1. Business Statistics for Managerial Decision Farideh Dehkordi-Vakil

2. Tests of Significance Confidence intervals are appropriate when our goal is to estimate a population parameter. The second type of inference is directed at assessing the evidence provided by the data in favor of some claim about the population. A significance test is a formal procedure for comparing observed data with a hypothesis whose truth we want to assess. The hypothesis is a statement about the parameters in a population or model. The results of a test are expressed in terms of a probability that measures how well the data and the hypothesis agree.

3. Example: Bank’s net income The community bank survey described in previous lecture also asked about net income and reported the percent change in net income between the first half of last year and the first half of this year. The mean change for the 110 banks in the sample is Because the sample size is large, we are willing to use the sample standard deviation s = 26.4% as if it were the population standard deviation . The large sample size also makes it reasonable to assume that is approximately normal.

4. Example: Bank’s net income Is the 8.1% mean increase in a sample good evidence that the net income for all banks has changed? The sample result might happen just by chance even if the true mean change for all banks is  = 0%. To answer this question we asks another Suppose that the truth about the population is that  = 0% (this is our hypothesis) What is the probability of observing a sample mean at least as far from zero as 8.1%?

5. Example: Bank’s net income The answer is: Because this probability is so small, we see that the sample mean is incompatible with a population mean of  = 0. We conclude that the income of community banks has changed since last year.

6. Example: Bank’s net income The fact that the calculated probability is very small leads us to conclude that the average percent change in income is not in fact zero. Here is why. If the true mean is  = 0, we would see a sample mean as far away as 8.1% only six times per 10000 samples. So there are only two possibilities:  = 0 and we have observed something very unusual, or  is not zero but has some other value that makes the observed data more probable

7. Example: Bank’s net income We calculated a probability taking the first of these choices as true (  = 0 ). That probability guides our final choice. If the probability is very small, the data don’t fit the first possibility and we conclude that the mean is not in fact zero.

8. Example:Is this percent change different from zero? Suppose that next year the percent change in net income for a sample of 110 banks is = 3.5%. (We assume that the standard deviation  = 26.4%.) This sample mean is closer to the value  = 0 corresponding to no mean change in income. What is the probability that the mean of a sample of size n = 110 from a normal population with  = 0 and standard deviation  = 26.4 is as far away or farther away from zero as ?

9. Example:Is this percent change different from zero? The answer is: A sample result this far from zero would happen just by chance in 8% of samples from a population having true mean zero. An outcome that could so easily happen by chance is not good evidence that the population mean is different from from zero.

10. Example:Is this percent change different from zero? The mean change in net assets for a sample of 110 banks will have this sampling distribution if the mean for the population of all banks is  = 0. A sample mean could easily happen by chance. A sample mean is far out on the curve that it would rarely happen just by chance.

11. Tests of Significance: Formal details The first step in a test of significance is to state a claim that we will try to find evidence against. Null Hypothesis H 0 The statement being tested in a test of significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference.” We abbreviate “null hypothesis” as H 0.

12. Tests of Significance: Formal details A null hypothesis is a statement about a population, expressed in terms of some parameter or parameters. The null hypothesis in our bank survey example is H 0 :  = 0 It is convenient also to give a name to the statement we hope or suspect is true instead of H 0. This is called the alternative hypothesis and is abbreviated as H a. In our bank survey example the alternative hypothesis states that the percent change in net income is not zero. We write this as H a :   0

13. Tests of Significance: Formal details Since H a expresses the effect that we hope to find evidence for we often begin with H a and then set up H 0 as the statement that the Hoped-for effect is not present. Stating H a is not always straight forward. It is not always clear whether H a should be one-sided or two-sided. The alternative H a :   0 in the bank net income example is two-sided. In any given year, income may increase or decrease, so we include both possibilities in the alternative hypothesis.

14. Example:Have we reduced processing time? Your company hopes to reduce the mean time  required to process customer orders. At present, this mean is 3.8 days. You study the process and eliminate some unnecessary steps. Did you succeed in decreasing the average process time? You hope to show that the mean is now less than 3.8 days, so the alternative hypothesis is one sided, H a :  < 3.8. The null hypothesis is as usual the “no change” value, H 0 :  = 3.8.

15. Tests of Significance: Formal details Test statistics We will learn the form of significance tests in a number of common situations. Here are some principles that apply to most tests and that help in understanding the form of tests: The test is based on a statistic that estimate the parameter appearing in the hypotheses. Values of the estimate far from the parameter value specified by H 0 gives evidence against H 0.

16. Tests of Significance: Formal details A test statistic measures compatibility between the null hypothesis and the data. Many test statistics can be thought of as a distance between a sample estimate of a parameter and the value of the parameter specified by the null hypothesis.

17. Example: bank’s income The hypotheses: H 0 :  = 0 H a :   0 The estimate of  is the sample mean. Because H a is two-sided, large positive and negative values of (large increases and decreases of net income in the sample) counts as evidence against the null hypothesis.

18. Example: bank’s income The test statistic The null hypothesis is H 0 :  = 0, and a sample gave the. The test statistic for this problem is the standardized version of : This statistic is the distance between the sample mean and the hypothesized population mean in the standard scale of z-scores.

19. Tests of Significance: Formal details The test of significance assesses the evidence against the null hypothesis and provides a numerical summary of this evidence in terms of probability. P-value The probability, computed assuming that H 0 is true, that the test statistic would take a value extreme or more extreme than that actually observed is called the P-value of the test. The smaller the p-value, the stronger the evidence against H 0 provided by the data. To calculate the P-value, we must use the sampling distribution of the test statistic.

20. Example: bank’s income The P-value In our banking example we found that the test statistic for testing H 0 :  = 0 versus H a :   0 is If the null hypothesis is true, we expect z to take a value not far from 0. Because the alternative is two-sided, values of z far from 0 in either direction count ass evidence against H 0. So the P-value is:

21. Example: bank’s income The p-value for bank’s income. The two-sided p-value is the probability (when H 0 is true) that takes a value at least as far from 0 as the actually observed value.

22. Tests of Significance: Formal details We know that smaller P-values indicate stronger evidence against the null hypothesis. But how strong is strong evidence? One approach is to announce in advance how much evidence against H 0 we will require to reject H 0. We compare the P-value with a level that says “this evidence is strong enough.” The decisive level is called the significance level. It is denoted be the Greek letter .

23. Tests of Significance: Formal details If we choose  = 0.05, we are requiring that the data give evidence against H 0 so strong that it would happen no more than 5% of the time (1 in 20) when H 0 is true. Statistical significance If the p-value is as small or smaller than , we say that the data are statistically significant at level .

24. Tests of Significance: Formal details You need not actually find the p-value to asses significance at a fixed level . You need only to compare the observed statistic z with a critical value that marks off area  in one or both tails of the standard Normal curve.

25. Test for a Population Mean