Hypothesis Testing Using z- and t-tests In hypothesis testing, one ...

B. Weaver (24-Aug-2010) z- and t-tests ... 6And from a table of the standard normal distribution (or using a computer program, as I did), wecan see that the probability of a z-score greater than or equal to 2.667 = 0.0038. Translating thatback to the original units, we could say that the probability of getting a sample mean of 108 (orgreater) is .0038 (assuming that the 25 children are a random sample from the generalpopulation).For part (b), do the same, but replace 108 with 92:X −µ X −µXX92 −100 −8z = = = = = − 2.667(1.7)σ σX15 3Xn 25Because the standard normal distribution is symmetrical about 0, the probability of a z-scoreequal to or less than -2.667 is the same as the probability of a z-score equal to or greater than2.667. So, the probability of a sample mean less than or equal to 92 is also equal to 0.0038. Hadwe asked for the probability of a sample mean that is either 108 or greater, or 92 or less, theanswer would be 0.0038 + 0.0038 = 0.0076.Part (c) above amounts to the same thing as asking, "What sample mean corresponds to a z-scoreof 1.645?", because we know that pz≥ ( 1.645) = 0.05. We can start out with the usual z-scoreformula, but need to rearrange the terms a bit, because we know that z = 1.645, and are trying todetermine the corresponding value of X .z =X − µσXX{ cross-multiply to get to next line }z σ = X −µ { add µ to both sides }X X Xz σ + µ = X { switch sides }XX(1.8)X( )= zσ + µ = 1.645 15 25 + 100 = 104.935XXSo, had we obtained a sample mean of 105, we could have concluded that the probability of amean that high or higher was .05 (or 5%).For part (d), because of the symmetry of the standard normal distribution about 0, we would usethe same method, but substituting -1.645 for 1.645. This would yield an answer of 100 - 4.935 =95.065. So the probability of a sample mean less than or equal to 95 is also 5%.The single sample z-testIt is now time to translate what we have just been doing into the formal terminology ofhypothesis testing. In hypothesis testing, one has two hypotheses: The null hypothesis, and the