Sampling Distribution of a Proportion (3 of 4)

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Using the normal approximation, a proportion of .50 is: (.50-.40)/.11 = 0.909 standard deviations above the mean. From a z table it can be calculated that 0.818 of the area is below a z of 0.909. Therefore the probability that 50% or more will pass the literacy test is only about 1 - 0.818 = 0.182.

Correction for Continuity
Since the normal distribution is a continuous distribution, the probability that a sample value will exactly equal any specific value is zero. However, this is not true when the normal distribution is used to approximate the sampling distribution of a proportion. A correction called the "correction for continuity" can be used to improve the approximation.

The basic idea is that to estimate the probability of, say, 10 successes out of 20 when π is 0.4, one should compute the area between 9.5 and 10.5 as shown below.

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