In the case of simple random sampling, the population proportion follows the mean exactly; that is, p = μ. If this idea is new to you, convince yourself by working through an example. Say we generate a sample of 10 elements, where 4 have a value of 1 and 6 have a value of 0 (1 = presence of a trait, 0 = absence of a trait). The proportion of the sample with the trait is 4/10 or 0.40 and so is the arithmetic mean, which = 0.40 ([1+1+1+1+0+0+0+0+0+0]/10 = 4/10). Cosmic.
In the case of simple random sampling, the population proportion follows the mean exactly; that is, p = μ. If this idea is new to you, convince yourself by working through an example. Say we generate a sample of 10 elements, where 4 have a value of 1 and 6 have a value of 0 (1 = presence of a trait, 0 = absence of a trait). The proportion of the sample with the trait is 4/10 or 0.40 and so is the arithmetic mean, which = 0.40 ([1+1+1+1+0+0+0+0+0+0]/10 = 4/10). Cosmic.
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