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STAT 231
Matthias Schonlau

Chapter 4 Testing Statistical Hypotheses Matthias Schonlau Stat 231 Overview • Introduction • Testing statistical hypotheses – Example T-test • Significance levels – Example: Spending in Coffee shops • One sided alternative Hypotheses – Example: Survivor 2011 – Example: Nadal vs Murray Introduction • The purpose of statistical inference is to draw conclusions about a population parameter – Point estimates • What is the “best” estimate ? (maximum likelihood) – Confidence intervals • What are plausible values for the population parameter? – Hypothesis tests • Test whether the population parameter could be a specific value Introduction • In law we are used to making hypotheses: H : Defendant is innocent 0 H1: Defendant is guilty • H zero is pronounced “H naught” • We find the defendant “guilty” or “not guilty” • We cannot prove the defendant is innocent. Introduction • Legal trials use the concept of “beyond reasonable doubt” – If a suspect is acquitted, this does not mean he/ she didn’t do it. – It means the jury was not certain “beyond a reasonable doubt” • In the US, for the same crime you can be found – not guilty in a criminal trial but – guilty in the civil trial The criminal trial requires a greater degree of evidence Testing statistical Hypotheses • Statistical hypotheses test whether a population parameter could be a specific value µ 0 H 0 0 µµ= H 1 0 µµ≠ Testing statistical hypotheses • In deductive logic we establish have whether something is true or not. • In statistics, we never know for sure whether a hypothesis is correct or not. • We can establish whether the hypothesis is very unlikely to occur by chance giventhe data • Given the hypothesis, what is the probability of observing dobsr something more extreme? p =D(d ≥ obs o Testing statistical hypotheses • The famous t-test • Assume YG~ (,)µσ • Calculate the probability or p-value PT tT≥t~ ) n− • where y −µ t = 0 s/ • t measures the distance of the hypothesized value to y Example T-test • A toy example with only 3 observations. • Test whether the population mean is 0. H 0 µ =0 H : µ ≠0 1 Example T-test • Compute the t-statistic: • n-1=2 degrees of freedom • What is the probability of getting a |t| value of 4.157 or larger? Example T-test t-dist->red (df = 2) normal dist->blue -5 -4 -3 -2 -1 0 1 2 3 4 5 Critical value of t = ±4.16 (alpha=.0533, 2-tail) • What is the probability of getting a |t| value of 4.157 or larger? Example T-test • To get the probability, we need a computer: • The probability is 5.3%. Example T-test • From the t-table in the notes, we know t2,α=0.054.3 t = 2.92 2,α=0.10 • The observed value is in the bounded by these two values. • Therefore we know the probability is between 5% and 10%. Example T-test • We reject the null-hypothesis, if the observed statistics is unlikely given the null hypothesis. • Small p-values are strong evidence against the hypothesis • Is 5.3% unlikely enough? Testing statistical hypotheses • There are thresholds that are typically used – those threshold are called significance levels Significance levels Evidence levels against the hypothesis Significance levels Interpretation p>0.1 No evidence 0.05 One –sided alternative hypotheses • If a new drug is introduced we would like to know whether the new drug performs better than the current one. H 0 effectiveness new drug ≤ effectiveness old drug H : effectivness new drug > effectieness old drug 1 • This is called a one sided hypotheses One –sided alternative hypotheses H 0 0 µµ= Shaded area is the probability of H : µµ≠ a more extreme sample average)or 1 0 under the null hypothesis One –sided alternative hypotheses H 0 0 µµ≤ Shaded area is the probability of H : µµ> a more extreme sample average)or 1 0 under the null hypothesis One –sided alternative hypotheses H 0 0 µµ≥ observing the sample average (orf H : µµ< a more extreme sample average) 1 0 under the null hypothesis One –sided alternative hypotheses Thes
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