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Births are approximately uniformly distributed between the 52 weeks of the year. They can be said to follow a uniform distribution from one to 53 (spread of 52 weeks).

a. X ~ _________

b. Graph the probability distribution.

c. f(x) = _________

d. ì = _________

e. ó = _________

f. Find the probability that a person is born at the exact moment week 19 starts. That is, find P(x = 19) = _________

g. P(2 < x < 31) = _________

h. Find the probability that a person is born after week 40.

i. P(12 < x|x < 28) = _________

j. Find the 70th percentile.

k. Find the minimum for the upper quarter

Short Answer

Expert verified

As per calculations

a. X~U(1,53)

b.

The probability distribution function will be;

localid="1649331118711" f(x)=1b-a=153-1=152

c.

f(x)=1520,1<X<53

d. 27

e. 15.01

f. 0

g. 0.5577

h. 0.25

i. 0.5926

j. 37.4

k. 40

Step by step solution

01

Definition of variables 

Variables are the terms that may have different values. Here in this distribution variable X is uniformly distributed between the 52 weeks of the year.

02

Findings the values of variables 

a. Uniform of all data

X~U(1,53)

b. The height of probability distribution

f(x)=1b-a=153-1=152

The graph of distribution is

c. The formulation of,

f(x)=1b-a,a<X<b0,1<X<53f(x)=152,1<X<530,Otherwise

d. The mean can be calculated as;

μ=a+b2=1+532=27

Hence the mean is 27

e. The standard deviation

σ=b-a212σ=53-1212σ=225.33σ=15.01

The standard Deviation is 15.01

f. For continuous probability, the point value is zero.

Hence the value of P(x=19)=0

g. The required probability is

P(2<x<31)=Base×Height=(32-2)×152=29×152=0.5577

h.

The value of

localid="1649333150490" P(x>40)=Base×Height=(53-40)×152=13×152=0.25

i. The probability of

P(12<xx<28)=P(12<x<28)P(x<28)=(28-12)×152(28-1)×152=1627=0.5926

j. For the 70 percentile

P(x<K)=Base×Height0.70=(K-1)×152k-1=0.70×52K=36.4+1=37.4

k. The required value can be calculated by calculating the 75th percentile of the data, which can be calculated as below;

P(x<K)=Base×Height0.75=(K-1)×152k-1=0.75×52K=39+1=40

The value of the minimum of the upper quarter is 40.

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