Same Sample Proportion with Different Sample Sizes for Chi-Squared Test for Goodness of Fit, Statistical Note 44

The level of confidence for statistical significance varies with the variation in the sample size of the same sample proportion.

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 respondents respond to one of two categories of response – Yes or No.

An expert tries with a sample size of 100 individuals and finds that 55 respondents are non-smokers and remaining 45 are smokers. He uses Chi-Squared test for goodness of fit to test whether the sample proportions of non-smokers and smokers represent the population proportions, using the formula for one degree of freedom as below:

Chi-square = Sum(O_i-E_i)²/E_i

where:

O_i = Sampled/ observed proportion for _ith category

E_i = population/ expected proportion for _ith category

Using above formula, an expert calculates Chi-squared value for 100 samples as:

Chi-square =Sum(O_i-E_i)²/E_i = (55-50)²/50+(45-50)²/50 = 1

An expert is curious and calculates chi-squared values with the same sample proportion of non-smokers but with increasing sample size as below:

Table 1: Sample size with Same Sample Proportion of Non-Smokers, Chi Squared Value and Level of Significance

An expert finds that upto 300 samples, an expert is less than 95 confident that the sample truly represents the population and there remains high sampling error. As the sample size increase from 400 to more, an expert is more than 95 percent confident and sampling error remains lower. Thus, at least 400 sample size is required for the sample proportion of non-smokers equal to 0.55 to significantly outnumber the sample proportion of smokers (0.45). In other words, 400 respondents need to be sampled for 55 percent non-smokers to significantly outnumber 45 percent smokers.