alternative hypothesis One-sided Alternative Hypotheses 4

5 9-1 Hypothesis Testing Statistical Hypotheses Test of a Hypothesis Hypotheses are statements about population or distribution under study. 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 (fail to reject hypothesis); if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false (reject hypothesis). 5

6 9-1 Hypothesis Testing Tests of Statistical Hypotheses Decision criteria for testing H 0 :μ = 50 cm/s versus H 1 :μ 50 cm/s. 6

7 9-1 Hypothesis Testing Tests of Statistical Hypotheses Critical Region Acceptance Region Critical Region critical values Decision criteria for testing H 0 :μ = 50 cm/s versus H 1 :μ 50 cm/s. 7

8 9-1 Hypothesis Testing Tests of Statistical Hypotheses Two wrong conclusions are possible: Type I Error Type II Error 8

9 9-1 Hypothesis Testing Tests of Statistical Hypotheses Sometimes the type I error probability is called the significance level, or the α-error, or the size of the test. 9

10 9-1 Hypothesis Testing Tests of Statistical Hypotheses Ex: σ =2.5 n=10 Standard deviation of the sample mean: σ 2.5 σ = = = X n μ x σ X 10

11 9-1 Hypothesis Testing Critical Region for H0 : μ = 50 H1: μ 50 Probability distribution of X Corresponding probability distribution of Z Z 11

12 9-1 Hypothesis Testing How can we reduce α? - by widening acceptance region (if take critical values 48 and 52, α =0.0114, Verify!) - by increasing sample size (if take n=16, α = Verify!) 12

13 9-1 Hypothesis Testing The probability of type II error when μ = 52 and n =

14 9-1 Hypothesis Testing 14

15 9-1 Hypothesis Testing The probability of type II error when μ = 50.5 and n =

16 9-1 Hypothesis Testing 16

17 9-1 Hypothesis Testing The probability of type II error when μ = 52 and n =

18 9-1 Hypothesis Testing 18

19 9-1 Hypothesis Testing Generally α is controlable when critical values are selected. Thus, rejection of null hypothesis H 0 is a strong conclusion. β is not constant but depends on the true value of the parameter and sample size. Accepting H 0 is a weak conclusion unless β is acceptably small. Prefer the terminology fail to reject H 0 ratherthan accept H 0 Fail to reject H 0 implies we have not found sufficient evidence to reject H 0. does not necessarily mean there is a high probability that H 0 is true. means more data are required to reach a strong conclusion. 19

20 9-1 Hypothesis Testing Power The power is computed as 1 - β, 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 when we are testing H 0 : μ = 50 cm/s against H 1 : μ not equal 50 cm/s. Suppose that the true value of the mean is μ = 52. When n = 10, we found that β = , so the power of this test is 1 - β = = when μ =

21 9-1 Hypothesis Testing One-Sided and Two-Sided Hypotheses Two-Sided Test: One-Sided Tests: 21

22 9-1 Hypothesis Testing Example 9-1 Suppose if the propellant burning rate is less than 50 cm/s Want to show this with a strong conclusion CLAIM Hypotheses should be stated as H 0 : μ = 50 cm/s H 1 : μ < 50 cm/s Since the rejection of H 0 is always a strong conclusion, this statement of the hypotheses will produce the desired outcome if H 0 is rejected. Although H 0 is stated with an equal sign, it is understood to include any value of μ not specified by H 1. Failing to reject H 0 does not mean μ = 50 cm/s exactly. Failing to reject H 0 means we do not have strong evidence in support of H 1. 22

23 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 Which is correct? Depends on the objective of the analysis. 23

24 9-1 Hypothesis Testing P-Values in Hypothesis Tests When H 0 is rejected at a specified α level, this gives no idea about whether the computed value of the test statistic is just barely in the rejection region or it is very far into this region. Thus, P-value has been adopted widely in practice 24

25 9-1 Hypothesis Testing P-Values in Hypothesis Tests Consider the two-sided hypothesis test for burning rate: H 0 : μ = 50 cm/s H 1 : μ 50 cm/s n=16, σ=2.5, P-value? x =

26 9-1 Hypothesis Testing Connection between Hypothesis Tests and Confidence Intervals Close relation between hypothesis tests and confidence intervals H 0 : μ = 50 cm/s n=16, σ=2.5, α=0.05, H 1 : μ 50 cm/s x = 51.3 Critical z values are z α/2 =z =1.96 and z =-1.96 which corresponds to Critical values ± 1.96 = [ ; ] 16 x = 51.3 is not in the acceptance region [ ; ]. So reject null hypothesis. Confidence interval for μ at α=0.05 is That is μ ± same conclusion! μ=50 is not in the confidence interval [ ; ]. So reject null hypothesis. 26

27 9-1 Hypothesis Testing Connection between Hypothesis Tests and Confidence Intervals 27

28 9-1 Hypothesis Testing General Procedure for Hypothesis Tests 1. From the problem context, identify the parameter of interest. 2. State the null hypothesis, H Specify an appropriate alternative hypothesis, H Choose a significance level, α. 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 H 0 should be rejected and report that in the problem context. 28

29 9-2 Tests on the Mean of a Normal Distribution, Variance Known Hypothesis Tests on the Mean We wish to test: The test statistic is: 29

30 9-2 Tests on the Mean of a Normal Distribution, Variance Known Hypothesis Tests on the Mean Reject H 0 if the observed value of the test statistic z 0 is either: Fail to reject H 0 if z 0 > z α/2 or z 0 < -z α/2 -z α/2 < z 0 < z α/2 30

31 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example

32 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example

33 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example

34 9-2 Tests on the Mean of a Normal Distribution, Variance Known Hypothesis Tests on the Mean 34

35 9-2 Tests on the Mean of a Normal Distribution, Variance Known Hypothesis Tests on the Mean (Continued) 35

36 9-2 Tests on the Mean of a Normal Distribution, Variance Known 36

37 9-2 Tests on the Mean of a Normal Distribution, Variance Known Hypothesis Tests on the Mean (Continued) 37

38 9-2 Tests on the Mean of a Normal Distribution, Variance Known P-Values in Hypothesis Tests 38

39 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Finding the Probability of Type II Error β 39

40 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Finding the Probability of Type II Error β The distribution of Z 0 under H 0 and H 1 40

41 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Finding the Probability of Type II Error β 0 if δ>0 Let z β be the 100β upper percentile of the standard normal distr. β = Φ(-z β ) z β z α/2 δ n σ 41

42 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Sample Size Formulas For a two-sided alternative hypothesis: 42

43 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Sample Size Formulas For a one-sided alternative hypothesis: 43

44 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-3 H 0 : µ=50 H 1 : µ 50 σ=2 α=0.05 n=25 If true µ=49, β=? Under H 1 : µ 50 Under H 0 : µ=50 z =1.96 Critical points are: 50 ± z σ/ n = 50 ± 1.96*2/ and

45 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-3 Under H 1 : µ 50 Under H 0 : µ=50 H 0 : µ=50 H 1 : µ 50 σ=2 α=0.05 n=25 If true µ=49, β=? β = P( X when μ=49) β = P Z 2 / 25 2 / 25 β = P = ( Z ) z = z = β

46 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example 9-3 Standardize the normal graphs N(-2.5,1) Under H 1 : µ 50 Under H 0 : µ=50 N(0,1) δ = 1 δ n 1* 25 = = 2.5 σ 2 With standard normal graph: With the formula: β = P( 1.96 X 1.96 when μ = -2.5) β = P Z 1 1 β = P = ( Z ) 1* 25 1* 25 β =Φ Φ 2 β =Φ(4.46) Φ (0.54) =

47 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example

48 9-2 Tests on the Mean of a Normal Distribution, Variance Known Type II Error and Choice of Sample Size Using Operating Characteristic Curves I I 48

49 9-2 Tests on the Mean of a Normal Distribution, Variance Known Example

50 9-2 Tests on the Mean of a Normal Distribution, Variance Known Large Sample Test If the distribution of the population is not known, but n>40 sample standard deviation s can be substituted for σ and test procedures in Section 9.2 are valid. 50

51 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Hypothesis Tests on the Mean One-Sample t-test 51

52 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Hypothesis Tests on the Mean The reference distribution for H 0 : μ = μ 0 with critical region for (a) H 1 : μ μ 0, (b) H 1 : μ > μ 0, and (c) H 1 : μ < μ 0. 52

53 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example

54 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example

55 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example 9-6 Normal probability plot of the coefficient of restitution data from Example

56 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example

57 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown 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. t 0 = 2.72, this is between two tabulated values, and Therefore, the P-value must be between 0.01 and <P-value<0.01. Suppose t 0 = 2.72 for a two-sided test, then 0.005*2<P-value<0.01*2 0.01<P-value<

58 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Type II Error and Choice of Sample Size The type II error of the two-sided alternative would be Where T 0 denotes the noncentral t random variable. 58

59 9-3 Tests on the Mean of a Normal Distribution, Variance Unknown Example

60 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Hypothesis Test on the Variance 60

61 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Hypothesis Test on the Variance The same test statistic is used for one-sided alternative hypotheses. For the one-sided hypothesis 61

62 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Hypothesis Test on the Variance 62

63 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example

64 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example

65 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Type II Error and Choice of Sample Size For the two-sided alternative hypothesis: Operating characteristic curves are provided in Charts VIIi and VIIj. 65

66 9-4 Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution Example

9-5 Tests on a Population Proportion 9-5.

67 9-5 Tests on a Population Proportion Large-Sample Tests on a Proportion Many engineering decision problems include hypothesis testing about p. An appropriate test statistic is 67

68 9-5 Tests on a Population Proportion Example

69 9-5 Tests on a Population Proportion Example

70 9-5 Tests on a Population Proportion Another form of the test statistic Z 0 is obtained by dividing the numerator and denominator by n or 70

71 9-5 Tests on a Population Proportion Type II Error and Choice of Sample Size For a two-sided alternative If the alternative is p < p 0 If the alternative is p > p 0 where p is true value of the population proportion 71

72 9-5 Tests on a Population Proportion Type II Error and Choice of Sample Size Less complex representation: σ H 1 p(1 p) n σ = H0 Critical values (C 1 and C 2 ) are: C = p z σ C = p + z = p (1 p ) n o α 2 H 2 o α 2 H Type II error is: β = PC ( < Pˆ < C) σ C μ = P < Z < σ 1 H ( ) H C μ σ 1 1 C2 μ H C1 μ H =Φ Φ 1 1 σ H σ 1 H1 H H 72

73 9-5 Tests on a Population Proportion Type II Error and Choice of Sample Size For a two-sided alternative For a one-sided alternative 73

74 9-5 Tests on a Population Proportion Example (0.97) 0.05(0.95) σ H = = σ H = = Critical value (C 1 ) is: C1 = po z α σ H = (0.0154) = β = PC ( < Pˆ ) = P( < Z) = 1 Φ 0.44 = 0.67 ( ) 74

75 9-5 Tests on a Population Proportion Example

76 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 O i be the observed frequency in the ith class interval. Let E i be the expected frequency in the ith class interval. The test statistic has approximately chi-square distribution with k-p-1 degrees of freedom. p: number of parameters of the hypothesized distribution estimated by sample statistics. 76

77 9-7 Testing for Goodness of Fit Example

78 9-7 Testing for Goodness of Fit Example

79 9-7 Testing for Goodness of Fit Example

80 9-7 Testing for Goodness of Fit Example

81 9-7 Testing for Goodness of Fit Example

82 9-7 Testing for Goodness of Fit Example

83 9-7 Testing for Goodness of Fit Example

84 9-7 Testing for Goodness of Fit Example 9-13 X=5.04+Z*0.08 Z X

85 9-7 Testing for Goodness of Fit Example

86 9-7 Testing for Goodness of Fit 86

280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE Tests of Statistical Hypotheses

280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE 9-1.2 Tests of Statistical Hypotheses To illustrate the general concepts, consider the propellant burning rate problem introduced earlier. The null