Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?

Annette Arroyo 2021-03-04 Answered
Consider the marks of all 1st-year students on a statistics test. If the marks have a normal distribution with a mean of 72 and a standard deviation of 9, then the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is?
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Dora
Answered 2021-03-05 Author has 98 answers
Step 1
Suppose, X be the marks of statistics students, X follows Normal distribution with mean 72 and standard deviation 9.
The average marks of 10 students,
X=110i=110Xi,
X follows Normal distributions with mean 72 and standard deviation 910.
Step 2
The probability that the sample mean between 71 and 73 is
P(71X73)
=P(71729/(10)(X729/(10)(73729/(10)),Z=(X729/(10)Normal(0,1)
=P(0.3514Z0.3514)
=P(Z0.3514)P(Z0.3514)
=Φ(0.3514)Φ(0.3514),Φ(z)=P(Zz)
=2Φ(0.3514)1,Φ(z)=1Φ(z)
=(2×0.6374)1
=0.2748
The values of Φ(0.3514) is taken from Normal distribution.
Therefore, the probability that a random sample of 10 students from this group have a sample mean between 71 and 73 is 0.2758.
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Jeffrey Jordon
Answered 2021-11-11 Author has 2064 answers

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