Test of Proportions

Quick reference

• 00:04 Hi, I'm Ray Sheen.
• 00:05 We've discussed hypothesis testing when both the independent and
• 00:09 dependent variables are continuous.
• 00:11 Now, it's time to look at the case when both are discrete.
• 00:15 We'll start with the test of proportions.
• 00:19 >> So let's look at our hypothesis testing decision tree.
• 00:22 We are considering the case when the data is discrete, both x and y values.
• 00:27 That means that we will be working with counts and instances, not measurements.
• 00:32 And in this lesson, we'll discuss the one-sample test of proportions and
• 00:36 the two-sample test of proportions.
• 00:39 Before we go into how to run these tests,
• 00:42 let's first explain what a test of proportions does.
• 00:45 Test of proportions is used with discrete or attribute data.
• 00:49 So the data will be counts of occurrences or true/false and on/off types of data.
• 00:56 The test of proportions will compare the percentage of items in a sample
• 01:00 that contains a particular attribute to another percentage,
• 01:05 either from another sample or from a baseline value.
• 01:08 The question is to determine if the proportions or percentages are the same.
• 01:13 Or more precisely, if there are differences in the proportion,
• 01:17 whether those differences are statistically significant.
• 01:20 This test is not testing means or standard deviation,
• 01:24 it's testing the percentage of counts in the category of interest.
• 01:29 The one sample test will compare the proportions to a target value that has
• 01:34 been set based upon historic measurements or
• 01:37 the known value of another large population.
• 01:40 The null hypothesis will be that the sample proportion equals the target value,
• 01:45 there is nothing unusual in the sample.
• 01:48 The alternative hypothesis will normally be that the sample
• 01:52 proportion value is larger or smaller than the target value.
• 01:57 The two sample test of proportion is similar in the comparison,
• 02:01 except that the target value is replaced with the proportion or
• 02:05 percentage in the second sample.
• 02:07 This test is normally used to determine if two samples are different or
• 02:11 if they're from the same population.
• 02:13 It's very common to use the tests with complaint or
• 02:17 defect data in a before and after comparison.
• 02:20 The two samples being before an improvement is introduced and
• 02:24 after it has been introduced.
• 02:26 The null hypothesis is always the proportion of the two samples is equal,
• 02:30 which means that subtracting one from the other results in zero.
• 02:34 The alternative hypothesis is that the proportions are not equal.
• 02:38 This set of hypothesis may be used in the analysis phase to discover differences
• 02:43 that will lead to an understanding of the problem.
• 02:45 If doing the before and after type of comparison,
• 02:48 this is normally done in the improve phase.
• 02:50 And we often want to show that the improved proportion is higher or
• 02:54 lower proportion than the original proportion.
• 02:57 In that case, the null hypothesis is that the two proportions are equal to or
• 03:02 less than or greater than the baseline of proportion.
• 03:06 And the alternative hypothesis is that the new proportion is higher or
• 03:10 lower than the baseline proportion.
• 03:12 And of course, the test to determine these differences is statistically significant.
• 03:18 Let's consider how we do one sample test of proportions.
• 03:22 For our example, we will consider the percentage of applicants to college for
• 03:26 the current college year that are accepted.
• 03:28 Historically, the college has been accepting 52% of applicants,
• 03:33 this year, it was 57%, is the change significant?
• 03:36 The null hypothesis would be that this year's percentage of applicants that
• 03:40 were accepted is equal to the historical percentage.
• 03:43 The alternative would be that this year's percentage is higher than
• 03:48 the historical percentage.
• 03:50 Excel does not perform this test.
• 03:52 In Minitab, select the Stat pulldown menu,
• 03:55 then select the Basic Statistics, and next, select 1 Proportion.
• 03:59 That will bring up this panel.
• 04:02 Now, select the column where the data is located.
• 04:05 We need to have the raw data so
• 04:06 that Minitab can also have a count of the sample size.
• 04:10 Recall that the sample size will impact the confidence interval.
• 04:14 Enter your target statistic in the window labeled Hypothesized proportion.
• 04:18 If you select the Options button, you can determine whether you want the test to be
• 04:23 for a relationship of equal, or greater than, or less than.
• 04:27 Then click OK and the results will be found in the session window of Minitab,
• 04:32 with a P Value that you can use to decide whether to reject or
• 04:36 fail to reject the null hypothesis.
• 04:38 Now, let's look at the two sample test of proportions.
• 04:42 We're comparing the proportions of an attribute between two samples of data.
• 04:46 In this illustration, the comparison is the rate of on-time
• 04:51 submittals of tax returns between 2016 and 2017.
• 04:55 Excel does not provide this as a standalone function, but
• 04:59 we can still do this analysis in Excel, it just will take a few steps.
• 05:03 So there are several functions in Excel that when we string them together,
• 05:08 can do this analysis.
• 05:10 One caution, make sure your data is in integer format because Excel will
• 05:14 be doing calculations with it.
• 05:16 Start with the standard VAR or variance function for each data set.
• 05:20 Be sure to record those values for future use.
• 05:24 Next, go to the data analysis pull down menu and
• 05:27 select the Z-test two samples for means function.
• 05:31 Enter the data range and the previously calculated variance for
• 05:34 each sample in the form that's displayed.
• 05:37 Also, enter any difference if there is one that you're expecting.
• 05:42 I normally set this to zero.
• 05:44 Excel will calculate a P value for both the greater than and
• 05:48 the less than conditions.
• 05:50 Minitab is simpler, select Stat, then Basic Statistics, and then 2 Proportions.
• 05:56 Select the format of your data, either all data is in one column with an adjacent
• 06:01 column that specifies which sample is associated with the data value, or
• 06:06 the data is in different columns with the sample as the column title.
• 06:11 Then select the appropriate data columns.
• 06:13 And finally, go to the Options button to change the confidence level or
• 06:17 to specify a relationship.
• 06:19 The default is to check if they are equal,
• 06:21 but you can check if one is greater than or less than the other.
• 06:24 The result is found in the session window with an associated P value.
• 06:29 Excel and Minitab will give slightly different results for P value.
• 06:34 But I find that the differences are out at the third or
• 06:37 fourth significant digit and are unlikely to impact the decision of whether or
• 06:42 not to reject the null hypothesis.
• 06:44 The last thing to discuss is the formulas involved.
• 06:47 This is for your reference if you're taking the IASSC exam.
• 06:51 Since Excel does not provide the test of proportions function,
• 06:55 you can do the analysis manually.
• 06:57 To do this, you will need the percentage in each sample and
• 07:01 the number of items in the sample.
• 07:03 These are the formulas for the one sample and two sample tests of proportions.
• 07:07 The p terms are the percentages,
• 07:09 and the n terms are the number of points in the sample.
• 07:12 The formula is a Z value, and that can be used to determine whether it is
• 07:17 an acceptable level depending upon the value of alpha selected.
• 07:22 >> The one sample and two sample test of proportions are quick and
• 07:26 easy tests that help us to understand the characteristics of the data sets that
• 07:30 are made up of discrete data.

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