I never accept someone telling me they have looked at the data and there is no difference between the variables. I am not comfortable with such a casual approach given that excellent analytical tools are available to provide a clear, repeatable, easily-performed, mathematical evaluation of the data. One way to do this is through a formal hypothesis test. There are five steps to a hypothesis as illustrated in Figure 1.

We will proceed to do a formal hypothesis test using the two RAS data sets referenced on the Exploratory Data Analysis page. The question we want to formally answer is: Are the two data sets, consisting of the return activated sludge concentrations from clarifiers A & B, the same or is there a statistically significant difference between them?

**Figure 1: The Five Steps to Conducting a Hypothesis Test**

Two software programs make doing hypothesis testing very easy. One program facilitating this testing is an Excel spreadsheet add-in product called Quantum XL and the second is Minitab. The use of Quantum is illustrated first. We will follow the five steps as closely as we can.

In Figure 2, using Quantum XL, the first three steps are identified.

**Step 1:** The null hypothesis is that there is no difference in the mean concentration of RAS from Clarifier A compared to the mean RAS concentration from Clarifier B. The alternative hypothesis is that there is a difference between the two mean values. This makes the hypothesis test a two-tailed test.

**Step 2:** The level of significance has been selected as being 0.05.

**Step 3:** The test statistic is shown as being -1.5832.

**Figure 2: The Five Steps to Conducting a Hypothesis Test**

The information shown in Figure 2 also takes into account steps 4 and 5 but step 4 requires some explanation. So here is the formal statement for the fourth step.

**Step 4:** If the test statistic is less than -1.96 or greater than 1.96, reject the null hypothesis and conclude that there is a statistically significant difference between the mean values of each data set. The formulation of the decision rule for this two-tailed hypothesis test is shown graphically in Figure 3. The values of ± 1.96 are derived from the selection of the significance level at 5% (α = 0.05). Quantum has already done the math for us and come to a conclusion as shown in Figure 2.

**Step 5:** The decision is that due to insufficient evidence we fail to reject the null hypothesis and conclude that there is no statistically significant difference between the mean concentration in the RAS from Clarifier A compared to the mean concentration in the concentration of the RAS from Clarifier B.

**Figure 3: The Fourth Step - The Decision Rule Portrayed**

Quantum XL provides additional information as part of its procedure for conducting a hypothesis test. Summary statistics and information about the test are shown in Figure 4. Figure 5 is a boxplot comparing the two sets of data. And Figure 6 shows dotplots of the two data sets. These are all just different ways Quantum has made it easy for you to look at how the data sets are shaped.

**Figure 4: Summary Statistics**

**Figure 5: Comparison of Data Sets Using Boxplots**

**Figure 6: Comparison of Data Sets Using Dotplots**

We will conclude our formal hypothesis testing by showing the output from having conducted the hypothesis test using Minitab. The result or summary page is shown in Figure 7. This is an automated procedure available within Minitab which is explained in more detail below. The Quantum procedure can also be run as an automated procedure. These two products make hypothesis testing really easy to do and generate an interpretation of the analysis that is so clear that whatever your skill level in statistics you will not make a mistake.

**Figure 7: Hypothesis Test Result Using Minitab - Page 1 of 3**

When you use the Minitab Assistant your options for running automated procedures are shown in Figure 8. This is a nice, handy feature in Minitab.

**Figure 8: Using the Minitab Assistant**

After picking the Hypothesis Tests option from the Minitab Assistant, you are then presented with the screen shown in Figure 9. This is a clear, easy-to-follow decision flow chart that guides you step-by-step through the hypothesis test.

**Figure 9: Using the Minitab Assistant for Hypothesis Testing**

The Minitab automated hypothesis test procedure actually generates a total of three pages. The summary page of the test results was shown in Figure 7. The other two pages generated as part of the complete report are shown in Figures 10 & 11.

**Figure 10: Minitab Assistant for Hypothesis Test Report Page 2 of 3**

**Figure 11: Minitab Assistant for Hypothesis Test Report Page 3 of 3**

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