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Testing the Value of t

We’ll use a two-tailed test for the problem here because we are only interested in whether our sample mean, 13.9, is significantly different from, the population mean of 18.16. Thus, the null hypothesis is that 13.9% could be found just by chance if we drew this sample of 10 countries from a population of countries where the mean is 18.16%.

Looking at the values in appendix B, we see that any t value above 2.262 is statistically significant at the .05 level with 9 degrees of freedom for a two-tailed test. With a t of .996 (we only look at the absolute value and ignore the minus sign), we can not reject the null hypothesis. The true mean percentage of female illiteracy rates across the 50 countries of the world is statistically indistinguishable from 13.9%.

Testing the Mean of Large Samples

Another way to see this is to apply what we learned about the normal distribution in chapter 6. We know that in any normal distribution for a large population, 68.26% of the statistics for estimating parameters will fall within one standard error of the actual parameter; 95% of the estimates will fall between the mean and 1.96 standard errors; and 99% of the estimates will fall between the mean and 2.58 standard errors.

In table 20.2, I showed you a sample of the data from the study that Ryan, Borgatti, and I did on attitudes about environmental activism (Bernard at al. 2009). In that study, we interviewed a random sample of 609 adults from across the United States. The mean age of respondents in our sample was 44.21, sd 15.75. Only 591 respondents agreed to tell us their age, so the standard error of the mean is:

Because we have a large sample, we can calculate the 95% confidence limits using the z distribution in appendix A:

In other words, we expect that 95% of all samples of 591 taken from the millions of adults in the United States will fall between 42.94 and 45.48. As we saw in chapter 6, these numbers are the 95% confidence limits of the mean. As it happens, we know from the U.S. Census Bureau that the real average age of the adult (over-18) population in the United States in 1997 (when the survey was done) was 44.98.

Thus: (1) the sample statistic (x = 44.21%) and (2) the parameter (^ = 44.98%) both fall within the 95% confidence limits, and we can not reject the null hypothesis that our sample comes from a population whose average age is equal to 44.98%.

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