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Section 10.2: Tests of Significance Hypothesis Testing Null and Alternative Hypothesis P-value Statistically Significant

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Goal of Tests of Significance To access the evidence provided by data about some claim concerning a population.

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Not sure if you believe me, you hand me a ball and ask me to serve 20 balls. I make 8 out of those 20 serves. Aha, you say. Someone who makes 80% of their serves would hardly ever only make 8 out of 20 shots, so I dont believe your claim! Suppose I claim to consistently make 80% of my volleyball serves.

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Your reasoning is based on what would happen if my claim were true and we repeated the sample of 20 serves many times. The probability of making 8/20 serves if I really make 80% is This probability is so small that you are convinced my statement cant be true.

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Basic Idea Behind Significance Tests An outcome that would rarely happen if a claim were true is good evidence that the claim is not true.

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Situation Read and summarize the situation given in: EXAMPLE 10.9, pg 560.

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Situation Summary Matched Pairs Experiment Design. Data: Sweetness Before score – Sweetness after score Is this good evidence that the cola lost sweetness in storage?

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Situation Continued = 1.02 What is the meaning of: a positive mean? a negative mean? a mean equal to 0?

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Test of Significance We dont know how conclusive or significant our results are. We must perform a significant test that asks the following questions: Does our reflect a real sweetness loss? Could we easily get our by chance?

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Step 1: Identify the Parameter We are trying to identify the mean µ loss in sweetness for the population. Note: we are always trying to draw conclusions about a parameter of a population, so we must always be stating tests in terms of the parameter.

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Step 2: State The Null Hypothesis The Null Hypothesis (H 0 ) says that there is no effect or change in the population. If H 0 is true, then the sample result is just chance at work. In our situation, the null hypothesis would be that the cola (population) does not lose sweetness. Written: H 0 : µ = 0

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Step 3: State the Alternative Hypothesis The effect we suspect is true, the alternative to no effect or no change is described by the Alternative Hypothesis (H a ) In our situation, the alternative hypothesis is that the cola does lose sweetness. Written: H a : µ > 0

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Significant Test Reasoning If H 0 is true, how surprising is our outcome of ? If the outcome is surprisingly large, then that is evidence against H 0 and for H a

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Sampling Distribution of if H 0 is true. Central Limit Theorem would dictate that the sampling distribution would be In our situation, the sampling distribution of x-bar would be N(0, 0.316). Assuming we know the population = 1

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Calculate P-Value If H 0 is true, calculate the probability that one would get an outcome of This probability is called the p-value. Standardize 1.02 if N(0,.316), then use Table A.

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P-Value Definition The probability, assuming H 0 is true, that the test statistic would take a value as extreme or more extreme than that actually observed is called the p-value. The smaller the p-value, the stronger the evidence against H 0.

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REMEMBER ! ! ! ! ! ! Hypothesis tests (or Significance Tests) find p-values. P-values describe how probable the NULL HYPOTHESIS is based on the sample statistic. WE ARE ALWAYS TESTING WHETHER THE NULL HYPOTHESIS IS PROBABLE OR NOT!

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Check for Understanding A hypothesis test was calculated. Answer whether there is significant evidence to reject the H 0 and accept the the H a. P-value =.45P-value =.0001 P-value =.03P-value =.21 P-value =.99P-value = E-12 P-value =.10 P-value =.11

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Writing Your Conclusion There is a (p-value) probability that our sample statistic of (sample mean or proportion) would occur if the Null Hypothesis were true and the population parameter was (pop. Mean or prop). This suggests that Null hypothesis is (likely, somewhat likely, not likely). If not likely, then describe the alternative hyp. As more likely.

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Statistically Significant To describe how significant the evidence against the H 0 is, we establish a significance level ( ). If the p-value is as small or smaller than the established, we say the evidence to reject the H 0 is statistically significant at the -level.

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Statistical Significance The smaller the significance level, the more evidence against the H 0 (or the more likely the H a ). Significance does not mean important; it means that the outcome is not likely to occur just by chance.

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Check for Understanding If the following p-values were calculated, state whether there is statistically significant evidence against the H 0 at the 0.10, 0.05, or 0.01 level. P-value =.45 P-value =.0001 P-value =.03 P-value =.21 P-value =.99 P-value = E-12 P-value =.10 P-value =.11

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Summary We stated a null hypothesis (no loss) We calculated a test statistic. We stated alternative hypothesis. We found the probability of getting the test statistic if H 0 was true. (p-value) Since p-value was very low, it was statistically significant evidence that the null hypothesis was false and the alternative true.

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Assignment Exercises: – 31, 33 – 37 Read pp for reference.

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