A company is buying new computers. The company needs no more than 220 desktops and at least 50 laptops. A total of at least 200 computers must be bought. Write a system of linear inequalities to represent the constraints of this situation. Let x represent the number of desktops, and let y represent the number of laptops.

The required correct answers are [tex]$$H_0: p = 0.08$$[/tex] , [tex]$$H_a: p < 0.08$$[/tex].

Let p be the proportion of residents in the town who are unemployed. The null hypothesis [tex]$H_0$[/tex] is that the proportion of unemployed residents in the town is the same as the national unemployment rate of 8%. The alternative hypothesis [tex]$H_a$[/tex] is that the proportion of unemployed residents in the town is significantly lower than the national unemployment rate.

Using the appropriate notation, the hypotheses can be expressed as:

$H_0: p = 0.08$

$H_a: p < 0.08$

Therefore, the appropriate set of hypotheses for the mayor's significance test are:

[tex]$$H_0: p = 0.08$$[/tex]

[tex]$$H_a: p < 0.08$$[/tex]

Note that this is a one-tailed test since the alternative hypothesis is only considering the possibility of the proportion being lower than the national unemployment rate

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Answer: To determine if Marco has made a mistake, we would need to know the expected mass of the contents of the beaker before the reaction took place. If the expected mass was 247 g or close to it, then Marco may not have made a mistake.

However, if the expected mass was significantly different from 247 g, then it is possible that Marco made a mistake in his experiment. It could be a measurement error, a calculation error, or a procedural error.

There are several reasons why Marco may have obtained a mass of 247 g at the end of the reaction. One possibility is that the reaction produced a product that was relatively volatile, and some of it was lost during the experiment. Another possibility is that Marco did not completely dry the product before weighing it, which could result in a higher measured mass due to the presence of residual moisture.

To determine the exact reason why Marco obtained a mass of 247 g, further investigation and experimentation would be needed.

Step-by-step explanation:

The length of the ladder is c = √(x^2 + 64) ft.

The question is asking to find the length of the ladder. We can use the Pythagorean Theorem to solve this problem. Let x be the length of the ladder.

Pythagoras theorem states that “In a right-angled triangle,  the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular, Base and Hypotenuse.

Here, the hypotenuse is the longest side, as it is opposite to the angle 90°. The sides of a right triangle (say a, b and c) which have positive integer values, when squared, are put into an equation, also called a Pythagorean triple.

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

The Pythagorean Theorem states that a^2 + b^2 = c^2.

Therefore, x^2 + 8^2 = c^2

Solving for c: x^2 + 64 = c^2

Taking the square root of both sides: √(x^2 + 64) = c

Therefore, the length of the ladder is c = √(x^2 + 64).

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a) For this experiment, the event that represents a "success" is: Adults not working during summer vacation. (Option a)

b) x is (approximately) a binomial random variable because the experiment consists of identical trials, there are only two possible outcomes on each trial (works or does not work), and the trials are independent. Therefore, (option B) is correct.

c) The value of p for this binomial experiment is 0.2, which is the probability of success (i.e., an adult not working during summer vacation) on a single trial.

d) P(x≥4) is approximately 0.121.
e) the probability that 2 or fewer of the 10 adults do not work during summer vacation is approximately 0.6777.

Probability is a measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

So with regards to the above:

a) For this experiment, the event that represents a "success" is: Adults not working during summer vacation. (Option a)

b) x is (approximately) a binomial random variable because the experiment consists of identical trials, there are only two possible outcomes on each trial (works or does not work), and the trials are independent. Therefore, (option B) is correct.

c) The value of p for this binomial experiment is 0.2, which is the probability of success (i.e., an adult not working during summer vacation) on a single trial.

d)  To find P(x≥4), we need to use the binomial probability formula:

P(x≥4) = 1 - P(x<4)

P(x<4) = P(x=0) + P(x=1) + P(x=2) + P(x=3)

Using the binomial probability formula, we can calculate:

P(x=0) = (10 choose 0) * (0.2)⁰ * (0.8)¹⁰ = 0.1074

P(x=1) = (10 choose 1) * (0.2)¹ * (0.8)⁹ = 0.2684

P(x=2) = (10 choose 2) * (0.2)² * (0.8)⁸ = 0.3019

P(x=3) = (10 choose 3) * (0.2)³ * (0.8)⁷ = 0.2013

Therefore,

P(x<4) = 0.1074 + 0.2684 + 0.3019 + 0.2013 = 0.879

And,

P(x≥4) = 1 - P(x<4) = 1 - 0.879 = 0.121

Hence, P(x≥4) is approximately 0.121.

e. To find the probability that 2 or fewer of the 10 adults do not work during summer vacation, we need to calculate P(x≤2). We can use the same formula as in part (d), but only sum up the probabilities for x=0, 1, and 2.

P(x=0) = 0.1074

P(x=1) = 0.2684

P(x=2) = 0.3019

Therefore,

P(x≤2) = P(x=0) + P(x=1) + P(x=2) = 0.1074 + 0.2684 + 0.3019
= 0.6777

Hence, the probability that 2 or fewer of the 10 adults do not work during summer vacation is approximately 0.6777.

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He bought 4.29 Kilos of fruit.

4+0.29=4.29

Luke bought 4.29 kilograms of fruit in all

Step-by-step explanation:

Simple addition will be used to find the total fruit Luke bought.

Given

Amount of apples he bought  = 4 kilograms

Amount of oranges he bought = 0.29 kilograms

so the total fruit will be:

[tex]\text{total fruit}=\text{Apples}+\text{oranges}[/tex]

[tex]=4+0.29[/tex]

[tex]=4.29[/tex]

So,

Luke bought 4.29 kilograms of fruit in all

Keywords: Measurement, addition

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It is important to be aware that the distribution of sample means x may not match the distribution of the original x values exactly, due to sampling variability.

if the original distribution of x values is known to be normalize size needs to be large, then the sample size does not need to be restricted in order to claim that the distribution of sample means x taken from random samples of a given size is normal. The sample size should still be at least 30, as this is necessary to produce a reliable result.

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Answer:

The slope is 5/4 or 1.25

Step-by-step explanation:

You can find the slope by using the equation (y2-y1)/(x2-x1)=Slope

This time it would be (12-7)/(8-4)

Simplified it would be 5/4 or 1.25

Answer:

x=-2.5

Step-by-step explanation:

3 4(2.5-5) = (1/27) 2(-2.5+10)

I hohonestly don't know how to explain it

The rate at which the value of the computer system is changing 4 years after it was purchased is "Declining at the rate of $8,803.21 per year". The correct option is D.

We can use the exponential decay formula [tex]V(t) = V0 * e^{-kt}[/tex], where V(t) is the value of the computer system after t years, V0 is the initial value, and k is the decay rate.

We know that V(1) = $15,700 and V(0) = $28,000, so we can solve for k:

[tex]$15,700 = $28,000 * e^{-k*1}[/tex]

[tex]e^{-k} = 0.5607[/tex]

-k = ln(0.5607) ≈ -0.5797

k ≈ 0.5797

Therefore, the decay rate is approximately 0.5797 per year.

To find the rate of change of the value of the computer system 4 years after it was purchased, we can take the derivative of V(t) with respect to t:

dV/dt = -k * V0 * [tex]e^{-kt}[/tex]

Substituting t = 4, V0 = $28,000, and k ≈ 0.5797, we get:

dV/dt = -0.5797 * $28,000 * [tex]e^{-0.5797*4}[/tex] ≈ -$8,803.21 per year

Therefore, the value of the computer system is declining at the rate of approximately $8,803.21 per year 4 years after it was purchased.

The correct answer is (D) Declining at the rate of $8,803.21 per year.

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The probability that the patients with a toothache has a cavity is 0.3073.

Probability is the extent to which an event is likely to occur.

The probability that the patient has a toothache given that they have a cavity is equal to the percentage of patients with cavities that have toothaches. So, it can be calculated as:

P(toothache | cavity) = 71% = 0.71

Also, the probability that a patient has a cavity is equal to the ratio of patients with cavities to total patients, which is given to be 1 in 23. So,

P(cavity) = 1/23 = 0.04347

The probability that the patient has a toothache can be determined from the fact that 1 in 10 patients have a toothache. So,

P(toothache) = 1/10 = 0.1

The probability that a patient has a toothache and a cavity is calculated as:

P(toothache and cavity) = P(toothache | cavity) * P(cavity) = 0.71 * 0.04347 = 0.0308

The probability that the patient has a toothache given that they have a cavity can be calculated using Bayes' theorem:

P(cavity | toothache) = P(toothache | cavity) * P(cavity) / P(toothache)= 0.71 * 0.04347 / 0.1 = 0.3073

Therefore, the probability that patients with a toothache have a cavity is 0.3073 or about 30.73%.

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Answer:

x=4/7

Step-by-step explanation:

To start, use the distributive property and multiply 2 by what is in the parenthesis. Do 2x3x as well as 2x-1. This will get you 6x-2 on the left side. On the right side, since there is a minus sign, change the x to -x and 3 to -3. You will ultimately get 6x-2=5-x-3. To solve this, you must isolate the variable by using inverse operations. The final answer is 4/7.

The p values that would reject the null hypothesis are (f) 0.0500 and (e) 0.0900

To reject the null hypothesis for a two-sided test at the 90% confidence level, we would compare the p-value to the significance level (alpha), which is equal to 1 - confidence level.

In this case, the confidence level is 90%, so alpha is 1 - 0.9 = 0.1.

Any p-value that is less than or equal to the significance level of 0.1 would lead to rejection of the null hypothesis.

So the p-values at which we would reject the null hypothesis are: 0.0500 and 0.0900

Therefore, the correct answers are options, (e), and (f).

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If m = 3/2, the two equations in the system are dependent and have an infinite number of solutions.

To do this, we can set the coefficients of x and y in the two equations equal to each other:

6 - 4m = 2k(2m - 1)

2n - 7 = 3k

Simplifying these equations, we get:

8m - 6 = 4km

2n - 7 = 3k

We can solve the first equation for k:

k = (8m - 6)/(4m)

Substituting this into the second equation and solving for n, we get:

n = (16m - 29)/(8m - 12)

Now we need to check if there are any values of m for which the equations are proportional. We can do this by checking if the expressions for k and n are the same for all values of m.

We find that the expressions for k and n are the same for m = 3/2.

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Complete Question:

Determine the values of m and n so that the following system of linear equations have infinite number of solutions:

(2m−1)x+3y−5=0

3x+(n−1)y−2=0

The new position of arrowhead graph ABCD is A'B'C'D' as shown in the below graph.

Graph rotation refers to the process of rotating a graph, which involves changing the orientation of the edges or nodes in the graph while preserving the relationships between them.

From the given arrowhead graph as below; the points labelled are

A (-2, 2), B (0, 5), C (2, 2) and D (0, 3)

On rotating the graph through 90° anticlockwise around the origin (0, 0), the new positions of the points are:

A (-2, 2)  ⇒  A' (-2, -2)

B (0, 5)   ⇒  B' (-5, 0)

C (2, 2)   ⇒  C' (-2, 2)

D (0, 3)   ⇒  D' (-3, 0)

Therefore, the new position of arrowhead graph ABCD is A'B'C'D' as shown in the below graph.

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X = 22

Y = 15

youre  welcome! Brainliest would be appreciated :)

The correct answer is the triangle

Triangle : 239 yards

Square : 224 yards

the given form of the rate function:[tex]tan² (t) + 1 = sec²[/tex](t)

Therefore, we can write the given function as:c (t) dtUsing integration by substitution, we haveu = tan (t) ⇒ du = sec² (t) dt

Therefore,S [tex]π/t tan (t) sec² (t) dt= S u du= ln |tan (t)| + C[/tex]Thus, the equivalent closed form of the given function is:S π/t 4tan (t) dt= 4 ln |tan (t)| + C

a. S π/π/4 5t+4/t² + 1 dt equivalent closed formThe question demands to find an equivalent closed form for each function. So let's find the equivalent closed form for the given functions:a. S π/π/4 5t+4/t² + 1 dt

Hint: begin by writing as a sum of two functionsNow, we need to write the given function as a sum of two functions. Let's first write the numerator of the function as a sum of two functions.

Using the formula, a²-b² = (a+b)(a-b), we have5t + 4 = (2 + √21)(√21 - 2)Therefore, we can write the numerator of the function as follows:5t + 4 = (√21 - 2)² - 17Using this in the given function,

we haveπ/π/4 [(√21 - 2)² - 17]/t² + 1 dtLet's further simplify the numeratorπ/π/4 [21 + 4 - 4√21 - 17t² + 34t - 17] / (t² + 1) dt= π/π/4 [-17t² + 34t + 8 - 4√21]/(t² + 1) dtLet's now find the closed form of this function using the integration formulaS f(x) dx = ln |f(x)| + C Therefore, the equivalent closed form of the function is:

S π/π/4 5t+4/t² + 1 dt= π/π/4 [-17t² + 34t + 8 - 4√21]/(t² + 1) dt= - π/2 ln |t² + 1| + 34 π/2 arctan (t) - 17 π/2 t + 2 π/√21 arctan [(2t-√21)/ √21] + Cb. S π/t 4tan (t) dt equivalent closed formNow, let's find the equivalent closed form of the second given function.b. S π/t 4tan (t)

dtHint: begin by using a trig identity to change the form of the rate function Let's now use the following trig identity to change

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Assumptions of the error terms in the simple linear regression model except (D) Error terms have a constant variance.

Linear Regression Model:

A linear regression model states that the relationship between a dependent variable y and one or more independent variables X. The dependent variable is also known as a response variable. Independent variables are also called explanatory variable. In statistics, linear regression is a linear method of modeling the relationship between a scalar response and one or more explanatory variables (also called dependent and independent variables). The case of one explanatory variable is called simple linear regression for more than one, the procedure is called multiple linear regression.

The term differs from multiple linear regression, which predicts multiple correlated dependent variables rather than a single scalar variable.

Error Term:

The error term represents the margin of error in a statistical model; it is the sum of the deviations inside the regression line and it provides an explanation of the difference between the theoretical value of the model and the actual observations.

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It pays to purchase the information for any value of c less than $13.25 then we buy the information for a price less than $13.25, we can increase our expected payoff above a loss of $1.

To calculate it Let C1 be the event that the first coin is selected and C2 be the event that the second coin is selected. Then, we have:

P(C1) = P(C2) = 1/2 (since one of the two coins is randomly chosen)

P(H|C1) = 0.6 (probability of getting heads when the first coin is flipped)

P(H|C2) = 0.3 (probability of getting heads when the second coin is flipped)

Let's consider the case when we do not buy the insider's information. Then, our expected payoff can be calculated as follows:

E(X) = P(C1) * P(H|C1) * (10) + P(C1) * P(T|C1) * (-10) + P(C2) * P(H|C2) * (10) + P(C2) * P(T|C2) * (-10)

= (1/2) * (0.610 - 0.410 + 0.310 - 0.710)

= -1

Therefore, if we do not buy the insider's information, our expected payoff is a loss of $1.

Now, let's consider the case when we buy the insider's information for an amount c.

If we buy the information, we will know which coin was selected and we can bet accordingly to maximize our expected payoff.

If we know that the first coin was selected, we should bet on heads since it has a higher probability of occurring. If we know that the second coin was selected, we should bet on tails since it has a higher probability of occurring.

Therefore, if we buy the insider's information, our expected payoff can be calculated as follows:

E(X|buying information) = P(C1) * P(H|C1) * (10-c) + P(C1) * P(T|C1) * (-c) + P(C2) * P(H|C2) * (-c) + P(C2) * P(T|C2) * (10-c)

= (1/2) * (0.6*(10-c) - 0.4c + 0.3(-c) - 0.7*(10-c))

= -0.2c + 1.5

To find the values of c for which it pays to purchase the information, we need to solve the inequality:

E(X|buying information) > E(X)

-0.2c + 1.5 > -1

Solving for c, we get:

c < 13.25

Therefore, it pays to purchase the information for any value of c less than $13.25. If we buy the information for a price less than $13.25, we can increase our expected payoff above a loss of $1.

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The probability of the following parts is: a. 0.33 b. 1.925 c. 0.176 d. 0.67 e. 0.99955416 f. 0.018318006 g. 0.4489

a. To find out the probability that none of the first three people is concerned about their credit scores, we need to find: P(none of the first three is concerned about their credit scores)we have been given the probability that a teenager is concerned about his/her credit score is 0.67. So the probability that a teenager is not concerned about his/her credit score is 0.33. Now we can say that this is a binomial distribution since we are repeating a procedure until success is achieved. So we can use the binomial distribution to find the above probability: P(none of the first three are concerned about their credit scores) = (0.33)^3 = 0.0359375

b. The expected value of X is given by: E(X) = 1/p = 1/0.67 = 1.4925

c. Variance of X is given by: Var(X) = (1-p)/p^2 = (0.33)/(0.67)^2 = 0.176

d. Since X is the number of teenagers we select who are not concerned about their credit scores before we find the first teenager who is concerned. The probability that X = 0 is the probability that the first teenager we select is concerned about his/her credit score, which is given by: P(X = 0) = p = 0.67

e. To find out the probability that X ≤ 4, we can use the complement rule: P(X ≤ 4) = 1 - P(X > 4) = 1 - [P(X = 5) + P(X = 6) + ....... to ∞] = 1 - (1 - p)^5 = 1 - (0.33)^5 = 0.99955416

f. To find out the probability that Y = 6, we need to find: P(Y = 6)We know that for Y = 6, we need to select 7 teenagers such that the first and the second teenager we select are not concerned about their credit scores, and the third to the seventh teenager we select are concerned about their credit scores. The probability that the first and the second teenager we select are not concerned about their credit scores is given by: P(selecting 2 teenagers not concerned about their credit scores) = (0.33)^2 = 0.1089 And the probability that the third to the seventh teenager we select are concerned about their credit scores is:

P(selecting 5 teenagers concerned about their credit scores) = (0.67)^5 = 0.16806957Therefore, P(Y = 6) = P(selecting 2 teenagers not concerned about their credit scores) * P(selecting 5 teenagers concerned about their credit scores) = 0.018318006

g. To find out the probability that Y = 0, we need to select the first two teenagers who are concerned about their credit scores. P(Y = 0) = P(selecting the first two teenagers who are concerned about their credit scores) = (0.67)^2 = 0.4489.

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Answer:

A

Step-by-step explanation:

The axis are counting by 2's.

If you start at (0,0) and to one unit to the right and one unit down.  You will be at the point (2,-2).  That point is on the line and in the table.  If you, again, start at (0,0) and go one unit to the lefts and 2 units down, you will be at the point (-2,-4).  Again you are on line line.  This is the only graph that shows those two points on the line.

Helping in the name of Jesus.

The smallest number of beads we can take so that the conditions for performing inference are met is 40.

Probability:

The probability of an event is a number that indicates the probability of the event occurring. Expressed as a number between 0 and 1 or as a percent sign between 0% and 100%. The more likely an event is to occur, the greater its probability. The probability of an impossible event is 0; the probability of a certain event occurring is 1. The probability of two complementary events A and B - A occurring or B occurring - adds up to 1.

According to the Question:

Given in the question,

Teacher prepares a large container filled with colored beads. She claims that 1/8 beads are red, 1/4are blue, and the rest are yellow. Your class will test the teacher's claim by randomly drawing a simple sample of beads from the container.

Quadrant                     Frequency

      1                                    18

      2                                   22

      3                                   39

      4                                   21

The proportions are 1/8 , 1/4 and 5/8

Here, the smallest probability is 1/8 , thus it would be used to compute the frequency.

Now,

The expected frequencies are calculated as:

E = np₁ = 15 (1/8) = 1.875

E = np₂ = 16(1/8) = 2

E = np₃ = 30(1/8) = 3.75

E = np₄ = 40(1/8) = 5

E = np₅ = 80(1/8) = 10

Here, conditions are fulfilling for 40 and 90 but the smallest sample size is contained by 40. Thus, the correct option is 40.

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Answer:

t = 7h

Step-by-step explanation:

lets have the total amount of helmets as t and helmets per box as h. Then it is t = 7h

The given integral is ∫6 to- 6 ∫36 to x^2 ∫36-y to 0 E)dz dX dy.

This integral can be rewritten as equivalent integrals in different orders of integration. Let's rewrite the integral in the following orders: a. dy dz dx b. dy dx dz c. dx dy dz d. dx dz dy e. dz dx dya. dy dz dx:In this case, we first integrate over z and then y and finally x.

Thus, the equivalent integral is ∫6 to- 6 ∫36 to x^2 ∫36-y to 0 dz dy dx= ∫6 to- 6 ∫36 to x^2 ∫0 to 36-y dz dy dx. Hence, the integral can be rewritten as ∫6 to- 6 ∫36 to x^2 ∫0 to 36-y dz dy dx.b. dy dx dz:In this case, we first integrate over z and then x and finally y.

Thus, the equivalent integral is ∫6 to- 6 ∫36 to x^2 ∫36-y to 0 dz dy dx= ∫6 to- 6 ∫0 to 36 ∫0 to y-36 dz dx dy. Hence, the integral can be rewritten as ∫6 to- 6 ∫0 to 36 ∫0 to y-36 dz dx dy.c. dx dy dz

In this case, we first integrate over z and then y and finally x. Thus, the equivalent integral is ∫6 to- 6 ∫36 to x^2 ∫36-y to 0 dz dy dx= ∫0 to 36 ∫-√(36-y) to √(36-y) ∫-√(36-x^2) to √(36-x^2) dz dx dy.

Hence, the integral can be rewritten as ∫0 to 36 ∫-√(36-y) to √(36-y) ∫-√(36-x^2) to √(36-x^2) dz dx dy.d. dx dz dy:In this case, we first integrate over y and then z and finally x.

Thus, the equivalent integral is ∫6 to- 6 ∫36 to x^2 ∫36-y to 0 dz dy dx= ∫0 to 36 ∫-√(36-y) to √(36-y) ∫-√(36-x^2) to √(36-x^2) dy dz dx.

Hence, the integral can be rewritten as ∫0 to 36 ∫-√(36-y) to √(36-y) ∫-√(36-x^2) to √(36-x^2) dy dz dx.e. dz dx dy:

In this case, we first integrate over y and then x and finally z.

Hence Option E is correct.

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Answer: $22.50

Step-by-step explanation:

Let x be the cost per pound of the secret-formula coffee beans.

The total cost of the secret-formula beans is 12x dollars.

The total cost of the other beans is 15 × 18 = 270 dollars.

The total cost of the mix is (12 + 15) × 20 = 540 dollars.

Since the barista mixed 12 pounds of the secret-formula beans with 15 pounds of the other beans, the total weight of the mix is 12 + 15 = 27 pounds.

We can set up an equation based on the total cost of the mix:

12x + 270 = 540

Subtracting 270 from both sides:

12x = 270

Dividing both sides by 12:

x = 22.5

Therefore, the barista's secret-formula coffee beans cost $22.50 per pound.

Answer: 576

Step-by-step explanation:

134+254+188= 576

Answer:

hjjjbcujnkknvcczynmcs

The probability that 15 people must be asked for Arya to find 5 people who voted for Frey is 0.0610.

a. The distribution of the random variable Z is Negative Binomial distribution. It has the support {5, 6, 7, ... } which represents the total number of trials until the fifth success. b. Let p = probability of success = 0.4 and q = probability of failure = 0.6Then, mean of Z, μ = 5/p = 5/0.4 = 12.5and variance of Z, σ² = (1-p)/p² = (0.6)/(0.4²) = 3.75So, standard deviation of Z, σ = √3.75 = 1.9365 (approx)

Therefore, the mean and standard deviation of Z are 12.5 and 1.9365 respectively.c. The probability that 15 people must be asked for Arya to find 5 people who voted for Frey is P(Z = 15).Using the Negative Binomial probability formula,P(Z = k) = (k-1)C(r-1) p^r q^(k-r)where k is the number of trials, r is the number of successes required,

p is the probability of success and q is the probability of failure.P(Z = 15) = (15-1)C(5-1) (0.4)^5 (0.6)^(15-5)= 14C4 (0.4)^5 (0.6)^10= (14!/4!10!) (0.4)^5 (0.6)^10= 1001 (0.01024) (0.00060466176)≈ 0.0610

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Answer:

171.21

Step-by-step explanation:

The value of the given expressions are as follows: a. 25 b. 4 c. 1/4 d. 1/3.

In mathematics, an expression is a combination of symbols and/or numbers that represents a mathematical object or quantity. Expressions can be written using variables, operations, functions, and mathematical symbols such as parentheses, exponents, and radicals. An expression can represent a value, an equation, or a formula, and can be evaluated or simplified using mathematical rules and properties. Examples of expressions include 2x + 5, sin(θ), and (a + b)².

Here,

a. [tex]125^{2/3}[/tex]

radical form:

[tex]\sqrt{ (125^2)} = 125^{1/2}[/tex]

[tex]125^{1/2}[/tex] = √125

= 5√5

Therefore, [tex]125^{2/3} = (125x^{1/3})^{2}[/tex]

= [tex](5^3)x^{2/3}[/tex]

= [tex]5x^{3*2/3}[/tex]

= 5²

= 25

b. √16

In radical form:

√16 = 4

Therefore, [tex]16x^{-1/2} = \sqrt{16}[/tex]

= 4

c. [tex]16^{1/2}[/tex]

In radical form:

1/√16 = 1/4

Therefore, [tex]16^{-1/2} = 1/\sqrt{16}[/tex]

= 1/4

d.[tex]\sqrt[4]{81}[/tex]

In radical form:

[tex]\sqrt[4]{81}[/tex] = √(√81)

= √9

= 3

Therefore, [tex]\sqrt[4]{81} = 1/\sqrt[4]{81}[/tex]

= 1/3

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