1 Lecture 6 Hypothesis Testing II: Proportions and 2 Populations Graduate School Quantitative Research Methods Gwilym Pryce

2 Notices: n Register

3 Aims & Objectives n Aim the aim of this lecture is to continue with our introduction of the method of hypothesis testing and to demonstrate a number of applications n Objectives –by the end of this lecture students should be able to carry out hypothesis tests on: two population means one population proportion two population proportions

4 Plan: n 1. Review of Significance n 2. Review of one sample tests on the mean n 3. Hypothesis tests about Two population means Homogenous variances Heterogeneous variances n 4. Deciding on whether variances are equal n 5. Hypothesis tests about proportions –One population –Two populations

5 Macro commands:

6 1.Review of Significance n P = significance level = chances of our observed sample mean occurring given that our assumption about the population (denoted by “H 0 ”) is true. n So if we find that this probability is small, it might lead us to question our assumption about the population mean. –I.e. if our sample mean is a long way from our assumed population mean then it is: either a freak sample or our assumption about the population mean is wrong.

7 If we draw the conclusion that it is our assumption re  that is wrong and reject H 0 then we have to bear in mind that there is a chance that H 0 was in fact true. –In other words: when P = 0.05, for every twenty times we reject H 0, then on one of those occasions we would have rejected H 0 when it was in fact true.

8 2. Review of one sample tests on the mean n We introduced a common framework for hypothesis testing: 4 Steps of Hypothesis testing: Step (1) state H 0 and H 1 Step (2) state  and formula Step (3) state decision rule Step (4) compute P & decide

9 We also looked at 2 specific tests: n Large sample sig. Test on one mean: Formula: Macro syntax: H_L1M n=(?) x_bar=(?) m=(?) s=(?). n Small sample sig. Test on one mean: Formula: Macro syntax: H_S1M n=(?) x_bar=(?) m=(?) s=(?).

10 3. Hypothesis tests about two population means n In SPSS: this is called the “Independent Sample t-test” go to Analyse, Compare Means... n Two different formulas for computing t: Equal Variances (formula has an exact t-distribution) Unequal Variances (does not have an exact t-distribution)

11 Example where variances are different: As part of your PhD, you want to test whether the new “Fun Phonics” reading method is better than the “Letterland” method. You examine the reading power of 6 year old children from two similar schools. –The first used the FP method and you found that this produced an average reading proficiency score of 53.7 (based on a sample of 22 children; s.d. = 11.5). –The second school used the Letterland method and you found that this produced an average reading proficiency score of (sample = 24; s.d. = 16.9). Test whether the FP method produces higher results at the 1% significance level.

12 n Use the 4 steps and the following formula to test whether the FP method produces higher results at the 1% significance level. n Can you use the canned SPSS procedure to do this problem? 4 Steps of Hypothesis testing: Step (1) state H 0 and H 1 Step (2) state  and formula Step (3) state decision rule Step (4) compute P & decide

13 (1) H 0 :  FP =  L (means are equal) H 1 :  FP >  L (upper tail test) (2)  = 0.01 (implies critical t value of 2.528), (3) Reject H 0 iff P <  I.e. if P < 0.01 n (4) P = Prob(t > 2.644) = , so reject H 0

14 Doing the calculation in SPSS: n You cannot use the canned SPSS procedure unless you have the original data. n But you can use the following macro commands: –Homogenous variances: H_S2Mp n1=(?) n2=(?) x_bar1=(?) x_bar2=(?) s1=(?) s2=(?). –Heterogeneous variances: H_S2Md n1=(?) n2=(?) x_bar1=(?) x_bar2=(?) s1=(?) s2=(?).

15 For the Letterland/FP example we would use the diff. Variances syntax: n H_S2Md n1=(22) n2=(24) x_bar1=(53.7) x_bar2=(42.51) s1=(11.5) s2=(16.9). n The upper tail sig. = I.e. less than 1% chance of false rejection, therefore reject H 0 of equal means in favour of the alternative hypothesis that Fun Phonics results in higher reading scores on average than Letterland.

16 4. How do we decide on whether the variances are similar? n Where variances are hugely different or exactly the same, the decision is simple. n When there is any ambiguity, we can use one of two tests to help us: Simple Ratio of Variances Test Levene’s Test

17 Simple Ratio of Variances test: n If we divide the ratio of variances of samples from two independent populations we find that that ratio has an F distribution in repeated samples: n where the denominator degrees of freedom calculated as n 1 –1 and the numerator degrees of freedom calculated as n 2 –1. NB Because the critical values for the F distribution are only calculated for the upper tail, if the F value you are have calculated is less than one, you need to invert it (i.e. swap round the numerator and denominator). F = s 1 2 / s 2 2

18 n This is the formula behind the following command: H8_S2VF n1=(?) n2=(?) s1(?) s2(?) n E.g. For the Letterland/FP example: H8_S2VF n1=(22) n2=(24) s1=(11.5) s2=(16.9). n Which tells us that there is less than a 5% chance of false rejection if we reject the null of equal variances. So reject the null I.e. we can be sure that the population variances are indeed different.

19 The Levene’s test n If we have the original data we can use Levene’s test which is a canned routine in SPSS. n The Levene’s test is more sophisticated & robust than the simple ratio of variances test: –If P (I.e. “sig.”) from the Levene’s test is small reject the H 0 of equal variances & use the 1st t- formula. – If P from the Levene’s test is large, accept H 1 & use the 2nd t-formula to compute the test statistic.

20 SPSS Output from test equal purchase prices between Cumberland and Durham (Nationwide data):

21 Two tails from one: n Along with the Levene’s test results, SPSS automatically supplies t-test results for both the equal and different variances formulas. n One problem with the SPSS t-test, however, is that it only gives the 2 tail sig., but you can work out the one tail sigs as follows: The two tailed significance is twice that of the smallest one tailed significance: 2 tailed sig. = 2  min[lower tail sig., upper tail sig.] But it can be a bit confusing working out which one tail significance level is the one you want (see notes).

22 Testing for 2 means summary: n If you’ve got the original data, First do the Levene’s test in SPSS Analyze, Compare Means, Independent Samples Then do the appropriate macro t-test to avoid confusion. H_S2Mp for equal variances or H_S2Md for different variances n If you don’t have the original data, First do the ratio of variances test H8_S2VF Then do the appropriate macro t-test H_S2Mp for equal variances or H_S2Md for different variances

Hypothesis tests on proportions One population (large samples only) n So far looked at: –how to make inferences about the population mean from our sample mean. n But sometimes the variable of interest is categorical household has or has not insurance; person is homosexual or not homosexual; a person has Aids or does not have Aids

24 n In such cases, what we are interested in is the proportion of cases that fall into a particular category: the proportion of households with insurance; the proportion of people who are homosexual; the proportion of people with Aids

25 n Calculating the sample proportion: p = x / n –where: x = cases with the attribute of interest e.g. the number of households with insurance n = sample size

26 CLT and Proportions: n Q/ Does the Central Limit Theorem apply to sample proportions? n A/ Yes. –Proportions from repeated random samples will be normally distributed around the population proportion . –We can then translate any sample proportion onto the standard normal curve by calculating its z score:

27 Example: –E.g. 1 As a historian, you want to find the proportion of citizens in medieval Scotland that contracted the plague. From a sample of 400 parish records, you find that 22 died of the plague. The assumption in the literature has been that 10% of the population had died. Test whether this assumption is valid using both 2 and 1 tailed tests.

28 Summary of data: n = 400 x = 22   = 0.1 –(1) H 0 :  = 10% H 1 :   10% (2-tailed test) –(2)  = 0.02, for example.

29 –(3) Reject H 0 iff P <  I.e. if P < 0.02 (this will happen if z c 2.33, where 2.33 is the z value associated with  = Since z c = , we know we can safely reject H 0 ). –(4) Calculate z: P = 2x(Prob(z < -3.00)) = 2x = since P < 0.02 (I.e. less than one in 50 chance of type I error) we can reject H 0. In fact, the chances of incorrect rejection of H 0 are less than one in 3,000. I.e. the chances of observing p (our sample proportion) assuming H 0 (  = 10%) to be true are so small that we are forced to question this assumption about  n = 400 x = 22   = 0.1

30 One tailed test: –(1) H 0 :  = 10% H 1 :  < 10% (lower tail test) –(2)  = 0.02 –(3) Reject H 0 iff P <  I.e. if P < 0.02 –(4) Lower tail sig. = P = Prob(z < -3.00) = since P < 0.02 we can reject H 0 knowing that the chances of incorrect rejection of H 0 are less than one in 740 »our cut-off rule for rejecting H 0 was no more than a one in 50 chance »one in 740 is a lot less than one in 50 so we can reject H 0 with confidence.

31 n The macro syntax for one proportion tests is as follows: n H6_L1P n=(400) x=(22) pi=(0.1). Which comes to the same result.

Hypothesis tests about Two population proportions n To test the hypothesis that the population proportions are equal: H 0 :  1 =  2 compute the z statistic: where: SE Dp is the pooled standard error = and

33 Example: n Two surveys of mortgage payment protection insurance (MPPI) are carried out, one on single parents with 1 child and one on single parents with 3 children. Amongst the first group, 67 out of a sample of 300 were found to have taken out MPPI, compared with 15 out of a sample of 101 in the second group. Is take-up significantly lower amongst the HHs with three children? p 1 = 67/300 = ; p 2 = 15/101 = ; p = ( )/(67+15) = ;

34 –(1) H 0 :  1 =  2 H 1:  1 >  2 –(2)  = 0.01 (z* =  2.33) –(3) Reject H 0 : if P < 0.01 –(4) P = Take-up is not significantly lower amongst HHs with 3 children at the1% sig. level; or even at 5% significance level. I.e. we cannot say that the difference in proportions is anything more than the effect of sampling variation.

35 n H7_L2P n1=(300) n2=(101) x1=(67) x2=(15).

36 Plan: n 1. Review of Significance n 2. Review of one sample tests on the mean n 3. Hypothesis tests about Two population means Homogenous variances Heterogeneous variances n 4. Deciding on whether variances are equal n 5. Hypothesis tests about proportions –One population –Two populations