Statistical Inference: What is it and how to do it?

Outline of survey data with at least two variables from a recent internet article 

This analysis focuses on a survey conducted in the US to ascertain the knowledge of the citizens on the components of the census questions. Among the questions asked in the survey is whether citizens had heard of the upcoming census (Cohn et al., 2020). The key variable here is awareness of the census. 95% of all adults reported to be aware of the census. Another question in the survey was to find out if citizens were aware that the citizenship questionnaire was to be included in the survey, 25%. The survey results showed that 17% of adults answered correctly that the citizenship questionnaire is not in the census questionnaire, while 56% believe it would be asked and 25% are not sure (Cohn et al., 2020). 

Brief explanation of why I selected the formula and why it matters 

In our case our survey, data has been presented in percentage scores per variable, which can be expressed in form of proportions. The formula evaluates whether there are significant differences between two proportions. The differences between the proportions can be ascertained whether they are due to chance or not. This can be evaluated using hypothesis testing for one sample difference between two proportions. Each of these scores can be evaluated against a predefined cutoff for significance. For instance, in our survey example, you might be interested to find out whether the difference of those who reported to be aware of the census (95%) and a defined cut off is due to chance or not. If it is due to chance, then the difference is not significant. Our hypothesis testing formula is explained in the next question. 

Explanation of the formula and definition of parameters 

The one sample hypothesis test for difference in proportion can be used at a given level of significance to decide whether the difference is significant. For instance at 95% level of significance level of (α=5%) 

Our various scores represent the point estimates to be compared to a given threshold or cut-off for significance. For example, one of the point estimates can be 0.95, representing 95% of those adults who reported to be aware of the census. Generally it can be expressed as: 

x is the number in the survey representing given group, and n is sample survey size. p 0 in this case is pre-specified cut off for significance 

Hypothesis Tests can be expressed as below 

Null hypothesis H 0 : p= p 0 

Alternative Hypothesis H 1 : p≠ p 0

Where  n  is the sample size. 

Decision Criteria: 

The null hypothesis is rejected if Z > Z α/2 , where Z α/2  is the 1-α/2 percentile of the standard normal distribution 

Confidence Intervals: 

Confidence intervals for the sample proportion can be established as follows; 

Reference 

Cohn, C., Brown, A., & Keeter, S. (2020). Most Adults Aware of 2020 Census and Ready to Respond, but Don’t Know Key Details. Retrieved from: https://www.pewsocialtrends.org/2020/02/20/most-adults-aware-of-2020-census-and-ready-to-respond-but-dont-know-key-details/