A population of values has a normal distribution with \mu=204.3 and \sigma=43. You intend to draw a random sample of size n=111.Find the probability that

geduiwelh 2021-02-08 Answered

A population of values has a normal distribution with μ=204.3 and σ=43. You intend to draw a random sample of size n=111.
Find the probability that a single randomly selected value is less than 191.2.
P(X<191.2)=?
Find the probability that a sample of size n=111 is randomly selected with a mean less than 191.2.
P(M<191.2)=?
Write your answers as numbers accurate to 4 decimal places.

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Expert Answer

l1koV
Answered 2021-02-09 Author has 100 answers

Step 1
Solution:
Let X be random variable.
From the given information, X follows normal distribution with mean μ=204.3 and a standard deviation is σ=43.
Step 2
Then, the probability that a single randomly selected value is less than 191.2 is
P(X<191.2)=P(Xμσ<191.2204.343)
=P(Z<0.305)
=0.3802[Using the excel function=NORM.DIST(0.305,0,1,TRUE)]
Thus, the probability that a single randomly selected value is less than 191.2 is 0.3802.
Step 3
From the given information, a sample size is n=111.
Then, the probability that a randomly selected with a mean less than 191.2 is
P(M<191.2)=P(Mμσn<191.2204.343111)
=P(Z<111(13.1)43)
=P(Z<3.210)
=0.0007[Using the excel function =NORM.DIST(3.210,0,1,TRUE)]
Thus, the probability that a randomly selected with a mean less than 191.2 is 0.0007.

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