Level of Confidence Increases with Sample Size: An Example of One Sample Proportion, Statistical Note 47

Level of confidence increases with an increase in sample size for one sample proportion. This note tries to exemplify this fact by using the process and data discussed in my statistical note 43 and 44 respectively.

For example, an expert is interested in knowing the proportion of non-smokers from the randomly sampled respondents (following randomization, the first rule of sample proportion). An expert assumes that half of adult population are smokers. An expert administers a question to the randomly adults – Are you a smoker? The respondents answer to one of two categories of response – Yes or No.

An expert tries with a sample size of 100 adults and finds that 55 are non-smokers and remaining 45 are smokers (following normality that non-/smokers need to be at least 10, second rule of sample proportion). An expert then retains the same sample proportion of smokers and hypothetically increases the sample size by 100 to 500. An expert assumes to follow independence in sampling without replacement that the population size is more than 10 times of sample size, third rule of sample proportion.

An expert estimates how confident he is in deciding that non-smokers statistically outnumber the smokers with the increase in sample size. 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

Table 1: Sample size with Same Sample Proportion of Non-Smokers and Level of Confidence to Conclude Non-Smokers Outnumber Smokers

An expert is 84 percent confident in deciding that the sample non-smoker proportion of 0.55 among 100 respondents statistically outnumbers the sample smoker proportion of 0.45. Usual threshold is that decisions are made at 95 percent level of confidence. Gradually, an expert increases sample size by 100 and finds that for a sample of 300 respondents or more an expert is more than 95 percent confident in deciding that the sample proportion of non-smokers equal to 0.55 is statistically higher than the sample proportion of smokers equal to 0.45. It proves that the level of confidence increases as sample size increases for one sample proportion.