Probability & Sampling:Question 1


Stephanie recorded the time, in minutes, she took to walk


from home to work.


{15, 16, 18, 20, 21)


She also recorded the time, in minutes, she took to walk


from work to home.


(14, 21, 21, 25, 27)


Based on the data she collected, what is the best


conclusion Stephanie can make?

1. x-3y -3=0, 3x-9y-2=0 has no solution

2. 2x+y=5,3x+2y=8 has a unique solution

3. 3x-5y=20,6x-10y=40 has infinitely many solution

4.  x-3y-7 =0,3x-3y-15=0 has No solution

5. 8x+5y=9,3x+2y=4 has a unique solution

(i) To solve using substitution method, we can rearrange the first equation to x=3y+3 and substitute it into the second equation:

3(3y+3) - 9y - 2 = 0

9y + 9 - 9y - 2 = 0

7 = 0

This is a contradiction, so the pair of equations has no solution.

(ii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 8

(4x + 2y = 10)

-x = -2

So, x = 2. Substituting this value into the first equation, we get:

2x + y = 5

2(2) + y = 5

y = 1

Therefore, the unique solution is (x,y) = (2,1).

(iii) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

6x - 10y = 40

(6x - 10y = 40)

0 = 0

This equation is true for any value of x and y, so the pair of equations has infinitely many solutions.

(iv) To solve using elimination method, we can subtract the first equation from the second equation:

3x - 3y - 15 - (x - 3y - 7) = 0

2x - 22 = 0

x = 11

Substituting this value into the first equation, we get:

11 - 3y - 7 = 0

-3y = -4

y = 4/3

Therefore, the unique solution is (x,y) = (11,4/3).

(v) To solve using elimination method, we can multiply the first equation by 2 and subtract it from the second equation:

3x + 2y = 4

(16x + 10y = 18)

-29x - 18y = -14

Solving for y, we get:

y = (29/18)x + (7/9)

Substituting this expression for y into the first equation, we get:

8x + 5((29/18)x + (7/9)) = 9

(143/18)x = 2/9

x = 2/13

Substituting this value into the expression for y, we get:

y = (29/18)(2/13) + (7/9) = 41/117

Therefore, the unique solution is (x,y) = (2/13,41/117).

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a) P(x) = R(x) - C(x) = 6x - (0.001x^2 + 18x + 40) = -0.001x^2 - 12x - 40

b) R(200) = 6(200) = 1200
  C(200) = 0.001(200)^2 + 18(200) + 40 = 4000
  P(200) = R(200) - C(200) = 1200 - 4000 = -2800

c) R'(x) = 6
  C'(x) = 0.002x + 18
  P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18) = -0.002x - 12

d) R'(200) = 6
  C'(200) = 0.002(200) + 18 = 18.4
  P'(200) = -0.002(200) - 12 = -12.4

Here are the answers to each part:

a) P(x) is the profit function, which is calculated as the difference between the revenue function and the cost function: P(x) = R(x) - C(x). In this case, P(x) = 6x - (0.001x^2 + 18x + 40).

b) To find R(200), C(200), and P(200), plug x = 200 into each function:
R(200) = 6(200) = 1200
C(200) = 0.001(200^2) + 18(200) + 40 = 7600
P(200) = 1200 - 7600 = -6400

c) To find R'(x), C'(x), and P'(x), we need to find the derivative of each function with respect to x:
R'(x) = d(6x)/dx = 6
C'(x) = d(0.001x^2 + 18x + 40)/dx = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18)

d) To find R'(200), C'(200), and P'(200), plug x = 200 into each derivative function:
R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = 6 - 18.4 = -12.4

I hope this helps! Let me know if you have any further questions.

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The number of people who like the plan is 1,590 people out of the 3,000 surveyed.

To determine how many people liked the plan, we'll need to use the percentage given and apply it to the total number of people surveyed.

Percentage is a way of expressing a proportion or a fraction as a whole number out of 100. In this case, the percentage we're working with is 53%, which means 53 out of every 100 people surveyed liked the plan. To find the number of people who liked the plan, we can multiply the total number of people surveyed (3,000) by the percentage who liked the plan (53%).

To do this calculation, first convert the percentage to a decimal by dividing 53 by 100, which gives us 0.53. Next, multiply 3,000 by 0.53:

3,000 * 0.53 = 1,590

So, 1,590 people out of the 3,000 surveyed liked the plan for the new park.

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Answer:

tan A: BC/AB

cos A: AB/AC

sin C: AB/AC

cos C: BC/AC

sin A: BC/AC

Step-by-step explanation:

sin of an angle: opposite/hypotenuse

cosine of an angle: adjacent/hypotenuse

tangent of an angle: opposite/adjacent

Answer:

Step-by-step explanation:

A reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.

Since the spinner has 8 congruent sections and is spun 24 times, we can use probability to make a reasonable prediction for the number of times it will land on the number 3.

1. Calculate the probability of landing on the number 3 for a single spin:
Since there are 8 congruent sections, the probability of landing on the number 3 is 1/8.

2. Determine the expected number of times the spinner will land on the number 3:
To do this, multiply the probability of landing on the number 3 (1/8) by the total number of spins (24).
Expected number of times = (1/8) * 24

3. Simplify the expression:
Expected number of times = 3

So, a reasonable prediction for the number of times the spinner will land on the number 3 is 3 times.

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A bag contains five red socks and eight blue socks. Lucky reaches into the bag and randomly selects two socks without replacement, the probability that Lucky will get different colored socks is 10/39.

We can divide the issue into two distinct possibilities and multiply them together to find a solution.

Let's start by thinking about the likelihood of choosing a red sock during the initial draw.

The likelihood of choosing a red sock on the first draw is 5/13 due to the fact that there are only five red socks among the total of thirteen socks (five red plus eight blue).

There are now twelve socks left in the bag after the first one is drawn, with four red and eight blue.

On the second draw, there is an 8/12 chance of choosing a blue sock, which is a different colour.

We add the probabilities together to determine the likelihood that both events (drawing a red sock first and a blue sock second) will occur:

(5/13) * (8/12) = 40/156 = 10/39

Therefore, the probability that Lucky will get different colored socks is 10/39.

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An angle of 0 radians is an angle along the positive x-axis of the unit circle. Its terminal point is (1, 0).

The tangent of 0 radians is defined as the ratio of the y-coordinate to the x-coordinate of the terminal point, which is 0/1 = 0.

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Answer: .04671 - 0.04679

Step-by-step explanation:

Answer:

0.04675

Step-by-step explanation:

0.04675 > 0.0467

0.04675 < 0.0468

The area of the triangle is 6 cm².

Let's denote the width of the triangle as "w." According to the given information, the length of the triangle is three times its width, so the length can be expressed as "3w."

The perimeter of a rectangle is given by the formula: Perimeter = 2(length + width). In this case, the perimeter of the rectangle is given as 24 cm.

We can set up the following equation based on the given information:

24 = 2(3w + w)

Simplifying the equation:

24 = 2(4w)

12w = 24

w = 24/12

w = 2 cm

Now that we have the width of the triangle, we can find the length:

Length = 3w = 3 * 2 = 6 cm

The area of a triangle is given by the formula: Area = (base * height) / 2. In this case, the base of the triangle is the width (2 cm) and the height is the length (6 cm).

Area = (2 * 6) / 2

Area = 12 / 2

Area = 6 cm²

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The probability that both events will occur is 0.22.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

The probability of Event A is 2/6 or 1/3

(since there are two ways to get a 1 or 2 on a six-sided die).

The probability of Event B is 4/6 or 2/3

(since there are four ways to get a number 4 or less on a six-sided die).

Using the formula for the probability of the intersection of two independent events.

P(A and B)

= P(A) x P(B)

= (1/3) x (2/3)

= 2/9

Rounded to the nearest hundredth,

The probability that both events will occur is 0.22.

Thus,

The probability that both events will occur is 0.22.

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The osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.

We can use the equation:

π = MRT

where:

π = osmotic pressure

M = molarity of the solution

R = gas constant

T = temperature in Kelvin

We must translate the proportion by mass to molarity in order to determine the molarity of the glucose solution. Glucose (C6H12O6) has a molecular weight of 180 g/mol.

So, for a 4.50% by mass solution of glucose, we have:

4.50 g glucose / 100 g solution = (4.50 g glucose / 180 g/mol) / (Molarity of solution)

Solving for molarity, we get:

Molarity of solution = 0.025 mol/L

Now we can plug in the values into the equation for osmotic pressure:

π = (0.025 mol/L) * (0.0821 L atm / mol K) * (300 K)

π = 0.616 atm

Therefore, the osmotic pressure for a 4.50% by mass aqueous solution of glucose (C6H12O6) at 300 K is 0.616 atm.

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a = 5

b = 4

c= 0

Therefore, the equation for graph C is Y = a ^b + c

Y = 5 ^4 + 0

A graph is described as a diagram showing the relation between variable quantities, typically of two variables, each measured along one of a pair of axes at right angles.

Graphs are a popular tool for graphically illuminating data relationships. A graph serves the purpose of presenting data that are either too numerous or complex to be properly described in the text while taking up less room.

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x²y' - 2xy' + (x² + 2)y = 0 is the associated homogeneous equation with non-homogeneous ODE x²yⁿ - 2xy' + (x² +2)y = 3x³.

It should be noted that the equation is same as the variable coefficient linear second order non-homogeneous ODE but just the right side zero.

The complementary solutions or homogeneous solutions to this homogeneous equation serve as the foundation for the space of all solutions to the non-homogeneous equation.

By assuming that y has the form y(x) = xr and substituting this into the homogeneous equation to create a characteristic equation, we can determine the complementary solutions. We may find the values of r that correspond to solutions of the type y(x) = xr by looking at the characteristic equation's roots.

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the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.

Let's break it down step-by-step:

1. Identify the function we want to approximate: f(x) = x^2 * (x+4)^3
2. Choose the point to approximate near Since we want to estimate f(-2.02), let's approximate near x = -2.
3. Compute the linear approximation (first-degree Taylor polynomial) at x = -2: f(-2) = (-2)^2 * (2)^3 = 4 * 8 = 32
4. Find the derivative of f(x): f'(x) = 2x(x+4)^3 + 3x^2(x+4)^2
5. Compute the derivative at x = -2: f'(-2) = 2(-2)(2)^3 + 3(-2)^2(2)^2 = -32 + 48 = 16
6. Use the linear approximation formula: f(-2.02) ≈ f(-2) + f'(-2)(-2.02 - (-2)) = 32 + 16(-0.02) = 32 - 0.32 = 31.68

Now, compare this approximation to the value given by a calculator: (-2.02)^2 * (2.02)^3 ≈ 31.741088. To compute the percentage error, use the formula:

Percentage Error = |(Approximate Value - Actual Value) / Actual Value| * 100%
Percentage Error = |(31.68 - 31.741088) / 31.741088| * 100% ≈ 0.1924%

So, using the linear approximation, we estimated the value of (-2.02)^2 * (2.02)^3 as 31.68, and the percentage error compared to the calculator's value is approximately 0.1924%.

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Box C has the smallest volume, followed by Box A, and Box B has the largest volume.

First, we need to find the volume of each box.

Recall that the formula for volume of a box is given as:

V = length x height x width

For Box A,

V = 3 cm x 2 cm x 4 1/2 cm = 27 cm³

For Box B,

V = 2 1/3 cm x 3 cm x 5 cm  = 7/3 cm x 3 cm x 5 cm = 35 cm³

For Box C,

V = 4 cm x 3 cm x 1 1/4 cm = 4 cm x 3 cm x 5/4 cm = 15 cm³

So, the order of the boxes by volume from least to greatest is: Box C, Box A, and Box B.

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Translation, rotation, and reflection, By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'.

To map triangle ABC to triangle A'B'C', you would need to follow a specific sequence of transformations, which may include translation, rotation, and reflection. Here's a step-by-step explanation:

Step 1: Translation
Translate triangle ABC by a specific vector (x, y) so that point A moves to point A'. The same vector will also move points B and C to their corresponding new positions.

Step 2: Rotation
If triangle A'B'C' is rotated compared to the translated triangle, rotate the translated triangle around point A' by a specific angle, either clockwise or counterclockwise, until point B aligns with point B'.

Step 3: Reflection
If triangle A'B'C' is a mirror image of the rotated triangle, reflect the rotated triangle across a line of symmetry (usually a line passing through A'). This will change the orientation of the triangle and align point C with point C'.

By following these transformations in sequence, you can map triangle ABC to triangle A'B'C'. Keep in mind that the specific details of translation, rotation, and reflection will depend on the coordinates and orientation of the given triangles.

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The Rounding to nearest hundredth, we get S₅ ≈ -12.25

The formula for a Riemann sum with n subintervals is:

[tex]S_n[/tex]= ∑ᵢ₌₁ⁿ f(cᵢ) Δx,

where Δx = (b - a)/n is the width of each subinterval and cᵢ is a point in the i-th subinterval. The value of cᵢ can be chosen arbitrarily, but here we are given specific values for c₁, c₂, c₃, c₄, and c₅.

In this problem, we have:

f(x) = 64 - x²

[a, b] = (-8, -3]

n = 5

Δx = (b - a)/n = (-3 - (-8))/5 = 1

Therefore, the width of each subinterval is 1.

The Riemann sum S₅ is:

S₅ = f(c₁) Δx + f(c₂) Δx + f(c₃) Δx + f(c₄) Δx + f(c₅) Δx

Substituting the given values for c₁, c₂, c₃, c₄, and c₅, we get:

S₅ = f(-7.5) + f(-6.5) + f(-5.5) + f(-4.5) + f(-3.5)

where f(x) = 64 - x².

Evaluating each term, we get:

f(-7.5) = 64 - (-7.5)² = 17.75

f(-6.5) = 64 - (-6.5)² = 5.75

f(-5.5) = 64 - (-5.5)² = -2.75

f(-4.5) = 64 - (-4.5)² = -12.25

f(-3.5) = 64 - (-3.5)² = -20.75

Therefore,

S₅ = 17.75(1) + 5.75(1) - 2.75(1) - 12.25(1) - 20.75(1) = -12.25.

Rounding to the nearest hundredth, we get S₅ ≈ -12.25.

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If f'(x) = 8x³ + 12x + 2 and f(1) = -4, f(-1) is equal to -18.

Given that f'(x) = 8x³ + 12x + 2, we can find the original function f(x) by integrating f'(x) with respect to x:

f(x) = 2x⁴ + 6x² + 2x + C, where C is an arbitrary constant.

We can then use the given initial condition f(1) = -4 to solve for C:

f(1) = 2(1)⁴ + 6(1)² + 2(1) + C = -4

Simplifying, we get:

C = -16

Therefore, the function f(x) is:

f(x) = 2x⁴ + 6x² + 2x - 16

To find f(-1), we substitute x = -1 into the expression for f(x):

f(-1) = 2(-1)⁴ + 6(-1)² + 2(-1) - 16 = -18

Thus, f(-1) equals -18.

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Answer:

Step-by-step explanation:

The probability of no heads is given as follows:

P(No Heads) = 17.5%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total number of outcomes is given as follows:

200.

The desired outcomes, those without heads, are Tails, Tails, which happened 35 times, hence the probability is given as follows:

p = 35/200

p = 0.175

p = 17.5%.

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We'll model Jack's anger in anger units after t weeks using an exponential decay equation, as he loses 5% of his anger every week.

To write an equation that models Jack's anger (let that be A(t)) after t weeks, we need to follow these steps:

1. Identify the initial amount of anger units (A0): Jack had 100 anger units at the beginning (t=0).
2. Determine the growth factor (1 - decay rate): Since Jack loses 5% of his anger every week, the growth factor is 1 - 0.05 = 0.95.
3. Set up the exponential decay equation: A(t) = A0 * (growth factor)^t.

By following these steps, the equation modeling Jack's anger after t weeks is:

A(t) = 100 * (0.95)^t

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(a) The weight's velocity as a function of time t in seconds is v(t) = 16 - 32.2t

(b) The height (in feet) of the weight above the ground as a function of time is s(t) = 186 + 16t - (1/2)(32.2)t^2

To find the weight's velocity and height as a function of time:

(a) The equation for velocity as a function of time is v(t) = v0 - gt,

where v0 is the initial velocity (in this case, 16 ft/s) and g is the acceleration due to gravity (32.2 ft/s^2).

Using this equation, we can find the weight's velocity as it travels upward:

v(t) = 16 - 32.2t

(b) The equation for height as a function of time is s(t) = s0 + v0t - (1/2)gt^2,

where s0 is the initial height (in this case, 186 ft).

Using this equation, we can find the height of the weight above the ground at any point in time:

s(t) = 186 + 16t - (1/2)(32.2)t^2

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The volume of the china cabinet is 21 cubic feet.

To find the volume of the china cabinet, we need to multiply its length, width, and height.

Since the dimensions are given in feet, we will use cubic feet as the unit of volume.

The length of the china cabinet is given as 1 ft, the width as 7 ft, and the height as 3 ft.

The volume can be calculated as follows:

Volume = length * width * height

Volume = 1 ft * 7 ft * 3 ft

Volume = 21 cubic feet

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The expression that represents the length (in feet) of the world's longest glass-bottom bridge is 4.3x+550.4.

Let's denote the length of the Coiling Dragon Cliff Skywalk as y (in feet). According to the given information, we have:

y = x + 128

The length of the world's longest glass-bottom bridge is 4.3 times longer than the Coiling Dragon Cliff Skywalk, so we can write an expression for it as:

Length of the longest glass-bottom bridge = 4.3 * y

Now, we can substitute the expression for y from the first equation:

Length of the longest glass-bottom bridge = 4.3 * (x + 128)

To simplify, distribute the 4.3:

Length of the longest glass-bottom bridge = 4.3x + 550.4

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The fare for this flight was $542.50 which is calculated using least-squares regression line equation. Therefore, the correct answer 542.50

To find the fare for this flight, we will first use the provided least-squares regression line equation to predict the fare and then account for the residual.

Step 1: Use the least-squares regression line equation to predict the fare.
ŷ = 102.50 + 0.65x, where ŷ is the predicted fare and x is the distance in miles.

Step 2: Substitute the given distance (x = 500 miles) into the equation.
ŷ = 102.50 + 0.65(500)

Step 3: Calculate the predicted fare.
ŷ = 102.50 + 325
ŷ = 427.50

The predicted fare for a 500-mile flight is $427.50.

Step 4: Adjust for the residual.
The residual for this flight is 115.0, which means the actual fare is $115 higher than the predicted fare.

Step 5: Add the residual to the predicted fare to find the actual fare.
Actual fare = Predicted fare + Residual
Actual fare = 427.50 + 115
Actual fare = 542.50

The fare for this flight was $542.50.

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Adding another 12 to the data set would increase the sum of the values by 12, resulting in a new sum of 239. To find the new mean, we divide the new sum by the total number of values in the set, which is now 8. So the new mean would be 29.875, which is slightly lower than the original mean of 31.6.

To find the new median, we first need to rearrange the values in ascending order: 12, 18, 26, 32, 41, 46, 46, 12. Since there are now an even number of values, we take the average of the middle two, which in this case is (26 + 32) / 2 = 29. So the new median would be 29, which is lower than the original median of 32.

In summary, adding another 12 to the data set would slightly decrease the mean and lower the median.

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