when testing the null h0 1 if x values that are

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Unformatted text preview: r test statistic does not necessarily mean right away that you are more likely to reject the null. Generally, a large (in absolute value) test statistic in the direction of the tail of the test is more likely to reject the null hypothesis. Utku Suleymanoglu (UMich) Hypothesis Testing 18 / 39 What determines test results? Why Left/Right Tail? Why do we focus on the left-tail if the null hypothesis is µ ≥ µ0 ?. When testing the null H0 : µ ≥ 1: if x values that are larger than 1 are surely supporting the null. ¯ We saw this in the previous example. Got x = 100? It is fine, because the null is µ ≥ 1, they are in agreement, so we should not reject ¯ the null. We focus on the left-tail, because values of x less than 1 is still possible. Notice when ¯ x < 1 = µ0 ⇒ z < 0. As x gets smaller, they also get more and more imporabable. ¯ ¯ If µ = 1, x = 0.9 is possible, so is x = 0.1. As we move to lower x ’s, the probability shrinks. We ¯ ¯ ¯ can calculate this probability, and we set a criteria for a low probability. This is a criteria for: x is too small to have come from a µ which is 1 or larger. ¯ Notice: If an x value is too small if µ = 1, it will be even less improbable if µ were larger. ¯ This is why we pay attention to the left tail. Exercise: For H0 : µ ≥ 1, and SE (¯ ) = 0.3, α = 0.05, find the largest x where we reject H0 . x ¯ Exercise: To make sure you understand this, do the right-tailed version yourself. Utku Suleymanoglu (UMich) Hypothesis Testing 19 / 39 What determines test results? Precision of the sampling distribution: Standard error of the sample statistic determines how much variation we should expect in x from sample to sample. If it is high, it is more likely to have ¯ samples with x that deviates largely from the hypothesized value. ¯ Consider the formula for test statistic: z= x − µ0 ¯ σ √ n = x − µ0 ¯ SE (¯ ) x Remember from Chapter 7: as the standard error of the mean (denominator above) decreases, we say we measure x more precisely. The same x − µ0 di...
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