Medication used to treat a certain condition is administered by syringe. The target dose in a particular application is µ. Because of the variations in the syringe, in reading the scale, and in mixing the fluid suspension, the actual dose administered is normally distributed with mean µ and variance 
.
a. What is the probability that the dose administered differs from the mean µ by less than ??
b. If X represents the dose administered, find the value of z so that P(X < µ + z?) = 0.90.
c. If the mean dose is 10 mg, the variance is 2.6 
, and a clinical overdose is defined as a dose larger than 15 mg, what is the probability that a patient will receive an overdose?
(a) The probability that a normal random variable is within one standard deviation of its mean is the area under the normal curve between ???? = ?1 and ???? = 1. This area is 0.8413 ? 0.1587 = 0.6826.
(b) The quantity + ???????? is the 90th percentile of the distribution of X. The 90th percentile of a normal distribution is 1.28 standard deviations above the mean. Therefore z = 1.28.
(c) The z-score of 15 is 
The area to the right of ???? = 3.10 is 1 ? 0.9990 = 0.0010.
P (X > 15) = 0.0010.
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