A researcher records the number of people who use a coupon when they dine at a certain restaurant. Identify the method of data collection.

Say on which days of the week you do the following things.

Je dine au restaurant $\rule{4cm}{1pt}$

In Outliers: The Story of Success (Little, Brown, 2008), the author notes that a disproportionate number of software millionaires were born around 1955. Is this a coincidence, or does birth year matter when gauging whether a software founder will be successful? On his Web blog (www.measuringusability .com), statistical consultant Jeff Sauro studied this question by analyzing the data shown in the table.

d. Fit a simple linear regression model relating the number $(y)$ of software millionaire birthdays in a decade to the number $(x)$ of CEO birthdays. Give the least squares prediction equation.

About 8% of males are colorblind. A researcher needs some colorblind subjects for an experiment and begins checking potential subjects. On average, how many men should the researcher expect to check to find one who is colorblind?

Use the given values to make a table of solutions. y = 3x + 9 for x = -4 , -3 , -2, -1

The life spans of car batteries are normally distributed, with a mean of 44 months and a standard deviation of 5 months. (a) Find the probability that the life span of a randomly selected battery is less than 36 months. (b) Find the probability that the life span of a randomly selected battery is between 42 and 60 months. (c) What is the shortest life expectancy a car battery can have and still be in the top 5% of life expectancies?

The scores received by juniors on the math portion of the PSAT are normally distributed with a mean of 48.6 and a standard deviation of 11.4. What is the probability that a randomly selected score is at least 76?

There are n distinct types of coupons, and each coupon obtained is, independently of prior types collected, of type i with probability pi,

$$\sum _ { i = 1 } ^ { n } p _ { i } = 1$$

. (a) If n coupons are collected, what is the probability that one of each type is obtained? (b) Now suppose that

$$p _ { 1 } = p _ { 2 } = \cdots = p _ { n } = 1 / n$$

. Let

$$E_i$$

be the event that there are no type i coupons among the n collected. Apply the inclusion–exclusion identity for the probability of the union of events to

$$P \left( \bigcup _ { i } E _ { i } \right)$$

to prove the identity

$$n ! = \sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \left( \begin{array} { l } { n } \\ { k } \end{array} \right) ( n - k ) ^ { n }$$

The chemicals in cigarette smoke are known to cause cancer. Propose a series of steps that could lead to development of lung cancer in a smoker.

(acostarse) Sandra ____________ __________________ temprano durante la semana.

In this exercise, the Laplace operator

$$\Delta$$

is defined by

$$\Delta \varphi = \frac { \partial ^ { 2 } \varphi } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } \varphi } { \partial y ^ { 2 } }$$

For any vector field

$$\mathbf { F } = \left\langle F _ { 1 } , F _ { 2 } \right\rangle,$$

define the conjugate vector field

$$\mathbf { F } ^ { * } = \left\langle - F _ { 2 } , F _ { 1 } \right\rangle.$$

Let

$$\mathbf { n }$$

be the outward-pointing unit normal vector to a simple closed curve

$$\mathcal { C }.$$

The normal derivative of a function

$$\varphi,$$

denoted

$$\frac { \partial \varphi } { \partial \mathbf { n } },$$

is the directional derivative

$$D _ { \mathbf { n } } ( \varphi ) = \nabla \varphi \cdot \mathbf { n }.$$

Prove that

$$\oint _ { \mathcal { C } } \frac { \partial \varphi } { \partial \mathbf { n } } d s = \iint _ { \mathcal { D } } \Delta \varphi d A$$

where

$$\mathcal { D }$$

is the domain enclosed by a simple closed curve

$$\mathcal { C }.$$

Hint: Let

$$\mathbf { F } = \nabla \varphi.$$

Show that

$$\frac { \partial \varphi } { \partial \mathbf { n } } = \mathbf { F } ^ { * } \cdot \mathbf { T },$$

where

$$\mathbf { T }$$

is the unit tangent vector, and apply Green's Theorem.

Why is a boundary surface diagram useful in representing an atomic orbital?

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. A linear combination of a set of eigenvectors of a matrix A is again an eigenvector of A.

Identify each $x$-value at which any absolute extreme value occurs. Explain how your answer is consistent with the Extreme Value Theorem.

$$y=f(x)$$

$$(a,b)$$