Exercise: estimation and confidence intervals
#Objective We did this exercise on the first day talking about estimation. It builds on the quiz from last week when the students had a problem asking for the mean and variance of $\frac{1}{2}(X+Y)$, where $X$ and $Y$ are independent random variables, each following a $N(2,3^2)$ distribution.
This exercise went really well. There were enough familiar elements that students didn’t get stuck often, but enough new stuff to keep the discussion going. Students were explaining concepts to each other and stretching for the meaning of the confidence interval, which was awesome to see.
By the way, I initiated this exercise before ever talking about confidence intervals. We’re moving that direction very soon, though…
Exercise
Overview
We will sample some data relating to the profit (in cents) per gallon of gas at gas stations nationally, and use the data to estimate the expected profit per gallon. This exercise is similar to a quiz question from last week.
Calculations
Assume that the population follows a normal distribution and that the variance is 9 ${\rm cents}^2$ (standard deviation is 3 cents). You will be pulling ten samples from the population, each independent and with identical distribution, and then calculating the mean of your ten samples.
- What are the variance and standard deviation of the sample mean?
Sampling
Now your group should come to the front and take ten samples (with replacement), writing each one down.
- Why do you think I had you replace each chip after sampling? (hint: what is the population for each sample, and what would it be if you didn’t return the chips?)
- The ink I used today is non-permanent, so be gentle with the chips to avoid wiping off data.
- There is writing on both sides - today’s data are the more delicate numbers.
Point estimation
Return to your seats and calculate the mean of your sample. Using your calculations from earlier, sketch the distribution of the sample mean (you may assume that it is Normal).
Interval estimation
Use the table to calculate the central interval with 50% probability mass. Shade this region on your sketch and label the end points.
- What is your interpretation of the meaning of the shaded interval? (hint: it has to do with confidence).