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Statistics for Business and Economics 7 th Edition Chapter 9 Hypothesis Testing: Single Population Statistics for Business and Economics 7 th Edition Chapter 9 Hypothesis Testing: Single Population Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -1

Chapter Goals After completing this chapter, you should be able to: n Formulate null Chapter Goals After completing this chapter, you should be able to: n Formulate null and alternative hypotheses for applications involving a single population mean from a normal distribution n a single population proportion (large samples) n the variance of a normal distribution n Formulate a decision rule for testing a hypothesis n Know how to use the critical value and p-value approaches to test the null hypothesis (for both mean and proportion problems) n Know what Type I and Type II errors are n Assess the power of a test n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -2

9. 1 n What is a Hypothesis? A hypothesis is a claim (assumption) about 9. 1 n What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: n population mean Example: The mean monthly cell phone bill of this city is μ = $42 n population proportion Example: The proportion of adults in this city with cell phones is p =. 68 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -3

The Null Hypothesis, H 0 n States the assumption (numerical) to be tested Example: The Null Hypothesis, H 0 n States the assumption (numerical) to be tested Example: The average number of TV sets in U. S. Homes is equal to three ( ) n Is always about a population parameter, not about a sample statistic Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -4

The Null Hypothesis, H 0 (continued) n n Begin with the assumption that the The Null Hypothesis, H 0 (continued) n n Begin with the assumption that the null hypothesis is true n Similar to the notion of innocent until proven guilty Refers to the status quo Always contains “=” , “≤” or “ ” sign May or may not be rejected Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -5

The Alternative Hypothesis, H 1 n Is the opposite of the null hypothesis n The Alternative Hypothesis, H 1 n Is the opposite of the null hypothesis n n n e. g. , The average number of TV sets in U. S. homes is not equal to 3 ( H 1: μ ≠ 3 ) Challenges the status quo Never contains the “=” , “≤” or “ ” sign May or may not be supported Is generally the hypothesis that the researcher is trying to support Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -6

Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: H 0: Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: H 0: μ = 50 ) Population Is X= 20 likely if μ = 50? If not likely, REJECT Null Hypothesis Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Suppose the sample mean age is 20: X = 20 Now select a random sample Sample Ch. 9 -7

Reason for Rejecting H 0 Sampling Distribution of X 20 If it is unlikely Reason for Rejecting H 0 Sampling Distribution of X 20 If it is unlikely that we would get a sample mean of this value. . . μ = 50 If H 0 is true . . . if in fact this were the population mean… Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall X . . . then we reject the null hypothesis that μ = 50. Ch. 9 -8

Level of Significance, n Defines the unlikely values of the sample statistic if the Level of Significance, n Defines the unlikely values of the sample statistic if the null hypothesis is true n n Defines rejection region of the sampling distribution Is designated by , (level of significance) n Typical values are. 01, . 05, or. 10 n Is selected by the researcher at the beginning n Provides the critical value(s) of the test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -9

Level of Significance and the Rejection Region Level of significance = H 0: μ Level of Significance and the Rejection Region Level of significance = H 0: μ = 3 H 1: μ ≠ 3 /2 Two-tail test /2 Represents critical value Rejection region is shaded 0 H 0: μ ≤ 3 H 1: μ > 3 Upper-tail test H 0: μ ≥ 3 H 1: μ < 3 0 Lower-tail test Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 0 Ch. 9 -10

Errors in Making Decisions n Type I Error n Reject a true null hypothesis Errors in Making Decisions n Type I Error n Reject a true null hypothesis n Considered a serious type of error The probability of Type I Error is n Called level of significance of the test n Set by researcher in advance Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -11

Errors in Making Decisions (continued) n Type II Error n Fail to reject a Errors in Making Decisions (continued) n Type II Error n Fail to reject a false null hypothesis The probability of Type II Error is β Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -12

Outcomes and Probabilities Possible Hypothesis Test Outcomes Actual Situation Decision Key: Outcome (Probability) H Outcomes and Probabilities Possible Hypothesis Test Outcomes Actual Situation Decision Key: Outcome (Probability) H 0 True Do Not Reject H 0 No Error (1 - ) Type II Error (β) Reject H 0 Type I Error ( ) No Error (1 -β) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall H 0 False Ch. 9 -13

Type I & II Error Relationship § Type I and Type II errors can Type I & II Error Relationship § Type I and Type II errors can not happen at the same time § Type I error can only occur if H 0 is true § Type II error can only occur if H 0 is false If Type I error probability ( ) , then Type II error probability ( β ) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -14

Factors Affecting Type II Error n All else equal, n β when the difference Factors Affecting Type II Error n All else equal, n β when the difference between hypothesized parameter and its true value n β when σ n β when n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -15

Power of the Test n n The power of a test is the probability Power of the Test n n The power of a test is the probability of rejecting a null hypothesis that is false i. e. , n Power = P(Reject H 0 | H 1 is true) Power of the test increases as the sample size increases Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -16

Hypothesis Tests for the Mean Hypothesis Tests for Known Copyright © 2010 Pearson Education, Hypothesis Tests for the Mean Hypothesis Tests for Known Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Unknown Ch. 9 -17

9. 2 n Test of Hypothesis for the Mean (σ Known) Convert sample result 9. 2 n Test of Hypothesis for the Mean (σ Known) Convert sample result ( ) to a z value Hypothesis Tests for σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -18

Decision Rule H 0: μ = μ 0 H 1: μ > μ 0 Decision Rule H 0: μ = μ 0 H 1: μ > μ 0 Alternate rule: Z Do not reject H 0 0 zα Reject H 0 μ 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Critical value Ch. 9 -19

p-Value Approach to Testing n p-value: Probability of obtaining a test statistic more extreme p-Value Approach to Testing n p-value: Probability of obtaining a test statistic more extreme ( ≤ or ) than the observed sample value given H 0 is true n n Also called observed level of significance Smallest value of for which H 0 can be rejected Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -20

p-Value Approach to Testing (continued) n n n Convert sample result (e. g. , p-Value Approach to Testing (continued) n n n Convert sample result (e. g. , statistic ) ) to test statistic (e. g. , z Obtain the p-value n For an upper tail test: Decision rule: compare the p-value to n If p-value < , reject H 0 n If p-value , do not reject H 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -21

Example: Upper-Tail Z Test for Mean ( Known) A phone industry manager thinks that Example: Upper-Tail Z Test for Mean ( Known) A phone industry manager thinks that customer monthly cell phone bill have increased, and now average over $52 per month. The company wishes to test this claim. (Assume = 10 is known) Form hypothesis test: H 0: μ ≤ 52 the average is not over $52 per month H 1: μ > 52 the average is greater than $52 per month (i. e. , sufficient evidence exists to support the manager’s claim) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -22

Example: Find Rejection Region (continued) n Suppose that =. 10 is chosen for this Example: Find Rejection Region (continued) n Suppose that =. 10 is chosen for this test Find the rejection region: Reject H 0 =. 10 Do not reject H 0 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 1. 28 Reject H 0 Ch. 9 -23

Example: Sample Results (continued) Obtain sample and compute the test statistic Suppose a sample Example: Sample Results (continued) Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 64, x = 53. 1 ( = 10 was assumed known) n Using the sample results, Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -24

Example: Decision (continued) Reach a decision and interpret the result: Reject H 0 =. Example: Decision (continued) Reach a decision and interpret the result: Reject H 0 =. 10 Do not reject H 0 1. 28 0 z = 0. 88 Reject H 0 Do not reject H 0 since z = 0. 88 < 1. 28 i. e. : there is not sufficient evidence that the mean bill is over $52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -25

Example: p-Value Solution Calculate the p-value and compare to (continued) (assuming that μ = Example: p-Value Solution Calculate the p-value and compare to (continued) (assuming that μ = 52. 0) p-value =. 1894 Reject H 0 =. 10 0 Do not reject H 0 1. 28 Z =. 88 Reject H 0 Do not reject H 0 since p-value =. 1894 > =. 10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -26

One-Tail Tests n In many cases, the alternative hypothesis focuses on one particular direction One-Tail Tests n In many cases, the alternative hypothesis focuses on one particular direction H 0: μ ≤ 3 H 1: μ > 3 H 0: μ ≥ 3 H 1: μ < 3 This is an upper-tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 This is a lower-tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -27

Upper-Tail Tests n There is only one critical value, since the rejection area is Upper-Tail Tests n There is only one critical value, since the rejection area is in only one tail H 0: μ ≤ 3 H 1: μ > 3 Do not reject H 0 Z 0 zα Reject H 0 μ Critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -28

Lower-Tail Tests n There is only one critical value, since the rejection area is Lower-Tail Tests n There is only one critical value, since the rejection area is in only one tail H 0: μ ≥ 3 H 1: μ < 3 Reject H 0 -z Do not reject H 0 0 Z μ Critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -29

Two-Tail Tests n n In some settings, the alternative hypothesis does not specify a Two-Tail Tests n n In some settings, the alternative hypothesis does not specify a unique direction There are two critical values, defining the two regions of rejection H 0: μ = 3 H 1: μ ¹ 3 /2 x 3 Reject H 0 Do not reject H 0 -z /2 Lower critical value Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 0 Reject H 0 +z /2 z Upper critical value Ch. 9 -30

Hypothesis Testing Example Test the claim that the true mean # of TV sets Hypothesis Testing Example Test the claim that the true mean # of TV sets in US homes is equal to 3. (Assume σ = 0. 8) n n n State the appropriate null and alternative hypotheses n H 0: μ = 3 , H 1: μ ≠ 3 (This is a two tailed test) Specify the desired level of significance n Suppose that =. 05 is chosen for this test Choose a sample size n Suppose a sample of size n = 100 is selected Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -31

Hypothesis Testing Example (continued) n n n Determine the appropriate technique n σ is Hypothesis Testing Example (continued) n n n Determine the appropriate technique n σ is known so this is a z test Set up the critical values n For =. 05 the critical z values are ± 1. 96 Collect the data and compute the test statistic n Suppose the sample results are n = 100, x = 2. 84 (σ = 0. 8 is assumed known) So the test statistic is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -32

Hypothesis Testing Example (continued) n Is the test statistic in the rejection region? Reject Hypothesis Testing Example (continued) n Is the test statistic in the rejection region? Reject H 0 if z < -1. 96 or z > 1. 96; otherwise do not reject H 0 =. 05/2 Reject H 0 -z = -1. 96 =. 05/2 Do not reject H 0 0 Reject H 0 +z = +1. 96 Here, z = -2. 0 < -1. 96, so the test statistic is in the rejection region Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -33

Hypothesis Testing Example (continued) n Reach a decision and interpret the result =. 05/2 Hypothesis Testing Example (continued) n Reach a decision and interpret the result =. 05/2 Reject H 0 -z = -1. 96 =. 05/2 Do not reject H 0 0 Reject H 0 +z = +1. 96 -2. 0 Since z = -2. 0 < -1. 96, we reject the null hypothesis and conclude that there is sufficient evidence that the mean number of TVs in US homes is not equal to 3 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -34

Example: p-Value n Example: How likely is it to see a sample mean of Example: p-Value n Example: How likely is it to see a sample mean of 2. 84 (or something further from the mean, in either direction) if the true mean is = 3. 0? x = 2. 84 is translated to a z score of z = -2. 0 /2 =. 025 . 0228 p-value =. 0228 +. 0228 =. 0456 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall -1. 96 -2. 0 0 1. 96 2. 0 Z Ch. 9 -35

Example: p-Value n (continued) Compare the p-value with n If p-value < , reject Example: p-Value n (continued) Compare the p-value with n If p-value < , reject H 0 n If p-value , do not reject H 0 Here: p-value =. 0456 =. 05 Since. 0456 <. 05, we reject the null hypothesis /2 =. 025 . 0228 -1. 96 -2. 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 0 1. 96 2. 0 Z Ch. 9 -36

9. 3 n t Test of Hypothesis for the Mean (σ Unknown) Convert sample 9. 3 n t Test of Hypothesis for the Mean (σ Unknown) Convert sample result ( ) to a t test statistic Hypothesis Tests for σ Known σ Unknown Consider the test The decision rule is: (Assume the population is normal) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -37

t Test of Hypothesis for the Mean (σ Unknown) (continued) n For a two-tailed t Test of Hypothesis for the Mean (σ Unknown) (continued) n For a two-tailed test: Consider the test (Assume the population is normal, and the population variance is unknown) The decision rule is: Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -38

Example: Two-Tail Test ( Unknown) The average cost of a hotel room in Chicago Example: Two-Tail Test ( Unknown) The average cost of a hotel room in Chicago is said to be $168 per night. A random sample of 25 hotels resulted in x = $172. 50 and s = $15. 40. Test at the = 0. 05 level. H 0: μ = 168 H 1: μ ¹ 168 (Assume the population distribution is normal) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -39

Example Solution: Two-Tail Test H 0: μ = 168 H 1: μ ¹ 168 Example Solution: Two-Tail Test H 0: μ = 168 H 1: μ ¹ 168 n = 0. 05 n n = 25 n is unknown, so use a t statistic n /2=. 025 Critical Value: t 24 , . 025 = ± 2. 0639 Reject H 0 -t n-1, α/2 -2. 0639 Do not reject H 0 0 1. 46 Reject H 0 t n-1, α/2 2. 0639 Do not reject H 0: not sufficient evidence that true mean cost is different than $168 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -40

9. 4 Tests of the Population Proportion n Involves categorical variables n Two possible 9. 4 Tests of the Population Proportion n Involves categorical variables n Two possible outcomes n n “Success” (a certain characteristic is present) “Failure” (the characteristic is not present) Fraction or proportion of the population in the “success” category is denoted by P Assume sample size is large Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -41

Proportions (continued) n Sample proportion in the success category is denoted by n n Proportions (continued) n Sample proportion in the success category is denoted by n n When n. P(1 – P) > 5, can be approximated by a normal distribution with mean and standard deviation n Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -42

Hypothesis Tests for Proportions n The sampling distribution of is Hypothesis approximately Tests for Hypothesis Tests for Proportions n The sampling distribution of is Hypothesis approximately Tests for P normal, so the test statistic is a z n. P(1 – P) < 5 n. P(1 – P) > 5 value: Not discussed in this chapter Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -43

Example: Z Test for Proportion A marketing company claims that it receives 8% responses Example: Z Test for Proportion A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 25 responses. Test at the =. 05 significance level. Check: Our approximation for P is = 25/500 =. 05 n. P(1 - P) = (500)(. 05)(. 95) = 23. 75 > 5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -44

Z Test for Proportion: Solution Test Statistic: H 0: P =. 08 H 1: Z Test for Proportion: Solution Test Statistic: H 0: P =. 08 H 1: P ¹. 08 =. 05 n = 500, =. 05 Decision: Critical Values: ± 1. 96 Reject . 025 -1. 96 0 1. 96 z -2. 47 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Reject H 0 at =. 05 Conclusion: There is sufficient evidence to reject the company’s claim of 8% response rate. Ch. 9 -45

p-Value Solution (continued) Calculate the p-value and compare to (For a two sided test p-Value Solution (continued) Calculate the p-value and compare to (For a two sided test the p-value is always two sided) Do not reject H 0 Reject H 0 /2 =. 025 p-value =. 0136: /2 =. 025 . 0068 -1. 96 0 1. 96 Z = -2. 47 Z = 2. 47 Reject H 0 since p-value =. 0136 < =. 05 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -46

9. 5 n Power of the Test Recall the possible hypothesis test outcomes: Actual 9. 5 n Power of the Test Recall the possible hypothesis test outcomes: Actual Situation n n H 0 True H 0 False Do Not Reject H 0 No error (1 - ) Type II Error (β) Reject H 0 Key: Outcome (Probability) Decision Type I Error ( ) No Error (1 -β) β denotes the probability of Type II Error 1 – β is defined as the power of the test Power = 1 – β = the probability that a false null hypothesis is rejected Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -47

Type II Error Assume the population is normal and the population variance is known. Type II Error Assume the population is normal and the population variance is known. Consider the test The decision rule is: or If the null hypothesis is false and the true mean is μ*, then the probability of type II error is Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -48

Type II Error Example n Type II error is the probability of failing to Type II Error Example n Type II error is the probability of failing to reject a false H 0 Suppose we fail to reject H 0: μ 52 when in fact the true mean is μ* = 50 Reject H 0: μ 52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 52 Do not reject H 0 : μ 52 Ch. 9 -49

Type II Error Example (continued) n Suppose we do not reject H 0: μ Type II Error Example (continued) n Suppose we do not reject H 0: μ 52 when in fact the true mean is μ* = 50 This is the range of x where H 0 is not rejected This is the true distribution of x if μ = 50 50 52 Reject H 0: μ 52 Do not reject H 0 : μ 52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -50

Type II Error Example (continued) n Suppose we do not reject H 0: μ Type II Error Example (continued) n Suppose we do not reject H 0: μ 52 when in fact the true mean is μ* = 50 Here, β = P( x ) if μ* = 50 β 50 52 Reject H 0: μ 52 Do not reject H 0 : μ 52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -51

Calculating β n Suppose n = 64 , σ = 6 , and =. Calculating β n Suppose n = 64 , σ = 6 , and =. 05 (for H 0 : μ 52) So β = P( x 50. 766 ) if μ* = 50 50. 766 Reject H 0: μ 52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 52 Do not reject H 0 : μ 52 Ch. 9 -52

Calculating β (continued) n Suppose n = 64 , σ = 6 , and Calculating β (continued) n Suppose n = 64 , σ = 6 , and =. 05 Probability of type II error: β =. 1539 50 52 Reject H 0: μ 52 Do not reject H 0 : μ 52 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -53

Power of the Test Example If the true mean is μ* = 50, n Power of the Test Example If the true mean is μ* = 50, n The probability of Type II Error = β = 0. 1539 n The power of the test = 1 – β = 1 – 0. 1539 = 0. 8461 Actual Situation Key: Outcome (Probability) Decision H 0 True Do Not Reject H 0 No error 1 - = 0. 95 Reject H 0 Type I Error = 0. 05 H 0 False Type II Error β = 0. 1539 No Error 1 - β = 0. 8461 (The value of β and the power will be different for each μ*) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -54

Hypothesis Tests of one Population Variance 9. 6 § Goal: Test hypotheses about the Hypothesis Tests of one Population Variance 9. 6 § Goal: Test hypotheses about the population variance, σ2 § If the population is normally distributed, has a chi-square distribution with (n – 1) degrees of freedom Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 11 -55

Hypothesis Tests of one Population Variance (continued) The test statistic for hypothesis tests about Hypothesis Tests of one Population Variance (continued) The test statistic for hypothesis tests about one population variance is Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 11 -56

Decision Rules: Variance Population variance Lower-tail test: Upper-tail test: Two-tail test: H 0: σ Decision Rules: Variance Population variance Lower-tail test: Upper-tail test: Two-tail test: H 0: σ 2 σ 02 H 1: σ 2 < σ 02 H 0: σ 2 ≤ σ 02 H 1: σ 2 > σ 02 H 0: σ 2 = σ 02 H 1: σ 2 ≠ σ 02 Reject H 0 if /2 Reject H 0 if or Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 11 -57

Chapter Summary n Addressed hypothesis testing methodology n Performed Z Test for the mean Chapter Summary n Addressed hypothesis testing methodology n Performed Z Test for the mean (σ known) n Discussed critical value and p-value approaches to hypothesis testing n Performed one-tail and two-tail tests n Performed t test for the mean (σ unknown) n Performed Z test for the proportion n Discussed type II error and power of the test n Performed a hypothesis test for the variance (χ2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 9 -58