We have a sample and a point estimate x 2 lets say we

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Unformatted text preview: e get a p-value 2 × P (Z > z = 0.33) = 2 × 0.3707 = 0.7414. This value is bigger than any reasonable α so reach at the same conclusion as before. We fail to reject the null hypothesis. Let’s go back one slide and see this on a picture. Utku Suleymanoglu (UMich) Hypothesis Testing 25 / 37 Testing Hypothesis about the Population Mean: σ known Hypothesis Testing Fundamentals Recap Before we go on to different cases, let’s repeat the general idea of hypothesis testing: We have an hypothetical value for a population parameter (µ = µ0 ) as a claim and we want to test this. We have a sample and a point estimate x = 2, let’s say. ¯ We know the sampling distribution of x assuming the claim is true from chapter 6. ¯ Then we can evaluate the probability of x or a similar draw from this sampling ¯ distribution. To do that we need to transform our normal random variable to standard normal, this gives us z-statistic. Then we can either calculate the probability associated with the z-statistic and see if it is small or big (p-value approach) compare it to some critical z-value so that we can assess how far off it is from the claimed value. (critical value approach) Either way, based on the assumption the claim is true, we assess the correctness of the claim by comparing it to what we observe in the data. If two are “different enough”, we say the claim is (probably) not correct. Utku Suleymanoglu (UMich) Hypothesis Testing 26 / 37 σ not Known Case 2: σ not known, population normal When σ is not known, we can use s , sample standard deviation, instead. Just like we did before. . . for CI’s. But remember, we need a modification to make this work. We need to use t-distribution instead of standard normal distribution. p-value approach is hard to perform with t-distribution, so we will just use the critical value approach. Let’s do a one tailed example first . . . Utku Suleymanoglu (UMich) Hypothesis Testing 27 / 37 σ not Known One-Tailed t-Tests For one tailed tests involving hypotheses: Left tailed: H 0 : µ ≥ µ0 H1 :µ < µ0 Right tailed: H 0 : µ ≤ µ0 H1 :µ > µ0 We reject the null the hypothesis if test statistic: t= x − µ0 ¯ √ s/ n is such that t < −tα,n−1 for left-tailed tests t > tα,n−1 for right-tailed tests where tα is the critical value with probability α in the upper tail. Utku Suleymanoglu (UMich) Hypothesis Testing 28 / 37 σ not Known Example Suppose you are interested in the labor supply of elderly. You have a data set that consists of 900 people aged 65. They are measu...
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