Medication used to treat a certain condition is administered by syringe. The target dose in a particular application is \(\mu\). 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 \(\mu\) and variance \(\sigma^{2}\).

a. What is the probability that the dose administered differs from the mean \(\mu\) by less than \(\sigma\) ?

b. If \(X\) represents the dose administered, find the value of \(z\) so that \(P(X<\mu+z \sigma)=0.90\).

c. If the mean dose is \(10 \mathrm{mg}\), the variance is \(2.6 \mathrm{mg}^{2}\), and a clinical overdose is defined as a dose larger than \(15 \mathrm{mg}\), what is the probability that a patient will receive an overdose?