For the decision algorithms in this exercise, find how many outcomes are possible.

$$\begin{matrix} \text{Step 1:} & \text{Step 2:}\\ \text{Alternative 1: 1 outcome} & \text{Alternative 1: 2 outcomes}\\ \text{Alternative 2: 2 outcomes} & \text{Alternative 2: 2 outcomes}\\ & \text{Alternative 3: 1 outcome}\\ \end{matrix}$$

Explain why the base b in $y=\log _{b} x$ cannot equal 1.

Identify the type I error and the type II error that correspond to the given hypothesis. The proportion of car crashes occurring less than a mile from home is 0.23.

Write on equation of the direct variation that includes the given point. (2, 2)

The following table shows crashworthiness ratings for 16 small cars. (3=Good, 2=Acceptable, 1=Marginal, 0=Poor)

$$\begin{matrix} \text{Frontal Crash Test Rating} & \text{3} & \text{2} & \text{1} & \text{0}\\ \text{Frequency} & \text{1} & \text{11} & \text{2} & \text{2}\\ \end{matrix}$$

a. Find the relative frequency distribution for the experiment of choosing a small car at random and determining its frontal crash rating. b. What is the relative frequency that a randomly selected small car will have a crash test rating of “Marginal” or worse?

The Clean Air Act prohibited the production of chlorofluorocarbons (CFCs) in the United States as of 1996. One widely used CFC, refrigerant R-12, has been replaced by R-134a in many applications because of their similar properties, including a low boiling point at atmospheric pressure, $T_{\mathrm{sat}}=243 \mathrm{K}$ and 246.9 K for R-12 and R-134a, respectively.

Compare the performance of these two refrigerants under the following conditions. The saturated refrigerant vapor at 310 K is condensed as it flows through a 30-mm diameter, 0.8-m-long tube whose wall temperature is maintained at 290 K. If vapor enters the tube at a flow rate of 0.010 kg/s, what is the rate of condensation and the flow rate of vapor leaving the tube?

The relevant properties of R-12 at $T_{\mathrm{sat}}=310 \mathrm{K}$ are $\rho_{v}=50.1 \mathrm{kg} / \mathrm{m}^{3}$, $h_{f g}=160 \mathrm{kJ} / \mathrm{kg}$, and $\mu_{v}=150 \times 10^{-7} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$ and those of liquid R-12 at $T_{f}=300 K$ are $\rho_{l}=1306 \mathrm{kg} / \mathrm{m}^{3}$, $c_{p, l}=978 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, \mu_{l}=2.54 \times 10^{-4} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$, $k_{l}=0.072 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$.

The properties of the saturated R-134a vapor are $\rho_{v}=46.1, \mathrm{kg} / \mathrm{m}^{3}, h_{f g}=166 \mathrm{kJ} / \mathrm{kg}$, and $\mu_{v}=136 \times 10^{-7} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$.

In a survey of 400 music downloads, 45 were country music, 16 were religious music, and 16 were jazz, while the remainder were other genres. Calculate the following relative frequencies: a. That a music download was jazz. b. That a music download was either religious music or jazz. c. That a music download was not country music.

Suppose you roll a number cube. Find each theoretical probability. P(8)

Solve each equation. Round to the nearest ten-thousandth. $5^{x}+3=12$

Calculate the relative frequency P(E) using the given information. N=500, fr(E)=300

Calculate the relative frequency P(E) using the given information. N=100, fr(E)=40

Let W be the event that you will use the Web site tonight, let I be the event that your math grade will improve, and let E be the event that you will use the Web site every night. Express the given event in symbols. You will use the Web site tonight or your math grade will not improve.

Describe the sample space S of the experiment and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) A sequence of two different letters is randomly chosen from those of the word sore; the first letter is a vowel.

Consider the general linear model

$$Y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \cdots + \beta _ { k } x _ { k } + \varepsilon$$

, where

$$E ( \varepsilon ) = 0 \text { and } V ( \varepsilon ) = \sigma ^ { 2 }$$

. Notice

$$\hat { \beta } _ { i } = \mathbf { a } ^ { \prime } \boldsymbol { \beta }$$

, where the vector a is defined by

$$a _ { j } = \left\{ \begin{array} { l l } { 1 , } & { \text { if } j = i } \\ { 0 , } & { \text { if } j \neq i } \end{array} \right.$$

. Use this to verify that

$$E \left( \hat { \beta } _ { i } \right) = \beta _ { i } \text { and } V \left( \hat { \beta } _ { i } \right) = c _ { i i } \sigma ^ { 2 }$$

, where

$$c _ { i i }$$

is the element in row i and column i of

$$\left( \mathbf { X } ^ { \prime } \mathbf { X } \right) ^ { - 1 }$$

.