Question

Answer the number 3 and 4 question

Worksheet 2: Chapters 3 and 8 (Normal Distributions; Sampling) Math 243, Sunmmer '19, Vargas (c) (3.5) What percentage of all MLB players are shorter than your team's shortest player, according to the Normal distribution? What about taller than your team's tallest player? 3. (3.4) Consider the N(73.7,2.3) distribution for the height in the population of all MLB players. (a) From the population, find the intervals of player heights within one standard deviation of the population mean, within two standard deviations of the mean, and within three standard deviations of the mean. (b) Order the data set for your team from least to greatest. (c) In each part below, consider how much data is within (including the endpoints) the spec- ified distance of the population mean in your team's data, how much is predicted by the 68-95-99.7 rule, and how much is guaranteed by Chebyshev's Inequality. You can round your answers in parts (ii) and (iii) to three decimal places if necessary One standard dev. Two standard dev. | Three standard dev. (i) Actual data (ii) 68-95-99.7 rule (iii) Chebyshev's Inequality 3 Math 243, Summer '19, Vargas Worksheet 2: Chapters 3 and 8 (Normal Distributions; Sampling) 4. (3.6) Find the first and third quartiles... (a). for the height of all MLB players, assuming the idealized Normal distribution. for the height of the players on your team. (b) 5. (8.2) Describe MLB players using the SOCR data. way to conduct a stratified random sample of 12 players from the population of 6. (8.3) What key type of bias might be present in using this data set that would prevent us using it to draw conclusions about the height of all adult individuals in the United States? In what way is this likely to skew the sample? 4

Answer 1

Solution

Back-up Theory

Chebyshev's inequality

If E(X) = µ and V(X) = σ², then P(|X - µ | ≥ kσ) ≤ 1/k² for a wide spectrum of distributions. ................ (1a)

(1a) => P(|X - µ | ≤ 1σ) ≥ 0; P(|X - µ | ≤ 2σ) ≥ 0.75; P(|X - µ | ≤ 3σ) ≥ 0,8889 ................................... (1b)

Now, to work out the solution,

Q3

Part (a)

Height Interval within 1 standard deviation of population mean: [71.4, 76.0] Answer 1

Height Interval within 2 standard deviation of population mean: [69.1, 78.3] Answer 2

Height Interval within 3 standard deviation of population mean: [66.8, 80.6] Answer 3

Part (b)

Orderd Set

70
71
71
72
72
72
72
72
72
72
72
72
73
73
73
73
74
74
74
74
74
74
74
74
75
75
75
75
75
75
76
76
76
76
78

Answer 4

Part (c)

	1 Standard deviation	2 Standard deviation	3 Standard deviation
Actual Data	88.6%	100%	100%
68 – 95 – 99.7 Rule	68%	95%	99.7%
Chebyshev’s Inequality [vide (1b) under Back-up Theory]	0%	75%	88.89%

Answer 5

DONE