SOLUTION: A sample of size 42 will be drawn from a population with mean 52 and standard deviation 9. a. Is it appropriate to use the normal distribution to find probabilities for x bar

Question 1178240: A sample of size 42 will be drawn from a population with mean 52 and standard deviation 9.
a.
Is it appropriate to use the normal distribution to find probabilities for x bar ?
b.
If appropriate find the probability that x bar will be between 53 and 54.
c.
If appropriate find the 45th percentile of x bar.
Found 2 solutions by Boreal, ewatrrr:
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
the short answer that is usually given is yes, because the sample size is > 30. One has to remember, however, that with a normal distribution, a sample size of 10 is sufficient and with a skewed population, a sample size of 50 may not be.
Anyway, given that, the sd of the sample means (the standard error) is sigma/sqrt(42)=1.389
The probability between 53 and 54 uses z=(53-52)/9/sqrt(42) or 0.72 and z=2/9/sqrt(42)=1.44
The probability of z being between 0.72 and 1.44 is 0.1608.
z(0.45)=-0.1257
z=(x bar-mean)/sd/sqrt(n)
-0.1257=(x bar-52/1.389
-0.1745=xbar-52
xbar=51.8255 or 51.8

Answer by ewatrrr(24785) (Show Source): You can put this solution on YOUR website!
Normal Distribution: n = 42 > 30 , μ = 52 σ =9
Using TI or similarly an inexpensive calculator like a Casio fx-115 ES plus
b. P(53 < x < 54) = normalcdf(53,54, 52, 9/√42) = .1608
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