Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-1 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 9-2 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 9-3 The Hypothesis  A hypothesis is a claim (assumption) about a population parameter:  population mean  population proportion Example: The mean monthly cell phone bill of this city is μ = $52 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 9-4  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.  Is always about a population parameter, not about a sample statistic.  May or may not be rejected The Null Hypothesis, H 0

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-5 The Alternative Hypothesis, H 1  Is the opposite of the null hypothesis  e.g., The mean 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 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 9-6 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 9-7 The Hypothesis Testing Process  Claim: The population mean age is 50.  H 0 : μ = 50, H 1 : μ ≠ 50  Sample the population and find sample mean. Population Sample

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-8 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 9-9 The Hypothesis Testing Process Sampling Distribution of X μ = 50 If H 0 is true If it is unlikely that you would get a sample mean of this value then you reject the null hypothesis that μ = if in fact this were the population mean… X

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-10 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 H 0 ?  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 9-11 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 9-12 The Test Statistic and Critical Values Critical Values Distribution of the test statistic Region of Rejection

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-13 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 9-14 Errors in Decision Making Possible Jury Trial Outcomes The Truth The VerdictInnocentGuilty InnocentNo ErrorType II Error GuiltyType I ErrorNo Error

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-15 Errors in Decision Making Possible Hypothesis Test Outcomes Actual Situation DecisionH 0 TrueH 0 False Do Not Reject H 0 No Error Probability 1 - α Type II Error Probability β Reject H 0 Type I Error Probability α No Error Probability 1 - β

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-16 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 )

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-17 Level of Significance, α H 0 : μ ≥ 50 H 1 : μ < 50 0 H 0 : μ ≤ 50 H 1 : μ > 50   Represents critical value Lower-tail test 0 Upper-tail test Two-tail test Rejection region is shaded 0 H 0 : μ = 50 H 1 : μ ≠ 50 Claim: The population mean age is 50. 

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap Steps in Hypothesis Testing 1. State the null hypothesis, H 0 State the alternative hypotheses, H 1 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 H 0 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 9-19 Hypothesis Tests for the Mean  Known Z Test Normal Distribution  Unknown t Test Student t Distribution Hypothesis Tests for 

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-20 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 9-21 Hypothesis Testing: σ Unknown  The t test statistic with n-1 degrees of freedom is: Hypothesized

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-22 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 = $ and S = Test at the  = 0.05 level.

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-23 Hypothesis Testing Problem 8 steps  1. State the appropriate null and alternative hypotheses H 0 : μ = 168 H 1 : μ ≠ 168 (This is a two tailed test)  2. Specify the desired level of significance  =.05 is chosen for this test  3. Choose a sample size sample of size n = 25 was selected  4. Determine the appropriate Test σ is unknown so this is a t test

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-24  5. Collect the data The sample results are n = 25, X= $ S = $15.40  6. So the test statistic is: The p value for n=25,  =.05, t= is.1570 Hypothesis Testing Problem (continued)

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-25  7. Is the test statistic in the rejection region? Reject H 0 if p is < alpha; otherwise do not reject H 0 Hypothesis Testing Problem (continued) 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 9-26  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 Hypothesis Testing Problem (continued) 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 9-27 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 9-28 One Tail Tests  In many cases, the alternative hypothesis focuses on a particular direction H 0 : μ ≥ 3 H 1 : μ < 3 H 0 : μ ≤ 3 H 1 : 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 9-29 Hypothesis Testing: Connection to Confidence Intervals  For X = 172.5, S = and n = 25, the 95% confidence interval is: ≤ μ ≤  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 9-30 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 9-31 Hypothesis Testing Proportions  Sample proportion in the success category is denoted by p  When both nπ and n(1-π) are at least 5, p can be approximated by a normal distribution with mean and standard deviation

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-32 Hypothesis Testing Proportions  The sampling distribution of p is approximately normal, so the test statistic is a Z value:

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-33 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 9-34 Hypothesis Testing Proportions Example H 0 : π =.08 H 1 : π ≠.08 α =.05 n = 500, p = 30/500 =.06 p-value for is.0497

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-35 Hypothesis Testing Proportions Example Reject H 0 at  =.05 Test Statistic: Decision: Conclusion:There is sufficient evidence to reject the company’s claim of 8% response rate. p-value for is.0497

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-36 Using PHStat Options

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-37 Sample PHStat Output Input Output

Statistics for Managers Using Microsoft Excel, 5e © 2008 Pearson Prentice-Hall, Inc.Chap 9-38 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 9-39 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 9-40 Answer Sheet for All Problems  ___________ Null Hypothesis  ___________ Alternate Hypothesis  ___________ Alpha  ___________ p-value  ___________ Decision (reject or do not reject)  Conclusion: