The amount filled by a certain bottling machine,X,is approximately normally distributed with unknown mean,μ,and standard deviation,σ =1.If n=16 bottles were randomly sampled,what is the probability that the mean fill in the sample is within 0.3 ounces of the population mean μ?
A)0.2358
B)0.6000
C)0.7698
D)0.2302
E)0.7642

Question

Quizplus · Accepted Answer

The standard error of the mean (SE) can be calculated as: 
SE = σ/sqrt(n) = 1/sqrt(16) = 0.25

The probability that the sample mean is within 0.3 ounces of the population mean can be calculated by standardizing the sample mean: 
P(μ-0.3 < X̄ < μ+0.3) = P(-1.2 < Z < 1.2) 
where Z = (X̄ - μ)/SE is a standard normal variable.

Using a standard normal table or calculator, we can find: 
P(-1.2 < Z < 1.2) ≈ 0.7698

Therefore, the answer is (C) 0.7698.

The Amount Filled by a Certain Bottling Machine,X,is Approximately Normally