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Tests of Significance Chapter 11

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Confidence intervals are used to estimate a population parameter. Tests of significance assess the evidence provided by the data about some claim concerning the population. Example: I claim that I make 80% of my free throws. You do not believe me, so we go to the gym and I shoot 20 free throws, making only 8! You use this result as backing up your claim that I do not make 80% of free throws, because if I truly made 80% of free throws I would be very unlikely to make only 8. What is the probability that I only make 8 if my claim was true? The reasoning of significance tests, like that of confidence intervals, is based on what we would expect to happen if we repeated an experiment/sample many times.

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Example: Diet cola uses artificial sweeteners to avoid sugar. These sweeteners gradually lose sweetness over time. To test a new diet drink we have 10 tasters sample the drink, store the drink 4 months, and have them taste the drink again. The drinks are scored on a sweetness scale and the difference between the first and second taste is our score for the loss of sweetness (matched pairs design). Large scores represent large sweetness losses. Below are the scores: 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3 Does the data provide good evidence that the cola lost sweetness in storage? How can you answer this question?

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Is this large enough to be significant. A test of significance asks: Does the sample mean 1.02 reflect a real loss of sweetness or could it just have occurred by chance? A test of significance starts with a careful statement of these two alternatives. First determine the parameter of interest. We always draw conclusions about a population parameter! In this case the population mean μ is the mean loss of sweetness. Our 10 tasters provide a sample of this population.

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Now we can state the null hypothesis. The null hypothesis states that there is no effect or no change in the population, i.e. drink did not lose sweetness: The effect that we suspect to be true will be our alternative hypothesis. The alternative hypothesis states that there is a change in the population, i.e. drink did lose sweetness one-sided alternative To answer the question we will use our knowledge of how would behave in repeated samples. To be able to do that we need to know the standard deviation of the population. Let’s assume

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Under the assumption that the null hypothesis is true, we can now draw the following normal curve: Label the curve based on the hypothesized mean, 0 and σ/sqrt(10)=0.32 0.32-0.320.64-0.640.96 -0.96

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Inference Toolbox The medical director of a large company looks at the medical records of 72 male executives aged 35-44. He finds that their average blood pressure is 126.07. If the average blood pressure of all males aged 35-44 is 128 with a standard deviation of 15, does this sample show evidence that the blood pressure of executives is different from the regular population? 72 126.07 n=72 128 15

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Step 1: Hypotheses State the population of interest & parameter being studied. Population: all male executives between 35-44 years of age in the company. Parameter: μ = mean blood pressure State your hypotheses. H 0 : μ=128 H a : μ 128

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Step 2: Conditions/Procedure State the procedure you will be using. One-sample z -test Verify the conditions required to use it. S RS I ndependence N ormality Once conditions are verified, state the distribution

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Step 3: Calculate test statistic & p-value To calculate the p-value, you are calculating, P(x >,<,or μ0)

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Step 4: Interpretation Connect Conclude Context Relate the p-value to the level. Reject or fail to reject H 0. How does the conclusion affect the claim?

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SRS A simple random sample is the cornerstone to making inferences. A srs is the necessary condition that ensures that as much variability between the individuals in the population is “cancelled” out when the individuals are randomly selected. This way, any conclusions can be “inferred” to the population.

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Normality Being able to satisfy the normality condition enables you to: Use a known and well established distribution. Calculate probabilities – a Chapter 2 topic To verify this condition: Sample Means, xx Sample Proportions, p

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Sample Means, x You KNOW the population is Normally Distributed. You DON’T KNOW the shape of the distribution of the population x x

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