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9-1 Statistical Tests Hypothesis Is the global temperature increasing Did the laws requiring hands free cell phone use result in a decrease in auto accidents? Is the housing market truly “cooling off”? These are all hypotheses.. Or suppositions. Hypothesis (cont) A Hypothesis proposes a model; if the data is consistent with the model, then there is no reason to think that the hypothesis is false. If the facts instead do not fit with the model, then perhaps the model might be wrong… The question is, how far off is acceptable? Hypothesis (cont) Hypotheses don’t involve estimating population parameters but instead are about the reasonableness of the value of the parameter. Null Hypotheses = Ho Parameter is correct as stated Alternative Hypotheses = H1 = HA Parameter is not correct as stated Hypothesis (cont) A Toyota salesman tells you that the Prius gets 45 mpg. You think he is exaggerating. Null Hypotheses = Ho : μ = .75 Alt. Hypotheses = H1(HA ) : μ < .75 Hypothesis (cont) An allergy drug has been tested and the claim is that 75% of patients in a large clinical trial find their symptoms significantly reduced. Null Hypotheses = Ho : p = po= .75 Alt. Hypotheses = H1(HA ) : p ≠ .75 Hypothesis (cont) Notice that the null is always an equal statement, while the alternate will be a greater than (right tailed), less than (left tailed) or not equal to (two tailed). This is important for reading the charts; the calculator will read the alternate hypothesis correctly without you saying left, right etc. Now… How are we going to decide whether to accept the null hypothesis or reject? That is, what are we really doing? How far off is data from the assumed statistic? What is the probability that the observed data is realistic with the assumed statistic? Test Statistic x μ z σ n With normal x and known σ, what is the probability that the value (or z value) exists with the assumed μ value? Four Steps 1. State the null hypothesis as well as the alternate. 2. Check the model (normal) 3. Calculate the test statistic. The goal is to get the P value (the probability that the observed statistic value could occur if the null hypothesis is correct). The smaller the P-value, the more likely the rejection of Ho. It suggests that results are less likely due to chance. 4. State the conclusion. Either reject or fail to reject the null hypothesis. Get in the habit of again drawing pictures to visualize the test (left, right, two) Mean Example Rosie, an aging sheep dog in Montana gets regular checkups from the local vet. Let x be a random variable that represents Rosie’s resting heart rate (beats per min). From past experience, the vet knows that x has a normal distribution with σ = 12 and, for dogs of this breed, μ = 115. Over the past six weeks, Rosie’s heart rate measured an average of 105.0 (six different measurements. The vet is concerned that Rosie’s heart rate may be slowing. Do the data indicate that this is the case? Steps Step 1: Ho : μ = 115 HA : μ < 115 Step 2: Independence? Likely. Randomization? indicates anything to the contrary. Nothing Step 3: find z -2.04 Step 4: As P( < 105.0) = P(z < -2.04)= 0.0207. That is, the probability of getting a sample mean below 105.0 is less than 2%, so reject Ho and accept HA. Note: we have NOT proved that the alternate is true. * Types of Errors Type 1: Rejecting the null hypothesis when it is actually true (false positive – diagnosing a healthy person with a disease, convicting an innocent person) Type 2: Accepting the null hypothesis when it is false. (false negative – diagnosing a sick person as free from disease, allowing a guilty person to go free) Levels of Significance α (alpha) = probability of rejecting Ho when it is true i.e. probability of a Type 1 error β (beta) = probability of accepting Ho when it is false i.e. probability of a Type 2 error Obviously we want α and β to be as small as possible Levels of significance (cont) The true situation What the evaluator does Reject Ho Accept Ho Ho true Ho false Type 1 = α Ok Ok Type 2 = β The power of the test is its ability to detect a false hypothesis. Power of the test = 1 – β The lower value for β, the more stringent the test. The power of the statistical test will increase as α increases, but a larger value of α increases the likelihood of a type 1 error. Level of Significance (cont) Typically α is decided in advance. Then the P-value is determined. P-value ≤ α then reject the null hypothesis and say that the data is statistically significant at the given level. P-value ≥ α then do not reject the null hypothesis. * The price to earnings ratio is an important tool in financial work. A random sample of 14 large US banks gave the following P/E ratios (source: Forbes) 24 16 22 14 12 13 17 22 15 19 23 13 11 18 The sample mean is approximately 17.1. Generally speaking, a low P/E ratio indicates a “value” stock. A recent copy of the Wall Street Journal indicated that the P/E ratio of the entire S&P 500 stock index is μ = 19. Let x b e a random variable representing the P/E ratio of all large U.S. bank stocks. We assume that x has a normal distribution and a σ = 4.5. Do these data indicate that the P/E ratio of all U.S. bank stocks is less than 19? Use α = 0.05.