Week 8 Fundamentals of Hypothesis Testing: One-Sample Tests Statistics Week 8 Fundamentals of Hypothesis Testing: One-Sample Tests

Goals of this note After completing this noe, you should be able to: Formulate null and alternative hypotheses for applications involving a single population mean or proportion Formulate a decision rule for testing a hypothesis Know how to use the p-value approaches to test the null hypothesis for both mean and proportion problems Know what Type I and Type II errors are

What is a Hypothesis? A hypothesis is a claim (assumption) about a population parameter: population mean population proportion The average number of TV sets in U.S. homes is equal to three ( ) A marketing company claims that it receives 8% responses from its mailing. ( p=.08 )

The Null Hypothesis, H0 States the assumption to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) Is always about a population parameter, not about a sample statistic

The Null Hypothesis, H0 (continued) Begins 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

The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g.: The average 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 Is generally the hypothesis that is believed (or needs to be supported) by the researcher

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

Hypothesis Testing Process Claim: the population mean age is 50. (Null Hypothesis: Population H0: μ = 50 ) Now select a random sample X Is = 20 likely if μ = 50? Suppose the sample If not likely, REJECT mean age is 20: X = 20 Sample Null Hypothesis

Sampling Distribution of H0: μ = 50 H1: μ ¹ 50 There are two cutoff values (critical values), defining the regions of rejection /2 /2 X 50 Reject H0 Do not reject H0 Reject H0 20 Likely Sample Results Lower critical value Upper critical value

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

Level of Significance and the Rejection Region Represents critical value a a H0: μ = 3 H1: μ ≠ 3 /2 /2 Rejection region is shaded Two tailed test H0: μ ≤ 3 H1: μ > 3 a Upper tail test H0: μ ≥ 3 H1: μ < 3 a Lower tail test

Errors in Making Decisions Type I Error When a true null hypothesis is rejected 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 a false null hypothesis The probability of a Type II Error is β

Possible Jury Trial Outcomes Example Possible Jury Trial Outcomes The Truth Verdict Innocent Guilty Innocent No error Type II Error Guilty Type I Error No Error

Outcomes and Probabilities Possible Hypothesis Test Outcomes Actual Situation Decision H0 True H0 False Key: Outcome (Probability) Do Not No error (1 - ) Type II Error ( β ) Reject a H Reject Type I Error ( ) No Error ( 1 - β ) H a

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 )

p-Value Approach to Testing p-value: Probability of obtaining a test statistic more extreme ( ≤ or  ) than the observed sample value given H0 is true Also called observed level of significance

p-Value Approach to Testing (continued) Convert Sample Statistic (e.g. ) to Test Statistic (e.g. t statistic ) Obtain the p-value from a table or computer Compare the p-value with  If p-value <  , reject H0 If p-value   , do not reject H0 X

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

Hypothesis Tests for the Mean  Known  Unknown

Hypothesis Testing Example Test the claim that the true mean # of TV sets in U.S. homes is equal to 3. 1. State the appropriate null and alternative hypotheses H0: μ = 3 H1: μ ≠ 3 (This is a two tailed test) 2. Specify the desired level of significance Suppose that  = .05 is chosen for this test 3. Choose a sample size Suppose a sample of size n = 100 is selected

Hypothesis Testing Example (continued) 4. Determine the appropriate Test σ is unknown so this is a t test 5. Collect the data Suppose the sample results are n = 100, = 2.84 s = 0.8 6. So the test statistic is: The p value for n=100, =.05, t=-2 is .048

Hypothesis Testing Example (continued) 7. Is the test statistic in the rejection region? Reject H0 if p is < alpha; otherwise do not reject H0 The p-value .048 is < alpha .05, we reject the null hypothesis

Hypothesis Testing Example (continued) 8. Express the conclusion in the context of the problem Since The p-value .048 is < alpha .05, we have rejected the null hypothesis Thereby proving the alternate hypothesis Conclusion: There is sufficient evidence that the mean number of TVs in U.S. homes is not equal to 3 If we had failed to reject the null hypothesis the conclusion would have been: There is not sufficient evidence to reject the claim that the mean number of TVs in U.S. home is 3

One Tail Tests In many cases, the alternative hypothesis focuses on a particular direction This is a lower tail test since the alternative hypothesis is focused on the lower tail below the mean of 3 H0: μ ≥ 3 H1: μ < 3 This is an upper tail test since the alternative hypothesis is focused on the upper tail above the mean of 3 H0: μ ≤ 3 H1: μ > 3

Lower Tail Tests a H0: μ ≥ 3 H1: μ < 3 There is only one critical value, since the rejection area is in only one tail a Reject H0 Do not reject H0 -t 3 Critical value

Upper Tail Tests H0: μ ≤ 3 H1: μ > 3 There is only one critical value, since the rejection area is in only one tail a Do not reject H0 Reject H0 t tα 3 Critical value

Assumptions of the One-Sample t Test The data is randomly selected The population is normally distributed or the sample size is over 30 and the population is not highly skewed

Hypothesis Tests for Proportions Involves categorical values 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 p

Proportions Sample proportion in the success category is denoted by ps (continued) Sample proportion in the success category is denoted by ps When both np and n(1-p) are at least 5, ps can be approximated by a normal distribution with mean and standard deviation

Hypothesis Tests for Proportions The sampling distribution of ps is approximately normal, so the test statistic is a Z value: Hypothesis Tests for p np  5 and n(1-p)  5 np < 5 or n(1-p) < 5 Not discussed in this chapter

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: n p = (500)(.08) = 40 n(1-p) = (500)(.92) = 460 

Z Test for Proportion: Solution Test Statistic: H0: p = .08 H1: p ¹ .08 a = .05 n = 500, ps = .05 Critical Values: ± 1.96 p-value for -2.47 is .0134 Decision: Reject H0 at  = .05 Reject Reject .025 .025 There is sufficient evidence to reject the company’s claim of 8% response rate. Conclusion: z -1.96 1.96 -2.47

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

Summary Addressed hypothesis testing methodology Discussed critical value and p–value approaches to hypothesis testing Discussed type 1 and Type2 errors Performed two tailed t test for the mean (σ unknown) Performed Z test for the proportion Discussed one-tail and two-tail tests Addressed pitfalls and ethical issues