Statistics Problem Set Solution: Probability, Independence, and EMV

This document presents a complete solution to a statistics problem set, tackling various concepts. The first question involves size classification, calculating probabilities, and testing for independence. The second question delves into decision-making under uncertainty, determining expected monetary value (EMV) for different scenarios. The final question focuses on descriptive statistics, including calculating z-scores and percentiles to analyze data from different slots. The solution provides step-by-step calculations and explanations for each problem, making it a valuable resource for students studying statistics.

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Statistical problem set
Name:
Institution:

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Question 1
Size Classification
Let
Full size = F Large engine = L
Midsize = M Small engine = S
Compact = C
P(F) = 0.2 P(S/F) = 0.2 P(L/F) = 0.8
P(M) = 0.4 P(S/M) = 0.7 P(L/M) = 0.3
P(C) = 0.4 P(S/C) = 0.95 P(L/C) = 0.05
a) P(L)
P(L) = P(L/F)*P(F) + P(L/M)*P(M) + P(L/C)*P(C)
= 0.8*0.2 + 0.3*0.4 + 0.05*0.4
= 0.16 + 0.12 + 0.02
= 0.3
b) Independence test M and L
P(L) = P(L/M)
0.3 = 0.3
Since the P(L) = P(L/M) then the events are independent.
c) P(C/S)
P(C/S) = [P(C)*P(S/C)]/P(S)
= [0.4*0.95]/0.7
= 0.38/07
= 0.5429

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Question 2
Expected monetary value for each decision
EV = ∑probability * payoff
Dennis = 1*200,000*0.5 = 100,000
Frank = (0.3*200,000 + 0.4*100,000 + 0.3*50,000) 0.3
=34,500
Question 3
Dennis
Frank
D= 200 p=0.3
D= 100 p=0.4
D= 50 p = 0.3
D= 200 p=1
D= 100
D= 50
D= 50
D= 100
D= 200
D= 50
D= 100
D= 200
D= 50
D= 100
D= 200

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Descriptive A B C
Mean 30000 27,000 27,000
SD 4000 6000 6000
a) Slot A Z= x −μ
σ
x=25,000 z = (25000 - 3000)/4000 = -1.25
p(-1.25) = 0.10565
x=28,000 z =(28000 - 30000)/4000 = -0.5
P(-0.5) = 0.30854
P(28000-25000) = 0.30854 – 0.10565 = 0.20289
b) 97 percentile number of slot C
Z(0.97) = 1.88
1.88= x – 27000
4000
1.88*4000 = x – 27000
7520+2700 = x
x = 34520
Therefore, the 97th percentile number of viewers in slot C is 34,520