26134 Business Statistics Autumn 2017

1. 26134 Business Statistics Autumn 2017 Tutorial 7: Hypothesis Testing for a single population parameterbstats@uts.edu.auB MathFin (Hons)M Stat (UNSW)PhD (UTS)mahritaharahap.wordpress.com/teaching-areas business.uts.edu.au

2. REVISION

3. Central Limit Theorem • Under CLT: Sampling distribution of sample mean will be normally distributed if: (1) the original distribution is normally distributed (sample size is irrelevant). (2) the original distribution is NOT normally distributed but the sample size is >30. • Note that only ONE of these conditions need to be satisfied for this conclusion to be reached. • In the case of the sampling distribution of sample proportion, sample size must be greater than 5/p and 5/q where q=1-p.

4. Standard Error For proportion:

5. Confidence Intervals • Mean (if σ known) • Mean (if σ unknown)

6. Estimating Sample Sizes MEAN: PROPORTION: p*q

7. In statistics we usually want to statistically analyse a population but collecting data for the whole population is usually impractical, expensive and unavailable. That is why we collect samples from the population (sampling) and make inferences about the population parameters using the statistics of the sample (inferencing) with some level of accuracy (confidence level) using an inference method called Hypothesis Testing. Std Deviation Sample Size N n p Proportion A populationis a collection of all possible individuals, objects, or measurements of interest. A sample is a subset of the population of interest. A parameter is a number that describes some aspect of a population. A statistic is a number that is computed from data in a sample.

8. Hypothesis Testing Process Hypothesis tests give us an objective way of assessing such questions. They are based on a proof by contradiction form of argument. • Specify the population parameter of interest. µ or p?We formulate a null hypothesis (H0)We formulate an alternative hypothesis (H1) – determine if it is a 1-tailed or 2-tailed test. • State the Level Of Significance. • We calculate a Test Statistic. Measures the compatibility of the sample statistic obtained, with the H0, assuming it is true. • Compute the Critical Value, and compare it to the test statistic. • Make a decision rule. If the test statistic is out side of the critical value, we reject H0. If the test statistic is on the other side, we do not reject H0. State the conclusion in context sopeople who don’t understand statistics can still understand your conclusions. H O T C D

9. Hypotheses Statistical tests are framed formally in terms of two competing hypotheses H0: null hypothesis H1: alternative hypothesis which are two competing claims about a population.

10. The Null Hypothesis (H0) H0 is the claim that there is no effect or difference (the status quo), and is always an equality(or could be expressed as one). • E.g., The mean height of UTS students is 160cm H0: µ = 160 The null hypothesis is always about a population parameter (e.g. µ = 160), and never about a sample statistic (e.g. = 160).

11. The Alternative Hypothesis (H1) The alternative hypothesis is the claim that we are trying to find evidence for. • It should reflect our hopes or suspicions. H1 can be one sidedor two sided. • The mean is not equal to 160 (H1: µ ≠ 160) 2-tailed test • The mean is greater than 160 (H1: µ > 160) 1-right tailed test • The mean is less than 160 (H1: µ < 160) 1-left tailed test How do we decide what H1 should be? • It depends on the question that we are trying to answer. Are we looking for differences, or are we looking for larger/smaller values.

12. Level of Significance • Confidence Level (100−α)% - How confident do you want to be? • Level of significance α% - we are willing to accept a α % chance that we incorrectly reject the null hypothesis

13. Test Statistic • Measures the compatibility of the sample statistic obtained, with the H0, assuming it is true. • The further away the test statistic is from H0, the more evidence against H0. The closer the test statistic is from H0, the less evidence against H0. • If population standard deviation given in the question, we use Z test statistic, if population standard deviation unknown, we use t test statistic.

14. Critical Value • The critical value can be seen as the separator between the reject H0 and do not reject H0 regions. • i.e 2 sided test H1: µ ≠ 160 • i.e 1 sided test H1: µ < 160 • This allowed the establishment of where the non-rejection region ends and the rejection region begins.

15. Statistical significance When results as extreme as the observed sample statistic are unlikely to occur by chance alone(assuming the null hypothesis is true), we say the sample results are statistically significant. • If our test statistic is as extreme as the critical value, this means our sample is statistically significant, we have convincing evidence against Ho, in favour of H1. • If our sample is not statistically significant, our test is inconclusive(we don’t have convincing evidence of H1).

16. Conclusion Make the conclusion in context to the problem. • We can reject H0, and therefore conclude that we have enough statistical evidence against H0. • We cannot reject H0, and therefore conclude that the data is consistent with H0 or we do not have enough statistical evidence against H0. The conclusion is not `Accept' H0 or H1, as we have assumed H0 to be true. That is, we have not proved H0 true, rather we look for evidence about whether it is false.

17. Activity 1: Impact of Sample Sizes

18. Errors in Hypothesis Testing Since we are using a sample to infer information about a population, we are going to get it wrong sometimes! • We need to quantify this risk. Type II Error No Error P(fail to reject H0|H0 True) = 1-a P(fail to reject H0|H0 False) = b No Error Type I Error P(reject H0|H0 False) = 1 - b P(reject H0|H0 True) = a

19. Errors in Hypothesis Testing The probability of a type I error is denoted by  and is called the level of significance of the test. • Thus, a test with  = 0.01 is said to have a level of significance of 1%. • Typical values are 0.01, 0.05, 0.10 and default value is 0.05 The probability of a type II error is denoted by . The value 1 –  is called the power of the test. The general procedure is to specify a value for the type I error α, then design the test procedure so that the probability of type II error β, has a suitably small value.

20. Errors in Hypothesis Testing: Analogy UTS has fire alarms throughout the campus (it’s the law) • The alarms will go off if they detect smoke (which usually means that there is a fire). • If there is a fire, then we want to evacuate the building. The default situation is that there is no fire • Null hypothesis: There is no fire. • Alternative hypothesis: There is a fire. We can set this problem up in terms of a hypothesis test.

21. Errors in Hypothesis Testing: Analogy No Error Type II Error Type I Error No Error

22. Activity 2: Type I Error

23. Good Luck!