Solve the equation.

$$\frac{0.05 x-9}{8}=0.2(x-25)$$

You are assigned to be part of a group of three students from the intro stats class described in the earlier exercise. What is the probability that of your other two groupmates,

at least one has had more than one semester of calculus?

A tray of electronic components contains 15 components, 4 of which are defective. If 4 components are selected, what is the possibility that (a) all 4 are defective? (b) 3 are defective and 1 is good? (c) exactly 2 are defective? (d) none are defective?

We cannot use the least-squares regression line to predict y from $x$. In these situations, statisticians often try to manipulate one or both of the variables to try to find a linear fit. This is called transforming the variables. For each data point, compute the cube of the wind speed to obtain $x^3$, then compute the least-squares regression line for predicting power $y$ from $x^3$.

Select the best answer. Which of the following is not a characteristic of the least-squares regression line? (a) The slope of the least-squares regression line is always between -1 and 1. (b) The least-squares regression line always goes through the point

$$(\overline{x}, \overline{y}).$$

(c) The least-squares regression line minimizes the sum of squared residuals. (d) The slope of the least-squares regression line will always have the same sign as the correlation. (e) The least-squares regression line is not resistant to outliers.

Do larger butterflies live longer? The wingspan (in millimeters) and the lifespan in the adult state (in days) were measured for $22$ species of butterfly. Following are the results.

$$\begin{array}{cccc} \hline \text{Wingspan} & \text{Lifespan} \hspace{25pt} & \text{Wingspan} & \text{Lifespan}\\ \hline 35.5 & 19.8 \hspace{25pt} & 25.9 & 32.5\\ 30.6 & 17.3 \hspace{25pt} & 31.3 & 27.5\\ 30.0 & 27.5 \hspace{25pt} & 23.0 & 31.0\\ 32.3 & 22.4 \hspace{25pt} & 26.3 & 37.4\\ 23.9 & 40.7 \hspace{25pt} & 23.7 & 22.6\\ 27.7 & 18.3 \hspace{25pt} & 27.1 & 23.1\\ 28.8 & 25.9 \hspace{25pt} & 28.1 & 18.5\\ 35.9 & 23.1 \hspace{25pt} & 25.9 & 32.3\\ 25.4 & 24.0 \hspace{25pt} & 28.8 & 29.1\\ 24.6 & 38.8 \hspace{25pt} & 31.4 & 37.0\\ 28.1 & 36.5 \hspace{25pt} & 28.5 & 33.7\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ Compute the least-squares regression line for predicting the lifespan from the wingspan.

$\textbf{b.}\hspace{10pt}$ Is it possible to interpret the $y$-intercept? Explain.

$\textbf{c.}\hspace{10pt}$ If the wingspans of two butterflies differ by $2$ millimeters, by how much would you predict their lifespans to differ?

$\textbf{d.}\hspace{10pt}$ Predict the lifespan for a butterfly whose wingspan is $28.5$ millimeters.

fill in the blanks. If P(E) = 0, then E is an ________ event, and if P(E) = 1, then E is a ________ event.

The chance that a customer will buy a new car and an extended warranty is $0.16$. If the probability that a customer will purchase a new car is $0.30$, find the probability that the customer will also purchase the extended warranty.

A car requires $5$ quarts of new motor oil when the engine oil is changed. What is the equivalent amount of motor oil measured in liters?

An automobile manufacturer found that the efficiency $E$ of a new engine as a function of the speed $S$ of the car was given by $E=0.864 S-0.00008 S^3$. Here $E$ is given in percent and $S$ is given in kilometers per hour. What is the maximum efficiency of the engine?

If you had to construct a mathematical model for events E and F, as described in parts (a) through (e), would you assume that they were independent events? Explain your reasoning. (a) E is the event that a businesswoman has blue eyes, and F is the event that her secretary has blue eyes. (b) E is the event that a professor owns a car, and F is the event that he is listed in the telephone book. (c) E is the event that a man is under 6 feet tall, and F is the event that he weighs over 200 pounds. (d) E is the event that a woman lives in the United States, and F is the event that she lives in the Western Hemisphere. (e) E is the event that it will rain tomorrow, and F is the event that it will rain the day after tomorrow.

Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event $E$ or event $F$ occurs. What does the sample space of this new super experiment look like? Show that the probability that event $E$ occurs before event $F$ is $P(E) /[P(E)+P(F)]$.

Let $E$ be the event that a new car requires engine work under warranty and let T be the event that the car requires transmission work under warranty. Suppose that $P (E) = 0.10$, $P (T) = 0.02$, and $P (E\ \text{and}\ T) = 0.01$.

$\textbf{a.}\hspace{10pt}$ Find the probability that the car needs work on either the engine, the transmission, or both.

$\textbf{b.}\hspace{10pt}$ Find the probability that the car needs no work on the engine.

A(n) ____ is any collection of outcomes from a probability experiment.