AP STATISTICS LESSON 10 – 2 DAY 1 TEST OF SIGNIFICANCE

ESSENTIAL QUESTION: What is a test of significance and how are they used? Objectives: To create tests of significance that will assess the evidence provided by data about some claim concerning a population. To draw conclusions from the tests of significance.

Tests of Significance The second common type of inference, called test of significance, has a different goal: To assess the evidence provided by data about some claim concerning a population.

Example 10.8 Page 559 I’m a Great Free-Throw Shooter I claim to be an 80% free throw shooter. I make 8 of 20 free throws. Is this possible if my claim is true? Significance tests use an elaborate vocabulary, but the basic idea is simple: An outcome that would rarely happen if a claim were true is good evidence that the claim is not true.

Example 10.9 Page 560 Sweetening Colas Significance test – A test of significance asks does the sample results x = 1.02 reflect a real loss of sweetness or could we easily get the outcome just by chance? Null hypothesis – state the null hypothesis. The null hypothesis says that there is no effect or no change in the population. If the null hypothesis is true, the sample result is just chance at work. H o : μ = 0 We write H o, read “H-nought,” to indicate the null hypothesis

Example 10.9 (continued…) The effect suspect is true, the alternative to “no effect” or “no change”, is described by the alternative hypothesis. We suspect that the cola does lose sweetness. In terms of the mean sweetness loss μ, the alternative hypothesis is H a : μ > 0 Suppose for the sake of argument that the null hypothesis is true, and that on the average there is no loss of sweetness. Is the sample outcome x = 1.02 surprisingly large under that supposition? If it is, that’s evidence against H o and in favor of H a ?.

How Does a Significance Test Work? A significance test works by asking how unlikely the observed outcome would be if the null hypothesis were really true? We measure the strength of the evidence against H o by the probability under the normal curve in figure to the right of the observed x. This probability is called the P-value. It is the probability of a result at least as far out as the result we actually got? The lower this probability, the more surprising our result, and the stronger the evidence against the null hypothesis.

P-values Small P-values are evidence against H o because they say that the observed result is unlikely to occur just by chance. Large P-values fail to give evidence against H o. How small must a P-value be in order to persuade us? There’s no fixed rule. But the level 0.05 (a result that would occur no more than once in 20 tries just by chance) is a common rule of thumb. A result with a small P-value, say less than 0.05 is called statistically significant.

Outline of a Test Here is the reasoning of a significance test in outline form: Describe the effect you are searching for in terms of a population parameter like the mean μ. (Never state a hypothesis of a sample statistic like x.) From the data, calculate a statistic like x that estimates the parameter. Is the value of this statistic far from the parameter value stated by the null hypothesis? If so, the data give evidence that the null hypothesis is false and that the effect you are looking for is really there. The P-value says how unlikely a result at least as extreme as the one we observed would be if the null hypothesis were true. Results with small P-value would rarely occur if the null hypothesis were true. We call such results statistically significant.