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Chapter 9

Chapter 9. Hypothesis Testing. Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze. Methods for Drawing Inferences. We can draw inferences on a population parameter in two ways: Estimation (Chapter 8) Hypothesis Testing (Chapter 9). Hypothesis Testing.

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Chapter 9

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  1. Chapter 9 Hypothesis Testing Understanding Basic Statistics Fifth Edition By Brase and Brase Prepared by Jon Booze

  2. Methods for Drawing Inferences We can draw inferences on a population parameter in two ways: Estimation (Chapter 8) Hypothesis Testing (Chapter 9)

  3. Hypothesis Testing Hypothesis testing is the process of making decisions about the value of a population parameter.

  4. Establishing the Hypotheses Null Hypothesis: A hypothesis about a parameter that often denotes a theoretical value, a historical value, or a production specification. Denoted as H0 Alternate Hypothesis: A hypothesis that differs from the null hypothesis, such that if we reject the null hypothesis, we will accept the alternate hypothesis. Denoted as H1 (in other sources HA).

  5. Hypotheses Restated

  6. Statistical Hypotheses The null hypothesis is always a statement of equality. H0: μ = k, where k is a specified value The alternate hypothesis states that the parameter (μor p) is less than, greater than, or not equal to a specified value.

  7. Statistical Hypotheses Which of the following is an acceptable null hypothesis? a). H0:   1.2 b). H0:  > 1.2 c). H0:  = 1.2 d). H0:   1.2

  8. Statistical Hypotheses Which of the following is an acceptable null hypothesis? a). H0:   1.2 b). H0:  > 1.2 c). H0:  = 1.2 d). H0:   1.2

  9. Types of Tests Left-Tailed Tests: H1: μ < k H1: p < k Right-Tailed Tests: H1: μ > k H1: p > k Two-Tailed Tests: H1: μ≠ k H1: p ≠ k

  10. Types of Tests A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim? a). H0:   150; H1:  > 150 b). H0:  > 150; H1:  = 150 c). H0:  = 150; H1:   150 d). H0:  = 150; H1:  > 150

  11. Types of Tests A production manager believes that a particular machine averages 150 or more parts produced per day. What would be the appropriate hypotheses for testing this claim? a). H0:   150; H1:  > 150 b). H0:  > 150; H1:  = 150 c). H0:  = 150; H1:   150 d). H0:  = 150; H1:  > 150

  12. Hypothesis Testing Procedure Select appropriate hypotheses. Draw a random sample. Calculate the test statistic. Assess the compatibility of the test statistic with H0. Make a conclusion in the context of the problem.

  13. Hypothesis Test of μx is Normal, σ is Known

  14. P-Value P-values are sometimes called the probability of chance. Low P-values are a good indication that your test results are not due to chance.

  15. P-Value for Left-Tailed Test

  16. P-Value for Right-Tailed Test

  17. P-Value for Two-Tailed Test

  18. Types of Errors in Statistical Testing Since we are making decisions with incomplete information (sample data), we can make the wrong conclusion. Type I Error: Rejecting the null hypothesis when the null hypothesis is true. Type II Error: Accepting the null hypothesis when the null hypothesis is false.

  19. Type I and Type II Errors

  20. Errors in Statistical Testing Unfortunately, we usually will not know when we have made an error. We can only talk about the probability of making an error. Decreasing the probability of making a type I error will increase the probability of making a type II error (and vice versa). We can only decrease the probability of both types of errors by increasing the sample size (obtaining more information), but this may not be feasible in practice.

  21. Level of Significance Good practice requires us to specify in advance the risk level of type I error we are willing to accept. The probability of type I error is the level of significance for the test, denoted by α (alpha).

  22. Type II Error The probability of making a type II error is denoted by β (Beta). 1 – β is called the power of the test. 1 – β is the probability of rejecting H0 when H0 is false (a correct decision).

  23. Type II Error The probability of making a type II error is denoted by β (Beta). 1 – β is called the power of the test. 1 – β is the probability of rejecting H0 when H0 is false (a correct decision).

  24. The ProbabilitiesAssociated with Testing

  25. Concluding a Statistical Test For our purposes, significant is defined as follows: At our predetermined level of risk α, the evidence against H0 is sufficient to reject H0. Thus we adopt H1.

  26. Concluding a Statistical Test For a particular experiment, P = 0.17 and  = 0.05. What is the appropriate conclusion? a). Reject the null hypothesis. b). Do not reject the null hypothesis. c). Reject both the null hypothesis and the alternative hypothesis. d). Accept both the null hypothesis and the alternative hypothesis.

  27. Concluding a Statistical Test For a particular experiment, P = 0.17 and  = 0.05. What is the appropriate conclusion? a). Reject the null hypothesis. b). Do not reject the null hypothesis. c). Reject both the null hypothesis and the alternative hypothesis. d). Accept both the null hypothesis and the alternative hypothesis.

  28. Statistical Testing Comments Frequently, the significance is set at α = 0.05 or α = 0.01. When we “accept” the null hypothesis, we are not proving the null hypothesis to be true. We are only saying that the sample evidence is not strong enough to justify the rejection of H0. Some statisticians prefer to say “fail to reject H0 ” rather than “accept H0 .”

  29. Interpretation of Testing Terms

  30. Testing µ When σ is Known State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

  31. Testing µ When σ is Known Use the standard normal table and the type of test (one or two-tailed) to determine the P-value. Make a statistical conclusion: If P-value≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

  32. Testing µ When σ is Known Suppose that the test statistic z = 1.85 for a right-tailed test. Use Table 3 in the Appendix to find the corresponding P-value. a). 0.2514 b). 0.0322 c). 0.9678 d). 0.0161

  33. Testing µ When σ is Known Suppose that the test statistic z = 1.85 for a right-tailed test. Use Table 3 in the Appendix to find the corresponding P-value. a). 0.2514 b). 0.0322 c). 0.9678 d). 0.0161

  34. Testing µ When σ is Unknown State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed (or mound-shaped), any sample size will suffice. If not, n ≥ 30 is required. Calculate:

  35. Testing µ When σ is Unknown Use the Student’s t table and the type of test (one or two-tailed) to determine (or estimate) the P-value. Make a statistical conclusion: If P-value≤ α, reject H0. If P-value > α, do not reject H0. Make a context-specific conclusion.

  36. Using Table 4 to Estimate P-values 0.025 < P-value < 0.050 Suppose we calculate t = 2.22 for a one-tailed test from a sample size of 6. df = n – 1 = 5.

  37. Testing µ Using theCritical Value Method The values of that will result in the rejection of the null hypothesis are called the critical region of the distribution. When we use a predetermined significance level α, the Critical Value Method and the P-Value Method are logically equivalent.

  38. Critical Regions for H0: µ = k

  39. Critical Regions for H0: µ = k

  40. Critical Regions for H0: µ = k

  41. Testing µ When σ is Known (Critical Region Method) State the null hypothesis, alternate hypothesis, and level of significance. If x is normally distributed, any sample size will suffice. If not, n ≥ 30 is required. Calculate:

  42. Testing µ When σ is Known (Critical Region Method) Show the critical region and critical value(s) on a graph (determined by the alternate hypothesis and α). Conclude in favor of the alternate hypothesis if z is in the critical region. State a conclusion within the context of the problem.

  43. Left-Tailed Tests

  44. Right-Tailed Tests

  45. Two-Tailed Tests

  46. Testing a Proportion p Binomial Experiments: r (# of successes) is a binomial variable n is the number of independent trials p is the probability of success on each trial Test Assumption: np > 5 and n(1 – p) > 5

  47. Testing a Proportion p Test Assumption: np > 5 and n(1 – p) > 5 The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution? a). n = 48, p = 0.39 b). n = 843, p = 0.09 c). n = 52, p = 0.93 d). n = 12, p = 0.51

  48. Testing a Proportion p Test Assumption: np > 5 and n(1 – p) > 5 The values of n and p for several experiments are shown below. Which experiment should not be tested using the normal distribution? a). n = 48, p = 0.39 b). n = 843, p = 0.09 c). n = 52, p = 0.93 d). n = 12, p = 0.51

  49. Types of Proportion Tests

  50. The Distributionof the Sample Proportion

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