{"id":2106,"date":"2019-11-02T19:43:39","date_gmt":"2019-11-02T23:43:39","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/simplestats\/?post_type=chapter&p=2106"},"modified":"2019-11-02T19:47:50","modified_gmt":"2019-11-02T23:47:50","slug":"9-2-the-f-test","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/simplestats\/chapter\/9-2-the-f-test\/","title":{"raw":"9.2 Between a Discrete and a Continuous Variable: The F-test","rendered":"9.2 Between a Discrete and a Continuous Variable: The F-test"},"content":{"raw":" \r\n\r\nWhen the discrete variable of interest has more than two categories, we can no longer use the simple t<\/em>-test presented in the previous section. While we can still use a boxplot chart for visualizing the association between the two variables -- where instead of two boxplots, we will have as many boxplots as there are groups (categories of the discrete variable) -- we no longer have only one difference to test.\r\n\r\n \r\n\r\nTesting multiple means for statistical significance is done through a version of a test called an F<\/em>-test. This F<\/em>-test tests whether the means of several groups[footnote]Note that \"several groups\" includes the two-groups case as well: you could<\/em> test the significance of a difference between the means of two groups with an F<\/em>-test too (it will just provide less information).[\/footnote] are all equal (versus at least one of them not being the same as the rest) through an analysis of variance (aka ANOVA).\r\n\r\n \r\n\r\nAt this point you might feel like a treatment of the topic of the kind I offered about the t<\/em>-test above would be a tad too much, and you will be correct: providing the full-on technical details and the formula of the F<\/em>-test is beyond the scope of this book.\r\n\r\n \r\n\r\nBriefly, the ANOVA F<\/em>-test calculates a ratio of variances (between groups to within groups, in terms of sums of squares): the larger the ratio, the more evidence there is against the null hypothesis, and vice versa. The F<\/em>-test statistic follows an F<\/em>-distribution (not discussed here), which provides the F<\/em>-value with its p<\/em>-value, which is then compared to the\u00a0\u03b1<\/em>-level and interpreted in the usual way. Example 9.2 illustrates.\r\n\r\n \r\n
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Example 9.2 Education Differences in Average Income, NHS 2011<\/em><\/p>\r\n\r\n<\/header>\r\n

\r\n\r\n \r\n\r\nPresumably, college is worth it. You delay your full entry into the labour force and instead invest in your education, with the hope that you will then be able to have a better -- and better<\/em>-paying<\/em> job.\r\n\r\n \r\n\r\nLet's examine this questions then -- do higher educational degrees translate into higher average income? -- using about 3 percent random sample of the NHS 2011<\/em> data. The variable income<\/em> is the same one I used in previous occasions (i.e.,\u00a0total income<\/em> in NHS 2011<\/em>). The groups to compare are the categories of a variable called (highest) degree<\/em>. The variable degree<\/em> is a recoded version of the NHS 2011<\/em>'s highest certificate, diploma or degree<\/em>. I recoded the original variable's thirteen categories in degree<\/em>'s six: 1) no high school, 2) high school, 3) certificate or diploma below Bachelor's, 4) Bachelor's, 5) Master's[footnote]This category includes\u00a0certificates above Bachelor's, and medical, dentistry, and veterinary degrees.<\/span>[\/footnote], and 6) PhD.<\/span>\r\n\r\n \r\n\r\nA brief descriptive investigation of the data reveals that the average income reported by the six education groups looks<\/em> different: \\$19,433 for respondents without a high school degree, \\$30,455 for respondents with a high school degree, \\$41,971 for respondents with more than a high school but less than a Bachelor's degree, \\$60,360 for respondents with a Bachelor's degree, \\$71,593 for respondents with a Master's degree, and \\$93,924 for respondents with a PhD. This potential positive association (more education, more income) is also reflected in the boxplots in Figure 9.1. While there are outliers with extremely high average income in all groups (the most extreme were even truncated at the top), the median and the outlier-less maximum income increase from left to right with the increase of highest degree.\r\n\r\n \r\n\r\nFigure 9.1 Average Income by Highest Degree, NHS 2011<\/em>\r\n\r\n\"\"\r\n\r\n \r\n\r\nAre these differences statistically significant? In other words, are the differences observed in the sample a result of regular sampling variation, or reflective of differences in the population?\r\n