KNR 445 Statistics t-tests Slide 1 Introduction to Hypothesis Testing The z-test

KNR 445 Statistics Effect sizes Slide 2 Stage 1: The null hypothesis  If you do research via the deductive method, then you develop hypotheses  From 497 (intro to research methods): Deduction

KNR 445 Statistics Effect sizes Slide 3 Stage 1: The null hypothesis  The null hypothesis  The hypothesis of no difference  Need for the null: in inferential stats, we test the empirical evidence for grounds to reject the null  Understanding this is the key to the whole thing…  The distribution of sample means, and its variation  Time for a digression…

KNR 445 Statistics Effect sizes Slide 4 The distribution of sampling means  Let’s look at this applet… This is the population from which you draw the sample Here’s one sample (n=5) Here’s the sample mean for the sample

KNR 445 Statistics Effect sizes Slide 5 The distribution of sampling means  Let’s look at this applet… If we take a 1,000 more samples, we get a distribution of sample means. Note that it looks normally distributed, but its variation alters with sample size (for later)

KNR 445 Statistics Effect sizes Slide 6 The distribution of sampling means  Let’s look at this applet… For now, the important thing to note is that some sample means are more likely than others, just as some scores are more likely than others in a normal distribution

KNR 445 Statistics Effect sizes Slide 7 Stage 1: The null hypothesis  Knowing that the distribution of sample means has certain characteristics (later, with the z-statistic) allows us to state with some certainty how likely it is that a particular sample mean is “different from” the population mean  Thus we test for this “statistical oddity”  If it’s sufficiently odd (different), we reject the null  If we reject the null, we conclude that our sample is not from the original population, and is in some way different to it (i.e. from another population)

KNR 445 Statistics Effect sizes Slide 8 Stage 1: The null hypothesis  Example of the null:  You’re looking for an overall population to compare to

KNR 445 Statistics Effect sizes Slide 9 Stage 1: The null hypothesis  Example of the null:  So the null is the assumption that our sample mean is equal to the overall population mean

KNR 445 Statistics Effect sizes Slide 10 Stage 2: The alternative hypothesis  Also known as the experimental hypothesis (H A, H 1 )  Two types:  1-tailed, or directional  Your sample is expected to be either more than, or less than, the population mean  Based on deduction from good research (must be justified)  2-tailed, or non-directional  You’re just looking for a difference  More exploratory in nature  Default in SPSS

KNR 445 Statistics Effect sizes Slide 11 Stage 2: The alternative hypothesis  Example of the alternative hypothesis H A can be that you expect the sample mean to be less than the null, greater than the null, or just different…which is it here?

KNR 445 Statistics Effect sizes Slide 12 Stage 2: The alternative hypothesis  So, here our H A : µ > Now, next… What the heck is that?

KNR 445 Statistics Effect sizes Slide 13 Stage 3: Significance threshold ( α )  How do we decide if our sample is “different”?  It’s based on probability  Recall normal distribution & z-scores

KNR 445 Statistics Effect sizes Slide 14 Stage 3: Significance threshold ( α )  Notice the fact that distances from the mean are marked by certain probabilities in a normal distribution

KNR 445 Statistics Effect sizes Slide 15 Stage 3: Significance threshold ( α )  Our distribution of sample means is similarly defined by probabilities  So, we can use this to make estimates of how likely certain sample means are to be derived from the null population  What we are saying here is that:  Sample means vary  The question is whether the variation is due to chance, or due to being from another population  When the variation exceeds a certain probability ( α ), we reject the null (see applet again)

KNR 445 Statistics Effect sizes Slide 16 Stage 3: Significance threshold ( α )  When the variation exceeds a certain probability ( α ), we reject the null… Sample means of these sizes are unusual. How unusual is dictated by the normal distribution’s pdf (probability density function)

KNR 445 Statistics Effect sizes Slide 17 Stage 3: Significance threshold ( α )  When the variation exceeds a certain probability ( α ), we reject the null… Convention in the social sciences has become to reject the null when the probability of the variation is less than This gives us our significance level ( α =.05)

KNR 445 Statistics Effect sizes Slide 18 Stage 4: The critical value of Z  How do we use this probability?  Every test uses a distribution  The z-test uses the z-distribution  So we use probabilities from the z distribution…  …and then we convert the difference between the sample and population means to a z-statistic for comparison  First, we need that probability – we can use tables for this…or an applet…let’s do the tables thing for now

KNR 445 Statistics Effect sizes Slide 19 Stage 4: The critical value of Z  For our example: This is α (=.10)

KNR 445 Statistics Effect sizes Slide 20 Stage 4: The critical value of Z  For our example:  α = 0.1, and the hypothesis is 1-tailed, so our distribution would look like this Rejection region α (=.10) Fail to reject the null 1 - α (=.90) Z score for the α (=.10) threshold

KNR 445 Statistics Effect sizes Slide 21 Stage 4: The critical value of Z  For our example:  However, the tables only show half the distribution (from the mean onwards), so we would have this: Area referred to in the table Rejection region α (=.10) Z score for the α (=.10) threshold

KNR 445 Statistics Effect sizes Slide 22 Stage 4: The critical value of Z So, we need to find a probability of Locate the number nearest to.4 in the table 2.Then look across to the “Z” column for the value of Z to the nearest tenth (= 1.2) 3.Then look up the column for the hundredths (.08) 4.So, z ≈ 1.285

KNR 445 Statistics Effect sizes Slide 23 Stage 5: The test statistic!  So, we insert that threshold value, and now we are asked for some more values… The sample mean The sample size The population SD

KNR 445 Statistics Effect sizes Slide 24 Stage 5: The test statistic!  Why do we need these three? Because now we have to convert our difference score to a score on the distribution of sample means  Remember this? The purpose of this statistic was to convert a raw score difference (from the mean) by scaling it according to the spread of raw scores in the distribution of raw scores

KNR 445 Statistics Effect sizes Slide 25 Stage 5: The test statistic! Sample mean Population mean Variability of sample means The purpose of this statistic is the same, but it converts a sample mean difference (from µ) by scaling it according to the spread of all sample means in the distribution of sample means

KNR 445 Statistics Effect sizes Slide 26 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…we’ll use the applet again Note sample size… & note spread of sample means

KNR 445 Statistics Effect sizes Slide 27 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…we’ll use the applet again As sample size goes up… Spread of sample means goes down

KNR 445 Statistics Effect sizes Slide 28 Stage 5: The test statistic!  Understanding influences on the distribution of sample means…  That means that the test statistic has to take sample size into account  Other influences are mean difference (sample – population) and variability in the population  How do you think each of these things influence the test statistic?  This will help you understand why the test statistic looks like it does

KNR 445 Statistics Effect sizes Slide 29 Stage 5: The test statistic! Sample mean Population mean Variability of sample means A closer look: to understand how the mean difference, population variance, and sample size affect the test statistic, we need to look at the SE M in more detail

KNR 445 Statistics Effect sizes Slide 30 Stage 5: The test statistic! Population standard deviation Sample size So…can you see the influences?

KNR 445 Statistics Effect sizes Slide 31 Stage 5: The test statistic!  To calculate, then…  First the standard error of the mean:  Now the test statistic itself:

KNR 445 Statistics Effect sizes Slide 32 Stage 5: The test statistic!  For you to practice, I’ve provided a simple excel file that does the calculation bit for you…

KNR 445 Statistics Effect sizes Slide 33 Stage 6: The comparison & decision  Do we fail to reject the null? Or reject the null?

KNR 445 Statistics Effect sizes Slide 34 3 ways of phrasing the decision…  What is the probability of obtaining a Z obs = if the difference is attributable only to random sampling error?  Is the observed probability (p) less than or equal to the  -level set?  Is p   ?

KNR 445 Statistics Effect sizes Slide 35 Reporting the Results  The observed mean of our treatment group was (  13.62) pages per employee per week. The z-test for one sample indicates that the difference between the observed mean of and the population average of was not statistically significant (Z obs = 1.27, p > 0.1). Our sample of employees did not use significantly more paper than the norm. Notice this would change if  changed

KNR 445 Statistics Effect sizes Slide 36 Do not reject H 0 vs. Accept H 0  Accept infers that we are sure Ho is valid  Do not reject implies that this time we are unable to say with a high enough degree of confidence that the difference observed is attributable to anything other than sampling error.

KNR 445 Statistics Effect sizes Slide 37 Note: Z & t-tests  Same concept, different assumptions  Can only use z-tests if you know population SD  You usually don’t – SPSS does not even provide the test  So SPSS uses t-test instead  t-test more robust against departures from normality (doesn’t affect the accuracy of the p-estimate as much)  T-test estimates population SD from sample SD

KNR 445 Statistics Effect sizes Slide 38 Note: Z & t-tests  To estimate pop SD from sample SD, the sample SD is inflated a little…

KNR 445 Statistics Effect sizes Slide 39 Note: Z & t-tests  To estimate standard error from sample SD, use the estimated SD again, thus…

KNR 445 Statistics Effect sizes Slide 40 Note: Z & t-tests  This is important  Size of estimated SE obviously depends on both SD of sample, and sample size

KNR 445 Statistics Effect sizes Slide 41 Testing in SPSS STEP 1: Choose the procedure. SPSS uses the one sample t-test instead of the z-test. It’s similar (see previous notes). I used the midterm data for this

KNR 445 Statistics Effect sizes Slide 42 Testing in SPSS STEP 2: Choose a variable to test STEP 3: Choose a population mean value to test it against (SPSS doesn’t have a clue what population your testing against, right?) STEP 4: Choose “OK” (you could also go into options and change the confidence interval size – the default is 95%)

KNR 445 Statistics Effect sizes Slide 43 And you get this… T-Test 1. Here’s the important bit – the statistical outcome (big difference) 2. Here’s the standard error 3. If you think of the equation, it’s obvious a mean difference this big would result in a significant difference, right?

KNR 445 Statistics Effect sizes Slide 44 Quittin’ time