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Statistics for Managers Using Microsoft® Excel 5th Edition Chapter 9 Fundamentals of Hypothesis Testing: One Sample Tests Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-1 Learning Objectives In this chapter, you will learn: The basic principles of hypothesis testing How to use hypothesis testing to test a mean or proportion The assumptions of each hypothesis-testing procedure, how to evaluate them, and the consequences if they are seriously violated How to avoid the pitfalls involved in hypothesis testing Ethical issues involved in hypothesis testing Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-2 The Hypothesis A hypothesis is a claim (assumption) about a population parameter: population mean Example: The mean monthly cell phone bill of this city is μ = $52 population proportion Example: The proportion of adults in this city with cell phones is π = .68 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-3 The Null Hypothesis, H0 States the assumption (numerical) to be tested Always contains “=” , “≤” or “” sign Example: The mean number of TV sets in U.S. Homes is equal to three. H 0 : μ 3 Is always about a population parameter, not about a sample statistic. H0 : μ 3 H0 : X 3 May or may not be rejected Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-4 The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g., The mean number of TV sets in U.S. homes is not equal to 3 ( H1: μ ≠ 3 ) Challenges the status quo Never contains the “=” , “≤” or “” sign May or may not be proven For one tail tests is generally the hypothesis that the researcher is trying to prove Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-5 Hypothesis Testing We assume the null hypothesis is true If the null hypothesis is rejected we have proven the alternate hypothesis If the null hypothesis is not rejected we have proven nothing as the sample size may have been to small Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-6 The Hypothesis Testing Process Claim: The population mean age is 50. H0: μ = 50, H1: μ ≠ 50 Sample the population and find sample mean. Population Sample Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-7 The Hypothesis Testing Process Suppose the sample mean age was X = 20. This is significantly lower than the claimed mean population age of 50. If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you reject the null hypothesis . In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-8 The Hypothesis Testing Process Sampling Distribution of X 20 If it is unlikely that you would get a sample mean of this value ... μ = 50 If H0 is true ... if in fact this were the population mean… Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. X ... then you reject the null hypothesis that μ = 50. Chap 9-9 The Test Statistic and Critical Values If the sample mean is close to the assumed population mean, the null hypothesis is not rejected. If the sample mean is far from the assumed population mean, the null hypothesis is rejected. How far is “far enough” to reject H0? The level of significance of the test statistic () creates a “line in the sand” for decision making. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-10 Level of Significance, Defines the unlikely values of the sample statistic if the null hypothesis is true Defines rejection region of the sampling distribution Is designated by , (level of significance) Typical values are .01, .05, or .10 Is the compliment of the confidence coefficient Is selected by the researcher before sampling Provides the critical value of the test Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-11 The Test Statistic and Critical Values Distribution of the test statistic Region of Rejection Region of Rejection /2 /2 /2 Critical Values Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-12 Errors in Decision Making Type I Error Reject a true null hypothesis Considered a serious type of error The probability of a Type I Error is Called level of significance of the test Set by researcher in advance Type II Error Failure to reject false null hypothesis The probability of a Type II Error is β Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-13 Errors in Decision Making Possible Jury Trial Outcomes The Truth The Verdict Innocent Guilty Innocent No Error Type II Error Guilty Type I Error No Error Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-14 Errors in Decision Making Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Do Not Reject H0 No Error Probability 1 - α Type II Error Probability β Reject H0 Type I Error Probability α No Error Probability 1 - β Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-15 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 H0 is true Type II error can only occur if H0 is false If Type I error probability ( ) , then Type II error probability ( β ) Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-16 Level of Significance, α Claim: The population mean age is 50. H0: μ = 50 H1: μ ≠ 50 H0: μ ≤ 50 H1: μ > 50 /2 Represents critical value /2 Two-tail test 0 Rejection region is shaded Upper-tail test 0 H0: μ ≥ 50 H1: μ < 50 Lower-tail test 0 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-17 8 Steps in Hypothesis Testing 1. State the null hypothesis, H0 State the alternative hypotheses, H1 2. Choose the level of significance, α 3. Choose the sample size, n 4. Determine the appropriate test statistic to use 5. Collect the data 6. Compute the p-value for the test statistic from the sample result 7. Make the statistical decision: Reject H0 if the p-value is less than alpha 8. Express the conclusion in the context of the problem Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-18 Hypothesis Tests for the Mean Hypothesis Tests for Known Z Test Normal Distribution Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Unknown t Test Student t Distribution Chap 9-19 Hypothesis Testing: σ Unknown If the population standard deviation is unknown, you use the sample standard deviation S instead of . Because of this change, you also use the t distribution instead of the Z distribution to test the null hypothesis about the mean. All other steps, concepts, and conclusions are the same as the known test. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-20 Hypothesis Testing: σ Unknown The t test statistic with n-1 degrees of freedom is: t n -1 Xμ Hypothesized S n Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-21 Hypothesis Testing: σ Unknown Problem The mean cost of a hotel room in New York 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. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-22 Hypothesis Testing Problem 8 steps 1. State the appropriate null and alternative hypotheses H0: μ = 168 H1: μ ≠ 168 (This is a two tailed test) 2. Specify the desired level of significance = .05 is chosen for this test Choose a sample size sample of size n = 25 was selected Determine the appropriate Test σ is unknown so this is a t test 3. 4. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-23 Hypothesis Testing Problem (continued) 5. Collect the data The sample results are n = 25, 6. X= $172.50 S = $15.40 So the test statistic is: t X μ S n 172.50 168 15.40 4.50 1.4610 3.08 25 The p value for n=25, =.05, t=1.4610 is .1570 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-24 Hypothesis Testing Problem (continued) 7. Is the test statistic in the rejection region? Reject H0 if p is < alpha; otherwise do not reject H0 The p-value .1570 is not < alpha .05, we do not reject the null hypothesis Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-25 Hypothesis Testing Problem (continued) 8. Express the conclusion in the context of the problem Since The p-value .1570 is > alpha .05, we have failed to reject the null hypothesis Thereby not proving the alternate hypothesis Conclusion: There is not sufficient evidence to reject the claim that the mean cost of a hotel room in NYC is $168 If we had rejected the null hypothesis the conclusion would have been: There is sufficient evidence to reject the claim that the mean cost of a hotel room in NYC is $168 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-26 Hypothesis Testing: σ Unknown Recall that you assume that the sample statistic comes from a random sample from a normal distribution. If the sample size is small (< 30), you should use a box-and-whisker plot or a normal probability plot to assess whether the assumption of normality is valid. If the sample size is large, the central limit theorem applies and the sampling distribution of the mean will be normal. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-27 One Tail Tests In many cases, the alternative hypothesis focuses on a particular direction H0: μ ≥ 3 H1: μ < 3 H0: μ ≤ 3 H1: μ > 3 This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-28 Hypothesis Testing: Connection to Confidence Intervals For X = 172.5, S = 15.40 and n = 25, the 95% confidence interval is: 172.5 - (2.0639) 15.4 to 172.5 (2.0639) 25 15.4 25 166.14 ≤ μ ≤ 178.86 Since this interval contains the hypothesized mean (168), you do not reject the null hypothesis at = .05 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-29 Hypothesis Testing Proportions Involves categorical variables Two possible outcomes “Success” (possesses a certain characteristic) “Failure” (does not possesses that characteristic) Fraction or proportion of the population in the “success” category is denoted by π Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-30 Hypothesis Testing Proportions Sample proportion in the success category is denoted by p p X n number of successes in sample sample size When both nπ and n(1-π) are at least 5, p can be approximated by a normal distribution with mean and standard deviation (1 ) σp μp n Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-31 Hypothesis Testing Proportions The sampling distribution of p is approximately normal, so the test statistic is a Z value: Z p (1 ) n Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-32 Hypothesis Testing Proportions Example A marketing company claims that it receives 8% responses from its mailing. To test this claim, a random sample of 500 were surveyed with 30 responses. Test at the = .05 significance level. First, check: n π = (500)(.08) = 40 n(1-π) = (500)(.92) = 460 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-33 Hypothesis Testing Proportions Example H0: π = .08 H1: π ≠ .08 α = .05 n = 500, p = 30/500 =.06 Z p (1 ) n .06 .08 .08(1 .08) 1.648 500 p-value for -1.648 is .0497 Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-34 Hypothesis Testing Proportions Example Test Statistic: p-value for -1.648 is .0497 Decision: Reject H0 at = .05 Conclusion: There is sufficient evidence to reject the company’s claim of 8% response rate. Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-35 Using PHStat Options Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-36 Sample PHStat Output Input Output Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-37 Potential Pitfalls and Ethical Considerations Use randomly collected data to reduce selection biases Do not use human subjects without informed consent Choose the level of significance, α, before data collection Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance Do not practice “data cleansing” to hide observations that do not support a stated hypothesis Report all pertinent findings Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-38 Chapter Summary In this chapter, we have Addressed hypothesis testing methodology Mentioned Z Test for the mean (σ known) Discussed the p–value approaches to hypothesis testing Discussed one-tail and two-tail tests Performed t test for the mean (σ unknown) Performed Z test for the proportion Discussed pitfalls and ethical issues Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-39 Answer Sheet for All Problems ___________ Null Hypothesis ___________ Alternate Hypothesis ___________ Alpha ___________ p-value ___________ Decision (reject or do not reject) Conclusion: Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc. Chap 9-40