Using picture words and specific details, explain what a confidence interval of 90% means. Your explanation should take sample standard deviation vs. population standard deviation into account, as well as z-scores.
A confidence interval of 90% represents the range of values within which we can be 90% confident that the true population parameter lies. It is a measure of the uncertainty associated with estimating the population parameter based on a sample.
To understand this concept, imagine a target with a bullseye at the center. The bullseye represents the true population parameter we are trying to estimate, such as the mean. The confidence interval is like a ring around the bullseye, indicating the range of values that are likely to contain the true parameter.
The width of the confidence interval is influenced by two factors: the sample standard deviation and the z-score. The sample standard deviation measures the variability within the sample data, while the z-score represents the number of standard deviations away from the mean.
If the sample standard deviation is large, it means that the data points in the sample are spread out, resulting in a wider confidence interval. On the other hand, if the sample standard deviation is small, the data points are closer together, leading to a narrower confidence interval.
The z-score determines the critical value that defines the width of the confidence interval. A higher z-score corresponds to a larger critical value, resulting in a wider interval. Conversely, a lower z-score leads to a smaller critical value and a narrower interval.
In summary, a confidence interval of 90% indicates that there is a 90% probability that the true population parameter falls within the range specified by the interval. The width of the interval is influenced by the sample standard deviation and the z-score, with a larger standard deviation and higher z-score leading to a wider interval.