Significance Tests / Hypothesis Testing

Just as when we test hypotheses concerning two population means, we take a random sample from each population, of sizes n1 and n2, and compute the sample standard deviations s1 and s2. In this context the samples are always independent. The populations themselves must be normally distributed.

In hypothesis testing does a hypothesis not get rejected despite

The , , is a statement of what a statistical hypothesis test is set up to establish.

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Now instead of testing 1000 plant extracts, imagine that you are testing just one. If you are testing it to see if it kills beetle larvae, you know (based on everything you know about plant and beetle biology) there's a pretty good chance it will work, so you can be pretty sure that a P value less than 0.05 is a true positive. But if you are testing that one plant extract to see if it grows hair, which you know is very unlikely (based on everything you know about plants and hair), a P value less than 0.05 is almost certainly a false positive. In other words, if you expect that the null hypothesis is probably true, a statistically significant result is probably a false positive. This is sad; the most exciting, amazing, unexpected results in your experiments are probably just your data trying to make you jump to ridiculous conclusions. You should require a much lower P value to reject a null hypothesis that you think is probably true.

Next section: to Inferential statistics (testing hypotheses)

Now imagine that you are testing those extracts from 1000 different tropical plants to try to find one that will make hair grow. The reality (which you don't know) is that one of the extracts makes hair grow, and the other 999 don't. You do the 1000 experiments and do the 1000 frequentist statistical tests, and you use the traditional significance level of PPPP values less than 0.05, but almost all of them are false positives.

The corresponding confidence interval can be derived from the Test Statistic

Hypothesis testing is the subject of this chapter.

The 0.95 quantile for the standard normal distribution is 1.645. Thiscorresponds to the critical value for the usual upper-tailed z-testwith a significance level of 0.05. In our case, we are using thesample mean, not z, as the test statistic. The 0.95 quantile for thesampling distribution of the mean under the null hypothesis is 0 +1.645*(1/sqrt(5)) = 0.736, i.e., the critical value is the null valueplus the z-value times the standard error of the mean. We rejectHo if the sample mean is greater than 0.736.

a statistic based upon the sample variance was used in testing hypotheses concerning


The level of statistical significance is often expressed as the so-called p-value. Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p-value) of observing your sample results (or more extreme) given that the null hypothesis is true. Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?


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The critical value (or values for a two-tailed test) is determined bythe null distribution, the significance level, and the direction ofthe alternative hypothesis. The critical value c (for a upper-tailedtest) is determined such that the probability the test statistic is aslarge or larger than c is equal to the significance level(alpha). Thus c is the 1-alpha quantile of the null distribution ofthe test statistic, i.e., the probability the test statistic (thesample mean in this example) is less than or equal to c is1-alpha. The cth quantile of the null distribution isvisualized by the yellow vertical line and the significance level(0.05 here) is the red-colored area to the right of the yellowline. Thus if the sample mean > c, we say that the test statistic is“too far” from the hypothesized value to support the nullhypothesis. How is c computed?

Hypothesis Testing for a population mean Small Sample YouTube ViSta The Visual Statistics System

Here is what we do for directional hypothesis testing:

the opposite of the research hypothesis. The null hypothesis states that any effects observed after treatment (or associated with a predictor variable) are due to chance alone. Statistically, the question that is being answered is "If these samples came from the same population with regard to the outcome, how likely is the obtained result?"