Download Hypothesis Testing: Understanding Null, Alternative Hypotheses and Test Statistic and more Exams Statistics in PDF only on Docsity! Single Population Tests of Hypothesis In statistics, a Hypothesis is an idea, an assumption or a theory about the, behavior of one or more populations. A Hypothesis Test is a statistical procedure that involves formulating a hypothesis and using sample data to decide on the validity of the hypothesis. The Null Hypothesis is a statement about the population. It is referred to as H0. The Alternative Hypothesis is a statement about the population that is opposite to the null hypothesis. It is referred to as Ha. A Type I Error is made when you reject the null hypothesis and the null h th i i t ll t I thypo es s s ac ua y rue. n o er words, you incorrectly reject a true null hypothesis due to sample variation. A Type II Error is made when you fail to reject the null hypothesis and the null hypothesis is false. In other words, you continue to believe a false null hypothesis. The probability of making a Type I Error is called (alpha) α . The probability of making a Type II Error is called β (beta). The choice of α and β are not generally independent. For example, choosing a very small value of α could make a statistical test too “insensitive” to detect errors in an assumed parameter value. A Two-Tail Test of the population mean h th f ll i ll d lt tias e o ow ng nu an a erna ve hypotheses: H0: μ = [a specific number] H [ ifi b ]a: μ ≠ a spec c num er 05.0=α 10: =μoHx . : =μo vs H : <μaH 05.0=α =μ:oH >μ:aH A Two-Tail Test Is used to test if the parameter differs from a certain number in either direction, higher or lower . A Two-Tail Test The null hypothesis must always be set up so the “=“ theory is in the null hypothesis. The two-tail test is usually applied when the problem statement has the key words changed or different in the problem statement . An Upper-Tail Test Isusedto test if theparameter has shifted to a value more than a certain number. Must always be set up so the “=“ in the “≤” i li i i l d d i h llnequa ty s nc u e n t e nu hypothesis. It is used when the problem h h k d i dstatement as t e ey wor s ncrease , greater than. An Upper-Tail Test The theory that you wish to “prove” is placed into the alternative hypothesis. For tests on population means with normal input populations and known σ , the test statistic is given by: Z = (Xbar-μ0)/(σ/√n) vs. Zα or Zα/2 Explanations…. For tests on population means based on small samples where the input population is normal and σ is unknown, the test statistic is given by: t = (Xbar-μ0)/(s/√n) vs. tα, n-1 or tα/2, n-1 For large sample tests on population ti th t t t ti ti i ipropor ons e es s a s c s g ven by: Z = (phat-p0) / (p0(1-p0)/n)1/2 Z Zvs. α, n-1 or α/2, n-1 • The p value is defined to be the smallest value of α for which you can reject H0. It provides a measure of the “strength” of the test conclusion.
