+ 4 and the count of successes is X+ 2. 4nsobservatioallofcount 2successesofcounts~++=p)4()~1(~**with,~:+−==±nppzSEzmmpCIThe “plus four” estimate of pis:And an approximate level Cconfidence interval is:Use this method when Cis at least 90% and sample size is at least 10.
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Significance test for pThe sampling distribution for is approximately normal for large sample sizes and its shape depends on p andn. Thus, we can easily test the null hypothesis: H0: p = p0(a given value we are testing).z=ˆ p −p0p0(1−p0)nIf H0is true, the sampling distribution is known The likelihood of our sample proportion given the null hypothesis depends on how far from p0our is in units of standard deviation.This is valid when both expected counts—expected successes np0and expected failures n(1 −p0)—are each 10 or larger.p0(1−p0)np0ˆ p pˆˆp
P-values and one- or two-sided hypotheses—reminderIf the P-value is as small or smaller than the chosen significance level a, then the difference is statistically significant and we reject H0.