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Our goal is to assess the evidence provided by the data in favor of some claim about the population. Section 6.2Tests of Significance

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Stating Hypotheses Hypotheses are statements about the parameter. H o is called the “Null Hypothesis”: It is the statement being tested. Usually a statement of “no effect” or “no difference.” Always includes equality. H a is called the “Alternative Hypothesis”: It is the statement we suspect or hope is true. It expresses the effect we hope to find evidence for. Never includes equality. May be one-sided ( “ ” ) or two-sided ( “ ≠ ” )

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We use the test statistic (in this case a z-score) to calculate the probability that we could get a statistic as extreme or more extreme as the one we got from our sample data if H o were true. This probability is called the p-value of the test. The smaller the p-value, the stronger the evidence against H o provided by the data.

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There will be five steps in doing such a test: 1.State the hypotheses 2.Calculate the value of the test statistic 3.Calculate the p-value of the test 4.Make a decision about the null hypothesis (we will decide whether or not to reject H o ) 5.State a conclusion (the conclusion will address H a )

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The One-Sample z Test for a Population Mean To test the hypothesis H o : µ = µ o based on a SRS of size n from a population with unknown mean µ and known standard deviation σ, compute the test statistic In terms of a standard normal random variable Z, the P-value for a test of H o against H a : µ > µ o is P(Z ≥ z) H a : µ < µ o is P(Z ≤ z) H a : µ ≠ µ o is 2 * P(Z ≥ |z| )

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the P-value for a test of H o against H a : µ > µ o is P(Z ≥ z) The shaded area to the right of z is the p-value of this “one-sided” test. z

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the P-value for a test of H o against H a : µ < µ o is P(Z ≤ z) The shaded area to the left of z is the p-value of this “one-sided” test. z

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the P-value for a test of H o against H a : µ ≠ µ o is 2 * P(Z ≥ |z| ) The total shaded area in both ends is the p-value of this “two-sided” test. – |z||z|

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P-values are exact when the population is normally distributed. They are approximate when n is large (at least 30) in other cases.

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Decision:Ho is TrueHo is False Reject HoType I errorCorrect decision Fail to Reject Ho Correct decision Type II error If we reject Ho when Ho is in fact true, this is a Type I error. If we fail to reject Ho when in fact Ho is false, this is a Type II error. The significance level, α, is the probability of making a Type I error – of rejecting Ho when Ho is really true. α = P(rejecting Ho when it’s really true)

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We never “prove” that H o is true – we only are unable to find enough evidence to indicate that H o should be rejected. In our court system, a defendant is considered to be innocent until proven guilty (beyond a reasonable doubt). O.J. Simpson was not convicted. Does this prove that he is innocent? No – it just means that the jury did not find enough evidence to convict him.

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Verdict: Defendant is Innocent Defendant is Guilty ConvictIncorrect verdictCorrect verdict Fail to Convict Correct verdict (although you have not actually “proved” that the defendant is innocent!) Incorrect verdict (but you have not “proved” that the defendant is innocent – s/he is guilty, but there was not enough evidence to convict them.)

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We never can “prove” anything – we can only assess probabilities and make decisions based on those probabilities.

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We use the sample data to draw a conclusion about the hypotheses. If the sample data results are quite different from what the null hypothesis H o claims, then we suspect that the difference is due to some other effect than just random chance. In this case, we reject H o and say that the data are statistically significant. If we do not reject the null hypothesis, we say that the data are not statistically significant.

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The p-value of the test gives us the smallest significance level α for which the sample data tell us to reject H o ; i.e., the smallest level at which the data are statistically significant. The advantage of knowing the p-value is that we know all levels of significance for which the observed sample statistic tells us to reject H o. Many research journals require authors to include the p-value of the observed sample statistic. Then readers will have more information and will know the test conclusion for any pre-set level of significance.

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p-value is ≤ α we say the data are statistically significant at level α, and we reject H o in favor of H a. Our conclusion: there is sufficient evidence at the α level of significance to support H a. p-value is > α we say the data are not statistically significant at level α, and we fail to reject Ho in favor of Ha. Our conclusion: there is not sufficient evidence at the α level of significance to support H a.

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Introduction to Biostatistics and Bioinformatics

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