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  See Problem Here

Solutions: (a) Let X be Judy's potassium level on a random day. We want to compute P[X < 3.5], given that X is normally distributed with mean $\mu = 3.8$ and standard deviation $\sigma = .2$. Letting

$$Z \; = \; \frac{X-\mu}{\sigma}$$

we want

$$P \left[ Z < \frac{3.5-3.8}{.2} \right] \; = \; P [ Z < -1.5 ] \; = \; 0.066807 .$$

(b) This just changes things to

$$P \left[ Z < \frac{3.5-3.8}{.3} \right] \; = \; P [ Z < -1.0 ] \; = \; 0.158655 .$$

(c) I see from the table that

$$P[ Z < -.5] \; = \; 0.308538 .$$

Since

$$P[ X < 3.5 ] \; = \; P \left[ \frac{X-\mu}{\sigma} < \frac{3.5-\mu}{.3} \right] \; = \; P[ Z < -.5] \; = \; 0.308538 ,$$

we have

$$\frac{3.5-\mu}{.3} \; = \; -.5$$

so

$$3.5-\mu \; = \; (-.5)*(.3) \quad \Longrightarrow \quad \mu \; = \; 3.5 - (-.5)*(.3) \; = \; 3.65 .$$

Problems? Questions? (I may have made a mistake, so don't hestitate to ask) dcox@stat.rice.edu