1. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-1 Chapter 9 Fundamentals of Hypothesis Testing: One-Sample Tests Business Statistics, A First Course 4 th Edition

2. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-2 Learning Objectives In this chapter, you 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  The ethical issues involved in hypothesis testing

3. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-3 What is a 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 = $42 Example: The proportion of adults in this city with cell phones is π = 0.68

4. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-4 The Null Hypothesis, H 0  States the claim or assertion to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( )  Is always about a population parameter, never about a sample statistic

5. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-5 The Null Hypothesis, H 0  Begin with the assumption that the null hypothesis is true  Similar to the notion of innocent until proven guilty  Refers to the status quo  Always contains “=”, “≤” or “  ” sign  May or may not be rejected (continued) Think about this: If we cannot reject the null, can we say the null is true? If not, what can we say?

6. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-6 The Alternative Hypothesis, H 1  Is the opposite of the null hypothesis  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 proven  Is generally the hypothesis that the researcher is trying to prove * Insight *: Rejecting the null simply provides evidence that the null is not true logically leaving only the alternative hypothesis. It is never “proof” only evidence base don the data given. Think about this: What if you wanted evidence in favor of the null hypothesis? How might you do this?

7. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Population Claim: the population mean age is 50. (Null Hypothesis: REJECT Suppose the sample mean age is 20: X = 20 Sample Null Hypothesis 20 likely if μ = 50?  Is Hypothesis Testing Process If not likely, Now select a random sample H 0 : μ = 50 ) X

8. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-8 Sampling Distribution of X μ = 50 If H 0 is true If it is unlikely that we would get a sample mean of this value...... then we reject the null hypothesis that μ = 50. Reason for Rejecting H 0 20... if in fact this were the population mean… X

9. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-9 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 0.01, 0.05, or 0.10  Is selected by the researcher at the beginning  1-  is the required confidence needed to conclude the alternative hypothesis H 1 * Insight * All we are doing is seeing if the parameter is inside the 1-  confidence interval for the point estimate. If inside, then H0, if outside then H1.

10. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-10 Hypothesis Tests for the Mean  Unknown Hypothesis Tests for  (t test) Think about this: Why do we assume  is unknown? Hint: think about CI’s for the unknown mean.

11. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-11 p-Value Approach to Testing  p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H 0 is true  Also called observed level of significance  We will compare this to our “required” level of significance   We can state our confidence in H 1 as 1- (p-value).  We can also compare this to our “required “ confidence level

12. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-12 p-Value Approach to Testing  Convert Sample Statistic (e.g., ) to Test Statistic (e.g., Z statistic )  Obtain the p-value from a table or computer  Compare the observed confidence (1-p) with the required confidence (1-  )  If confidence > 1- , conclude H 1  If confidence < 1- , H 0 is acceptable X (continued)

13. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-13 0.0228  /2 = 0.025 p-Value Example  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? -1.960 -2.0 Z1.96 2.0 X = 2.84 is translated to a Z score of Z = -2.0 p-value = 0.0228 + 0.0228 = 0.0456 0.0228  /2 = 0.025

14. Chap 9-14  Compare the p-value with , or the observed confidence (1-p) with the required confidence (1-  ). Easy way to remember: “If p is low the null has to go” Here: p-value = 0.0456  = 0.05 Since 95.44% > 95%, we reject the null hypothesis (continued) p-Value Example 0.0228  /2 = 0.025 -1.960 -2.0 Z1.96 2.0 0.0228  /2 = 0.025 Think about this: Can you see how this is exactly the same as looking at the confidence interval?

15. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-15 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 * insight * You can always determine what test based on the comparator in H1. “less-than” means “lower”

16. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-16 Example: Two-Tail Test (  Unknown) The average 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. (Assume the population distribution is normal) H 0 : μ  = 168 H 1 : μ  168

17. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-17   = 0.05  n = 25   is unknown, so use a PHStat t test sigma unknown Example Solution: Two-Tail Test Do not conclude H 1 : not sufficient evidence that true mean cost is different than $168. We can only be 84.3% confident that the mean differs from 168 and we needed to be 95%.  /2=.025 H 0 : μ  = 168 H 1 : μ  168 P-Value =.157

18. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. 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/ 25 to 172.5 + (2.0639) 15.4/ 25 166.14 ≤ μ ≤ 178.86  Since this interval contains the Hypothesized mean (168), we do not reject the null hypothesis at  = 0.05

19. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-19 Hypothesis Tests for 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

20. Chap 9-20 Proportions  Sample proportion in the success category is denoted by p   RoT: When both n π and n(1- π ) are at least 5, p can be approximated by a normal distribution with mean and standard deviation  (continued) * Insight * We will use this like the normality assumption for t-tests

21. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-21 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  = 0.05 significance level. Check: n π = (500)(.08) = 40 n(1- π ) = (500)(.92) = 460

22. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-22 Calculate the p-value and compare to  (For a two-tail test the p-value is always two-tail) (continued) p-Value Solution Conclude H 1 since p-value = 0.0136 <  = 0.05. Alternatively we can be 98.64% confident that the population proportion differs from.08. Use PHStat Z test for a Proportion The P-value from PHStat is.0136

23. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-23 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, and the type of test (one-tail or two-tail) 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

24. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-24 Chapter Summary  Addressed hypothesis testing methodology  Discussed p–value approach to hypothesis testing  Performed one-tail and two-tail tests

25. Business Statistics, A First Course (4e) © 2006 Prentice-Hall, Inc. Chap 9-25 Chapter Summary  Performed t test for the mean (σ unknown)  Performed Z test for the proportion  Discussed pitfalls and ethical issues (continued)