What is the relationship between Sample Size and the width of the Confidence Interval? Please explain further as to why
The relationship between sample size and the width of the confidence interval is inverse or opposite. In other words, as the sample size increases, the width of the confidence interval decreases, and vice versa.
To understand why this relationship exists, we need to first understand what a confidence interval is. A confidence interval is a range of values within which we estimate the true population parameter lies, based on the sample data.
When we compute a confidence interval, we take into consideration two main factors: the level of confidence we want (e.g., 95% confidence) and the sample data. The level of confidence indicates the likelihood or probability that the true population parameter falls within the confidence interval. Common levels of confidence include 90%, 95%, and 99%.
Now, let's consider the impact of sample size on the width of the confidence interval. When we have a larger sample size, we have more data points or observations to work with. This additional information allows us to have a more precise estimate of the population parameter.
As the sample size increases, the variability or spread of the sample mean decreases. Consequently, the standard error (which measures the uncertainty of the estimate) decreases as well. The standard error is a critical component of calculating the width of the confidence interval.
A smaller standard error results in a narrower confidence interval because it indicates the estimate is more precise. On the other hand, a larger sample size (holding other factors constant) will result in a larger standard error, leading to a wider confidence interval.
To summarize, increasing the sample size improves the precision of our estimate, leading to a narrower confidence interval. Conversely, a smaller sample size will result in a wider confidence interval due to the increased uncertainty in estimating the true population parameter.
It's important to note that while increasing the sample size generally results in a narrower confidence interval, there may be other factors at play. For instance, if the population variability is extremely high, a larger sample size may not lead to as significant a reduction in the width of the confidence interval.