Find the interval $\left[\mu-z \frac{\sigma}{\sqrt{n}}, \mu+z \frac{\sigma}{\sqrt{n}}\right]$ within which $95$ percent of the sample means would be expected to fall, assuming that each sample is from a normal population.

$$\mu=1,000, \sigma=15, n=9$$

Are the following statements true or false? Explain your reasoning.

a. "If we see a standardized $z$-value beyond $\pm 3$, the variable cannot be normally distributed."

b. "If $X$ and $Y$ are two normally distributed random variables measured in different units (e.g., $X$ is in pounds and $Y$ is in kilograms), then it is not meaningful to compare the standardized $z$-values."

c. "Two machines fill 2 -liter soft drink bottles by using a similar process. Machine $A$ has $\mu=1,990 \mathrm{ml}$ and $\sigma=5 \mathrm{ml}$ while Machine $B$ has $\mu=1,995 \mathrm{ml}$ and $\sigma=3 \mathrm{ml}$. The variables cannot both be normally distributed since they have different standard deviations."

(a) Find the standard error of the mean for each sampling situation (assuming a normal population). (b) What happens to the standard error each time you quadruple the sample size?

$$\sigma=32, n=64$$

The amount of fill in a half-liter (500 ml) soft drink bottle is normally distributed. The process has a standard deviation of 5 ml. The mean is adjustable.

(a) Where should the mean be set to ensure a 95 percent probability that a half-liter bottle will not be underfilled?

(b) A 99 percent probability?

(c) A 99.9 percent probability? Explain.

The amount of fill in a half-liter ( $500 \mathrm{ml})$ soft drink bottle is normally distributed. The process has a standard deviation of $5 \mathrm{ml}$. The mean is adjustable. (a) Where should the mean be set to ensure a 95 percent probability that a half-liter bottle will not be underfilled? (b) A 99 percent probability? (c) A $99.9$ percent probability? Explain.

Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used.

$$P(-0.50<Z<0)$$

Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used.

$$P(Z>0)$$

Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used.

a. P$(−1.22 < Z < 2.15)$

b. P$(−3.00 < Z < 2.00)$

c. P$(Z < 2.00)$

d. P$(Z = 0)$

(a) At what x value does f (x) reach a maximum for a normal distribution N(75, 5)?

(b) Does f (x) touch the X-axis at $\mu$ $\pm$ 3$\sigma$?

The length of a Colorado brook trout is normally distributed.

(a) What is the probability that a brook trout’s length exceeds the mean?

(b) Exceeds the mean by at least 1 standard deviation?

(c) Exceeds the mean by at least 2 standard deviations?

(d) Is within 2 standard deviations?

Jim’s systolic blood pressure is a random variable with a mean of 145 mmHg and a standard deviation of 20 mmHg. For Jim’s age group, 140 is the threshold for high blood pressure.

(a) If Jim’s systolic blood pressure is taken at a randomly chosen moment, what is the probability that it will be 135 or less?

(b) 175 or more?

(c) Between 125 and 165?

(d) Discuss the implications of variability for physicians who are trying to identify patients with high blood pressure.

If the weight (in grams) of cereal in a box of Lucky Charms is N(470,5), what is the probability that the box will contain less than the advertised weight of 453 g?

The fracture strength of a certain type of manufactured glass is normally distributed with a mean of 579 MPa with a standard deviation of 14 MPa.

(a) What is the probability that a randomly chosen sample of glass will break at less than 579 MPa?

(b) More than 590 MPa?

(c) Less than 600 MPa? (Data are from Science 283 [February 26, 1999], p. 1296.)

For a large Internet service provider (ISP), Web virus attacks occur at a mean rate of 150 per day. (a) Estimate the probability of at least 175 attacks in a given day. (b) Estimate the probability of fewer than 125 attacks. (c) Is the normal approximation justified? Show all calculations. (d) Use Excel to calculate the actual Poisson probabilities. How close were your approximations?