Question

A U.S. government survey in 2007 said that the proportion of young Americans that earn a high school diploma is 0.87. a) Suppose you took a simple random sample of 100 young Americans. We know because of sampling variability that the sample proportion of those who earned a high school diploma would vary each time you took a sample. Assuming the value above can be taken as the population proportion (i.e. as a parameter), what model can be used to describe how these sample proportions would vary? Please be sure to include the name of the distribution and the parameter values. b) Find the probability that 89% or more in the sample will have earned a high school diploma.

Answer 1

The probability that 89% or more in the sample will have earned a high school diploma is p ~ N = ( 0.87 , 0.001131 )

The normal distribution can be used to model the sample proportions.

The Central Limit Theorem and the importance of the normal distribution model in statistics (CLT). According to this theory, regardless of the type of distribution, the variables are sampled from, averages calculated from independent, identically distributed random variables have approximately normal distributions (provided it has finite variance).

Given,

N ( sample size ) = 100

probability of earning a high school diploma (p)  = 0.87

probability of not earning a high school diploma ( q ) = 0.13

hence ;

∈( P ) = p = 0.87

Var ( p ) = $\frac{pq}{n}$  = $\frac{0.86*0.13}{100}$  = 0.001131

therefore ; p ~ N = ( 0.87 , 0.001131 )

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