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Learn about the distribution of sample proportion as a point estimate of population proportion, including the conditions for normality, mean, and standard deviation.

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  1. Section 8.2 Distribution of the Sample Proportion

  2. Point Estimate of a Population Proportion Suppose that a random sample of size n is obtained from a population in which each individual either does or does not have a certain characteristic. The sample proportion, denoted (read “p-hat”) is given by where x is the number of individuals in the sample with the specified characteristic. The sample proportion is a statistic that estimates the population proportion, p.

  3. Sampling Distribution of For a simple random sample of size n with population proportion p: • The shape of the sampling distribution of is approximately normal provided np(1-p)≥10. • The mean of the sampling distribution of is . • The standard deviation of the sampling distribution of is

  4. Sampling Distribution of • When sampling from finite populations, this assumption is verified by checking that the sample size n is no more than 5% of the population size N (n ≤ 0.05N).

  5. Parallel Example 4: Compute Probabilities of a Sample Proportion According to the Centers for Disease Control and Prevention, 18.8% of school-aged children, aged 6-11 years, were overweight in 2004. • In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight? • Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude?

  6. Solution • n=90 is less than 5% of the population size • np(1-p)=90(.188)(1-.188)≈13.7≥10 • is approximately normal with mean=0.188 and standard deviation = • In a random sample of 90 school-aged children, aged 6-11 years, what is the probability that at least 19% are overweight? , P(Z>0.05)=1-0.5199=0.4801

  7. Solution • is approximately normal with mean=0.188 and standard deviation = 0.0412 • Suppose a random sample of 90 school-aged children, aged 6-11 years, results in 24 overweight children. What might you conclude? , P(Z>1.91)=1-0.9719=0.028. We would only expect to see about 3 samples in 100 resulting in a sample proportion of 0.2667 or more. This is an unusual sample if the true population proportion is 0.188.

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