Lecture 04b Activity

Unit B · Chapter 04

An in-class activity. Nothing to turn in and no answer key – this one needs other people, which is why it happens in class rather than at home.

The idea. The 95% in a 95% confidence interval is a property of the procedure, not of the one interval in front of you. One person cannot see that. A roomful of people, each building one interval, can – the board is the sampling distribution.

Format. About 15 minutes · individually · one laptop each (sharing works too).

1. Commit, on your own (2 minutes)

iqpop.csv (on the chapter page) holds IQ scores for a whole population of 10,000 people. Because we have everybody, we know the population mean exactly: it is 100. Everyone will draw one sample of \(n = 25\) from it and build one 95% interval, using exactly the formula from today’s slides. Before running anything, write down:

Will your interval contain 100?
Out of all the intervals this room is about to make, how many will miss?

2. Build your interval (5 minutes)

Sampling is random, so – as in the last lecture – we each get a different sample without having to coordinate. Everything below is sample() from last lecture, then today’s formula one piece at a time.

library(tidyverse)

pop <- read_csv("iqpop.csv")

x <- sample(pop$iq, size = 25)   # your sample, n = 25

mean(x) - qt(0.975, df = 24) * sd(x) / sqrt(25)   # lower bound
mean(x) + qt(0.975, df = 24) * sd(x) / sqrt(25)   # upper bound

Now do it again with \(n = 100\), changing the three numbers that depend on sample size. Bring four things to the board: at each sample size, did your interval contain 100 (yes or no), and how wide was it?

3. Read the board (6 minutes)

Two columns per person: \(n = 25\) and \(n = 100\), each marked hit or miss, each with a width.

Count the misses across the whole board – both columns, since coverage does not care about \(n\). How many should there be?
Point at someone who missed. Did they do anything wrong?
Which changed between the columns: the miss rate, or the widths?
In real research you get one interval, and nobody tells you the true mean. Looking at the board, could anyone here have known theirs was a miss?

If you have more time: re-run yours a few times with new seeds while the board fills in. Watch which numbers in your two formula lines change from run to run, and which never do.

If you were not in class

About 5% of the board should miss – one or two intervals per thirty – and the people who missed did nothing wrong. A room this size seeing zero misses is also common and also consistent: 95% is a long-run rate, and a run of thirty often comes up clean. Moving to \(n = 100\) leaves the miss rate alone and cuts the widths roughly in half. Coverage is what the method promises; width is what your sample size buys.

The board is the argument. From inside one interval there is no signal at all about whether yours is a hit or a miss, which is exactly why “there is a 95% chance the true value is in this interval” does not follow from the procedure’s guarantee – and why reading it off a single plot in a textbook never quite convinces anyone.

Notice, too, what you typed: two lines of arithmetic you already knew. The interval on the board came from the formula on the slides, not from anything hidden.