Tests of significance: The basics BPS chapter 15 © 2006 W.H. Freeman and Company

Objectives (BPS chapter 15) Tests of significance: the basics  The reasoning of tests of significance  Stating hypotheses  Test statistics  P-values  Statistical significance  Tests for a population mean  Using tables of critical values  Tests from confidence intervals

We have seen that the properties of the sampling distribution of the sample mean help us estimate a range of likely values for population mean . (This is what we did when we found a confidence interval for .) We can also rely on the properties of the sampling distribution to test hypotheses. Example: You are in charge of quality control in your food company. You sample randomly four packs of cherry tomatoes, each labeled 1/2 lb. (227 g). Suppose you find that the average weight from your four boxes is 225 g. Obviously, we cannot expect boxes filled with whole tomatoes to all weigh exactly half a pound. The question is this:  Is the somewhat smaller weight simply due to chance variation, or  is it evidence that the calibrating machine that sorts cherry tomatoes into packs needs revision?

Hypotheses tests A test of statistical significance tests a specific hypothesis using sample data to decide on the validity of the hypothesis. In statistics, a hypothesis is an assumption, or a theory about the characteristics of one or more variables in one or more populations. What you want to know: Is the calibrating machine that sorts cherry tomatoes into packs out of order? The same question reframed statistically: Is the population mean µ for the distribution of weights of cherry tomato packages equal to 227 g (i.e., half a pound)?

The null hypothesis is the statement being tested. It is a statement of “no effect” or “no difference,” and it is labeled H 0. The significance test will assess the weight of the evidence against H 0 The alternative hypothesis is the claim we are trying to find evidence for, and it is labeled H a. Weight of cherry tomato packs: H 0 : µ = 227 g (µ is the average weight of the population of packs) H a : µ ≠ 227 g (µ is either larger or smaller) x Bar is the evidence that we will use to decide between H 0 and H a.

The Logic of Hypothesis Testing  Assume the null hypothesis is true (although it may not be)!  Determine how likely it is to get data as extreme as what you got IF this is the case (i.e. IF the null hypothesis is TRUE).  If it is very unlikely to get data as extreme as what you actually got, then you begin to doubt the assumption that the null hypothesis is true (i.e. you will reject the null hypothesis).  If it is not very unlikely to get data as extreme as what you got, then you have no reason to doubt the null hypothesis (i.e. you will fail to reject the null hypothesis).  What does “data as extreme as what you got” mean?  It means all data that favors the alternative hypothesis at least as strongly as yours does!

Does the packaging machine need revision?  H 0 : µ = 227 g versus H a : µ ≠ 227 g  What is the probability of drawing a random sample such as yours if H 0 is true?  Recall: x Bar = 225, n = 4  We will need to make an assumption (for now) If we assume the null hypothesis is true (i.e. the packaging machine is working correctly), what is the sampling distribution of the sample means for samples of size 4? What is the z-score of the sample mean you got? This value is called the test statistic. X = population variable = weight of a box of c.t.’s X is assumed normal

Does the packaging machine need revision?  H 0 : µ = 227 g versus H a : µ ≠ 227 g  What is the probability of drawing a random sample such as yours if H 0 is true?  Recall:  We will need to make an assumption (for now) What are other z-scores as extreme as your data’s z-score in the direction of the alternative hypothesis? The alternative hypothesis is that  = 227, so z-scores that are as extreme as ours in the direction of Ha are z-scores that are at least as far from 0 as ours. X = basic measurement = weight of a box of c.t.’s 0 z = 0.8z = -0.8 Density of z (std. normal) Remember, the null hypothesis is  = 227, which in standard coordinates is z = 0.

Does the packaging machine need revision?  H 0 : µ = 227 g versus H a : µ ≠ 227 g  What is the probability of drawing a random sample such as yours if H 0 is true?  Recall:  We will need to make an assumption (for now) How likely is it to get a sample mean as extreme as yours? This probability is called the P -value of the test. Do we reject or fail to reject the null hypothesis? Do you think the machine needs recalibration? X = basic measurement = weight of a box of c.t.’s Density of z (std. normal) 0 z = 0.8z = -0.8 P = 1 – normCDF(-0.8,0.8) = Need to discuss the interpretation of the P -value, and significance levels

Interpreting a P-value P is the probability that random variation alone (coming from the sampling process) accounts for the difference between the null hypothesis and the observed x Bar value.  A small P-value implies that random variation because of the sampling process alone is not likely to account for the observed difference.  With a small P-value, we reject H 0. The true property of the population is significantly different from what was stated in H 0. Thus small P-values are strong evidence AGAINST H 0. But how small is small enough?

The significance level  The significance level, α, is the largest P-value tolerated for rejecting a true null hypothesis (how much evidence against H 0 we require). This value is decided on before conducting the test.  If the P-value is equal to or less than α (P ≤ α), then we reject H 0.  If the P-value is greater than α (P > α), then we fail to reject H 0. Does the packaging machine need revision? Two-sided test. The P-value is 42.37%. *If α had been set to 10%, then we would fail to reject H a. *If α had been set to 5%, then we would fail to reject H a. *If α had been set to 1%, then we would fail to reject H a. Common  values are  = 10%,  = 5%, and  = 1% At any of these significance levels, our evidence is not significant that  is different from 227.

One-sided and two-sided tests  A two-tail or two-sided test of the population mean has these null and alternative hypotheses: H 0 : µ = [a specific number] H a : µ  [a specific number]  A one-tail or one-sided test of a population mean has these null and alternative hypotheses: H 0 : µ = [a specific number] H a : µ < [a specific number] OR H 0 : µ = [a specific number] H a : µ > [a specific number] The FDA tests whether a generic drug has an absorption extent similar to the known absorption extent of the brand-name drug it is copying. Higher or lower absorption would both be problematic, thus we test: H 0 : µ generic = µ brand H a : µ generic  µ brand two-sided

How to determine hypotheses? What determines the choice of a one-sided versus two-sided test is what we know about the problem before we perform a test of statistical significance. A health advocacy group tests whether the mean nicotine content of a brand of cigarettes is greater than the advertised value of 1.4 mg. Here, the health advocacy group suspects that cigarette manufacturers sell cigarettes with a nicotine content higher than what they advertise in order to better addict consumers to their products and maintain revenues. Thus, this is a one-sided test:H 0 : µ = 1.4 mgH a : µ > 1.4 mg It is important to make that choice before performing the test or else you could make a choice of “convenience” or fall in circular logic. Let’s work problems 15.5, 15.6, and 15.7!

P-value in one-sided and two-sided tests To calculate the P-value for a two-sided test, you can use the symmetry of the normal curve. Find the P-value for a one-sided test and double it. One-sided (one-tailed) test Two-sided (two-tailed) test

Let’s Work Problems 15.2, 15.4, 15.10, 15.14, and State teacher’s hypotheses Determine the test statistic for the z-test Determine the p-value of the test Make a decision regarding the hypotheses Interpret your results in the context of this problem

Confidence intervals to test hypotheses Because a two-sided test is symmetrical, you can also use a confidence interval to test a two-sided hypothesis. α / 2 In a two-sided test, C = 1 – α. C confidence level α significance level Packs of cherry tomatoes (σ  = 5 g): H 0 : µ = 227 g versus H a : µ ≠ 227 g Sample average 222 g. 95% CI for µ = 222 ± 1.96*5/√4 = 222 g ± 4.9 g xBar = 225 g does belong to the 95% CI (217.1 to g). Thus, we accept H 0.

Ex: Your sample gives a 99% confidence interval of. With 99% confidence, could samples be from populations with µ =0.86? µ =0.85? 99% C.I. Logic of confidence interval test Cannot reject H 0 :  = 0.85 Reject H 0 :  = 0.86 A confidence interval gives a black and white answer: Reject or don’t reject H 0. But it also estimates a range of likely values for the true population mean µ. A P-value quantifies how strong the evidence is against the H 0. But if you reject H 0, it doesn’t provide any information about the true population mean µ.

Tests for a population mean µ defined by H 0 Sampling distribution σ/√n To test the hypothesis H 0 : µ = µ 0 based on an SRS of size n from a Normal population with unknown mean µ and known standard deviation σ, we rely on the properties of the sampling distribution N(µ 0, σ√n). The P-value is the area under the sampling distribution for values at least as extreme, in the direction of H a, as that of our random sample. We can calculate a z-score for the data or use the normalcdf key directly to find the p-value