Binary Outcome Using Proportion - Assignment Example

Central Limit Theorem (CLT) for a Proportion states that “As sample size increases, the distribution of the sample proportion p = x/n approaches a normal distribution with mean π and standard deviation.” The statistic p = x/n is assumed normally distributed when the sample is large. A conservative rule of thumb that normality may be assumed whenever nπ ≥ 10 and n(1 − π) ≥ 10. This rule requires a very large sample size to assume normality when π differs greatly from 0.

50 (Doane & Seward 2007). Using the Central Limit Theorem, the probability that a sample proportion will fall within a given interval can be stated. The confidence interval for a population proportion, π at a given confidence level (1 – α) is given by The value of z can be obtained using a normal table (Z table) or using Excel function NORMINV(α/2). The width of the confidence interval for a population proportion, π depends on the sample size, confidence level (1 – α), and the sample proportion p.

The estimate of difference and standard deviation of two-population proportion can be given by and, respectively. Using this estimate, a confidence interval for the difference of two population proportions, (π1− π2), is given by For normal sampling distribution, the test statistic for the hypothesis test will be z score. This test statistic is compared with the critical value of z score at the selected level of significance, α for retaining or rejecting the null hypothesis (H0).

The test statistic for a population proportion with hypothesized proportion π0 is the difference between the sample proportion p and the hypothesized proportion π0 divided by the estimated standard error of the proportion (denoted σp) as given below