It is a three-part discussion that should be completed in sequential order.

Part One - Hypothesis Testing

ReadLecture Four. Lecture Four starts out with the five-step procedure for hypothesis testing. What is this? What does it do for us? Why do we need to follow these steps in making a judgement about the populations our samples came from? What are the "tricky" parts of developing appropriate hypotheses to test? What examples can you suggest where this process might be appropriate in your personal or professional lives? (This should be started on Day 1.)

Part Two - T-tests

ReadLecture Five. Lecture Five illustrates several t-tests on the data set. What conclusions can you draw from these tests about our research question on equal pay for equal work? What is missing from these results to give us a comprehensive answer to the question? Why? (This should be started on Day 3.)

Part Three - F-test

ReadLecture Six. Lecture Six introduces you to the F-test for variance equality. Last week, we discussed how adding a variation measure to reports of means was a smart thing to do. Why does variation make our analysis of the equal pay for equal work question more complicated? What causes of variation impact salary that we have not discussed yet? How can you relate this issue to measures used in your personal or professional lives? (This should be completed by Day 5.)

Your responses should be separated in the initial post, addressing each part individually, similar to what you see here.

Lecture 4 (Sampling basics and Hypothesis test) This week we turn from descriptive statistics to inferential statistics and making decisions about our populations based on the samples we have. For example, our class case research question is really asking if in the entire company population of employees, do males and females receive the same pay for doing equal work. However, we are not analyzing the entire population, instead we have a sample of 25 males and 25 females to work with. This brings us to the idea of sampling - taking a small group/sample from a larger population. To paraphrase, not all samples are created equal. For example, if you wanted to study religious feelings in the United States, would you only sample those leaving a fundamentalist church on a Wednesday? While this is a legitimate element of US religions, it does not represent the entire range of religious views - it is representative of only a portion of the US population, and not the entire population. The key to ensuring that sample descriptive statistics can be used as inferential statistics - sample results that can be used to infer the characteristics (AKA parameters) of a population - is have a random sample of the entire population. A random sample is one where, at the start, everyone in the population has the same chance of being selected. There are numerous ways to design a random sampling process, but these are more of a research class concern than a statistical class issue. For now, we just need to make certain that the samples we use are randomly selected rather than selected with an intent of ensuring desired outcomes are achieved. The issue about using samples that students often new to statistics is that the sample statistic values/outcomes will rarely be exactly the same as the population parameters we are trying to estimate. We will have, for each sample, some sampling error, the difference between the actual and the sample result. Researchers feel that this sampling error is generally small enough to use the data to make decisions about the population (Lind, Marchel, & Wathen, 2008). While we cannot tell for any given sample exactly what this difference is, we can estimate the maximum amount of the error. Later, we will look at doing this; for now, we just need to know that this error is incorporated into the statistical test outcomes that we will be studying. Once we have our random sample (and we will assume that our class equal pay case sample was selected randomly), we can start with our analysis. After developing the descriptive statistics, we start to ask questions about them. In examining a data set, we need to not only identify if important differences exist or not but also to identify reasons differences might exist. For our equal pay question, it would be legal to pay males and females different salaries if, for example, one gender performed the duties better, or had more required education, or have more seniority, etc. Equal pay for equal work, as we are beginning to see, is more complex than a simple single question about salary equality. As we go thru the class, we will be able to answer increasingly more complex questions. For this week, we will stay with questions about involving ways to sort our salary results - looking for differences might exist. Some of these questions for this week with our equal pay case could include: Could the means for both males and females be the same, and the observed difference be due to sampling error only? Could the variances for the males and female be the same (AKA statistically equal)? Could salaries per grade be statistically equal? Could salaries per degree (undergraduate and graduate) be the same? Etc. Hypothesis Testing As we might expect, research and statistics have a set procedure/process on how to go about answering these questions. The hypothesis testing procedure is designed to ensure that data is analyzed in a consistent and recognized fashion so everyone can accept the outcome. Statistical tests focus on differences - is this difference large enough to be significant, that is not simply a sampling error? If so, we say the difference is statistically significant; if not, the difference is not considered statistically significant. This phrasing is important as it is easy to measure a difference from some point, it is much harder to measure \"things are different.\" It is that pesky sampling error that interferes with assessing differences directly. Before starting the hypothesis test, we need to have a clear research question. The questions above are good examples, as each clearly asks if some comparison is statistically equal or not. Once we have a clear question - and a randomly drawn sample - we can start the hypothesis testing procedure. The procedure itself has five steps: Step 1: State the null and alternate hypothesis Step 2: Form the decision rule Step 3: Select the appropriate statistical test Step 4: Perform the analysis Step 5: Make the decision, and translate the outcome into an answer to the initial research question. Step 1. The null hypothesis is the \"testable\" claim about the relationship between the variables. It always makes the claim of no difference exists in the populations. For the question of male and female salary equality, it would be: Ho: Male mean salary = Female mean salary. If this claim is found not to be correct, then we would accept the alternate hypothesis claim: Ha: Male salary mean =/= (not equal) Female salary mean. (Note, some alternate ways of phrasing these exist, and we will cover them shortly. For now, let's just go with this format.) Step 2. This step involves selecting the decision rule for rejecting the null hypothesis claim. This will be constant for our class - we will reject the null hypothesis when the p-value is equal to or less than 0.05 (this probability is called alpha). Other common values are .1, and .01 - the more severe the consequences of being wrong if we reject the null, the smaller the value of alpha we select. Recall that we defined the p-value last week as the probability of exceeding a value, the value in this case would be the statistical outcome from our test. Step 3. Selecting the appropriate statistical test is the next step. We start with a question about mean equality, so we will be using the T-test - the most appropriate test to determine if two population means are equal based upon sample results. Step 4. Performing the analysis comes next. Fortunately for us, we can do all the arithmetic involved with Excel. We will go over how to select and run the appropriate T-test below. Step 5. Interpret the test results, making a decision on rejecting or not rejecting the null hypothesis, and using this outcome to answer the research question is the final step. Excel output tables provide all the information we need to make our decision in this step. Step 1: Setting up the hypothesis statements In setting up a hypothesis test for looking at the male and female means, there are actually three questions we could ask and associated hypothesis statements in step 1. 1. Are male and female mean salaries equal? a. Ho: Male mean salary = Female mean salary b. Ha: Male mean salary =/= Female mean salary 2. Is the male mean salary equal to or greater than the Female mean salary? a. Ho: Male mean salary => Female mean salary b. Ha: Male mean salary Female mean salary While they appear similar each answers a different question. We cannot, for example, take the first question, determine the means are not equal and then say that, for example, the male mean is greater than the female mean because the sample results show this. Our statistical test did not test for this condition. If we are interested in a directional difference, we need to use a directional set of hypothesis statements as shown in statements 2 and 3 above. Rules. There are several rules or guidelines in developing the hypothesis statements for any statistical test. 1. The variables must be listed in the same order in both claims. 2. The null hypothesis must always contain the equal (=) sign. 3. The null can contain an equal (=), equal to or less than () claim. 4. The null and alternate hypothesis statement must, between them, account for all possible actual comparisons outcomes. So, if the null has the equal (=) claim, the alternate must contain the not equal (=/= or ) statement. If the null has the equal or less than () claim. Finally, if the null has the equal to or greater (=> or ) claim, the null must contain the less than ( Female salary mean) and the opposite null (Male salary mean Female mean salary. The arrow in the alternate points to the positive/right tail and that is where the calculated t-statistic is. So, even if the p-value is smaller than alpha in a one tail test, we need to ensure the t-statistic is in the correct tail for rejection. References Lind, D. A., Marchel, W. G., & Wathen, S. A. (2008). Statistical Techniques in Business & Finance. (13th Ed.) Boston: McGraw-Hill Irwin. Lecture 6 (Additional information on t-tests and hypothesis testing) Lecture 5 focused on perhaps the most common of the t-tests, the two sample assuming equal variance. There are other versions as well; Excel lists two others, the two sample assuming unequal variance and the paired t-test. We will end with some comments about rejecting the null hypothesis. Choosing between the t-test options As the names imply each of the three forms of the t-test deal with different types of data sets. The simplest distinction is between the equal and unequal variance tests. Both require that the data be at least interval in nature, come from a normally distributed population, and be independent of each other - that is, collected from different subjects. The F-test for variance. To determine if the population variances of two groups are statistically equal - in order to correctly choose the equal variance version of the t-test - we use the F statistic, which is calculated by dividing one variance by the other variance. If the outcome is less than 1.0, the rejection region is in the left tail; if the value is greater than 1.0, the rejection region is in the right tail. In either case, Excel provides the information we need. To perform a hypothesis test for variance equality we use Excel's F-Test Two-Sample for Variances found in the Data Analysis section under the Data tab. The test set-up is very similar to that of the t-test, entering data ranges, checking Labels box if they are included in the data ranges, and identifying the start of the output range. The only unique element in this test is the identification of our alpha level. Since we are testing for equality of variances, we have a two sample test and the rejection region is again in both tails. This means that our rejection region in each tail is 0.25. The F-test identifies the p-value for the tail the result is in, but does not give us a one and two tail value, only the one tail value. So, compare the calculated p-value against .025 to make the rejection decision. If the p-value is greater than this, we fail to reject the null; if smaller, we reject the null of equal variances. Excel Example. To test for equality between the male and female salaries in the population, we set up the following hypothesis test. Research question: Are the male and female population variances for salary equal? Step 1: Ho: Male salary variance = Female salary variance Ha: Male salary variance Female salary variance Step 2: Reject Ho if p-value is less than Alpha = 0.025 for one tail. Step 3: Selected test is the F-test for variance Step 4: Conduct the test Step 5: Conclusion and interpretation. The test resulted in an F-value less than 1.0, so the statistic is in the left tail. Had we put Females as the first variable we would have gotten a right tail F-value greater than 1.0. This has no bearing on the decision. The F value is larger than the critical F (which is the value for a 1-tail probability of 0.25 - as that was entered for the alpha value). So, since our p-value (.44 rounded) is > .025 and/or our F (0.94 rounded) is greater than our F Critical, we fail to reject the null hypothesis of no differences in variance. The correct ttest would be the two-sample T-test assuming equal variances. Other T-tests. We mentioned that Excel has three versions of the t-test. The equal and unequal variance versions are set up in the same way and produce very similar output tables. The only difference is that the equal variance version provides an estimate of the common variation called pooled variance while this row is missing in the unequal variance version. A third form of the t-test is the T-Test: Paired Two Sample for Means. A key requirement for the other versions of the t-test is that the data are independent - that means the data are collected on different groups. In the paired t-test, we generally collect two measures on each subject. An example of paired data would be a pre- and post-test given to students in a statistics class. Another example, using our class case study would the comparing the salary and midpoint for each employee - both are measured in dollars and taken from each person. An example of NON-pared data, would the grades of males and females at the end of a statistics class. The paired t-test is set up in the same way as the other two versions. It provides the correlation (a measure of how closely one variable changes when another does - to be covered later in the class) coefficient as part of its output. An Excel Trick. You may have noticed that all of the Excel t-tests are for two samples, yet at times we might want to perform a one-sample test, for example quality control might want to test a sample against a quality standard to see if things have changed or not. Excel does not expressly allow this. BUT, we can do a one-sample test using Excel. The reason is a bit technical, but boils down to the fact that the two-sample unequal variance formula will reduce to the one-sample formula when one of the variables has a variance equal to 0. So using the unequal variance t-test, we enter the variable we are interested - such as salary - as variable one and the hypothesized value we are testing against - such as 45 for our case - as variable two, ensuring that we have the same number of variables in each column. Here is an example of this outcome. Research question: Is the female population salary mean = 45? Step 1: Ho: Female salary mean = 45 Ha: Female salary mean 45 Step 2: Reject the null hypothesis is less than Alpha = 0.05 Step 3: Selected test is the two sample unequal variance t-test Step 4: Conduct the test Step 5: Conclusions and Interpretation. Since the two tail p-value is greater than (>) .025 and/or the absolute value of the t-statistic is less than the critical two tail t value, we fail to reject the null hypothesis. Our research question answer is that, based upon this sample, the overall female salary average could equal 45. Miscellaneous Issues on Hypothesis Testing Errors. Statistical tests are based on probabilities, there is a possibility that we could make the wrong decision in either rejecting or failing to reject the null hypothesis. Rejecting the null hypothesis when it is true is called a Type I error. Accepting (failing to reject) the null when it is false is called a Type II error. Both errors are minimized somewhat by increasing the sample size we work with. A type I error is generally considered the more severe of the two (imagine saying a new medicine works when it does not), and is managed by the selection of our alpha value - the smaller the alpha, the harder it is to reject the null hypothesis (or, put another way, the more evidence is needed to convince us to reject the null). Managing the Type II error probability is slightly more complicated and is dealt with in more advanced statistics class. Choosing an alpha of .05 for most test situations has been found to provide a good balance between these two errors. Reason for Rejection. While we are not spending time on the formulas behind our statistical outcomes, there is one general issue with virtually all statistical tests. A larger sample size makes it easier to reject the null hypothesis. What is a non-statistically significant outcome based upon a sample size of 25, could very easily be found significant with a sample size of, for example, 25,000. This is one reason to be cautious of very large sample studies - far from meaning the results are better, it could mean the rejection of the null was due to the sample size and not the variables that were being tested. The effect size measure helps us investigate the cause of rejecting the null. The name is somewhat misleading to those just learning about it; it does NOT mean the size of the difference being tested. The significance of that difference is tested with our statistical test. What it does measure is the effect the variables had on the rejection (that is, is the outcome practically significant and one we should make decisions using) versus the impact of the sample size on the rejection (meaning the result is not particularly meaningful in the real world). For the two-sample t-test, either equal or unequal variance, the effect size is measured by Cohen's D. Unfortunately, Excel does not yet provide this calculation automatically, however it is fairly easy to generate. Cohen's D = (absolute value of the difference between the means)/the standard deviation of both samples combined. Note: the total standard deviation is not given in the t-test outputs, and is not the same as the square root of the pooled variance estimate. To get this value, use the fx function stdev.s on the entire data set - both samples at the same time. Interpreting the effect size outcome is fairly simple. Effect sizes are generally between 0 and 1. A large effect (a value around .8 or larger) means the variables and their interactions caused the rejection of the null, and the result has a lot of practical significance for decision making. A small effect (a value around .2 or less) means the sample size was more responsible for the rejection decision than the variable outcomes. The medium effect (values around .5) are harder to interpret and would suggest additional study (Tanner & Youssef-Morgan, 2013). References Lind, D. A., Marchel, W. G., & Wathen, S. A. (2008). Statistical Techniques in Business & Finance. (13th Ed.) Boston: McGraw-Hill Irwin. Tanner, D. E. & Youssef-Morgan, C. M. (2013). Statistics for Managers. San Deigeo, CA: Bridgepoint Education