Understanding Confidence Intervals

Let’s start this section with an analogy involving fishing, and assume we’re trying to catch a fish. On one hand, we can use a spear, while on the other, we can use a net. Using the net will probably allow us to catch more fish!

Now think back to our pennies exercise where we were trying to estimate the true population mean year $\mu$ of all US pennies. Think of the value of $\mu$ as a fish.

On one hand, we can use the appropriate point estimate/sample statistic to estimate $\mu$, which we saw in the Pennies Activity lesson is the sample mean $\bar{x}$. Based on our sample of 50 pennies from the bank, the sample mean was 1995.44. We can think of using this value as “fishing with a spear.”

What will “fishing with a net” correspond to? Look at the bootstrap distribution once more. Between which two years will we say that most sample means lie? This question is somewhat subjective, so saying that most sample means lie between 1992 and 2000 won’t be unreasonable. Think of this interval as the net.

What we’ve just illustrated is the concept of a confidence interval, which we’ll abbreviate with CI. As opposed to a point estimate/sample statistic that estimates the value of an unknown population parameter with a single value, a confidence interval gives what can be interpreted as a range of plausible values. Going back to our analogy, point estimates/sample statistics can be thought of as spears, whereas confidence intervals can be thought of as nets.