Sampling Distribution

STAT 200 - Lecture 15

Review: 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;

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(8, 62)}%; top: ${getRandom(0, 65)}%; 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

Review: 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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        appendAxes(svg, xAxis, yAxis);
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    /**********************************/
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     **** of Sampling Distribution 
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    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 */
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        .attr("fill", "steelblue")
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        .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
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                    .querySelectorAll("td")
                    .forEach(entry => {
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                    .querySelectorAll("td")
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            else {
                d3.select(d.target).attr("fill", "steelblue");
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    /** 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  */
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        .on("mouseover", function (d) {
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        .on("mouseout", function (d) {
            d3.select(this).style("cursor", "default");
        })
        .on("click", (d, i, e) => {
            d.target.parentElement.clicked = !d.target.parentElement.clicked;
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                    .attr("fill", "#c45149");
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                    .attr("y1", d => y_fish(0))
                    .attr("x2", d => x_fish(43.45))
                    .attr("y2", d => y_fish(185));
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                    .attr("fill", "gray");
                g_element.select('text')
                    .text('Show true mean');

                g_element.select("line").remove()
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    g.append("rect")
        .attr("x", x_fish(28))
        .attr("y", y_fish(190))
        .attr("width", 122)
        .attr("height", 32)
        .attr("fill", "gray")
    
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        .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');
        


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        .on("click", (d, i, e) => {
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        .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.

• Exercise (discuss with your neighbor): What would you like this distribution to look like? Why?

Sampling Distribution

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

• 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

• 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.

• But how is this useful if we need to consider all possible samples?
  • We would be better off checking out the entire population.
  • Compare the sizes of Table 1 and Table 2.

Sampling Distribution

• Luckily, we do not need to have access to all possible samples.

• 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

• We shall now discuss how to approximate the sampling distribution of the following two statistics using only one sample:
  • \(\hat{p}\): sample proportion;
  • \(\overline{x}\): sample mean;
• In both cases, given some conditions, we can approximate the sampling distributions using a Normal model.

The case of \(\hat{p}\)

Proportion: mean and statistics

• The true proportion of elements sharing a certain characteristic in a population is denoted by \(p, 0<p<1\);

• For a sample of \(n\) elements randomly selected from the population, the sample proportion is given by:

\[ \hat{p} = \frac{\text{# individuals sampled who have the characteristic}}{n} \]

Sampling distribution of \(\hat{p}\)

We can approximate the sampling distribution of \(\hat{p}\) using Normal curve, if the following conditions are satisfied:

1. the sample is randomly drawn from the population.

2. the sample is drawn in an independent fashion.
   • in practice, check whether the sample size is no greater than 10% of the population size.

3. the sample size is large, such that \(np\geq 10\) and \(n(1-p)\geq 10\).

Sampling distribution of \(\hat{p}\)

• The mean of \(\hat{p}\) across all possible samples of size \(n\) is \(E\left[\hat{p}\right] = p\), where \(p\) is the population proportion.

• The variance of \(\hat{p}\) is given by \(Var\left(\hat{p}\right)=\frac{p(1-p)}{n}\).

• We can use \(\hat{p} \sim N\left(p, \frac{p(1-p)}{n}\right)\).

Exercise

Research shows that 1.3% of patients who had eye laser surgery experienced post-surgical complications. Independent of the research study, an eye healthcare center wants to estimate the proportion of patients undergoing laser surgeries that experienced post-surgical complications. It interviews a random sample of 1000 patients who had eye laser surgery and records for each patient whether he/she experienced post-surgical complications.

1. Identify the population, sample, variable of interest, parameter and statistic.

• Population:

• Sample:

• Variable of interest:

• Parameter:

• Statistic:

2. Describe the sampling distribution model for the sample proportion of patients experiencing post-surgical complications by naming the model and telling its mean and standard deviation.

\(\hat{p} =\) sample proportion of patients who had complications;
\(n =\)

\(E(\hat{p}) =\)
\(SD(\hat{p}) =\)

\(\hat{p}\sim N(\text{mean }=\quad\quad\quad, \text{Var } = \quad\quad\quad )\)

3. What is the probability that the sample proportion of patients who experienced post-surgical complications lies within 0.7% of the population proportion?
4. What is the probability that in this group over 2% of the patients will be found having experienced post-surgical complications?
5. What is the probability that in this group over 20 patients will be found having experienced post-surgical complications?
6. Suppose that in the sample of 1000 patients obtained by the healthcare center, 25 patients had post-surgical complications. Is this result compatible with the research finding?

If the result from the sample is compatible with the research finding, it should be quite usual to obtain a sample that has 25 patients with post-surgical complications. We will find out if this is the case. We will assume the research finding do accurately describe the population and evaluate how usual it is to obtain a sample proportion \(\hat{p}\) that falls __________________ by using probability.

Since the probability is __________________, it is __________________ to obtain a sample that has 25 patients with post-surgical complications. The result from the sample is __________________ with the research finding.

The case of the sample mean

Mean: parameter and statistic

• The population mean \(\mu\) is a parameter, which is fixed but usually unknown.
  • We estimate the mean using the sample mean, \(\overline{Y}\).
• For a sample of \(n\) individuals randomly selected from the population, the sample mean is given by:

\[ \overline{Y} = \frac{Y_1 + Y_2 + \ldots + Y_n}{n} \]

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

We can approximate the sampling distribution of \(\overline{Y}\) using Normal curve, if the following conditions are satisfied:

1. the sample is randomly drawn from the population.

2. the sample is drawn in an independent fashion.
   • in practice, check whether the sample size is no greater than 10% of the population size.

3. the sample size is sufficiently large.

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

• The mean of \(\overline{Y}\) across all possible samples of size \(n\) is \(E\left[\overline{Y}\right] = \mu\) (the population mean).

• The variance of \(\overline{Y}\) is given by \(Var\left[\overline{Y}\right]=\frac{\sigma^2}{n}\).

• We can use \(\overline{Y} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\).

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.

Note on sampling distribution of \(\overline{Y}\)

• Both approximations, \(\hat{p} \sim N\left(p, \frac{p(1-p)}{n}\right)\) and \(\bar{Y}\sim N\left(\mu, \frac{\sigma^2}{n}\right)\), are consequences of the Central Limit Theorem.

Important

If the population is Normal, \(\overline{Y} \sim N\left(\mu, \frac{\sigma^2}{n}\right)\) is not an approximation – it is an exact result and valid for any value of \(n\).

Central Limit Theorem

Exercise 2

A manufacturer of automobile batteries claims that the distribution of the battery life has a mean of 54 months and a standard deviation of 6 months. Suppose a consumer group decides to check the claim by purchasing a random sample of 50 of these batteries and subjecting them to tests that determine their lives. Assuming that the manufacturer’s claim is true:

(a) describe the sampling distribution of the mean lifetime of a random sample of 50 batteries.

\(\overline{Y} =\) sample mean lifetime of 50 randomly chosen batteries
\(n =\)

\(E\left[\overline{Y}\right] =\)
\(Var\left(\overline{Y}\right) =\)

\(\overline{Y}\sim approx. N(\text{mean }=\quad\quad, \text{Var } = \quad\quad )\)

2. What is the probability that the consumer group’s sample has a mean lifetime of 52 or fewer months?

3. within what range of values does the mean lifetime of 50 batteries have a 95% chance of falling?

Exercise 3

A bottling company uses a filling machine to fill plastic bottles with cola. A bottle should contain 300 mL. In fact, the contents vary according to the Normal model with mean 298 mL and standard deviation 3 mL

(a) What is the probability that an individual bottle contains less than 295 mL

2. What is the probability that the mean content of bottles in a six-pack is less than 295 mL?

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