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Sample size plays a crucial role in determining the width of a confidence interval. Think of it as the number of observations or pieces of data you collect from a larger group. When you increase the sample size, you get more data, which usually makes your estimates more precise.
This precision is reflected in a reduction of the standard error, which is the measure of how much your sample mean is expected to vary from the true population mean. Thus, by increasing your sample size, the confidence interval becomes narrower.
This happens because in the formula for confidence interval width \( ext{Width} = z \times \frac{\sigma}{\sqrt{n}} \), the standard error (\( \frac{\sigma}{\sqrt{n}} \)) decreases as \( n \) (the sample size) increases.
When the sample size is small, your estimates can be quite variable. Increasing the sample size reduces this variability and consequently, the confidence interval shrinks. This makes your interval more reliable in estimating the true population parameter.

Population size refers to the total number of individual units you are interested in studying. A common misconception is that larger populations impact the width of confidence intervals. In reality, the population size itself does not directly influence the confidence interval width. This is a surprise to many given that populations can be enormous.
Instead, it is the sample size, rather than the population size, that mainly determines the confidence interval width. As long as your sample is a small fraction relative to the entire population, its size or growth (like from 25 million to 50 million) doesn’t significantly affect your ability to make inferences about the population. This is because confidence intervals are calculated assuming an infinite or very large population. This is why statistical methods designed for random sampling work the same regardless of the population size.
The independence of confidence interval width from population size simplifies data analysis, especially in cases where the population is so large that studying every single individual is impractical.

The confidence level determines how sure you are that the true population parameter lies within your confidence interval. It is usually represented as a percentage, such as 90%, 95%, or 99%. More intuitively, a 95% confidence level means that if we repeated a study numerous times, 95% of the confidence intervals calculated from those samples would contain the true population parameter.
Changing the confidence level alters the z-score in the confidence interval formula. When the confidence level decreases from say 95% to 90%, the z-score also decreases. This results in a narrower confidence interval. This is because a lower confidence level implies that you are accepting less certainty that the true population parameter lies within the interval. Hence, you don't need the interval to cover as wide a range of possibilities.
Adjusting your confidence level is a balancing act between certainty and specificity in your data analysis. Lower confidence levels yield tighter intervals, possibly offering more concise insights at the cost of certainty.