This Exam P sample reference tests Normal Distribution. The mean of 25 normal claims has standard deviation 1000. Standardizing 20,000 gives z=0.6 and upper-tail probability 0.2743, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
DThe value 0.33 is a rough approximation not obtained from the standard normal upper tail at z=0.6.
EThe value 0.45 results approximately from using the individual standard deviation 5,000, which gives z=0.12, instead of the sample-mean standard deviation 1,000.
Original practice · fully worked
Original variant: sample-average fill volume
Bottle fill volume is normal with mean 502 milliliters and standard deviation 8 milliliters. For a random sample of 16 bottles, find the probability that the sample mean is below 499 milliliters.
A 0.0228
B 0.0668
C 0.1587
D 0.3085
E 0.6915
Variant answer in brief
The sample-mean standard deviation is 2. Standardizing 499 gives -1.5, whose normal lower tail is 0.0668.
Setup
Setup
For 16 normal bottle fills, the sample mean remains normal with mean 502 ml and standard deviation 8 divided by 4, or 2 ml.
Xˉ∼N(502,1682)
Model
Model
Standardize 499 relative to the sample-mean distribution, giving a lower-tail cutoff of -1.5.
z=8/4499−502=−1.5
Compute
Compute
The standard normal lower-tail probability at -1.5 is 0.0668072.
P(Xˉ<499)=Φ(−1.5)=0.0668072
Answer
Answer
Thus the probability that the sample mean is below 499 ml is 0.0668, corresponding to choice B.
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