Do Now 60: What are the formulas to find area for a square, triangle, rectangle, parallelogram, trapezoid, circle, ellipse and equilaterial triangle?

Answer:

1. 20-15/0-8= -5/8

Step-by-step explanation:

2. In a positive slope the Y value increases as the X value increases (as the line moves towards the right it gets higher). In a negative slope the Y value decreases as the X value increases (as the line moves towards the right it gets lower). (Question 3 is in here)

4. It depends on the amount of time she's been driving

Answer:

D = -4

E = 2

Step-by-step explanation:

The procedure for completing the square method is given as:

x2 + bx + c = 0

x2 + bx = -c

x2 + bx + (b/2)2 = -c + (b/2)2

Aolving the LHS simultaneously,

(x + b)2 = -c + (b/2)2

Find the square root of both sides,

(x + b) = sqrt(-c + (b/2)2)

So for the equation, x2 - 8x + 12

x2 - 8x = -12

x2 - 8x + 16 = -12 + 16

(x - 4)2 = 4

So square root both sides,

(x - 4) = 2

Comparing this to the equation given,

(x - D) = E;

D = -4

E = 2

Answer:

x = 45.

Step-by-step explanation:

As the triangles are similar the corresponding sides are in the same proportion. Therefore:

18 / 2 = x / 5

x / 5 = 9

x = 5*9

x = 45.

Answer:

I think the correct answer is feet

Answer:

Feet

Step-by-step explanation:

Answer:

a) f(x) = (x + 6)(x + 2)(x – 3)

b) [tex]f(x) = x^3 + 5x^2 - 12x - 36[/tex]

Step-by-step explanation:

We have to find the polynomial with zeroes as -6, -2, and 3.

Roots are those values of x for which the polynomial is zero.

a) f(x) = (x + 6)(x + 2)(x – 3)

[tex]f(x) = (x + 6)(x + 2)(x -3)=0\\\Rightarrow x+6 = 0, x+2 = 0, x-3 = 0\\\Rightarrow x = -6,x = -2, x = 3[/tex]

b)

[tex]f(x) = x^3 + 5x^2 - 12x - 36 \\\text{It can be factored as}\\f(x) = (x+6)(x-3)(x+2) = 0\\\Rightarrow (x+6)=0,(x-3)=0,(x+2)=0\\\Rightarrow x = -6, x = 3, x = -2[/tex]

c) f(x) = (x – 6)(x – 2)(x + 3)

[tex]f(x) = (x - 6)(x - 2)(x + 3) =0\\\Rightarrow (x -6)(x -2)(x + 3) = 0\\\Rightarrow (x -6)=0,(x -2)=0,(x + 3)=0\\\Rightarrow x = 6, x = 2, x = -3[/tex]

d)

[tex]f(x) = x^3 - 5x^2 - 12x + 36\\f(x) = (x-2)(x+3)(x-6)=0\\\Rightarrow (x-2) = 0, (x+3)=0,(x-6)=0\\\Rightarrow x =2, x = -3, x = 6[/tex]

Answer:

Step-by-step explanation:

Final answer:

It takes 35 hours for the blood concentration of tadalafil to fall to 25% of its initial strength, given a half-life of 17.5 hours.

Explanation:

We can determine how long it takes for the blood concentration of a drug to fall to a certain percentage of its initial strength by knowing the drug's half-life. In this case, a half-life is the time required for the concentration of the drug to reduce to half its original value. For tadalafil 10mg/tab with a half-life of 17.5 hours, we want to calculate when the concentration falls to 25% of its initial strength.

To reach 25% of the original concentration, the drug needs to go through two half-lives (each half-life reduces the concentration by half):

Therefore, it will take 35 hours for the blood concentration of tadalafil to fall to 25% of its initial dosage, which corresponds to Option C.

Answer:

B) tenants in common.

Explanation:

There were two men, first was James Smith, and second was Bill Ross who bought a property mutually for their business venture by declared proportionate interest of 1/3 as well as 2/3 sequentially without the right regarding survivorship. Both of them want their shares to be acquired by their spouses'. Both hold a partnership but not a company. James, as well as Bill, most likely took the title as tenants in common. Because tenants in common possess versatility in whence they divide control as well as assign the rights concerning survivorship. Joint tenants, on the different hand, keep the property in similar shares while a common partnership is a business unit serving separate sharers. Severalty, as asserted beforehand, is individual ownership.

Answer:

The decomposition is as shown in the attachment below

Step-by-step explanation:

Where A, B, C, D, E, F, G, H, I, J are numerical value of the coefficients that can be determined by method of comparing coefficients. Hence , there are two methods for solving partial fractions ; Cover up method and method of comparing coefficients.

The form of the partial fraction decomposition is as shown in the attachment.

Answer:

x = 3.5.

Step-by-step explanation:

This is an equilateral triangle ( as all angles = 60 degrees) so all 3 sides are of equal length.

Thus 2x - 3 = 4

2x = 7

x = 3.5.

Answer:

hi its me

Step-by-step explanation:

Option D is the correct answer.

Step-by-step explanation:

The given system of equations are [tex]y=2x[/tex] and [tex]y=x^{2} +1[/tex]

To plot the equations in the graph, if we know the values of x and y coordinates.

To plot the equation [tex]y=2x[/tex] in the graph, let us substitute the value for x to find the value of y-coordinate.

For, [tex]x=0[/tex] ⇒ [tex]y=0[/tex]

For, [tex]x=1[/tex] ⇒ [tex]y=2[/tex]

For, [tex]x=2[/tex] ⇒ [tex]y=4[/tex]

Similarly, to plot the equation [tex]y=x^{2} +1[/tex], let us substitute the value for x,

For, [tex]x=0[/tex] ⇒ [tex]y=1[/tex]

For, [tex]x=1[/tex] ⇒ [tex]y=2[/tex]

For, [tex]x=2[/tex] ⇒ [tex]y=5[/tex]

These values are plotted in the graph. The graph that represents the system of equations is attached below.

Thus, the correct answer is option D.

Answer:

d

Step-by-step explanation:

Answer:

[tex]y = mx+b[/tex]

[tex] m =\frac{y_2 -y_2}{x_2 -x_1}[/tex]

[tex] b = y_1 -m x_1[/tex]

Or equivalently:

[tex] b = y_2 - m x_2[/tex]

Step-by-step explanation:

If we are assuming that we have:

x independent variable

y dependent variable

And we want to find an equation of the line, we have the following general expression:

[tex]y = mx+b[/tex]

Where m represent the slope and b the y intercept. The general formula for the slope is given by:

[tex] m =\frac{y_2 -y_2}{x_2 -x_1}[/tex]

Where [tex] (x_1,y_1) , (x_2,y_2)[/tex] are the minimum required points in order to estimate the slope.

In order to find the y intercept we just need to use one of the points selected [tex] (x_1,y_1) , (x_2,y_2)[/tex] and we can solve for b like this:

[tex] b = y_1 -m x_1[/tex]

Or equivalently:

[tex] b = y_2 - m x_2[/tex]

Answer:

c i did the test

Step-by-step explanation:

Answer: The system of inequalities representing the situation are

x + y ≥ 20

10x + 20y ≤ 280

Step-by-step explanation:

Let x represent the number of $10 tickets that the radio station is giving away.

Let y represent the number of $20 tickets that the radio station is giving away.

They want to give away at least 20 tickets. This means that

x + y ≥ 20

The total cost of all the tickets they give away can be no more then $280. This means that

10x + 20y ≤ 280

Part A

This line contains point G (as an endpoint) and it runs perpendicular to line FE. This line is in the bottom face of the rectangular prism. This line is also in the front face as well.

===================================================

Part B

AB is parallel to CD, GH and FE.

===================================================

Part C

Skew lines are lines that aren't parallel nor perpendicular nor intersecting. Skew lines can only be found in 3D space (not 2D space). Think of a line on the ground running from north to south. If we had a straight wire that did not bend at all run from north to south, then we would have two parallel lines. However, if we aim the wire to go from east to west, then the line on the ground and the wire will be skew lines. The line on the ground and the wire line do not intersect. The same story happens with AB and FE, along with AB and AG.

===================================================

Part D

The front face is plane AGF. We simply locate the three points A, G, F and note this is the front face. The back face is parallel to the front face. This face is plane GCH, and it contains point C.

note: we could say "plane AGFD" but we only need 3 points at minimum to uniquely determine a plane.

Any flat surface (plane) of a rectangular prism fits the description as they each have corners making right angles. Each edge of these faces can be seen as a line segment that would form a straight line if extended.

In a rectangular prism, each segment can be thought of as a part of a line. The plane that fits this description could be any of the flat surfaces of the prism, as they have each corner making a right angle. For example, consider you have a box (a rectangular prism). The front face of the box forms a plane, and each edge of this face can be seen as a line segment. Similarly, the top face of the box also forms a plane with all right angles. The edges of each face are line segments that are part of a straight line if extended. Therefore, all faces (planes) and edges (line segments) of the rectangular prism fit the description.

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Answer:

32-35 Week case

[tex]z = \frac{2025-2500}{800}=-0.594[/tex]

40 week case

[tex]z = \frac{2225-2700}{375}=-1.27[/tex]

For this case since the z score is lower (-1.27<-0.574) for the baby of 40 week this one would be the baby that weighs less relative to the gestation period

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

32-35 Week case

Let X the random variable that represent the weight, and for this case we know the distribution for X is given by:

[tex]X \sim N(2500,800)[/tex]  

Where [tex]\mu=2500[/tex] and [tex]\sigma=800[/tex]

The z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

And if w replace for the value of x=2025 we got:

[tex]z = \frac{2025-2500}{800}=-0.594[/tex]

40 week case

Let X the random variable that represent the weight, and for this case we know the distribution for X is given by:

[tex]X \sim N(2700,375)[/tex]  

Where [tex]\mu=2700[/tex] and [tex]\sigma=375[/tex]

The z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

And if w replace for the value of x=2225 we got:

[tex]z = \frac{2225-2700}{375}=-1.27[/tex]

For this case since the z score is lower (-1.27<-0.574) for the baby of 40 week this one would be the baby that weighs less relative to the gestation period

Answer:

The answer to your question is y = -11/2x - 19

Step-by-step explanation:

Data

Point (-4, 3)

⊥ to the line x- intercept of 22 and y-intercept of -4.

Process

1.- Find the equation of the perpendicular line

This line has passes through the points  (22, 0) and (0, -4)

Slope = [tex]\frac{-4 - 0}{0 - 22} = \frac{-4}{-22} = \frac{2}{11}[/tex]

2.- Find the slope of the new equation

Slope = [tex]\frac{-11}{2}[/tex]    because the lines are perpendicular

3.- Find the equation of the new line

            y - 3 = -11/2 (x + 4)

            y - 3 = -11/2x - 44/2

            y - 3 = -11/2 x - 22

            y = -11/2 x - 22 + 3

            y = -11/2x - 19

See the graph below

Answer: the total value of the investment in 10 years is $42500

Step-by-step explanation:

The formula for simple interest is expressed as

I = PRT/100

Where

P represents the principal

R represents interest rate

T represents time in years

I = interest after t years

From the information given

T =10 years

P = $25000

R = 7%

Therefore

I = (25000 × 7 × 10)/100

I = 1750000/100

I = $17500

Therefore, the total value of the investment in 10 years would be

25000 + 17500 = $42500

Answer:

a. The ball enters the green at Q (-13.46, -32.31)

b. 3.189s

c. 5.824s

d. Closest point =  (10.77, -16.15), it occurs 6.102s after the ball starts its journey

Step-by-step explanation:

he rightmost edge is taken as the right end of the horizontal diameter

Coordinates of the hole at H:(0,0)

Coordinates of the ball B:(-40,-50)

Coordinates of the point the ball exits green P:(35,0)

So, the line joining B and P is given by

Looking for the gradient of the line B P we have

(x2-x1)/(y1-y2) =(35 –(-40))/(0-(-50))

 = 75/50=3/2

The line equation therefore = (y-(-50))/(x-(-50)) = 2/3

(y + 50)/(x+40) =2/3

y+50 = 2/3 (x+40)

The equation of a circle is given by

x^2+y^2 = 35^2

 solving the two equations we have

x = ((3y+150)/2)-40 = 1.5y-35

Substituting the value of x into the above equations we have (1.5y-35)2 + y2 = 352

3.25×y2+105×y = 0

From where we have y×(3.25×y + 105) = 0

From where y = 0 or y = -105/3.25 = -32.31m

Also x = 35 or x = -13.46

The line intersects the circle at

Q:(-13.46,-32.31) and P:(35,0)

a. The ball enters the green at Q (-13.46, -32.31)

b. The ball enters the green after (length BQ)/Velocity

Where length BQ is given as

√((-13.46-(-40))2+(-32.31-(-50))2) = √(26.532+17.692)  = 31.89ft

The ball enters after 31.89ft/10ft/s or 3.189s

c. To find out how long the ball spends inside the green we have

Q:(-13.46,-32.31) and P:(35,0)

Length QP2 = (-13.46-(35))2 + (-32.31- 0)2 = 3392.31 = 58.24m

Time spent in the green = 58.24/10 = 5.824s

d. The ball is closest to the cup at the point when the perpendicular from the cup intersects the line PQ

and this is at the center of the line PQ hence the locus of the point is

((-13.46 + (35-(-13.46))/2), (-32.31+(0-(-32.31))/2) =  (10.77, -16.15)

This occurs at (√( (10.77-(-40))2 + (-16.15(-50))2))/10 = 61.02ft/10ft/s = 6.102s after the ball starts its journey

As the cup of the 9th hole in the golf course is located as the dead in the center of the milled green circular shape that is 70 feet. A ball follows the straight-line path and exists at the fright most edge.  

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Answer:

[tex] A_n = [2^{n-1} , 3(2^{n-1}), 5(2^{n-1}), 7(2^{n-1}),....][/tex]

Step-by-step explanation:

For this case we need to produce an infinite collection of sets [tex] A_a, A_2, A_3,....[/tex] with the property that every [tex] A_i[/tex] has an infinite number of elements with [tex] A_i \cap A_j , i\neq j[/tex] and [tex] \cup_{i=1}^{\infty} A_i = N[/tex]

So for this case we need to create a set A who satisfy 3 conditions (Infinite number of elements, Disjoint and Union represent the natural numbers)

So if we define the nth term for the set A:

[tex] A_n = [2^{n-1} , 3(2^{n-1}), 5(2^{n-1}), 7(2^{n-1}),....][/tex]

We see that the set A represent all the odd multiplies of [tex] 2^{n-1}[/tex] and if we check the properties we have this:

Disjoint

If we select [tex] A_n , A_m[/tex] with [tex] n\neq m[/tex] and we can assume for example that [tex] n<m[/tex]  and if we have an element a in the intersection of the sets [tex] a \in A_n \cap A_m[/tex], so then needs to exists some odd numbers k and l such that

[tex] a = 2^{n-1} k = 2^{m-1}l[/tex]

And since we assume that [tex] n<m[/tex] then we have that [tex] n\leq m-1[/tex] and we can write:

[tex] 2^{m-1} =2^m 2^i , i\geq 0[/tex]

And then [tex] 2^{n-1} k= 2^n 2^i l[/tex] and if we divide by [tex] 2^{n-1}[/tex] we got:

[tex] k = 2 2^i l[/tex] so then k is not odd since the last statement contradicts this. So then we can conclude that [tex] A_n \cap A_m = \emptyset[/tex]

Union

For this case we need to show that  [tex] \cup_{i=1}^{\infty} A_i = N[/tex]

Since each element [tex] A_n[/tex] is a subset of the natural numbers then the unision of the sets represent N

For the other side of the explanation if we assume that [tex] a\in N[/tex] we can write [tex] a= 2^{n-1}k[/tex] for any [tex] n\in N[/tex] and k odd, and by this [tex] a\in A_n[/tex] and we chaek the property.

Infinite condition

For this case [tex] A_n = [2^{n-1} , 3(2^{n-1}), 5(2^{n-1}), 7(2^{n-1}),....][/tex]

is an infinite set since we don't have a limit for n so then we have infinite elements for this case.

And since all the properties are satisfied we end the problem.

Answer:

Step-by-step explanation:

Distance covered= 195miles

Time = 180min = 3hours

Speed = distance/time

Speed = 195/3

Speed = 65miles/hour

Travis is driving at an average rate of speed of 65 miles per hour.

Speed in mathematics is one of the measurable quantity which is defined as the ratio of the distance covered by an object to the time taken to cover that distance.

Or in other words, speed of an object can be defined as the rate of change of the position of that object in any direction.

Speed is a scalar quantity, as it only has magnitude and no direction.

Speed can be calculated by the formula,

Speed = Distance / Time

Distance Travis travelled = 195 miles

Time taken to travel that distance = 180 minutes

                                                        = 180 / 60 hours

                                                        = 3 hours.

Speed = 195 / 3

           = 65 miles per hour

Hence the average rate of speed of Travis in the drive is 65 miles per hour.

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Answer: the number of hot dogs that were purchased was 27

the number of bags of popcorn that were purchased was 13

Step-by-step explanation:

Let x represent the number of hot dogs that were purchased.

Let y represent the number of bags of popcorn that were purchased.

A group of 40 children attend a baseball game on a field trip. Each child received either a hot dog or a bag of popcorn. It means that

x + y = 40

Hot dogs were $2.25 and popcorn was $1.75. If the total bill was $83.50. It means that

2.25x + 1.75y = 83.50 - - - - - - - - - -1

Substituting x = 40 - y into equation 1, it becomes

2.25(40 - y) + 1.75y = 83.50

90 - 2.25y + 1.75y = 83.50

- 2.25y + 1.75y = 83.50 - 90

- 0.5y = - 6.5.y = - 6.5/ -0.5

y = 13

x = 40 - y = 40 - 13

x = 27

Answer: Explanatory variable = " number of times the owner has an advertisement played on the radio"

Response variable = "number of new customers who will visit a shop"

Step-by-step explanation:

In the given situation , the business owner is predicting the number of new customers who will visit a shop based on the number of times the owner has an advertisement played on the radio.

Here , He is controlling the advertisement played on the radio to see the response of customers.

Therefore ,

Explanatory variable = " number of times the owner has an advertisement played on the radio"

Response variable = "number of new customers who will visit a shop"

In the given linear regression model, the 'number of times an advertisement is played on the radio' is the explanatory variable, and the 'number of new customers who will visit the shop' is the response variable.

In a linear regression model, there are two types of variables: the explanatory variable and the response variable. In this particular context of predicting the number of new customers who will visit a shop based on the number of times an advertisement is played on the radio, the explanatory variable is the 'number of times the advertisement is played on the radio'. This is because it is the variable that we are changing in hopes of influencing the outcome.

The response variable, on the other hand, is the 'number of new customers who will visit the shop'. This is the outcome we are interested in predicting or explaining.

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The value of the function is 6

Step-by-step explanation:

The function that we have in this problem is

[tex]f(x)=\left \{ {{3x^2+1}, -4<x<6 \atop {6}, 6\leq x <9} \right.[/tex]

This means that:

when x is between -4 (excluded) and 6 (excluded), the value of the function is [tex]3x^1+2[/tex]

when x is between 6 (included) and 9 (excluded), the value of the function is 6

In this problem, we are asked to evaluate f(x) when x = 6. Therefore, we are in the second case: therefore, the value of the function is 6,

[tex]f(x)=6[/tex]

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Answer:

the value is 6

Step-by-step explanation:

Answer:

The answer is B.

Step-by-step explanation:

The variable A in the I=PAT equation stands for affluence. It represents the average consumption of each person in the population. As the consumption of each person increases, the total environmental impact increases as well. The choice B is the best choice that represents average consumption of each person in the population.

Answer:

The answer to your question is   3(25t² - 4)

Step-by-step explanation:

To factor this binomial, just look for the common factor between both terms.

                                75t² + 12

                                                      75     12      2

                                                      75      6     2

                                                      75      3     3

                                                      25      3     3

                                                      25      1      5

                                                        5       1      5

                                                         1

The greatest common factor is 3, that is the common factor.

Now, factor the polynomial

                                  3(25t² - 4)

Answer: 3(25t 2 + 4)

Step-by-step explanation:

If you want to find the area of the living room minus 1, it could be 8 m², 14 m², 19 m², or 24 m², depending on what you are looking for.

To find the area of the living room, you can use the provided information:

The area of the living room is:

9 m²

15 m²

20 m²

25 m²

You mentioned "The area of the living room - 1," which seems to imply you want to find the area of the living room, subtracting 1 from it. To do that, simply subtract 1 from each of the given options:

9 m² - 1 = 8 m²

15 m² - 1 = 14 m²

20 m² - 1 = 19 m²

25 m² - 1 = 24 m²

So, if you want to find the area of the living room minus 1, it could be 8 m², 14 m², 19 m², or 24 m², depending on what you are looking for.

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Answer:

what do you need help with?

Step-by-step explanation:

Answer:

It would be 15 cm. because if the triangles are similar then BC = EF

Answer:

The inequality represents the scenario is [tex]t\leq 30[/tex].

Step-by-step explanation:

Given:

Number of minutes used to make cookie written on box = 30 minutes.

Number of minutes used to make cookies = [tex]t \ minutes[/tex]

We need to write inequality represents the scenario

Solution:

We can say that;

Number of minutes used to make cookies should be less than or equal to Number of minutes used to make cookie written on box.

framing in equation form we get;

[tex]t\leq 30[/tex]

Hence The inequality represents the scenario is [tex]t\leq 30[/tex].

The inequality representing the scenario where cookie mix instructions indicate a maximum of 30 minutes to make cookies is t ≤ 30.

The inequality that represents the scenario where the cookie mix instructions say it will take no longer than 30 minutes to make cookies is t ≤ 30. This inequality means that the time taken to make the cookies, represented by t, is less than or equal to 30 minutes. If we consider that one option is to bake the cookies for 10 minutes, that fits well within the constraint given by the inequality since 10 is less than 30.

Answer:

Step-by-step explanation:

No of ways in which 5 cards are chosen is [tex]N=^{52}C_5[/tex]

No of ways in which at most  4 cards are spades [tex]N_0=[/tex]Total ways-5 spades

[tex]=^{52}C_5-^{13}C_5[/tex]

Probability that at most 4 cards are spade [tex]P=\frac{N_0}{N}[/tex]

[tex]P=\frac{^{52}C_5-^{13}C_5}{^{52}C_5}[/tex]

[tex]P=\frac{2598960-1287}{2598960}[/tex]

[tex]P=\frac{2597673}{2598960}=0.9995[/tex]

The probability of being dealt 5 cards with at most 4 of them being spades from a standard deck is approximately 0.9995.

To determine the probability of being dealt 5 random cards with at most 4 of them being spades from a standard deck of 52 cards, follow these steps:

1. Understanding the total number of possible hands

⇒ C(52, 5) = 2,598,960

2. Calculating the number of desired outcomes

0 Spades: C(13, 0) × C(39, 5)

1 Spade: C(13, 1) × C(39, 4)

2 Spades: C(13, 2) × C(39, 3)

3 Spades: C(13, 3) × C(39, 2)

4 Spades: C(13, 4) × C(39, 1)

0 Spades:

⇒ 1 × 575,757 = 575,757

1 Spade:

⇒ 13 × 82,251 = 1,069,263

2 Spades:

⇒ 78 × 9,139 = 712,842

3 Spades:

⇒ 286 × 741 = 211,926

4 Spades:

⇒ 715 × 39 = 27,885

⇒ 575,757 + 1,069,263 + 712,842 + 211,926 + 27,885 = 2,597,673

3. Calculating the probability

⇒ P(at most 4 spades) = 2,597,673 ÷ 2,598,960

≈ 0.9995

Answer:

a) [tex] P(Y\geq 10) = PX \leq 2) = 0.558[/tex]

b) [tex] E(X) = \mu_X = np = 12*0.2 = 2.4[/tex]

[tex] \sigma_X = \sqrt{np(1-p)}=\sqrt{12*0.2*(1-0.2)}=1.386[/tex]

[tex] E(Y) = \mu_Y = np = 12*0.8 = 9.6[/tex]

[tex] \sigma_Y = \sqrt{np(1-p)}=\sqrt{12*0.8*(1-0.8)}=1.386[/tex]

For this case the expected value of people lying is 2.4 and the complement is 9.6 and that makes sense since we have a total of 12 poeple.

And the deviation for both variables are the same.

Step-by-step explanation:

Assuming this previous info : "A federal report finds that lie detector tests given to truthful persons have probability about 0.2 of  suggesting that the person is deceptive. A company asks 12 job applicants about thefts from previous employers, using  a lie detector to assess their truthfulness. Suppose that all 12 answer truthfully. Let X = the number of people who the lie  detector says are being deceptive"

For this case the distribution of X is binomial [tex] X \sim N(n=12, p=0.2)

And we define the new random variable Y="the number of people who the lie detector says are telling the truth" so as we can see y is the oppose of the random variable X, and the distribution for Y would be given by:

[tex] Y \sim Bin (n=12,p=1-0.2=0.8)[/tex]

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

Part a

For this case we want to find this probability:

[tex] P(Y \geq 10) = P(Y=10)+P(Y=11) +P(Y=12)[/tex]

And if we find the individual probabilites we got:

[tex]P(Y=10)=(12C10)(0.8)^{10} (1-0.8)^{12-10}=0.283[/tex]

[tex]P(Y=11)=(12C11)(0.8)^{11} (1-0.8)^{12-11}=0.206[/tex]

[tex]P(Y=12)=(12C12)(0.8)^{12} (1-0.8)^{12-12}=0.069[/tex]

And if we replace we got:

[tex] P(Y \geq 10) =0.283+0.206+0.069=0.558[/tex]

And for this case if we find [tex] P(X\leq 2)=P(X=0) +P(X=1)+P(X=2)[/tex]  for the individual probabilites we got:

[tex]P(X=0)=(12C0)(0.2)^0 (1-0.2)^{12-0}=0.069[/tex]

[tex]P(X=1)=(12C1)(0.2)^1 (1-0.2)^{12-1}=0.206[/tex]

[tex]P(X=2)=(12C2)(0.2)^2 (1-0.2)^{12-2}=0.283[/tex]

[tex] P(X\leq 2)=0.283+0.206+0.069=0.558[/tex]

So as we can see we have [tex] P(Y\geq 10) = P(X \leq 2) = 0.558[/tex]

Part b

Random variable X

For this case the expected value is given by:

[tex] E(X) = \mu_X = np = 12*0.2 = 2.4[/tex]

And the deviation is given by:

[tex] \sigma_X = \sqrt{np(1-p)}=\sqrt{12*0.2*(1-0.2)}=1.386[/tex]

Random variable Y

For this case the expected value is given by:

[tex] E(Y) = \mu_Y = np = 12*0.8 = 9.6[/tex]

And the deviation is given by:

[tex] \sigma_Y = \sqrt{np(1-p)}=\sqrt{12*0.8*(1-0.8)}=1.386[/tex]

For this case the expected value of people lying is 2.4 and the complement is 9.6 and that makes sense since we have a total of 12 poeple.

And the deviation for both variables are the same.

Answer:

dfbustos is correct! But I wanted to add a bit more to part b, specifically why the standard deviation makes sense.

Step-by-step explanation:

Y = # of people who the lie detector says are telling the truth.

X = # of people who the lie detector says are telling being deceptive.

This leads to the following conclusion:

Y = 12 - X

(We are given that 12 people answered.)

Because the standard deviation is not affected by adding or subtracting a constant, it makes sense for the σY = σX.