Find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. $P(z < -1.22$ or $z > 1.32)$

The mass of a radioactive isotope at time t (in years) is $M(t)=200 e^{-0.14 t} \mathrm{g}$. What is the mass of the isotope initially? How fast is the mass of the isotope changing 2 years later?

Complete the chart. Review each word, its root, and the mentor sentences. Use the information and your own knowledge to predict the meaning of each word. For each word, list at least two related words. Refer to a dictionary or other resources if needed.

$$\begin{array}{|l|l|l|l|} \hline \text { WORD } & \text { MENTOR SENTENCES } & \text { PREDICT MEANING } & \text { RELATED WORDS } \\ \hline \text { depiction} & \text { 1. The new president's vision for America is a } & & \\ \text {ROOT} & \text {depiction of peace, equality, and prosperity. } & & \\ \text {-pict -} & \text {2. Your proposal for the new playground should include some} & & \\ \text { "paint"} & \text { sort of depiction, such as a drawing or map.} & & \\ \hline \end{array}$$

A z-score is 0. Which of these statements must be true? Explain your reasoning.

The mean is 0.

Let m, g, and L be positive constants. (a) Show that $V(\theta, \omega)=\frac{1}{2} m L^{2} \omega^{2}+m g L(1-\cos \theta)$ is positive definite in an open neighborhood containing $(\theta, \omega)=(0,0)$. (b) Show that $\dot{V}(\theta, \omega)$ is negative semidefinite on solution trajectories of

$$\begin{array}{l} \dot{\theta}=\omega \\ \dot{\omega}=-\frac{\mu}{m} \omega-\frac{g}{L} \sin \theta \end{array}$$

on an open neighborhood containing $(\theta, \omega)=(0,0)$.

Find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(-1.54 < z < 1.54)

Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.

Draw two normal curves that have different means but the same standard deviation. Describe the similarities and differences.

Integration by parts is required. The probability density function for the diameter of a drilled hole in millimeters is

$$10e-^10(x-5)$$

for x > 5 mm. Although the target diameter is 5 millimeters, vibrations, tool wear, and other nuisances produce diameters greater than 5 millimeters. a. Determine the mean and variance of the diameter of the holes. b. Determine the probability that a diameter exceeds 5.1 millimeters.

Find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = 1.365

The thickness of a conductive coating in micrometers has a density function of $$ 600 x-^2 for 100 μm < x < 120 μm. a. Determine the mean and variance of the coating thickness. b. If the coating costs $0.50 per micrometer of thickness on each part, what is the average cost of the coating per part? $$

(a) find the z-score that corresponds to each value and (b) determine whether any of the values are unusual. The test scores for the Law School Admission Test (LSAT) are normally distributed. In a recent year, the mean test score was 155.69 and the standard deviation was 5.05. The test scores of four students selected at random are 168, 161, 144, and 155.

Sample annual salaries (in thousands of dollars) for employees at a company are listed.

$$\begin{matrix} \text{42} & \text{36} & \text{48} & \text{51} & \text{39} & \text{39} & \text{42}\\ \text{36} & \text{48} & \text{33} & \text{39} & \text{42} & \text{45} & \text{50}\\ \end{matrix}$$

Find the sample mean and the sample standard deviation.

Suppose that f(x)=1.5 x^2 for -1<x<1. Determine the mean and variance of X.