4 and the count of successes is X 2 4 ns observatio all of count 2 successes of

# 4 and the count of successes is x 2 4 ns observatio

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+ 4 and the count of successes is X + 2. 4 ns observatio all of count 2 successes of counts ~ + + = p ) 4 ( ) ~ 1 ( ~ * * with , ~ : + = = ± n p p z SE z m m p CI The “plus four” estimate of p is: And an approximate level C confidence interval is: Use this method when C is at least 90% and sample size is at least 10.

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Significance test for p The sampling distribution for is approximately normal for large sample sizes and its shape depends on p and n . Thus, we can easily test the null hypothesis: H 0 : p = p 0 (a given value we are testing). z = ˆ p p 0 p 0 (1 p 0 ) n If H 0 is true, the sampling distribution is known The likelihood of our sample proportion given the null hypothesis depends on how far from p 0 our is in units of standard deviation. This is valid when both expected counts—expected successes np 0 and expected failures n (1 p 0 )—are each 10 or larger. p 0 (1 p 0 ) n p 0 ˆ p p ˆ ˆ p
P -values and one- or two-sided hypotheses reminder If the P -value is as small or smaller than the chosen significance level a , then the difference is statistically significant and we reject H 0 .

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