ABSTRACT
The simplest example would be the tossing of a fair coin. We would expect the proportion of heads to be equal to the proportion of tails. Therefore, we would expect a head to occur 50% of the time, or have a proportion of 0.50. Our null hypothesis is that we are presented with a fair coin: H0: Pheads = 0.50
The only alternative is that the likelihood of tossing a head is something other than 50%. H1: Pheads ≠ 0.50 If we toss the coin 100 times and this results in 50 heads and 50 tails, the numerator of the above ratio (Eq. 15.1) would be zero, resulting in z = 0. As the discrepancy between what we observe and what we expect (50% heads) increases, the resultant zvalue will increase until it eventually becomes large enough to be significant. Significance is determined using the critical z-values for a normalized distribution, previously discussed in Chapter 6. For example, from Table B2 in Appendix B, +1.96 or −1.96 are the critical values in the case of a 95% level of confidence. For a 99% level of confidence the critical z-values would be +2.58 or –2.58. In the one-sample case the proportions found for a single sample are compared to a theoretical population to determine if the sample is selected from that same population. H0: pˆ = P0 H1: pˆ ≠ P0 The test statistic is as follows:
n )P(1P
P pˆ =z 00
−
Eq. 15.2
where P0 is the expected proportion for the outcome, 1 − P0 is the complement proportion for the “not” outcome, pˆ is the observed proportion of outcomes in the sample, and n is the number of observations (sample size). The decision rule is
with α = ___, reject H0 if z > z(1-α/2) or z < −z(1-α/2) where z(1-α/2) = 1.96 for α = 0.05 or 2.58 for α = 0.01. Like the t-test, this is a twotailed test and modifications can be made in the decision rule to test directional hypotheses with a one-tailed test. The one-sample case can be used to test the previous question about fairness of a particular coin. If a coin is tossed 20 times and 13 heads are the result, is it a fair coin? As seen earlier the hypotheses are: H0: Pheads = 0.50 H1: Pheads ≠ 0.50 In this case the pˆ is 13/20 or 0.65, P0 equals 0.50 and n is 20. The calculation would be as follows:
Figure 15.1 Effects of sample size on z-test results.
