A bowler is bragging that his average is ``at least 180''. We observe him play three games, his scores are 125, 155, 140 ( , S=15). Should we accept or reject his claim? We should reject it. Why? Because a sample average as low as 140 is unlikely from a 180 bowler. How unlikely? A 180 bowler will bowl a 3-game average of 140 or lower only 2 percent of the time. Is 2 percent of the time unlikely? In statistics, yes. 5 percent or less is called statistically significant.
The decision making process above is called a test of significance . Here is the way a statistical report would formally present the test, in numbered stages.
Here is a more detailed description of each component of the test of significance above.
H0 and H1 are called the null hypothesis and alternative hypothesis , respectively. The two hypotheses describe the two possibilities: the claim is true ( ), or the claim is false (). Note that
(i) the two hypotheses are statements about the population (ii) the two hypotheses are complementary; if one occurs the other does not (iii) the hypothesis with the equal sign is the null hypothesisA test of significance rejects (population statement) H0 and concludes H1 if the sample values are ``significantly far from H0 and inside H1''. Hence, we reject and conclude if is some significant distance below 180. How far below 180 is ``significant''? The test statistic helps us determine where to draw the line in the sand.
Using the t curve with n-1=2 degrees of freedom, the likelihood of chance variation resulting in a t-value as low as -4.62 is .02.
Since this likelihood is less than .05 (the standard for statistical significance), we declare that ``t=-4.62 is significantly below 0'', or that `` is significantly below 180'', and reject . In general, the P-value is the total area under the curve more extreme than t in support of H1. If t is ``deep in H1 territory'', then the P-value is small. If P-value .05, we reject H0 with statistical significance. If P-value .01, we reject H0 with high statistical significance. If P-value is larger than .05, we accept H0.
Summary of the lower-tailed t-test for :