\[ \begin{array}{c} A=\left[\begin{array}{lll} -5 & 1 & -7 \end{array}\right] \\ B=\left[\begin{array}{llll} -8 & 7 & 5 & -5 \end{array}\right] \\ C=\left[\begin{array}{ll} -4 & -2 \end{array}\right]

The probability that a randomly selected student is female or a physics major would be 0.65.

Probability refers to a possibility that deals with the occurrence of random events.

The probability of all the events occurring needs to be 1.

The probability that a randomly selected student is female or a physics major can be calculated:

P(PM or F) = P(PM) + P(F) - P(PMF)

Where P(PM or F) = Probability of student selected is Physics Major or Female needed to find.

P(PM) = Probability of student selected is Physics Major

P(PM) = Number of Physics Major / Total number of students in the Physics class

P(PM)  = 14 / 40 = 0.35

P(F) = Probability of student selected is Female = Number of female students in the Physics class / Total number students in the Physics class = 18 / 40 = 0.45

P(PMF) = Probability of student selected is Physics Major and Female  = 6 / 40 = 0.15

Substituting the values into equation (1);

P(PM or F) = 0.35 + 0.45 - 0.15

P(PM or F) = 0.65

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The following solutions with regard to resolving or simplifying the radii of the circular pond are given below.

Part A: 5√100 + √164 meters = 5 × 10 + 2√41 meters

Part B: 5 × 10 + 2√41 meters = 50 + 2√41 meters

Part C: The radius of Pond B is 10√2 meters.

Part D: the radius of Pond B is greater than the radius of Pond A.

Part A: The mistake is in Step 1. To simplify the square root of 164, we need to find its prime factors:

164 = 2 × 2 × 41

So, we can write:

√164 = √(2 × 2 × 41) = 2√41

Using this, we can rewrite Step 1 as:

5√100 + √164 meters = 5 × 10 + 2√41 meters

Part B: Using the corrected step from Part A, we can simplify the radius of Pond A as:

5 × 10 + 2√41 meters = 50 + 2√41 meters

So, the radius of Pond A is 50 + 2√41 meters.

Part C: The radius of Pond B is already simplified, so we don't need to do any additional steps. It is:

25√200/5 meters = 5√200 meters

We can simplify this further by finding the prime factorization of 200:

200 = 2 × 2 × 2 × 5 × 5

So, we can write:

√200 = √(2 × 2 × 2 × 5 × 5)

= 2 × 5√2

Using this, we can rewrite the expression for the radius of Pond B as:

5 × 2 × √2 meters

= 10√2 meters

Thus, the radius of Pond B is 10√2 meters.

Part D: We can compare the radii of the ponds using the original expressions by writing:

√164 < 25√200/5

Simplifying both sides:

2√41 < 10√2

Dividing both sides by 2:

√41 < 5√2

Squaring both sides (since both sides are positive):

41 < 25 × 2

41 < 50

So, the inequality is true. Therefore, the radius of Pond B is greater than the radius of Pond A.

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Full Question:

Two circular ponds at a botanical garden have the following radii:

Pond A: Sqrt(164) meters;

Pond B: 25Sqrt(200)/5 meters:

Todd simplified the radius of pond A this way:

5sqrt(164)

Step 1: 5sqrt(100) + Sqrt(164) meters

Step 2: 5(10 + 8)

Step 3: 5(18)

Step 4: 90

One of the steps above is incorrect.

Part A: Rewrite the steps so that it is correct

Part B: Using the corrected step from Part A, simplify the radius of Pond A.

Part C: Simplify the expression for the radius of pond B.

Part D: Write an inequality to compare the radii of the ponds, using the original expressions.

Answer:

x=4, y=-2

Step-by-step explanation:

[tex]2y=x-8\\2y-x=-8\\2(2x-10)-x=-8\\4x-20-x=-8\\3x-20=-8\\3x=-8+20\\x=12/3\\x=4\\\\\\y=2x-10\\y=2(4)-10\\y=8-10\\y=-2[/tex]

None of the expressions given is equivalent to (2 - ³√24)²..

Mathematical statements must contain a sentence, at least one mathematical operation, and at least two numbers or factors. With this mathematical operation, you can increase, split, add, or take something away. The shape of a phrase is as follows: Expression: (Number/Variable, , Math Operator)

Let's first simplify the expression (2 - ³√24)²:

(2 - ³√24)² = (2 - 24^(1/3))² = (2 - 2.29)² = (-0.29)² = 0.0841

Now we can check which expressions are equivalent to 0.0841 when (2 - ³√24)² is evaluated:

4 = 4.0000... (not equivalent to 0.0841)

04 = 0.04 (not equivalent to 0.0841)

26.2 - 8 = 18.2 (not equivalent to 0.0841)

02 - 5.22 = -3.22 (not equivalent to 0.0841)

Therefore, none of the expressions given is equivalent to (2 - ³√24)².

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The first step in solving by elimination is (4) Multiply top equation by 2

From the question, we have the following parameters that can be used in our computation:

5x + y = 9

10x - 7y = -18

To eliminate x the coefficient of x in both equations muet be the same i.e 10

Using the above as a guide, we have the following:

Multiply (1) by 2

So, we have

10x + 2y = 18

10x - 7y = -18

Hence, the first step is (4)

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139

because you do 5 dollars times 11 cats equals 55 and 12 dollars times 7 dogs equals 84 add it together is 139

The vector x whose image under T is b is x = [23/19 -129/19​].

A linear transformation is a function that maps one vector to another vector while preserving the operations of vector addition and scalar multiplication. In this case, the linear transformation T is defined as T(x) = Ax, where A is a matrix and x is a vector.

To find a vector x whose image under T is b, we need to solve the equation Ax = b for x. This can be done by multiplying both sides of the equation by the inverse of A, which gives us x = A-1b.

Using the formula for the inverse of a 2x2 matrix, we can find A-1:

A-1 = (1/(6*4 - 5*1)) * [4 -5​-1 6​] = [4/19 -5/19​-1/19 6/19​]

Now we can multiply A-1 by b to get x:

x = A-1b = [4/19 -5/19​-1/19 6/19​] * [−9−17​] = [(4/19)*(-9) + (-5/19)*(-17) (−1/19)*(-9) + (6/19)*(-17)​] = [23/19 -129/19​]

Therefore, the vector x whose image under T is b is x = [23/19 -129/19​].

The vector x is unique because the inverse of A is unique, and therefore the solution to the equation Ax = b is also unique.

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1. Independent Variable: Group (No News or watch news)

Dependent Variable: Depression Scores

2. Null Hypothesis: μ = 0

Alternative Hypothesis: μ < 0

3. Yes, you can reject the null hypothesis at a = 0.05 because there is a considerable difference in the depression ratings of the two groups

1. According to the information provided, the score for depression depends on whether a group watches the news or not.

Group is an independent variable (No News or watch news).

Depression scores are a dependent variable.

2. Null Hypothesis: The two groups do not differ in their depression levels, which is the null hypothesis.

Alternate Hypothesis: Those who don't read or watch the news will score less depressed than people who receive therapy as usual.

μ = 0 is the null hypothesis.

Between the two groups, there is no difference in depression scores.

Differential Hypothesis:  μ < 0

Those who don't read or watch the news will score less depressed than people who receive therapy as usual.

3. Due to the p-value of (0.024) < α(0.05) at α = 0.05, we can rule out the null hypothesis.

The rejection zone of the t-distribution with 12 degrees of freedom has the t-score of -2.58.

As a result, it is clear that there is a considerable difference in the depression ratings of the two groups, with the group that does not watch or read the news scoring significantly lower on the depression scale than the group that continues to get therapy as usual.

The proper response is "Yes".

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The complete question is:

Dr. Z is interested in discovering if there is a difference in depression scores between those who do not watch or read the news and those who continue with therapy as normal. She divides her clients with depression into 2 groups. She asks Group 1 not to watch or read any news for two weeks while in therapy and asks Group 2 to continue with therapy as normal. The dataset score is a record of the results of the measure, administered after 2 weeks.

Independent Samples T-Test

                                   t                          df                          p

Score                      -2.580                    12                      0.024

1. Identify IV and DV.

2. State the null hypothesis and the directional (one-tailed) alternative hypothesis

3. Can you reject the null hypothesis at a = 0.05?

The value of (g times f)(x) is 6x² + x - 2. So correct is C.

The input values of a function are called the domain, and the output values are called the range. The domain can be any set of values, but each input value must have a unique corresponding output value. If there are multiple input values that produce the same output value, the function is not considered to be well-defined. Functions are used to model a wide range of phenomena in many different fields, including science, engineering, and economics. They are also used in calculus to study rates of change and slopes of curves.

Overall, functions are a fundamental concept in mathematics, and have many practical applications in the real world.

To find g(x) times f(x), we need to multiply the two functions together.

(g times f)(x) = g(x) * f(x)

First, we need to find g(f(x)):

g(f(x)) = g(3x+2) = 2(3x+2) - 1 = 6x + 3

Now we can substitute this into the expression for (g times f)(x):

(g times f)(x) = g(x) * f(x) = (2x-1) * (3x+2)

Using the distributive property, we get:

(g times f)(x) = 6x² + 4x - 3x - 2 = 6x² + x - 2

Therefore, (g times f)(x) = 6x² + x - 2.

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Answer:

1 trillion times larger

Step-by-step explanation:

One kilogram (kg) is equal to 1,000,000,000,000 (1 trillion) nanograms (ng).

Therefore, one kilogram is 1 trillion times larger than one nanogram.

According to the given information, the probabilites are a) 0.159, b)0.107, d) 0.893 e) 0.662 c) expected value is 12.192 f) standard deviation is 3.0121.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

a) Using the binomial probability formula, we have:

P(X = 13) = (48)^13 * (0.254)^13 * (0.746)^35

= 0.159

So the probability that 13 out of 48 flowers will bloom early is 0.159.

b) To find the probability that fewer than 13 flowers will bloom early, we can find the cumulative probability up to 12:

P(X < 13) = P(X = 0) + P(X = 1) + ... + P(X = 12)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=0 to 12

= 0.107

So the probability that fewer than 13 flowers will bloom early is 0.107.

c) The expected value of a binomial distribution is given by n*p, where n is the number of trials and p is the probability of success. In this case, we have:

E(X) = 48 * 0.254

= 12.192

So on average, we expect to observe about 12.192 early blooming flowers.

d) To find the probability that at least 13 flowers will bloom early, we can use the complement rule and find the probability that 12 or fewer flowers will bloom early, and subtract that from 1:

P(X ≥ 13) = 1 - P(X < 13)

= 1 - 0.107

= 0.893

So the probability that at least 13 flowers will bloom early is 0.893.

e) To find the probability that between 9 and 14 flowers (inclusive) will bloom early, we can find the cumulative probability from 9 to 14:

P(9 ≤ X ≤ 14) = P(X = 9) + P(X = 10) + ... + P(X = 14)

= ∑(48 choose k) * (0.254)^k * (0.746)^(48-k) from k=9 to 14

= 0.662

So the probability that between 9 and 14 flowers (inclusive) will bloom early is 0.662.

f) The variance of a binomial distribution is given by np(1-p), and the standard deviation is the square root of the variance. In this case, we have:

Var(X) = 48 * 0.254 * (1-0.254)

= 9.078

SD(X) = sqrt(Var(X))

= sqrt(9.078)

= 3.0121 (rounded to 4 decimal places)

So the standard deviation of the number of flowers that bloom early in a row of 48 flowers is approximately 3.0121.

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The gradient of line A is 4/72% or 0.0555.The gradient of line B is -2/6 or -0.3333.

Value is the worth of something, or the amount of importance, good qualities, or benefits that something has. It is determined by the amount of effort, time, and resources that have been put into it. Value can also refer to a person's moral or ethical principles and standards. It is the measure of worth that someone places on something or someone. Ultimately, value is subjective, as it is based on individual opinion.

The gradients of lines A and B indicate the rate of change of the line in relation to the x-axis. The gradient of line A is positive, which indicates that the line is increasing as it moves along the x-axis. This means that the y-value of the line is increasing as the x-value increases. On the other hand, line B has a negative gradient, which means that the y-value of the line is decreasing as the x-value increases. This means that the line is decreasing as it moves along the x-axis. This is why the gradients of lines A and B are different.

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Answer:

$1.50

Step-by-step explanation:

3 boxes = $4.50

1 box= $4.50÷3

1box= $1.50

Answer:$1.50

Step-by-step explanation:

The simplified form of the rational expression is given as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)] = 1/[(x + 4)(x - 3)]

The rational expression for this problem is defined as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)].

To simplify the rational expression, first we must factor the quadratic equations at the denominator of each of the fractions, hence:

The simplification of the first quadratic function, x² - 3x - 10 = (x - 5)(x + 2), means that the numerators of each of the fractions can be simplified.

As the numerators of each of the fractions are simplified along with x² - 3x - 10, the remaining part is x² + x - 12, hence the simplified expression is given as follows:

[(x - 5)/(x² - 3x - 10)] x [(x + 2)/(x² + x - 12)] = 1/[(x + 4)(x - 3)]

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In response to the supplied query, we may state that Ben's speed, equation rounded to one decimal place, is x + 0.8 = 2.5 mph.

Using the equals symbol (=) to indicate equivalence, a math equation links two statements. Algebraic equations prove the equality of two mathematical expressions by a mathematical assertion. The equal sign, for example, provides a gap between the numbers 3x + 5 and 14 in the equation 3x + 5 = 14. You can use a mathematical formula to understand the connection between the two phrases that are written on opposite sides of a letter. Most of the time, the logo and the particular software match. e.g., 2x - 4 = 2 is an example.

Rate times distance

Let x represent Alan's speed in mph. Ben's speed will then be x + 0.8 mph.

They will have travelled 12.6 kilometres in total when they finally cross paths. The formula is as follows:

[tex]12.6 = (x + x + 0.8) × 3\\12.6 = 6x + 2.4 \s10.2 = 6x \sx = 1.7[/tex]

Alan is moving at a speed of 1.7 mph, decimal place included.

Ben's speed, rounded to one decimal place, is x + 0.8 = 2.5 mph.

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Answer:

​Alan's rate of speed is 1.7 mph.

​Ben's rate of speed is 2.5 mph.

Step-by-step explanation:

We can use the distance-rate-time formula to solve the given problem.

[tex]\boxed{\sf Distance=rate \times time}[/tex]

Create a distance equation for each of the hikers.

Therefore, the equation for Alan's distance in terms of r is:

[tex]\implies \sf d = 3r[/tex]

Therefore, the equation for Ben's distance in terms of r is:

[tex]\implies \sf d = 3(r+0.8)[/tex]

Alan and Ben are 12.6 miles apart and walking toward each other.  

Therefore, their combined distances will equal 12.6 miles.

Set the sum of the found expressions for distance equal to 12.6 and solve for r:

[tex]\implies \sf d_{Alan}+d_{Ben}=12.6[/tex]

[tex]\implies \sf 3r+3(r+0.8)=12.6[/tex]

[tex]\implies \sf 3r+3r+2.4=12.6[/tex]

[tex]\implies \sf 6r+2.4=12.6[/tex]

[tex]\implies \sf 6r=10.2[/tex]

[tex]\implies \sf r=1.7[/tex]

Therefore:

​Alan's rate of speed is 1.7 mph.

​Ben's rate of speed is 2.5 mph.

Local Minima is 5 and 80

The critical numbers, a and b, of f'(x)=x^4−19x^3+135x^2+2 can be found by setting the derivative f'(x) = 0 and solving for x.

The derivative is: f'(x) = 4x^3 − 57x^2 + 270x

By setting the derivative equal to 0, we get:
4x^3 − 57x^2 + 270x = 0

Solving for x, we get:
x = 0, x = 5, x = 9

The critical numbers of f'(x) are 0, 5 and 9. The second derivative f''(x) is undefined at x = 0, so this is a local maximum point, while x = 5 and 9 are local minimum points.

Table:
Critical number | Second derivative (f''(x)) |  Value
0 | Undefined | Local Maximum
5 | -80 | Local Minimum
9 | 80 | Local Minimum

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The volume of the cone that will fill the cone halfway is [tex]198.72 cm3[/tex] by the given base radius of 9/2cm and 14cm deep.

The  formula for the volume of the cone is

[tex]V = (1/3)πr^2h[/tex]

where h is the height of the cone, r is its base's radius and is a mathematical constant roughly equal to 3.14159.

We must first determine the height of the cone that is filled halfway in to calculate the volume of the cone that will fill it halfway.

Half of the cone's entire height, or 14 cm divided by 2, or 7 cm, corresponds to the height of the half-filled cone.

The cone's base has a radius of 9/2 cm.

We can calculate the volume of the cone that will fill the cone halfway using the formula for a cone's volume as follows:

[tex]V = (1/3)πr^2h[/tex]

[tex]V = (1/3)π(9/2)^2(7) (7)[/tex]

[tex]V ≈ 198.72 cm^3[/tex] (rounded to 2 decimal places) (rounded to 2 decimal places)

Therefore, [tex]198.72 cm3[/tex] is the volume of the cone that will fill it halfway.

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There are 6 faces in a cube, the total surface area of the toy box is 6 x 4 = 24 square feet. So correct option is B.

In geometry, a cube is a three-dimensional shape that is bounded by six square faces, with each face meeting at right angles. A cube is a regular polyhedron, which means that all of its faces are congruent and all of its edges have the same length.

The cube has a total of eight vertices (corners) and twelve edges, with each vertex connecting three edges and each edge connecting two vertices. The volume of a cube can be calculated using the formula V = s^3, where s is the length of one edge.

Cubes are commonly used in mathematics, physics, and engineering to represent three-dimensional objects and to model various phenomena. For example, cubes can be used to represent the atoms in a crystal lattice, the cells in a grid, or the pixels in a digital image

The toy box is a cube, and each edge measures 2 feet. So, the surface area of one face of the cube is 2 x 2 = 4 square feet. Since there are 6 faces in a cube, the total surface area of the toy box is 6 x 4 = 24 square feet.

Therefore, the answer is (B) 24 feet squared.

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The complete question is:

The manufacturer should ship 0 refrigerators to warehouse A and 80 refrigerators to warehouse B to minimize costs. The minimum cost is $800.

The objective of the problem is to minimize the cost of shipping the refrigerators to the two warehouses while also meeting the requirements of shipping at least 100 refrigerators and hiring at least 300 workers. This can be formulated as a linear programming problem with the following decision variables:

x1 = the number of refrigerators shipped to warehouse A
x2 = the number of refrigerators shipped to warehouse B

The objective function is the total cost of shipping the refrigerators:

minimize 12x1 + 10x2

The constraints are the requirements of shipping at least 100 refrigerators, hiring at least 300 workers, and the maximum capacity of each warehouse:

x1 + x2 >= 100
4x1 + 2x2 >= 300
x1 <= 75 (100 - 25)
x2 <= 80 (100 - 20)

Using the graphical method, we can plot the constraints and find the feasible region. The optimal solution will be at one of the corner points of the feasible region.

After plotting the constraints and finding the feasible region, we can evaluate the objective function at each of the corner points to find the minimum cost. The corner points and their corresponding costs are:

(75, 0) -> 12(75) + 10(0) = $900
(0, 80) -> 12(0) + 10(80) = $800
(50, 100) -> 12(50) + 10(100) = $1700
(60, 90) -> 12(60) + 10(90) = $1740

The minimum cost is $800, which occurs when 0 refrigerators are shipped to warehouse A and 80 refrigerators are shipped to warehouse B.

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Answer:

[tex]\left(x-\frac{5}{2}\right)^2=\frac{25}{2}[/tex]

Step-by-step explanation:

I am not completely sure if you just wanted me to write it in square form, so I will show the process of writing it in square form and how to find the solution.

[tex]-4x^2+20x=-25[/tex]    start by moving 25 to the right side

[tex]\frac{-4x^2+20x}{-4}=\frac{-25}{-4}[/tex]         divide both sides by -4

= [tex]x^2-5x=\frac{25}{4}[/tex]    

Now we rewrite the equation in the form [tex]\:x^2+2ax+a^2[/tex]

= [tex]x^2-5x+\left(-\frac{5}{2}\right)^2=\frac{25}{2}[/tex]  apply perfect square formula: [tex]\left(a-b\right)^2=a^2-2ab+b^2[/tex]

= [tex]\left(x-\frac{5}{2}\right)^2=\frac{25}{2}[/tex]   -- square form

Solutions to the quadratic equation:

Possible solutions: [tex]x=\frac{5}{\sqrt{2}}+\frac{5}{2}[/tex]    and [tex]\:x=-\frac{5}{\sqrt{2}}+\frac{5}{2}[/tex]

Answer:

$9 an hour

Step-by-step explanation:

A) 1

B) 18

C) 4

D) 90

To show that d/dx (csc(x)) = −csc(x)cot(x), we will use the chain rule and the derivative of the inverse function.

The chain rule states that the derivative of a composite function is the product of the derivatives of the individual functions. The derivative of the inverse function is the reciprocal of the derivative of the original function. Here are the steps:

1. Start with the given function: d/dx (csc(x))
2. Rewrite csc(x) as 1/sin(x): d/dx (1/sin(x))
3. Use the chain rule to find the derivative: (d/dx 1)(d/dx sin(x))
4. The derivative of 1 is 0, so the first term becomes 0: (0)(d/dx sin(x))
5. The derivative of sin(x) is cos(x), so the second term becomes cos(x): (0)(cos(x))
6. Simplify the expression: 0
7. Use the derivative of the inverse function to find the derivative of 1/sin(x): -1/sin^2(x)
8. Rewrite sin^2(x) as (sin(x))(sin(x)): -1/(sin(x))(sin(x))
9. Simplify the expression by canceling out one of the sin(x) terms: -1/sin(x)
10. Rewrite 1/sin(x) as csc(x): -csc(x)
11. Rewrite -1/sin(x) as -cot(x): -cot(x)
12. Combine the two terms to get the final answer: −csc(x)cot(x)

Therefore, d/dx (csc(x)) = −csc(x)cot(x).

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Answer:

Heterozygous matches up with C

Homozygous dominant is D

Homozygous recessive is E

Genotype is A

and Phenotype is B

Answer:

0.011

Step-by-step explanation:

Answer:9.05

Step-by-step explanation: uh why? its easy

Based on the given information, the sample space for the event can be represented using a tree diagram, where the first level consists of the three pants (black, blue, and tan) and the second level consists of the five shirts (white, black, green, blue, and red). Each branch represents a possible combination of a pair of pants and a shirt.

To find the total number of outcomes in the sample space, we can simply multiply the number of options for pants (3) by the number of options for shirts (5), giving us a total of 15 possible outcomes.

For Natalie selecting a pair of pants and shirt that are the same color, the favourable outcomes would be the combinations where the pants and shirt are either both black or both blue. These can be listed as:

Black pants and black shirt

Blue pants and blue shirt

For Natalie selecting a black shirt, the favourable outcomes would be the combinations where the shirt is black. These can be listed as:

Black pants and black shirt

Blue pants and black shirt

Tan pants and black shirt

For Natalie selecting a blue article of clothing, the favourable outcomes would be the combinations where either the pants or the shirt (or both) are blue. These can be listed as:

Black pants and blue shirt

Blue pants and white shirt

Blue pants and black shirt

Blue pants and green shirt

Blue pants and blue shirt

Tan pants and blue shirt

The probability that there are not greater than 859 defective products in 359 packages is 0.9999.

We are given that;

Distribution: X 0 3 4 1 2 Р 0.02 0.14 0.34 0.36 0.14 1

0(X) = 0.9635 2

Now,

We need to know the probability of success for each package, which is the probability of having a defective product. Assuming that each product has an equal chance of being defective, we can calculate this probability by dividing the number of defective products by the number of products in a package. In this case, we have:

[tex]p = \frac{0.02 + 0.14 + 0.34 + 0.36 + 0.14}{4} = 0.25[/tex]

This means that each product has a 25% chance of being defective, and each package has a binomial distribution with n=4 and p=0.25.

To find the average and the standard deviation of the number of defective products in a package, we can use these formulas³:

[tex]$$E(X) = np$$$$\sigma(X) = \sqrt{np(1-p)}$$[/tex]

Plugging in the values of $n$ and $p$, we get:

[tex]$$E(X) = 4 \times 0.25 = 1$$$$\sigma(X) = \sqrt{4 \times 0.25 \times (1-0.25)} = \sqrt{0.75} \approx 0.866$$[/tex]

Therefore, the average number of defective products in a package is 1, and the standard deviation is 0.866.

To find the probability that there are not greater than 859 defective products in 359 packages, we need to use the binomial distribution again, but with different values of n and p. In this case, we have:

[tex]$$n = 359 \times 4 = 1436$$$$p = 0.25$$[/tex]

We want to find the probability that [tex]$X \leq 859$[/tex], where [tex]$X$[/tex] is the number of defective products in 1436 trials. We can use this formula:

[tex]$$P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i}p^i(1-p)^{n-i}$$Plugging in the values of $n$, $p$, and $k$, we get:$$P(X \leq 859) = \sum_{i=0}^{859} \binom{1436}{i}0.25^i(1-0.25)^{1436-i}$$[/tex]

Therefore, by probability the answer will be 0.9999.

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In a right-angled triangle, the sum of two acute angles is equal to 90°.

So if α and β are the two acute angles of a right-angled triangle, then α + β = 90°.Using the formula for sin of the difference of two angles, we can write:sin2α = sin(90° − β) = sin90°cosβ − cos90°sinβ = cosβsin2β = sin(90° − α) = sin90°cosα − cos90°sinα = cosαSince sin2α = cosβ and sin2β = cosα, we can conclude that sin2α = sin2β.Now, let's write 3sinx − 5cosx in the form kcos(x+a). Using the formula for cos of the sum of two angles, we can write:3sinx − 5cosx = kcos(x+a) = k(cosxcos a − sinxsin a)Equating the coefficients of sinx and cosx, we get:−5 = kcos a3 = −ksin aSquaring and adding these equations, we get:25 + 9 = k^2(cos^2a + sin^2a) = k^2So k = √34.Taking the ratio of the two equations, we get:−5/3 = cos a/sin a = 1/tan aSo tan a = −3/5.Using the inverse tangent function, we get:a = tan^−1(−3/5)So the final answer is:kcos(x+a) = √34cos(x + tan^−1(−3/5)).

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a. False. If we form the table with aspirin in the first row (instead of placebo), the confidence interval is (1/2.33, 1/1.44) = (0.43, 0.69).

b. True. We may get a different odds ratio value if we treat marijuana use as the response variable than if we treat alcohol use as the response variable.

c. False. Interchanging two rows or interchanging two columns in a contingency table does have an effect on the value of the X2 or G-chi-squared statistics. Thus, these tests treat both the rows and the columns of the contingency table as nominal scale, and if either or both variables are ordinal, the test takes that information into account.

d. False. Income and gender are not necessarily independent in the 2 x 2 marginal table, even if they are conditionally independent given the type of job.

e. False. The estimated odds ratio between opinion on global warming and whether one is a Democrat or a Republican does not equal 2.96 x 2.08 = 6.2, it equals (2.96/2.08) = 1.42.

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