Rodolfo Lourenzutti

Learning objectives

Define what a confidence interval is and why we want to generate one
Explain how the bootstrap sampling distribution can be used to create confidence intervals
Write a computer script to calculate confidence intervals for a population parameter using bootstrapping
Effectively visualize point estimates and confidence intervals
Interpret and explain results from confidence intervals
Discuss the potential limitations of these methods
Define what percentiles are and write a computer script to calculate them

Review: bootstrap approximation of the sampling distribution

Today’s goal

Our goal: define confidence intervals, and compute them using the bootstrap distribution

Point Estimation vs. Confidence Intervals

image source: Modern Dive by Ismay & Kim

Using the bootstrap to calculate a confidence interval (plausible range)

95% confidence interval via bootstrapping

confidence interval: range from lower to upper

lower: \(2.5^{\text{th}}\) percentile
- value such that \(2.5\%\) of the bootstrap point estimates fall below
upper: \(97.5^{\text{th}}\) percentile
- value such that \(97.5\%\) of the bootstrap point estimates fall below

Visualizing confidence intervals

attribution: zoology.ubc.ca/~whitlock/…

Interpretation of a 95% confidence interval (CI)

We are \(95\%\) confident that the true population parameter lies within our calculated interval.

But what does “\(95\%\) confident” actually mean?
- It is easy to confuse confidence with probability.
- Specifically, it means that if we collect many independent samples of size \(n\) and construct a \(95\%\) CI from each, about \(95\%\) of these intervals will contain the true population parameter.
- The uncertainty is about our sample: is our sample one of the successful \(95\%\) that captures the parameter, or one of the unlucky \(5\%\)?

Misconception: Probability vs. Confidence

Common Misconception: CI as a Probability

It is incorrect to say: “There is a \(95\%\) probability that the population parameter lies within our interval.”

The population parameter is a fixed constant, not a random variable. It either lies within the calculated interval, or it does not.
The probability is either \(0\) or \(1\), we just do not know which.
The \(95\%\) confidence level describes the process: if we repeat the sampling process many times, \(95\%\) of the calculated intervals will contain the true parameter.

Revisiting the Aquarium Example

Recall from Module 01: we have a population of \(20\) fish in an aquarium, with a true mean weight of \(\mu = 43.45\) dkg.
We draw random samples of size \(n = 3\) to estimate \(\mu\). There are \(1{,}140\) possible samples.

Let’s construct a confidence interval for every single one of the \(1{,}140\) possible samples.
To start, we will define the interval as the sample mean plus/minus some margin of error (\(M\)): \[\bar{x} \pm M\]
Let’s see what happens to our coverage rate when we change the size of the margin \(M\)!

All 1,140 Possible Intervals

Set Margin of Error (\(M\)): 5.0 dkg Coverage: 63.8% (727 / 1140 CIs)

**Table 2:** *All possible samples of three fish from the aquarium and their respective sample mean.*
Sample	Sample Mean (dkg)	Sample	Sample Mean (dkg)	Sample	Sample Mean (dkg)	Sample	Sample Mean (dkg)
(#1, #2, #3)	45.33	(#1, #2, #4)	49.33	(#1, #2, #5)	37.67	(#1, #2, #6)	44.33
(#1, #2, #7)	43.33	(#1, #2, #8)	43.00	(#1, #2, #9)	44.00	(#1, #2, #10)	49.00
(#1, #2, #11)	38.33	(#1, #2, #12)	45.33	(#1, #2, #13)	42.00	(#1, #2, #14)	43.67
(#1, #2, #15)	49.67	(#1, #2, #16)	43.67	(#1, #2, #17)	41.67	(#1, #2, #18)	41.67
(#1, #2, #19)	50.00	(#1, #2, #20)	42.00	(#1, #3, #4)	49.67	(#1, #3, #5)	38.00
(#1, #3, #6)	44.67	(#1, #3, #7)	43.67	(#1, #3, #8)	43.33	(#1, #3, #9)	44.33
(#1, #3, #10)	49.33	(#1, #3, #11)	38.67	(#1, #3, #12)	45.67	(#1, #3, #13)	42.33
(#1, #3, #14)	44.00	(#1, #3, #15)	50.00	(#1, #3, #16)	44.00	(#1, #3, #17)	42.00
(#1, #3, #18)	42.00	(#1, #3, #19)	50.33	(#1, #3, #20)	42.33	(#1, #4, #5)	42.00
(#1, #4, #6)	48.67	(#1, #4, #7)	47.67	(#1, #4, #8)	47.33	(#1, #4, #9)	48.33
(#1, #4, #10)	53.33	(#1, #4, #11)	42.67	(#1, #4, #12)	49.67	(#1, #4, #13)	46.33
(#1, #4, #14)	48.00	(#1, #4, #15)	54.00	(#1, #4, #16)	48.00	(#1, #4, #17)	46.00
(#1, #4, #18)	46.00	(#1, #4, #19)	54.33	(#1, #4, #20)	46.33	(#1, #5, #6)	37.00
(#1, #5, #7)	36.00	(#1, #5, #8)	35.67	(#1, #5, #9)	36.67	(#1, #5, #10)	41.67
(#1, #5, #11)	31.00	(#1, #5, #12)	38.00	(#1, #5, #13)	34.67	(#1, #5, #14)	36.33
(#1, #5, #15)	42.33	(#1, #5, #16)	36.33	(#1, #5, #17)	34.33	(#1, #5, #18)	34.33
(#1, #5, #19)	42.67	(#1, #5, #20)	34.67	(#1, #6, #7)	42.67	(#1, #6, #8)	42.33
(#1, #6, #9)	43.33	(#1, #6, #10)	48.33	(#1, #6, #11)	37.67	(#1, #6, #12)	44.67
(#1, #6, #13)	41.33	(#1, #6, #14)	43.00	(#1, #6, #15)	49.00	(#1, #6, #16)	43.00
(#1, #6, #17)	41.00	(#1, #6, #18)	41.00	(#1, #6, #19)	49.33	(#1, #6, #20)	41.33
(#1, #7, #8)	41.33	(#1, #7, #9)	42.33	(#1, #7, #10)	47.33	(#1, #7, #11)	36.67
(#1, #7, #12)	43.67	(#1, #7, #13)	40.33	(#1, #7, #14)	42.00	(#1, #7, #15)	48.00
(#1, #7, #16)	42.00	(#1, #7, #17)	40.00	(#1, #7, #18)	40.00	(#1, #7, #19)	48.33
(#1, #7, #20)	40.33	(#1, #8, #9)	42.00	(#1, #8, #10)	47.00	(#1, #8, #11)	36.33
(#1, #8, #12)	43.33	(#1, #8, #13)	40.00	(#1, #8, #14)	41.67	(#1, #8, #15)	47.67
(#1, #8, #16)	41.67	(#1, #8, #17)	39.67	(#1, #8, #18)	39.67	(#1, #8, #19)	48.00
(#1, #8, #20)	40.00	(#1, #9, #10)	48.00	(#1, #9, #11)	37.33	(#1, #9, #12)	44.33
(#1, #9, #13)	41.00	(#1, #9, #14)	42.67	(#1, #9, #15)	48.67	(#1, #9, #16)	42.67
(#1, #9, #17)	40.67	(#1, #9, #18)	40.67	(#1, #9, #19)	49.00	(#1, #9, #20)	41.00
(#1, #10, #11)	42.33	(#1, #10, #12)	49.33	(#1, #10, #13)	46.00	(#1, #10, #14)	47.67
(#1, #10, #15)	53.67	(#1, #10, #16)	47.67	(#1, #10, #17)	45.67	(#1, #10, #18)	45.67
(#1, #10, #19)	54.00	(#1, #10, #20)	46.00	(#1, #11, #12)	38.67	(#1, #11, #13)	35.33
(#1, #11, #14)	37.00	(#1, #11, #15)	43.00	(#1, #11, #16)	37.00	(#1, #11, #17)	35.00
(#1, #11, #18)	35.00	(#1, #11, #19)	43.33	(#1, #11, #20)	35.33	(#1, #12, #13)	42.33
(#1, #12, #14)	44.00	(#1, #12, #15)	50.00	(#1, #12, #16)	44.00	(#1, #12, #17)	42.00
(#1, #12, #18)	42.00	(#1, #12, #19)	50.33	(#1, #12, #20)	42.33	(#1, #13, #14)	40.67
(#1, #13, #15)	46.67	(#1, #13, #16)	40.67	(#1, #13, #17)	38.67	(#1, #13, #18)	38.67
(#1, #13, #19)	47.00	(#1, #13, #20)	39.00	(#1, #14, #15)	48.33	(#1, #14, #16)	42.33
(#1, #14, #17)	40.33	(#1, #14, #18)	40.33	(#1, #14, #19)	48.67	(#1, #14, #20)	40.67
(#1, #15, #16)	48.33	(#1, #15, #17)	46.33	(#1, #15, #18)	46.33	(#1, #15, #19)	54.67
(#1, #15, #20)	46.67	(#1, #16, #17)	40.33	(#1, #16, #18)	40.33	(#1, #16, #19)	48.67
(#1, #16, #20)	40.67	(#1, #17, #18)	38.33	(#1, #17, #19)	46.67	(#1, #17, #20)	38.67
(#1, #18, #19)	46.67	(#1, #18, #20)	38.67	(#1, #19, #20)	47.00	(#2, #3, #4)	50.67
(#2, #3, #5)	39.00	(#2, #3, #6)	45.67	(#2, #3, #7)	44.67	(#2, #3, #8)	44.33
(#2, #3, #9)	45.33	(#2, #3, #10)	50.33	(#2, #3, #11)	39.67	(#2, #3, #12)	46.67
(#2, #3, #13)	43.33	(#2, #3, #14)	45.00	(#2, #3, #15)	51.00	(#2, #3, #16)	45.00
(#2, #3, #17)	43.00	(#2, #3, #18)	43.00	(#2, #3, #19)	51.33	(#2, #3, #20)	43.33
(#2, #4, #5)	43.00	(#2, #4, #6)	49.67	(#2, #4, #7)	48.67	(#2, #4, #8)	48.33
(#2, #4, #9)	49.33	(#2, #4, #10)	54.33	(#2, #4, #11)	43.67	(#2, #4, #12)	50.67
(#2, #4, #13)	47.33	(#2, #4, #14)	49.00	(#2, #4, #15)	55.00	(#2, #4, #16)	49.00
(#2, #4, #17)	47.00	(#2, #4, #18)	47.00	(#2, #4, #19)	55.33	(#2, #4, #20)	47.33
(#2, #5, #6)	38.00	(#2, #5, #7)	37.00	(#2, #5, #8)	36.67	(#2, #5, #9)	37.67
(#2, #5, #10)	42.67	(#2, #5, #11)	32.00	(#2, #5, #12)	39.00	(#2, #5, #13)	35.67
(#2, #5, #14)	37.33	(#2, #5, #15)	43.33	(#2, #5, #16)	37.33	(#2, #5, #17)	35.33
(#2, #5, #18)	35.33	(#2, #5, #19)	43.67	(#2, #5, #20)	35.67	(#2, #6, #7)	43.67
(#2, #6, #8)	43.33	(#2, #6, #9)	44.33	(#2, #6, #10)	49.33	(#2, #6, #11)	38.67
(#2, #6, #12)	45.67	(#2, #6, #13)	42.33	(#2, #6, #14)	44.00	(#2, #6, #15)	50.00
(#2, #6, #16)	44.00	(#2, #6, #17)	42.00	(#2, #6, #18)	42.00	(#2, #6, #19)	50.33
(#2, #6, #20)	42.33	(#2, #7, #8)	42.33	(#2, #7, #9)	43.33	(#2, #7, #10)	48.33
(#2, #7, #11)	37.67	(#2, #7, #12)	44.67	(#2, #7, #13)	41.33	(#2, #7, #14)	43.00
(#2, #7, #15)	49.00	(#2, #7, #16)	43.00	(#2, #7, #17)	41.00	(#2, #7, #18)	41.00
(#2, #7, #19)	49.33	(#2, #7, #20)	41.33	(#2, #8, #9)	43.00	(#2, #8, #10)	48.00
(#2, #8, #11)	37.33	(#2, #8, #12)	44.33	(#2, #8, #13)	41.00	(#2, #8, #14)	42.67
(#2, #8, #15)	48.67	(#2, #8, #16)	42.67	(#2, #8, #17)	40.67	(#2, #8, #18)	40.67
(#2, #8, #19)	49.00	(#2, #8, #20)	41.00	(#2, #9, #10)	49.00	(#2, #9, #11)	38.33
(#2, #9, #12)	45.33	(#2, #9, #13)	42.00	(#2, #9, #14)	43.67	(#2, #9, #15)	49.67
(#2, #9, #16)	43.67	(#2, #9, #17)	41.67	(#2, #9, #18)	41.67	(#2, #9, #19)	50.00
(#2, #9, #20)	42.00	(#2, #10, #11)	43.33	(#2, #10, #12)	50.33	(#2, #10, #13)	47.00
(#2, #10, #14)	48.67	(#2, #10, #15)	54.67	(#2, #10, #16)	48.67	(#2, #10, #17)	46.67
(#2, #10, #18)	46.67	(#2, #10, #19)	55.00	(#2, #10, #20)	47.00	(#2, #11, #12)	39.67
(#2, #11, #13)	36.33	(#2, #11, #14)	38.00	(#2, #11, #15)	44.00	(#2, #11, #16)	38.00
(#2, #11, #17)	36.00	(#2, #11, #18)	36.00	(#2, #11, #19)	44.33	(#2, #11, #20)	36.33
(#2, #12, #13)	43.33	(#2, #12, #14)	45.00	(#2, #12, #15)	51.00	(#2, #12, #16)	45.00
(#2, #12, #17)	43.00	(#2, #12, #18)	43.00	(#2, #12, #19)	51.33	(#2, #12, #20)	43.33
(#2, #13, #14)	41.67	(#2, #13, #15)	47.67	(#2, #13, #16)	41.67	(#2, #13, #17)	39.67
(#2, #13, #18)	39.67	(#2, #13, #19)	48.00	(#2, #13, #20)	40.00	(#2, #14, #15)	49.33
(#2, #14, #16)	43.33	(#2, #14, #17)	41.33	(#2, #14, #18)	41.33	(#2, #14, #19)	49.67
(#2, #14, #20)	41.67	(#2, #15, #16)	49.33	(#2, #15, #17)	47.33	(#2, #15, #18)	47.33
(#2, #15, #19)	55.67	(#2, #15, #20)	47.67	(#2, #16, #17)	41.33	(#2, #16, #18)	41.33
(#2, #16, #19)	49.67	(#2, #16, #20)	41.67	(#2, #17, #18)	39.33	(#2, #17, #19)	47.67
(#2, #17, #20)	39.67	(#2, #18, #19)	47.67	(#2, #18, #20)	39.67	(#2, #19, #20)	48.00
(#3, #4, #5)	43.33	(#3, #4, #6)	50.00	(#3, #4, #7)	49.00	(#3, #4, #8)	48.67
(#3, #4, #9)	49.67	(#3, #4, #10)	54.67	(#3, #4, #11)	44.00	(#3, #4, #12)	51.00
(#3, #4, #13)	47.67	(#3, #4, #14)	49.33	(#3, #4, #15)	55.33	(#3, #4, #16)	49.33
(#3, #4, #17)	47.33	(#3, #4, #18)	47.33	(#3, #4, #19)	55.67	(#3, #4, #20)	47.67
(#3, #5, #6)	38.33	(#3, #5, #7)	37.33	(#3, #5, #8)	37.00	(#3, #5, #9)	38.00
(#3, #5, #10)	43.00	(#3, #5, #11)	32.33	(#3, #5, #12)	39.33	(#3, #5, #13)	36.00
(#3, #5, #14)	37.67	(#3, #5, #15)	43.67	(#3, #5, #16)	37.67	(#3, #5, #17)	35.67
(#3, #5, #18)	35.67	(#3, #5, #19)	44.00	(#3, #5, #20)	36.00	(#3, #6, #7)	44.00
(#3, #6, #8)	43.67	(#3, #6, #9)	44.67	(#3, #6, #10)	49.67	(#3, #6, #11)	39.00
(#3, #6, #12)	46.00	(#3, #6, #13)	42.67	(#3, #6, #14)	44.33	(#3, #6, #15)	50.33
(#3, #6, #16)	44.33	(#3, #6, #17)	42.33	(#3, #6, #18)	42.33	(#3, #6, #19)	50.67
(#3, #6, #20)	42.67	(#3, #7, #8)	42.67	(#3, #7, #9)	43.67	(#3, #7, #10)	48.67
(#3, #7, #11)	38.00	(#3, #7, #12)	45.00	(#3, #7, #13)	41.67	(#3, #7, #14)	43.33
(#3, #7, #15)	49.33	(#3, #7, #16)	43.33	(#3, #7, #17)	41.33	(#3, #7, #18)	41.33
(#3, #7, #19)	49.67	(#3, #7, #20)	41.67	(#3, #8, #9)	43.33	(#3, #8, #10)	48.33
(#3, #8, #11)	37.67	(#3, #8, #12)	44.67	(#3, #8, #13)	41.33	(#3, #8, #14)	43.00
(#3, #8, #15)	49.00	(#3, #8, #16)	43.00	(#3, #8, #17)	41.00	(#3, #8, #18)	41.00
(#3, #8, #19)	49.33	(#3, #8, #20)	41.33	(#3, #9, #10)	49.33	(#3, #9, #11)	38.67
(#3, #9, #12)	45.67	(#3, #9, #13)	42.33	(#3, #9, #14)	44.00	(#3, #9, #15)	50.00
(#3, #9, #16)	44.00	(#3, #9, #17)	42.00	(#3, #9, #18)	42.00	(#3, #9, #19)	50.33
(#3, #9, #20)	42.33	(#3, #10, #11)	43.67	(#3, #10, #12)	50.67	(#3, #10, #13)	47.33
(#3, #10, #14)	49.00	(#3, #10, #15)	55.00	(#3, #10, #16)	49.00	(#3, #10, #17)	47.00
(#3, #10, #18)	47.00	(#3, #10, #19)	55.33	(#3, #10, #20)	47.33	(#3, #11, #12)	40.00
(#3, #11, #13)	36.67	(#3, #11, #14)	38.33	(#3, #11, #15)	44.33	(#3, #11, #16)	38.33
(#3, #11, #17)	36.33	(#3, #11, #18)	36.33	(#3, #11, #19)	44.67	(#3, #11, #20)	36.67
(#3, #12, #13)	43.67	(#3, #12, #14)	45.33	(#3, #12, #15)	51.33	(#3, #12, #16)	45.33
(#3, #12, #17)	43.33	(#3, #12, #18)	43.33	(#3, #12, #19)	51.67	(#3, #12, #20)	43.67
(#3, #13, #14)	42.00	(#3, #13, #15)	48.00	(#3, #13, #16)	42.00	(#3, #13, #17)	40.00
(#3, #13, #18)	40.00	(#3, #13, #19)	48.33	(#3, #13, #20)	40.33	(#3, #14, #15)	49.67
(#3, #14, #16)	43.67	(#3, #14, #17)	41.67	(#3, #14, #18)	41.67	(#3, #14, #19)	50.00
(#3, #14, #20)	42.00	(#3, #15, #16)	49.67	(#3, #15, #17)	47.67	(#3, #15, #18)	47.67
(#3, #15, #19)	56.00	(#3, #15, #20)	48.00	(#3, #16, #17)	41.67	(#3, #16, #18)	41.67
(#3, #16, #19)	50.00	(#3, #16, #20)	42.00	(#3, #17, #18)	39.67	(#3, #17, #19)	48.00
(#3, #17, #20)	40.00	(#3, #18, #19)	48.00	(#3, #18, #20)	40.00	(#3, #19, #20)	48.33
(#4, #5, #6)	42.33	(#4, #5, #7)	41.33	(#4, #5, #8)	41.00	(#4, #5, #9)	42.00
(#4, #5, #10)	47.00	(#4, #5, #11)	36.33	(#4, #5, #12)	43.33	(#4, #5, #13)	40.00
(#4, #5, #14)	41.67	(#4, #5, #15)	47.67	(#4, #5, #16)	41.67	(#4, #5, #17)	39.67
(#4, #5, #18)	39.67	(#4, #5, #19)	48.00	(#4, #5, #20)	40.00	(#4, #6, #7)	48.00
(#4, #6, #8)	47.67	(#4, #6, #9)	48.67	(#4, #6, #10)	53.67	(#4, #6, #11)	43.00
(#4, #6, #12)	50.00	(#4, #6, #13)	46.67	(#4, #6, #14)	48.33	(#4, #6, #15)	54.33
(#4, #6, #16)	48.33	(#4, #6, #17)	46.33	(#4, #6, #18)	46.33	(#4, #6, #19)	54.67
(#4, #6, #20)	46.67	(#4, #7, #8)	46.67	(#4, #7, #9)	47.67	(#4, #7, #10)	52.67
(#4, #7, #11)	42.00	(#4, #7, #12)	49.00	(#4, #7, #13)	45.67	(#4, #7, #14)	47.33
(#4, #7, #15)	53.33	(#4, #7, #16)	47.33	(#4, #7, #17)	45.33	(#4, #7, #18)	45.33
(#4, #7, #19)	53.67	(#4, #7, #20)	45.67	(#4, #8, #9)	47.33	(#4, #8, #10)	52.33
(#4, #8, #11)	41.67	(#4, #8, #12)	48.67	(#4, #8, #13)	45.33	(#4, #8, #14)	47.00
(#4, #8, #15)	53.00	(#4, #8, #16)	47.00	(#4, #8, #17)	45.00	(#4, #8, #18)	45.00
(#4, #8, #19)	53.33	(#4, #8, #20)	45.33	(#4, #9, #10)	53.33	(#4, #9, #11)	42.67
(#4, #9, #12)	49.67	(#4, #9, #13)	46.33	(#4, #9, #14)	48.00	(#4, #9, #15)	54.00
(#4, #9, #16)	48.00	(#4, #9, #17)	46.00	(#4, #9, #18)	46.00	(#4, #9, #19)	54.33
(#4, #9, #20)	46.33	(#4, #10, #11)	47.67	(#4, #10, #12)	54.67	(#4, #10, #13)	51.33
(#4, #10, #14)	53.00	(#4, #10, #15)	59.00	(#4, #10, #16)	53.00	(#4, #10, #17)	51.00
(#4, #10, #18)	51.00	(#4, #10, #19)	59.33	(#4, #10, #20)	51.33	(#4, #11, #12)	44.00
(#4, #11, #13)	40.67	(#4, #11, #14)	42.33	(#4, #11, #15)	48.33	(#4, #11, #16)	42.33
(#4, #11, #17)	40.33	(#4, #11, #18)	40.33	(#4, #11, #19)	48.67	(#4, #11, #20)	40.67
(#4, #12, #13)	47.67	(#4, #12, #14)	49.33	(#4, #12, #15)	55.33	(#4, #12, #16)	49.33
(#4, #12, #17)	47.33	(#4, #12, #18)	47.33	(#4, #12, #19)	55.67	(#4, #12, #20)	47.67
(#4, #13, #14)	46.00	(#4, #13, #15)	52.00	(#4, #13, #16)	46.00	(#4, #13, #17)	44.00
(#4, #13, #18)	44.00	(#4, #13, #19)	52.33	(#4, #13, #20)	44.33	(#4, #14, #15)	53.67
(#4, #14, #16)	47.67	(#4, #14, #17)	45.67	(#4, #14, #18)	45.67	(#4, #14, #19)	54.00
(#4, #14, #20)	46.00	(#4, #15, #16)	53.67	(#4, #15, #17)	51.67	(#4, #15, #18)	51.67
(#4, #15, #19)	60.00	(#4, #15, #20)	52.00	(#4, #16, #17)	45.67	(#4, #16, #18)	45.67
(#4, #16, #19)	54.00	(#4, #16, #20)	46.00	(#4, #17, #18)	43.67	(#4, #17, #19)	52.00
(#4, #17, #20)	44.00	(#4, #18, #19)	52.00	(#4, #18, #20)	44.00	(#4, #19, #20)	52.33
(#5, #6, #7)	36.33	(#5, #6, #8)	36.00	(#5, #6, #9)	37.00	(#5, #6, #10)	42.00
(#5, #6, #11)	31.33	(#5, #6, #12)	38.33	(#5, #6, #13)	35.00	(#5, #6, #14)	36.67
(#5, #6, #15)	42.67	(#5, #6, #16)	36.67	(#5, #6, #17)	34.67	(#5, #6, #18)	34.67
(#5, #6, #19)	43.00	(#5, #6, #20)	35.00	(#5, #7, #8)	35.00	(#5, #7, #9)	36.00
(#5, #7, #10)	41.00	(#5, #7, #11)	30.33	(#5, #7, #12)	37.33	(#5, #7, #13)	34.00
(#5, #7, #14)	35.67	(#5, #7, #15)	41.67	(#5, #7, #16)	35.67	(#5, #7, #17)	33.67
(#5, #7, #18)	33.67	(#5, #7, #19)	42.00	(#5, #7, #20)	34.00	(#5, #8, #9)	35.67
(#5, #8, #10)	40.67	(#5, #8, #11)	30.00	(#5, #8, #12)	37.00	(#5, #8, #13)	33.67
(#5, #8, #14)	35.33	(#5, #8, #15)	41.33	(#5, #8, #16)	35.33	(#5, #8, #17)	33.33
(#5, #8, #18)	33.33	(#5, #8, #19)	41.67	(#5, #8, #20)	33.67	(#5, #9, #10)	41.67
(#5, #9, #11)	31.00	(#5, #9, #12)	38.00	(#5, #9, #13)	34.67	(#5, #9, #14)	36.33
(#5, #9, #15)	42.33	(#5, #9, #16)	36.33	(#5, #9, #17)	34.33	(#5, #9, #18)	34.33
(#5, #9, #19)	42.67	(#5, #9, #20)	34.67	(#5, #10, #11)	36.00	(#5, #10, #12)	43.00
(#5, #10, #13)	39.67	(#5, #10, #14)	41.33	(#5, #10, #15)	47.33	(#5, #10, #16)	41.33
(#5, #10, #17)	39.33	(#5, #10, #18)	39.33	(#5, #10, #19)	47.67	(#5, #10, #20)	39.67
(#5, #11, #12)	32.33	(#5, #11, #13)	29.00	(#5, #11, #14)	30.67	(#5, #11, #15)	36.67
(#5, #11, #16)	30.67	(#5, #11, #17)	28.67	(#5, #11, #18)	28.67	(#5, #11, #19)	37.00
(#5, #11, #20)	29.00	(#5, #12, #13)	36.00	(#5, #12, #14)	37.67	(#5, #12, #15)	43.67
(#5, #12, #16)	37.67	(#5, #12, #17)	35.67	(#5, #12, #18)	35.67	(#5, #12, #19)	44.00
(#5, #12, #20)	36.00	(#5, #13, #14)	34.33	(#5, #13, #15)	40.33	(#5, #13, #16)	34.33
(#5, #13, #17)	32.33	(#5, #13, #18)	32.33	(#5, #13, #19)	40.67	(#5, #13, #20)	32.67
(#5, #14, #15)	42.00	(#5, #14, #16)	36.00	(#5, #14, #17)	34.00	(#5, #14, #18)	34.00
(#5, #14, #19)	42.33	(#5, #14, #20)	34.33	(#5, #15, #16)	42.00	(#5, #15, #17)	40.00
(#5, #15, #18)	40.00	(#5, #15, #19)	48.33	(#5, #15, #20)	40.33	(#5, #16, #17)	34.00
(#5, #16, #18)	34.00	(#5, #16, #19)	42.33	(#5, #16, #20)	34.33	(#5, #17, #18)	32.00
(#5, #17, #19)	40.33	(#5, #17, #20)	32.33	(#5, #18, #19)	40.33	(#5, #18, #20)	32.33
(#5, #19, #20)	40.67	(#6, #7, #8)	41.67	(#6, #7, #9)	42.67	(#6, #7, #10)	47.67
(#6, #7, #11)	37.00	(#6, #7, #12)	44.00	(#6, #7, #13)	40.67	(#6, #7, #14)	42.33
(#6, #7, #15)	48.33	(#6, #7, #16)	42.33	(#6, #7, #17)	40.33	(#6, #7, #18)	40.33
(#6, #7, #19)	48.67	(#6, #7, #20)	40.67	(#6, #8, #9)	42.33	(#6, #8, #10)	47.33
(#6, #8, #11)	36.67	(#6, #8, #12)	43.67	(#6, #8, #13)	40.33	(#6, #8, #14)	42.00
(#6, #8, #15)	48.00	(#6, #8, #16)	42.00	(#6, #8, #17)	40.00	(#6, #8, #18)	40.00
(#6, #8, #19)	48.33	(#6, #8, #20)	40.33	(#6, #9, #10)	48.33	(#6, #9, #11)	37.67
(#6, #9, #12)	44.67	(#6, #9, #13)	41.33	(#6, #9, #14)	43.00	(#6, #9, #15)	49.00
(#6, #9, #16)	43.00	(#6, #9, #17)	41.00	(#6, #9, #18)	41.00	(#6, #9, #19)	49.33
(#6, #9, #20)	41.33	(#6, #10, #11)	42.67	(#6, #10, #12)	49.67	(#6, #10, #13)	46.33
(#6, #10, #14)	48.00	(#6, #10, #15)	54.00	(#6, #10, #16)	48.00	(#6, #10, #17)	46.00
(#6, #10, #18)	46.00	(#6, #10, #19)	54.33	(#6, #10, #20)	46.33	(#6, #11, #12)	39.00
(#6, #11, #13)	35.67	(#6, #11, #14)	37.33	(#6, #11, #15)	43.33	(#6, #11, #16)	37.33
(#6, #11, #17)	35.33	(#6, #11, #18)	35.33	(#6, #11, #19)	43.67	(#6, #11, #20)	35.67
(#6, #12, #13)	42.67	(#6, #12, #14)	44.33	(#6, #12, #15)	50.33	(#6, #12, #16)	44.33
(#6, #12, #17)	42.33	(#6, #12, #18)	42.33	(#6, #12, #19)	50.67	(#6, #12, #20)	42.67
(#6, #13, #14)	41.00	(#6, #13, #15)	47.00	(#6, #13, #16)	41.00	(#6, #13, #17)	39.00
(#6, #13, #18)	39.00	(#6, #13, #19)	47.33	(#6, #13, #20)	39.33	(#6, #14, #15)	48.67
(#6, #14, #16)	42.67	(#6, #14, #17)	40.67	(#6, #14, #18)	40.67	(#6, #14, #19)	49.00
(#6, #14, #20)	41.00	(#6, #15, #16)	48.67	(#6, #15, #17)	46.67	(#6, #15, #18)	46.67
(#6, #15, #19)	55.00	(#6, #15, #20)	47.00	(#6, #16, #17)	40.67	(#6, #16, #18)	40.67
(#6, #16, #19)	49.00	(#6, #16, #20)	41.00	(#6, #17, #18)	38.67	(#6, #17, #19)	47.00
(#6, #17, #20)	39.00	(#6, #18, #19)	47.00	(#6, #18, #20)	39.00	(#6, #19, #20)	47.33
(#7, #8, #9)	41.33	(#7, #8, #10)	46.33	(#7, #8, #11)	35.67	(#7, #8, #12)	42.67
(#7, #8, #13)	39.33	(#7, #8, #14)	41.00	(#7, #8, #15)	47.00	(#7, #8, #16)	41.00
(#7, #8, #17)	39.00	(#7, #8, #18)	39.00	(#7, #8, #19)	47.33	(#7, #8, #20)	39.33
(#7, #9, #10)	47.33	(#7, #9, #11)	36.67	(#7, #9, #12)	43.67	(#7, #9, #13)	40.33
(#7, #9, #14)	42.00	(#7, #9, #15)	48.00	(#7, #9, #16)	42.00	(#7, #9, #17)	40.00
(#7, #9, #18)	40.00	(#7, #9, #19)	48.33	(#7, #9, #20)	40.33	(#7, #10, #11)	41.67
(#7, #10, #12)	48.67	(#7, #10, #13)	45.33	(#7, #10, #14)	47.00	(#7, #10, #15)	53.00
(#7, #10, #16)	47.00	(#7, #10, #17)	45.00	(#7, #10, #18)	45.00	(#7, #10, #19)	53.33
(#7, #10, #20)	45.33	(#7, #11, #12)	38.00	(#7, #11, #13)	34.67	(#7, #11, #14)	36.33
(#7, #11, #15)	42.33	(#7, #11, #16)	36.33	(#7, #11, #17)	34.33	(#7, #11, #18)	34.33
(#7, #11, #19)	42.67	(#7, #11, #20)	34.67	(#7, #12, #13)	41.67	(#7, #12, #14)	43.33
(#7, #12, #15)	49.33	(#7, #12, #16)	43.33	(#7, #12, #17)	41.33	(#7, #12, #18)	41.33
(#7, #12, #19)	49.67	(#7, #12, #20)	41.67	(#7, #13, #14)	40.00	(#7, #13, #15)	46.00
(#7, #13, #16)	40.00	(#7, #13, #17)	38.00	(#7, #13, #18)	38.00	(#7, #13, #19)	46.33
(#7, #13, #20)	38.33	(#7, #14, #15)	47.67	(#7, #14, #16)	41.67	(#7, #14, #17)	39.67
(#7, #14, #18)	39.67	(#7, #14, #19)	48.00	(#7, #14, #20)	40.00	(#7, #15, #16)	47.67
(#7, #15, #17)	45.67	(#7, #15, #18)	45.67	(#7, #15, #19)	54.00	(#7, #15, #20)	46.00
(#7, #16, #17)	39.67	(#7, #16, #18)	39.67	(#7, #16, #19)	48.00	(#7, #16, #20)	40.00
(#7, #17, #18)	37.67	(#7, #17, #19)	46.00	(#7, #17, #20)	38.00	(#7, #18, #19)	46.00
(#7, #18, #20)	38.00	(#7, #19, #20)	46.33	(#8, #9, #10)	47.00	(#8, #9, #11)	36.33
(#8, #9, #12)	43.33	(#8, #9, #13)	40.00	(#8, #9, #14)	41.67	(#8, #9, #15)	47.67
(#8, #9, #16)	41.67	(#8, #9, #17)	39.67	(#8, #9, #18)	39.67	(#8, #9, #19)	48.00
(#8, #9, #20)	40.00	(#8, #10, #11)	41.33	(#8, #10, #12)	48.33	(#8, #10, #13)	45.00
(#8, #10, #14)	46.67	(#8, #10, #15)	52.67	(#8, #10, #16)	46.67	(#8, #10, #17)	44.67
(#8, #10, #18)	44.67	(#8, #10, #19)	53.00	(#8, #10, #20)	45.00	(#8, #11, #12)	37.67
(#8, #11, #13)	34.33	(#8, #11, #14)	36.00	(#8, #11, #15)	42.00	(#8, #11, #16)	36.00
(#8, #11, #17)	34.00	(#8, #11, #18)	34.00	(#8, #11, #19)	42.33	(#8, #11, #20)	34.33
(#8, #12, #13)	41.33	(#8, #12, #14)	43.00	(#8, #12, #15)	49.00	(#8, #12, #16)	43.00
(#8, #12, #17)	41.00	(#8, #12, #18)	41.00	(#8, #12, #19)	49.33	(#8, #12, #20)	41.33
(#8, #13, #14)	39.67	(#8, #13, #15)	45.67	(#8, #13, #16)	39.67	(#8, #13, #17)	37.67
(#8, #13, #18)	37.67	(#8, #13, #19)	46.00	(#8, #13, #20)	38.00	(#8, #14, #15)	47.33
(#8, #14, #16)	41.33	(#8, #14, #17)	39.33	(#8, #14, #18)	39.33	(#8, #14, #19)	47.67
(#8, #14, #20)	39.67	(#8, #15, #16)	47.33	(#8, #15, #17)	45.33	(#8, #15, #18)	45.33
(#8, #15, #19)	53.67	(#8, #15, #20)	45.67	(#8, #16, #17)	39.33	(#8, #16, #18)	39.33
(#8, #16, #19)	47.67	(#8, #16, #20)	39.67	(#8, #17, #18)	37.33	(#8, #17, #19)	45.67
(#8, #17, #20)	37.67	(#8, #18, #19)	45.67	(#8, #18, #20)	37.67	(#8, #19, #20)	46.00
(#9, #10, #11)	42.33	(#9, #10, #12)	49.33	(#9, #10, #13)	46.00	(#9, #10, #14)	47.67
(#9, #10, #15)	53.67	(#9, #10, #16)	47.67	(#9, #10, #17)	45.67	(#9, #10, #18)	45.67
(#9, #10, #19)	54.00	(#9, #10, #20)	46.00	(#9, #11, #12)	38.67	(#9, #11, #13)	35.33
(#9, #11, #14)	37.00	(#9, #11, #15)	43.00	(#9, #11, #16)	37.00	(#9, #11, #17)	35.00
(#9, #11, #18)	35.00	(#9, #11, #19)	43.33	(#9, #11, #20)	35.33	(#9, #12, #13)	42.33
(#9, #12, #14)	44.00	(#9, #12, #15)	50.00	(#9, #12, #16)	44.00	(#9, #12, #17)	42.00
(#9, #12, #18)	42.00	(#9, #12, #19)	50.33	(#9, #12, #20)	42.33	(#9, #13, #14)	40.67
(#9, #13, #15)	46.67	(#9, #13, #16)	40.67	(#9, #13, #17)	38.67	(#9, #13, #18)	38.67
(#9, #13, #19)	47.00	(#9, #13, #20)	39.00	(#9, #14, #15)	48.33	(#9, #14, #16)	42.33
(#9, #14, #17)	40.33	(#9, #14, #18)	40.33	(#9, #14, #19)	48.67	(#9, #14, #20)	40.67
(#9, #15, #16)	48.33	(#9, #15, #17)	46.33	(#9, #15, #18)	46.33	(#9, #15, #19)	54.67
(#9, #15, #20)	46.67	(#9, #16, #17)	40.33	(#9, #16, #18)	40.33	(#9, #16, #19)	48.67
(#9, #16, #20)	40.67	(#9, #17, #18)	38.33	(#9, #17, #19)	46.67	(#9, #17, #20)	38.67
(#9, #18, #19)	46.67	(#9, #18, #20)	38.67	(#9, #19, #20)	47.00	(#10, #11, #12)	43.67
(#10, #11, #13)	40.33	(#10, #11, #14)	42.00	(#10, #11, #15)	48.00	(#10, #11, #16)	42.00
(#10, #11, #17)	40.00	(#10, #11, #18)	40.00	(#10, #11, #19)	48.33	(#10, #11, #20)	40.33
(#10, #12, #13)	47.33	(#10, #12, #14)	49.00	(#10, #12, #15)	55.00	(#10, #12, #16)	49.00
(#10, #12, #17)	47.00	(#10, #12, #18)	47.00	(#10, #12, #19)	55.33	(#10, #12, #20)	47.33
(#10, #13, #14)	45.67	(#10, #13, #15)	51.67	(#10, #13, #16)	45.67	(#10, #13, #17)	43.67
(#10, #13, #18)	43.67	(#10, #13, #19)	52.00	(#10, #13, #20)	44.00	(#10, #14, #15)	53.33
(#10, #14, #16)	47.33	(#10, #14, #17)	45.33	(#10, #14, #18)	45.33	(#10, #14, #19)	53.67
(#10, #14, #20)	45.67	(#10, #15, #16)	53.33	(#10, #15, #17)	51.33	(#10, #15, #18)	51.33
(#10, #15, #19)	59.67	(#10, #15, #20)	51.67	(#10, #16, #17)	45.33	(#10, #16, #18)	45.33
(#10, #16, #19)	53.67	(#10, #16, #20)	45.67	(#10, #17, #18)	43.33	(#10, #17, #19)	51.67
(#10, #17, #20)	43.67	(#10, #18, #19)	51.67	(#10, #18, #20)	43.67	(#10, #19, #20)	52.00
(#11, #12, #13)	36.67	(#11, #12, #14)	38.33	(#11, #12, #15)	44.33	(#11, #12, #16)	38.33
(#11, #12, #17)	36.33	(#11, #12, #18)	36.33	(#11, #12, #19)	44.67	(#11, #12, #20)	36.67
(#11, #13, #14)	35.00	(#11, #13, #15)	41.00	(#11, #13, #16)	35.00	(#11, #13, #17)	33.00
(#11, #13, #18)	33.00	(#11, #13, #19)	41.33	(#11, #13, #20)	33.33	(#11, #14, #15)	42.67
(#11, #14, #16)	36.67	(#11, #14, #17)	34.67	(#11, #14, #18)	34.67	(#11, #14, #19)	43.00
(#11, #14, #20)	35.00	(#11, #15, #16)	42.67	(#11, #15, #17)	40.67	(#11, #15, #18)	40.67
(#11, #15, #19)	49.00	(#11, #15, #20)	41.00	(#11, #16, #17)	34.67	(#11, #16, #18)	34.67
(#11, #16, #19)	43.00	(#11, #16, #20)	35.00	(#11, #17, #18)	32.67	(#11, #17, #19)	41.00
(#11, #17, #20)	33.00	(#11, #18, #19)	41.00	(#11, #18, #20)	33.00	(#11, #19, #20)	41.33
(#12, #13, #14)	42.00	(#12, #13, #15)	48.00	(#12, #13, #16)	42.00	(#12, #13, #17)	40.00
(#12, #13, #18)	40.00	(#12, #13, #19)	48.33	(#12, #13, #20)	40.33	(#12, #14, #15)	49.67
(#12, #14, #16)	43.67	(#12, #14, #17)	41.67	(#12, #14, #18)	41.67	(#12, #14, #19)	50.00
(#12, #14, #20)	42.00	(#12, #15, #16)	49.67	(#12, #15, #17)	47.67	(#12, #15, #18)	47.67
(#12, #15, #19)	56.00	(#12, #15, #20)	48.00	(#12, #16, #17)	41.67	(#12, #16, #18)	41.67
(#12, #16, #19)	50.00	(#12, #16, #20)	42.00	(#12, #17, #18)	39.67	(#12, #17, #19)	48.00
(#12, #17, #20)	40.00	(#12, #18, #19)	48.00	(#12, #18, #20)	40.00	(#12, #19, #20)	48.33
(#13, #14, #15)	46.33	(#13, #14, #16)	40.33	(#13, #14, #17)	38.33	(#13, #14, #18)	38.33
(#13, #14, #19)	46.67	(#13, #14, #20)	38.67	(#13, #15, #16)	46.33	(#13, #15, #17)	44.33
(#13, #15, #18)	44.33	(#13, #15, #19)	52.67	(#13, #15, #20)	44.67	(#13, #16, #17)	38.33
(#13, #16, #18)	38.33	(#13, #16, #19)	46.67	(#13, #16, #20)	38.67	(#13, #17, #18)	36.33
(#13, #17, #19)	44.67	(#13, #17, #20)	36.67	(#13, #18, #19)	44.67	(#13, #18, #20)	36.67
(#13, #19, #20)	45.00	(#14, #15, #16)	48.00	(#14, #15, #17)	46.00	(#14, #15, #18)	46.00
(#14, #15, #19)	54.33	(#14, #15, #20)	46.33	(#14, #16, #17)	40.00	(#14, #16, #18)	40.00
(#14, #16, #19)	48.33	(#14, #16, #20)	40.33	(#14, #17, #18)	38.00	(#14, #17, #19)	46.33
(#14, #17, #20)	38.33	(#14, #18, #19)	46.33	(#14, #18, #20)	38.33	(#14, #19, #20)	46.67
(#15, #16, #17)	46.00	(#15, #16, #18)	46.00	(#15, #16, #19)	54.33	(#15, #16, #20)	46.33
(#15, #17, #18)	44.00	(#15, #17, #19)	52.33	(#15, #17, #20)	44.33	(#15, #18, #19)	52.33
(#15, #18, #20)	44.33	(#15, #19, #20)	52.67	(#16, #17, #18)	38.00	(#16, #17, #19)	46.33
(#16, #17, #20)	38.33	(#16, #18, #19)	46.33	(#16, #18, #20)	38.33	(#16, #19, #20)	46.67
(#17, #18, #19)	44.33	(#17, #18, #20)	36.33	(#17, #19, #20)	44.67	(#18, #19, #20)	44.67

{
  const slider = document.getElementById("margin-slider");
  const marginVal = document.getElementById("margin-val-display");
  const coverageDisplay = document.getElementById("coverage-display");
  const mu = 43.45;

  function updateTable() {
    const table = document.getElementById("all-samples-fish");
    if (!table || !slider || !marginVal || !coverageDisplay) return;

    const M = parseFloat(slider.value);
    marginVal.textContent = M.toFixed(1) + " dkg";

    // Update headers from "Sample Mean (dkg)" to "CI"
    const headers = table.querySelectorAll("thead th");
    headers.forEach((th, index) => {
      if (index % 2 === 1 && th.textContent !== "CI") {
        th.textContent = "CI";
      }
    });

    // Update caption dynamically to remind students of the true mean
    const caption = table.querySelector("caption");
    if (caption && !caption.dataset.updated) {
      caption.dataset.updated = "true";
      caption.innerHTML = "<strong>Table 2:</strong> <em>All possible samples of three fish from the aquarium and their respective confidence intervals. (Reminder: true mean is &mu; = 43.45 dkg).</em>";
    }

    const cells = table.querySelectorAll("tbody td");
    let totalCount = 0;
    let coverCount = 0;

    for (let i = 0; i < cells.length; i += 2) {
      if (i + 1 >= cells.length) break;
      const labelCell = cells[i];
      const valCell = cells[i+1];

      // Store original mean value if not already done
      if (!valCell.dataset.originalMean) {
        valCell.dataset.originalMean = valCell.textContent.trim();
      }

      const originalMeanStr = valCell.dataset.originalMean;
      const meanVal = parseFloat(originalMeanStr);

      if (!isNaN(meanVal)) {
        totalCount++;
        const covers = Math.abs(meanVal - mu) <= M;

        // Update the cell content to show "mean ± M"
        valCell.textContent = `${originalMeanStr} ± ${M.toFixed(1)}`;

        if (covers) {
          coverCount++;
          labelCell.style.backgroundColor = "#e8f5e9"; // light green background
          labelCell.style.color = "#2e7d32"; // dark green text
          valCell.style.backgroundColor = "#e8f5e9";
          valCell.style.color = "#2e7d32";
        } else {
          labelCell.style.backgroundColor = "#ffebee"; // light red background
          labelCell.style.color = "#c62828"; // dark red text
          valCell.style.backgroundColor = "#ffebee";
          valCell.style.color = "#c62828";
        }
      }
    }

    const percentage = (coverCount / totalCount) * 100;
    coverageDisplay.textContent = `Coverage: ${percentage.toFixed(1)}% (${coverCount} / ${totalCount} CIs)`;
  }

  if (slider) {
    slider.addEventListener("input", updateTable);
    // Run update multiple times on load/reveal events to ensure correct rendering in Revealjs slides
    updateTable();
    setTimeout(updateTable, 100);
    setTimeout(updateTable, 500);

    // Also update whenever Reveal.js slide changes (to handle lazy rendering)
    if (window.Reveal) {
      window.Reveal.on('slidechanged', updateTable);
    }
  }
}

Play a Game: Capture the Mean! 🎣

Seed:

Guesses this match: 0

Set Margin (\(M\)): 5.0 dkg

Coverage: –

Correct guesses: 0 / 0

Streak: 0 🔥

{
  // Quarto/Reveal.js can re-run this OJS cell when the slide is revisited
  // (the coverage-table cell has the same issue, but is idempotent so it
  // doesn't matter there). Here each run would create a fresh closure with
  // its own hidden population and re-register click listeners on top of the
  // old ones, so the same "seed" could silently produce different games.
  // Guard so the whole setup only ever runs once per container.
  const gameContainer = document.getElementById("game-container");
  if (gameContainer && gameContainer.dataset.gameInitialized) {
    // Already initialized in an earlier run of this cell -- do nothing.
  } else {
    if (gameContainer) gameContainer.dataset.gameInitialized = "true";

    const seedInput = document.getElementById("game-seed-input");
    const newGameBtn = document.getElementById("new-game-btn");
    const guessCounterEl = document.getElementById("game-guess-counter");
    const slider = document.getElementById("game-margin-slider");
    const marginVal = document.getElementById("game-margin-val");
    const coverageDisplay = document.getElementById("game-coverage-display");
    const drawBtn = document.getElementById("draw-btn");
    const playzone = document.getElementById("game-playzone");
    const fishDisplay = document.getElementById("fish-display");
    const questionEl = document.getElementById("game-question");
    const guessYesBtn = document.getElementById("guess-yes");
    const guessNoBtn = document.getElementById("guess-no");
    const feedbackEl = document.getElementById("game-feedback");
    const scoreCorrectEl = document.getElementById("score-correct");
    const scoreTotalEl = document.getElementById("score-total");
    const streakEl = document.getElementById("score-streak");
    const revealBtn = document.getElementById("reveal-mean-btn");
    const trueMeanDisplay = document.getElementById("true-mean-display");

    // Number of guesses in the current match before the reveal button appears.
    const REVEAL_AFTER_GUESSES = 3;

    // Overall score, kept across matches (only reset on page load).
    let totalGuesses = 0;
    let correctGuesses = 0;
    let streak = 0;

    // Per-match state (reset every time a new hidden population is generated).
    let weights = [];
    let mu = 0;
    let allMeans = [];
    let matchGuessCount = 0;
    let currentIndices = null;
    let currentMean = 0;

    // Seeded PRNG (mulberry32) so a given seed always reproduces the same population.
    function mulberry32(seed) {
      let a = seed;
      return function () {
        a |= 0; a = (a + 0x6D2B79F5) | 0;
        let t = Math.imul(a ^ (a >>> 15), 1 | a);
        t = (t + Math.imul(t ^ (t >>> 7), 61 | t)) ^ t;
        return ((t ^ (t >>> 14)) >>> 0) / 4294967296;
      };
    }

    // Turns whatever the student typed (a number or arbitrary text) into a 32-bit integer seed.
    function seedFromInput(str) {
      if (!str) return Math.floor(Math.random() * 1e9);
      const trimmed = str.trim();
      if (/^\d+$/.test(trimmed)) return parseInt(trimmed, 10) >>> 0;
      let h = 0;
      for (let i = 0; i < trimmed.length; i++) {
        h = (Math.imul(31, h) + trimmed.charCodeAt(i)) | 0;
      }
      return h >>> 0;
    }

    function randNormal(rng, mean, sd) {
      const u1 = rng();
      const u2 = rng();
      const z = Math.sqrt(-2 * Math.log(u1)) * Math.cos(2 * Math.PI * u2);
      return mean + z * sd;
    }

    // Generates a hidden population of 20 "fish" from a seed. The true mean is
    // never derived from a fixed constant -- it is whatever the generated
    // population's mean turns out to be, so it is genuinely unknown up front.
    function generatePopulation(seed) {
      const rng = mulberry32(seed);
      const baseMean = 30 + rng() * 30; // somewhere between 30 and 60 dkg
      const pop = [];
      for (let i = 0; i < 20; i++) {
        const w = Math.max(5, randNormal(rng, baseMean, 8));
        pop.push(Math.round(w * 10) / 10);
      }
      const trueMean = pop.reduce((a, b) => a + b, 0) / pop.length;
      return { weights: pop, mu: trueMean };
    }

    function computeAllMeans(pop) {
      const means = [];
      for (let i = 0; i < 20; i++) {
        for (let j = i + 1; j < 20; j++) {
          for (let k = j + 1; k < 20; k++) {
            means.push((pop[i] + pop[j] + pop[k]) / 3);
          }
        }
      }
      return means;
    }

    function getCoverage(M) {
      let coverCount = 0;
      allMeans.forEach(meanVal => {
        if (Math.abs(meanVal - mu) <= M) {
          coverCount++;
        }
      });
      const pct = ((coverCount / allMeans.length) * 100).toFixed(1);
      return { pct, count: coverCount, total: allMeans.length };
    }

    function updateSliderDisplay() {
      if (!slider || !marginVal || !coverageDisplay) return;
      const M = parseFloat(slider.value);
      marginVal.textContent = M.toFixed(1) + " dkg";

      const cov = getCoverage(M);
      coverageDisplay.textContent = `Coverage: ${cov.pct}% (${cov.count} / ${cov.total} CIs)`;

      // Update the question text dynamically if a sample is active and we haven't guessed yet
      if (currentIndices && feedbackEl && feedbackEl.style.display === "none") {
        updateQuestionPrompt(M);
      }
    }

    function updateQuestionPrompt(M) {
      questionEl.innerHTML = `Does this sample yield a confidence interval <em>x̄</em> &plusmn; <em>M</em> ` +
        `(i.e., <span style="font-size: 1.1em; color: #1a237e; background: #e8eaf6; padding: 0.15em 0.4em; border-radius: 4px; font-weight: 800; font-family: monospace;">` +
        `${currentMean.toFixed(2)} &plusmn; ${M.toFixed(1)}</span>) that captures the true parameter?`;
    }

    // Starts a fresh match: regenerates the hidden population/mean from the
    // seed box (writing back a random seed if the box was left blank so it
    // can be copied down to replay this exact match later).
    function startNewMatch() {
      const seed = seedFromInput(seedInput.value);
      seedInput.value = String(seed);

      const generated = generatePopulation(seed);
      weights = generated.weights;
      mu = generated.mu;
      allMeans = computeAllMeans(weights);

      matchGuessCount = 0;
      currentIndices = null;
      currentMean = 0;

      guessCounterEl.textContent = `Guesses this match: ${matchGuessCount}`;
      revealBtn.style.display = "none";
      trueMeanDisplay.textContent = "";
      playzone.style.display = "none";
      feedbackEl.style.display = "none";

      updateSliderDisplay();
    }

    function handleDraw() {
      // Select 3 random unique indices
      let indices = [];
      while (indices.length < 3) {
        const idx = Math.floor(Math.random() * 20);
        if (!indices.includes(idx)) {
          indices.push(idx);
        }
      }

      currentIndices = indices;
      const sampleWeights = indices.map(i => weights[i]);
      const sum = sampleWeights.reduce((a, b) => a + b, 0);
      currentMean = sum / 3;

      // Display fish visuals scaled by weight
      fishDisplay.innerHTML = "";
      indices.forEach(idx => {
        const wt = weights[idx];
        const fishNum = (idx % 4) + 1;
        const height = 20 + ((wt - 24) / (61 - 24)) * 20; // scale between 20px and 40px

        const wrapper = document.createElement("div");
        wrapper.style.display = "flex";
        wrapper.style.flexDirection = "column";
        wrapper.style.alignItems = "center";
        wrapper.style.gap = "0.2em";

        const img = document.createElement("img");
        img.src = `../slides-module01/imgs/fish${fishNum}.svg`;
        img.style.height = `${height}px`;
        img.style.width = "auto";
        img.style.filter = "drop-shadow(0px 2px 4px rgba(0,0,0,0.15))";

        const label = document.createElement("span");
        label.textContent = `Fish #${idx+1}`;
        label.style.fontSize = "0.75em";
        label.style.fontWeight = "bold";
        label.style.color = "#37474f";

        wrapper.appendChild(img);
        wrapper.appendChild(label);
        fishDisplay.appendChild(wrapper);
      });

      // Update text and show gameplay zone
      const M = parseFloat(slider.value);
      updateQuestionPrompt(M);

      playzone.style.display = "block";
      feedbackEl.style.display = "none";
      document.getElementById("guess-buttons").style.display = "flex";

      // Scroll slider slide to keep the game in view
      playzone.scrollIntoView({ behavior: 'smooth', block: 'nearest' });
    }

    function handleGuess(guessYes) {
      if (!currentIndices) return;

      const M = parseFloat(slider.value);
      const lower = currentMean - M;
      const upper = currentMean + M;
      const covers = (mu >= lower) && (mu <= upper);

      const isCorrect = (guessYes && covers) || (!guessYes && !covers);

      totalGuesses++;
      matchGuessCount++;
      guessCounterEl.textContent = `Guesses this match: ${matchGuessCount}`;

      if (isCorrect) {
        correctGuesses++;
        streak++;
        feedbackEl.style.backgroundColor = "#e8f5e9";
        feedbackEl.style.color = "#2e7d32";
        feedbackEl.innerHTML = "Correct!";
      } else {
        streak = 0;
        feedbackEl.style.backgroundColor = "#ffebee";
        feedbackEl.style.color = "#c62828";
        feedbackEl.innerHTML = "Not quite.";
      }

      // Update scoreboard
      scoreCorrectEl.textContent = correctGuesses;
      scoreTotalEl.textContent = totalGuesses;
      streakEl.textContent = `${streak} ${streak > 0 ? "🔥" : ""}`;

      // Hide buttons and show feedback
      document.getElementById("guess-buttons").style.display = "none";
      feedbackEl.style.display = "block";

      // Clear sample state
      currentIndices = null;

      // Only offer the reveal after enough guesses in this match, so students
      // cannot just draw once and immediately check the answer.
      if (matchGuessCount >= REVEAL_AFTER_GUESSES) {
        revealBtn.style.display = "inline-block";
      }
    }

    function handleReveal() {
      trueMeanDisplay.textContent = `True mean for this seed: ${mu.toFixed(2)} dkg`;
    }

    if (newGameBtn) {
      newGameBtn.addEventListener("click", startNewMatch);
    }

    if (slider) {
      slider.addEventListener("input", updateSliderDisplay);
    }

    if (drawBtn) {
      drawBtn.addEventListener("click", handleDraw);
    }

    if (guessYesBtn) {
      guessYesBtn.addEventListener("click", () => handleGuess(true));
    }

    if (guessNoBtn) {
      guessNoBtn.addEventListener("click", () => handleGuess(false));
    }

    if (revealBtn) {
      revealBtn.addEventListener("click", handleReveal);
    }

    startNewMatch();
  }
}

The Trade-Off: Margin vs. Confidence

We want to find intervals that are narrow and have high coverage.
How can we find those intervals when we only have one sample?
We can use our estimated sampling distribution:
- via bootstrapping
- via CLT (next module)

How does the sample affect the interval?

A confidence interval depends entirely on the sample collected.
- The boundaries of the interval are statistics.
If we collect a different sample, our confidence interval will almost certainly be different.

Visualizing a 95% confidence interval

Changing the confidence level to 99%

How sample size affects the bootstrap distribution

Shading the intervals: \(n = 10\) vs. \(n = 100\)

What should we report in our findings?

We can report both our sample point estimate and the confidence interval.

Point estimate: Our best estimate of the population parameter.
Confidence interval: A plausible range of values where we expect the true population parameter to fall.

Coding it up: the bootstrap distribution

The infer package builds the bootstrap distribution with a pipeline of four verbs:

bootstrap_dist <- sample |>
  specify(response = var) |>
  generate(reps = 1000, type = "bootstrap") |>
  calculate(stat = "mean")

specify(): which variable are we studying?
generate(): resample (with replacement) 1000 times
calculate(): compute the statistic for each resample

Coding it up: the bootstrap distribution

bootstrap_dist is a tibble with one row per resample:

Response: var (numeric)
# A tibble: 1,000 × 2
   replicate  stat
       <int> <dbl>
 1         1  14.9
 2         2  14.8
 3         3  15.2
 4         4  15.7
 5         5  16.2
 6         6  13.6
 7         7  14.8
 8         8  14.5
 9         9  13.8
10        10  15.8
# ℹ 990 more rows

Coding it up: the confidence interval

ci_95 <- bootstrap_dist |>
  get_confidence_interval(level = 0.95, type = "percentile")

level: the confidence level (e.g. \(0.95\) for a \(95\%\) CI)
type = "percentile": use the percentile method we just learned

# A tibble: 1 × 2
  lower_ci upper_ci
     <dbl>    <dbl>
1     12.3     16.8

Coding it up: visualizing the interval

visualize(bootstrap_dist) +
  shade_confidence_interval(ci_95)

The same workflow, four different statistics

The specify() |> generate() |> calculate() pipeline does not change – only what we specify and calculate does.

A mean: how long does a food delivery take, on average?
A proportion: what fraction of shipped packages arrive damaged?
A difference in means: do two phone models really differ in battery life?
A difference in proportions: did a website redesign really improve conversion?

Try editing and running the code on the next slides yourself!

Example: a mean

A delivery app wants to know the average delivery time for orders placed downtown.

Population: all downtown orders
Variable: delivery time (minutes)
Parameter: \(\mu\), the true mean delivery time

Example: a mean – the CI

Example: a proportion

A shipping company wants to know what fraction of packages in a shipment arrive damaged.

Population: all packages in the shipment
Variable: condition (damaged / ok)
Parameter: \(p\), the true proportion damaged

Example: a proportion – the CI

Note

For a categorical variable, specify() needs success to say which level we are counting.

Example: a difference in means

A reviewer wants to know if two phone models really differ in average battery life.

Population: all units of each phone model
Variable: battery life (hours), and which model
Parameter: \(\mu_A - \mu_B\), the true difference in mean battery life

Example: a difference in means – the CI

Note

Comparing two groups uses a formula, response ~ explanatory, and order fixes the direction of the subtraction.

Example: a difference in proportions

A company wants to know if a website redesign really improved the conversion rate.

Population: all visitors to each version of the site
Variable: converted (yes / no), and which design they saw
Parameter: \(p_{\text{new}} - p_{\text{old}}\), the true difference in conversion rate

Example: a difference in proportions – the CI

Today’s worksheet

Calculate confidence intervals using the bootstrap distribution
Interpret confidence intervals
Explore its properties and the effect of sample size

Take-home points

Scroll down

A confidence interval (CI) gives a plausible range where we expect our true population parameter to fall.
We can calculate the \(C\%\) CI by taking the \(\bigl(\frac{100-C}{2} \bigr)^{\text{th}}\) and \(\bigl( \frac{100+C}{2} \bigr)^{\text{th}}\) percentiles from the bootstrap distribution.
- For example, for a \(90\%\) CI, we take the \(5^{\text{th}}\) and \(95^{\text{th}}\) percentiles.
Interpretation: We are \(C\%\) confident that the true population parameter lies within our interval.
- But always make sure to interpret in the context of the problem at hand.
Confidence vs. precision trade-off: A higher level of confidence yields a wider (less precise) interval.

Now it’s your turn!

Navigate to Canvas, open worksheet_04

We are here to help!

Confidence Intervals via Bootstrapping

Learning objectives

Review: bootstrap approximation of the sampling distribution

Review: bootstrap approximation of the sampling distribution

Review: bootstrap approximation of the sampling distribution

Review: bootstrap approximation of the sampling distribution

Review: bootstrap approximation of the sampling distribution

Today’s goal

Point Estimation vs. Confidence Intervals

Using the bootstrap to calculate a confidence interval (plausible range)

Using the bootstrap to calculate a confidence interval (plausible range)

Using the bootstrap to calculate a confidence interval (plausible range)

Using the bootstrap to calculate a confidence interval (plausible range)

95% confidence interval via bootstrapping

Visualizing confidence intervals

Interpretation of a 95% confidence interval (CI)

Misconception: Probability vs. Confidence

Revisiting the Aquarium Example

All 1,140 Possible Intervals

Play a Game: Capture the Mean! 🎣

The Trade-Off: Margin vs. Confidence

How does the sample affect the interval?

Visualizing a 95% confidence interval

Changing the confidence level to 99%

How sample size affects the bootstrap distribution

Shading the intervals: \(n = 10\) vs. \(n = 100\)

What should we report in our findings?

Coding it up: the bootstrap distribution

Coding it up: the bootstrap distribution

Coding it up: the confidence interval

Coding it up: visualizing the interval

The same workflow, four different statistics

Example: a mean

Example: a mean – the CI

Example: a proportion

Example: a proportion – the CI

Example: a difference in means

Example: a difference in means – the CI

Example: a difference in proportions

Example: a difference in proportions – the CI

Today’s worksheet

Take-home points

Now it’s your turn!