Introduction to Inference for Data Science

STAT 201 - Lecture 01

Population and parameters

• Population: The set of all units we want to study;

• Parameter: A quantity that summarizes the population (e.g., mean, median, std. dev., quantiles, min, max, proportion);

  • Parameters are constant (not random!);
  • Parameters are usually unknown;

• Variable: A variable is an attribute of the elements in the population.

Example: Aquarium

An aquarium has 20 fish. We are interested in the average weight of the fish in the aquarium.

• Population: 20 fish in the aquarium;
• Variable of interest: weight;
• Parameter: average weight of the 20 fish in the aquarium (denoted by \(\mu\));

Example: Aquarium

{

  function getRandom(min, max) {
    //Generates a number from U(min, max)
    return Math.random() * (max - min) + min;
  }

  // This code append the images to the aquarium
  const N = 20; // how many images to append
  const div = document.querySelector("#aquarium");
  
  for (let i=0; i < N; i++){
     let img_file;
     if (i % 4 == 0) img_file = 'imgs/fish1.svg';
     if (i % 4 == 1) img_file = 'imgs/fish2.svg';
     if (i % 4 == 2) img_file = 'imgs/fish3.svg';
     if (i % 4 == 3) img_file = 'imgs/fish4.svg';
     let img = html`<img src="${img_file}" height="${getRandom(20,75)}px" width=auto style='position: absolute; left: ${getRandom(13, 67)}%; top: ${getRandom(0, 75)}%; padding:0; margin:0;'></img>`;
     div.append(img);
  }
}

Table 1: The weight of the 20 fish in the aquarium. (Note 1dkg = 10g).

Fish	Weight (dkg)	Fish	Weight (dkg)	Fish	Weight (dkg)	Fish	Weight (dkg)
Fish #1	43	Fish #6	44	Fish #11	26	Fish #16	42
Fish #2	46	Fish #7	41	Fish #12	47	Fish #17	36
Fish #3	47	Fish #8	40	Fish #13	37	Fish #18	36
Fish #4	59	Fish #9	43	Fish #14	42	Fish #19	61
Fish #5	24	Fish #10	58	Fish #15	60	Fish #20	37

Sample and statistics

• Random Sample: a randomly selected subset of the population;

• Statistic: a quantity calculated based on a sample (e.g., sample mean and sample std. dev.);

  • statistics used to estimate parameters;
  • statistics are random quantities (they depend on the sample drawn);

Example: Aquarium

• Suppose we randomly choose 3 fish from the aquarium to weigh them;

• We can calculate the sample mean (denoted by \(\overline{y}\)), to estimate the population mean, \(\mu\).

• The problem is: there are 1140 possible samples of 3 fish we could select.
  • How close will \(\overline{y}\) be to \(\mu\)?

Example: Sampling Distribution

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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

Figure 1: Sampling distribution of the sample average of weight (dkg) for samples of size 3 fish from the aquarium.

{
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        const table = document.getElementById(table_name);
        const entries_pop = table.querySelectorAll("td");
        let data = Array(1140);
        
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    }

    const makeXAxis = (bins, xLabel, height = 400) => {
        let x = d3.scaleLinear()
        .domain([bins[0].x0 - 1, bins[bins.length - 1].x1 + 5])
        .range([margin.left, width - margin.right]);

        let xAxis = g => g
        .attr("transform", `translate(0, ${height - margin.bottom})`)
        .call(d3.axisBottom(x).ticks(width / 100).tickSizeOuter(0))
        .call(g => g.append("text")
            .attr("x", width / 2)
            .attr("y", 7)
            .attr("fill", "currentColor")
            .attr("font-weight", "bold")
            .attr("text-anchor", "bottom")
            .attr('font-size', '16px')
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            .attr("dy", "2.5em")
            .text(xLabel)
            .attr("class","axes-label"));

            return [x, xAxis];

    }

    const makeYAxis = (bins, height = 400) => {
        let y = d3.scaleLinear()
            .domain([0, d3.max(bins, d => d.length) + 10]).nice()
            .range([height - margin.bottom, margin.top]);

        let yAxis = g => g
            .attr("transform", `translate(${margin.left},0)`)
            .call(d3.axisLeft(y).ticks(height / 100))
            .call(g => g.select(".tick:last-of-type text").clone()
            .attr("x", -(height - margin.bottom)/3)
            .attr("y", -45)
            .attr("font-weight", "bold")
            .attr('font-size', '16px')
            .attr('transform', 'rotate(270)')
            .attr("text-anchor", "middle")
            .text("Frequency")
            .attr("class","axes-label"));

            return [y, yAxis];
    }

    const appendBars = (svg, bins, x, y) => {
        svg.append("g")
            .attr("fill", "steelblue")
            .selectAll("rect")
            .data(bins)
            .join("rect")
            .attr("x", d => x(d.x0) + 1)
            .attr("width", d => Math.max(0, x(d.x1) - x(d.x0) - 1))
            .attr("y", d => y(d.length))
            .attr("height", d => y(0) - y(d.length))
            .attr("class", 'bin');
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    const appendTitle = (svg, title) => {
        svg.append("text")
        .attr("x", width / 2)
        .attr("y", 50)
        .attr("text-anchor", "middle")
        .text(title)
        .attr("dy", "-15px")
        .attr("class", "plot-title");
    }

    const appendAxes = (svg, xAxis, yAxis) => {
        svg.append("g")
            .call(xAxis);
        
        svg.append("g")
            .call(yAxis);
    }

    const makeHistogram = (id, data, nbins, title, xLabel, height = 400) => {
        let svg = selectConfigureSVG(id, height);
        
        let bins = d3.bin()
            .thresholds(nbins)(data);

        let [x, xAxis] = makeXAxis(bins, xLabel, height);
        let [y, yAxis] = makeYAxis(bins, height);
        appendBars(svg, bins, x, y);
        appendTitle(svg, title);
        appendAxes(svg, xAxis, yAxis);
    }


    /** Getting the calculated style properties */
    let margin = { top: 25, right: 5, bottom: 25, left: 60 },
        height = 400,
        //width = 800 - margin.left - margin.right;
        width = document.querySelector("#all-samples-fish").clientWidth - margin.left - margin.right;


    /**********************************/
    /****  Creates the histogram 
     **** of Sampling Distribution 
    /**********************************/

    /** Collecting the data from the table */
    let sample_means_fish = get_data_from_table("all-samples-fish", true);
    let sampling_dist_fish = selectConfigureSVG("sampling-dist-fish", height = 500);

    /** creates the bin with thresholds of 20 */
    let bins = d3.bin().thresholds(20)(sample_means_fish);

    let [x_fish, xAxis] = makeXAxis(bins, "Average Fish Weight (dkg)", height = 500);
    let [y_fish, yAxis] = makeYAxis(bins, height = 500);
    appendAxes(sampling_dist_fish, xAxis, yAxis);

    /** appending the bars and adding event handlers */
    sampling_dist_fish.append("g")
        .attr("fill", "steelblue")
        .selectAll("rect")
        .data(bins)
        .join("rect")
        .attr("x", d => x_fish(d.x0) + 1)
        .attr("width", d => Math.max(0, x_fish(d.x1) - x_fish(d.x0) - 1))
        .attr("y", d => y_fish(d.length))
        .attr("height", d => y_fish(0) - y_fish(d.length))
        .attr("class", 'bin')
        .on("mouseenter", (d, i, nodes) => { 
            
            // Mouse-over event: turns the bin red and add the number of data points in the bin to the top of the bin
            d3.select(d.target).attr("fill", "red");
            d3.select(d.target.parentNode)
                .append("text")
                .attr("x", (x_fish(i.x0) + x_fish(i.x1)) / 2)
                .attr("text-anchor", "middle")
                .attr("y", y_fish(i.length + 3))
                .attr("class", "freq")
                .text(i.length)
                .property("bar", d.target)
                .style('font-size','0.7em');

            d3.select(d.target).style("cursor", "pointer"); // change the cursor
            document.getElementById("all-samples-fish")
                    .querySelectorAll("td")
                    .forEach(entry => {
                        if (entry.value >= d.target.__data__.x0 &&
                            entry.value < d.target.__data__.x1){
                            entry.style.color = 'red';
                    }
            });
        })
        .on("mouseout", (d, i, nodes) => { 
            // Mouse-out event: returns to the original configuration
            if (!d.target.flag) {
                d3.select(d.target).attr("fill", "steelblue")
                d3.selectAll(".freq")
                    .filter((e, j, texts) => {
                        return texts[j].bar === d.target;
                    }).remove();
                d3.select(d.target).style("cursor", "default");

                document.getElementById("all-samples-fish")
                    .querySelectorAll("td")
                    .forEach(entry => {
                        if (entry.value >= d.target.__data__.x0 &&
                            entry.value < d.target.__data__.x1){
                            entry.style.color = '';
                    }
                });
            }
        })
        .on("click", (d, i, nodes) => { 
            // click event: lock/unlock the mouse over changes.
            d.target.flag = !d.target.flag;
            if (d.target.flag) {
                sampling_dist_fish.dispatch("mouseenter");
            }
            else {
                d3.select(d.target).attr("fill", "steelblue");
            }
        });


    /** Append Subtitle */
    sampling_dist_fish.append("text")
        .text("Population Mean: 43.45")
        .attr("x", x_fish(28))
        .attr("y", y_fish(197))
        .attr("text-anchor", "start")
        .attr("dy", "6px")
        .attr("class", "plot-subtitle");

    /** Button to Show/Hide Mean  */
    let g = sampling_dist_fish.append("g")
        .on("mouseover", function (d) {
            d3.select(this).style("cursor", "pointer"); // change the cursor
        })
        .on("mouseout", function (d) {
            d3.select(this).style("cursor", "default");
        })
        .on("click", (d, i, e) => {
            d.target.parentElement.clicked = !d.target.parentElement.clicked;
            let g_element = d3.select(d.target.parentElement);
            if (d.target.parentElement.clicked) {
                g_element.select("rect")
                    .attr("fill", "#c45149");
                g_element.select('text')
                    .text('Hide true mean');

                g_element
                    .append('line')
                    .style("stroke", "black")
                    .style("stroke-width", 3)
                    .attr("x1", d => x_fish(43.45))
                    .attr("y1", d => y_fish(0))
                    .attr("x2", d => x_fish(43.45))
                    .attr("y2", d => y_fish(185));
            }
            else {
                g_element.select("rect")
                    .attr("fill", "gray");
                g_element.select('text')
                    .text('Show true mean');

                g_element.select("line").remove()
            }
        });

    /** Creates the rectangle of the button */
    g.append("rect")
        .attr("x", x_fish(28))
        .attr("y", y_fish(190))
        .attr("width", 122)
        .attr("height", 32)
        .attr("fill", "gray")
    
    g.append("text")
        .text("Show true mean")
        .attr("text-anchor", "left")
        .attr("x", x_fish(28.5))
        .attr("y", y_fish(181))
        .attr("fill", "white")
        .style("font-size", '15px');
        


    /**  Button to Reset */
    let reset = sampling_dist_fish.append("g")
        .on("mouseover", function (d) {
            d3.select(this).style("cursor", "pointer"); // change the cursor
        })
        .on("click", (d, i, e) => {
            var bins = d3.selectAll(".bin");
            for (var i = 0; i < bins._groups[0].length; i++) {
                let bin = bins._groups[0][i];
                bin.flag = false;
            }
            bins.dispatch("mouseout");
        });

    /** Creates the rectangle of the button */
    reset.append("rect")
        .attr("x", x_fish(28))
        .attr("y", y_fish(170))
        .attr("width", 122)
        .attr("height", 32)
        .attr("fill", "#e3732d");

    /** Add text to the button */
    reset.append("text")
        .text("Reset plot")
        .attr("x", x_fish(28) + 61)
        .attr("y", y_fish(161))
        .attr("text-anchor", "middle")
        .style("font-size", '15px')
        .attr("fill", "white");

}

• Exercise: If you take a sample at random, what would be the probability that the mean of your sample is fairly close to the true mean, say between 40 and 46?

• Exercise: True or False: Each time we take a random sample we will have a different sampling distribution.

Sampling Distribution

The sampling distribution of a statistic, like \(\overline{y}\) and \(\hat{p}\), is the distribution of the value of the statistic across all possible random samples of size \(n\) that could be selected from a given population.

Sampling Distribution

• Some key questions:
  • What is the centre of the sampling distribution?
  • How spread is the sampling distribution?
  • What is the shape of the sampling distribution?

Sampling Distribution

• As you can see, the sampling distribution tells us a lot about our statistic.
  • We could, for example, calculate the probability that our statistic would be to a certain range around the parameter.

Sampling Distribution

• But how is this useful if we need to have access to all possible samples?
  • In this case, we would be better off just checking out the entire population.

Sampling Distribution

• Luckily, we do not need to have access to all possible samples to know the sampling distribution.

• All we need is a sample! The same sample you use to calculate the statistic will also give you information on the uncertainty of your estimate.

Sampling distribution of \(\overline{Y}\)

• Exercise: Using the applet above, study what happens to the spread of the sampling distribution when the sample size increases.

References

• Sampling Distribution and CLT applets:
  • M. Whitlock, B. Dalstein, Z. Almasslawi, D. Tan, University of British Columbia
• Fish images: Mrmw, CC0, via Wikimedia Commons.
  • Fish 1 || Fish 2 || Fish 3 || Fish 4