One Sample Proportion for Statistical Significance and Sample Size, Statistical Note 43

As the sample size increases, even slightly bigger proportion of category of interest could significantly outnumber another category of binary response.

For example, an expert is interested in knowing the proportion of smokers from the randomly selected sampled respondents. An expert assumes that half of adult population are smokers. An expert administers a question to the adults – Are you a smoker? The respondent responds to the two categories response – Yes or No.

An expert estimates that how many non-smokers would statistically outnumber the smokers to draw valid conclusion. An expert uses the following formula to calculate the minimum number of non-smokers from the given sample size to outnumber the smokers:

 z=(p’-p)/√(pq/n)

  where ,

z=Test statistic, standard normal variate, with a value of 1.96 at 95% level of confidence

p’=Sample proportion of non-smokers

p=Population proportion of non-smokers, equal to 0.50

q= Population proportion of smokers, equal to 0.50

n=Sample size

An expert tries with a sample size of 10 individuals and calculates the minimum sample proportion or number of non-smokers required to statistically significant outnumber the smokers. Gradually he increases the sample size and calculates the minimum sample proportion and number of non-smokers required to statistically outnumber the smokers as shown in the table below:

Table 1: Number of Non-Smokers Required to Statistically Significant Outnumber the Smokers

An expert assumes whether six out of 10 non-smokers or the sample non-smoker proportion of 0.60 is enough for statistically significance to outnumber smokers. An expert then used the above formula and finds that the sample non-smoker proportion of 0.8099 or eight non-smokers out of 10 respondents are required for statistically significant outnumber the smokers. Gradually, an expert tries with one hundred thousand hypothetical sample size of respondents with the assumption that 50,001 non-smokers would outnumber 49,999 smokers. Unlike, using the formula an expert finds that the sample non-smoker proportion of 0.5030 or 50,300 non-smokers are required for statistically significant outnumber the smokers. An expert finally understands that as the sample size increases, the smaller sample proportion of non-smokers than the assumed sample proportions presented in the realtime column in the table could significantly outnumber the smokers.