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.
