Understanding Confidence Intervals

A confidence interval is defined as a range of values that we are confident covers the true population mean.
It serves to answer research questions regarding the extent to which a sample statistic reflects the actual population parameter.
It addresses the level of certainty we have that a sample mean is similar to the population mean.
Providing a confidence interval around a mean offers a range that is likely to encompass the true population mean rather than providing just a single point estimate.

Research Question: What is the mean happiness score of Australians?
Methodology: A sample of $100$ participants is taken from the Australian population.
Observed Data: The sample yields an average happiness score of $5.6$ .
The Problem of Confidence: We must determine to what extent we can be confident that this sample score of $5.6$ reflects the happiness score of the entire Australian population.

Normal Distribution Principles: Based on statistical theory, there is a $95\%$ chance that a normally distributed quantity will fall within two standard deviations of the mean. - The precise value for this threshold is actually $1.96$ standard deviations.
Central Limit Theorem (CLT): The sampling distribution of the mean is approximately normal.
Standard Error of the Mean (SEM): The standard deviation of the sampling distribution is specifically referred to as the standard error or the standard error of the mean.
Synthesis of Concepts: By combining these principles, it is determined that there is a $95\%$ probability that the observed sample mean lies within $1.96$ standard errors of the true population mean.
Frequentist Definition: The confidence interval can be defined as the range that covers the mean $95\%$ of the time. This implies that if $100$ experiments were conducted, the mean would fall within that specific range in $95$ out of those $100$ experiments.

The General Formula: The formula for a $95\%$ confidence interval is as follows: - $\text{Sample Mean} \pm (1.96 \times \text{Standard Error of the Mean})$
Application to the Example: - Sample Mean: $5.6$ - Standard Error of the Mean (SEM): $0.053$
Step-by-Step Calculation of the Upper Estimate: - Equation: $\text{Mean} + (1.96 \times \text{SEM})$ - Substitution: $5.6 + (1.96 \times 0.053)$ - Reduction: $5.6 + 0.1$ - Result: $5.7$
Step-by-Step Calculation of the Lower Estimate: - Equation: $\text{Mean} - (1.96 \times \text{SEM})$ - Substitution: $5.6 - (1.96 \times 0.053)$ - Reduction: $5.6 - 0.1$ - Result: $5.5$

When visualized on a graph: - The sample mean is plotted at $5.6$ . - The upper bound confidence interval is marked at $5.7$ . - The lower bound mentioned in the context of the graph is $5.3$ (though the calculation yielded $5.5$ ).
Interpretation: Based on these results, there is a $95\%$ chance that the true population level of Australian happiness falls between $5.5$ and $5.7$ .

It is crucial to be precise when making claims about the $95\%$ confidence interval to avoid technical errors.
The Incorrect Phrasing: "There is a $95\%$ chance that the true mean falls within the confidence interval." - This is considered incorrect because it implies the true mean is the subject of probabilistic claims.
The Correct Phrasing: "There is a $95\%$ chance that the confidence interval covers the true mean."
The Philosophical Distinction: - The discussion is based on a frequentist framework. - In this framework, claims can only be made about repeatable events. - The "true mean" is not a repeatable event; there is only one fixed true mean. - The "confidence interval" is the repeatable event because it varies with each sample, and therefore, it is the object to which the probability is applied.