Assignment 2: Connection between Confidence Intervalsand Sampling Distributions:

The purpose of this activity is to help give you a betterunderstanding of the underlying reasoning behindthe interpretation of confidence intervals. In particular, you willgain a deeper understanding of why we say that we are “95%confidentthat the population mean iscovered by the interval.”

When the simulation loads you will see a normal-shapeddistribution, which represents the sampling distribution ofthe mean (x-bar) for random samples of a particular fixedsample size, from a population with a fixed standard deviation ofσ.

The green line marks the value of the population mean, μ.

To begin the simulation, click the very top“sample” button at the topmost right of thesimulation. You will see a line segment appear underneath thedistribution; you should see that the line segment has a tiny reddot in the middle.

You have used the simulation to select a single sample from thepopulation; the simulation has automatically computed the mean(x-bar) of your sample; your x-bar value is represented by thelittle red dot in the middle of the line segment. The line segmentrepresents a confidence interval. Notice that, by default, thesimulation used a 95% confidence level.

Question 1:

Did your 95% confidence interval contain (or “cover”) thepopulation mean μ (the green line)?

If your confidence interval did cover the populationmean μ, then the simulation will have recorded 1 “hit” on the rightside of the simulation.

Now, click to select another single sample.

Question 2:

Was your second sample mean x-bar (the new red dot) the samevalue as your 1st sample mean? (i.e., is it in the same relativelocation along the axis?) Why is this result to be expected?

Question 3:

A new 95% confidence interval has also been constructed (the newline segment, centered at the location of your second x-bar). Doesthe new interval cover the population mean μ?

Notice, under “total” on the right side of the simulation, thenumber of total selected samples has been tallied.

Now click “sample 50” repeatedly until thesimulation tallies a “total” of around 1,000 samples. You will seethat the simulation computes the “percent hit” for all theintervals.

Question 4:

What percentage of the many 95% confidence intervals shouldcover the population mean μ?

Question 5:

Now let’s summarize some key ideas.

Based on what you’ve seen on the simulation (with the level setat 95%), decide which of the following statements are true andwhich are false.

1. Each interval is centered at the population mean (μ).
2. Each interval is centered at the sample mean (x-bar).
3. The population mean (μ) changes when different samples areselected.
4. The sample mean (x-bar) changes when different samples areselected.
5. In the long run, 95% of the intervals will contain (or “cover”)the sample mean (x-bar).
6. In the long run, 95% of the intervals will contain (or “cover”)the population mean (μ).