What is the equation of the graph below?

A. y = − (x − 3)^2 + 1
B. y = − (x + 3)^2 + 1
C. y = (x − 3)^2 − 1
D. y = (x + 3)^2 − 1

The area of the trapezoid is 81 in² (optionC)

A trapezoid is a quadrilateral with one pair of opposite sides parallel. The area of a trapezoid is expressed as;

A = 1/2(a+b)h

Where a and b are the two opposite sides and h is the height.

The height = 9 in

The two opposite sides are 5 and 13

Therefore ;

A = 1/2( 13+5) 9

A = 1/2 × 18 × 9

A = 9 × 9

A= 81 in²

therefore the area of the trapezoid is 81 in²

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Since the sample space contains  [tex]36[/tex] equally likely outcomes, each outcome has a probability of [tex]1/36[/tex]

The sample space consists of all possible outcomes when rolling two number cubes. Each cube has six equally likely outcomes, so the total number of outcomes is[tex]6\times6=36.[/tex]

P(1 and 2) refers to the probability of rolling a 1 on the first cube and a 2 on the second cube. There is only one outcome that satisfies this condition, which is [tex](1, 2)[/tex] .

Therefore, the probability of rolling a 1 and 2 is:

P(1 and 2) = (number of outcomes that satisfy the condition)/(total number of possible outcomes)

P(1 and 2) [tex]= 1/36[/tex]

Therefore, the probability of rolling a [tex]1[/tex] and  [tex]2[/tex] is  [tex]1/36[/tex] .

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Answer:

Since the sample space contains 36 equally likely outcomes, each outcome has a probability of 1 / 36.

We know that probability is defined as number of favourable outcomes divided by total number of outcomes.

Step-by-step explanation:

An experiment occurs of rolling two fair number cubes.

We have given number of favourable outcomes = 2 {(2,1), (1,2)}

Number of Total Outcomes = 36

So ,

Probability = Number of favourable outcomes / Total Outcomes

P(1 and 2) refers to the probability of rolling a 1 on the first cube and a 2 on the second cube. There is only one outcome that satisfies this condition, which is (1, 2).

P(1 and 2) = 2 / 36

P(1 and 2) = 1 / 18

Hence , 1 / 18 is the answer as a fraction in simplest form.

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Answer:

  140 cm²

Step-by-step explanation:

You want the total surface area of a triangular prism as represented by the given net.

The total area is the sum of the areas of the two triangular bases and the three rectangular faces. The relevant formulas are ...

  A = 1/2bh . . . . . . area of triangle with base b, height h

  A = bh . . . . . . . area of rectangle with base b, height h

The total area is ...

  SA = 2(1/2bh) +(b1)h +(b2)h +(b3)h . . . . . . . triangle h=5, rectangle h=8

  SA = 2(1/2)(4 cm)(5 cm) + (8 cm)(6 cm +4 cm +5 cm)

  = 20 cm² +(8 cm)(15 cm)

  = 140 cm²

The total surface area of the triangular prism is about 140 cm².

the original number of movies Ronnie had was 60, which is option A.

Let's first understand the given problem. Ronnie had a certain number of movies, which we don't know and is represented by the variable 'm'. One-fourth of the total movies (m/4) were action movies. Now, Ronnie sold eleven action movies and ended up with only four action movies left. This means that he sold m/4 - 11 action movies, and the remaining action movies were 4. So, we can write the equation as:

m/4 - 11 = 4

Let's solve this equation to find the value of m. First, we will add 11 to both sides of the equation to get:

m/4 = 4 + 11

Simplifying this, we get:

m/4 = 15

Now, we will multiply both sides of the equation by 4 to get rid of the fraction:

m = 15 x 4

m = 60

Therefore, the original number of movies Ronnie had was 60, which is option A.

To explain this in more detail, we can use algebraic reasoning. We started with the equation m/4 - 11 = 4 and solved for m. To do so, we first added 11 to both sides of the equation to isolate the variable on one side. This gave us m/4 = 15. To solve for m, we multiplied both sides by 4 to cancel out the denominator and get m = 60.

Therefore, Ronnie originally had 60 movies, and one-fourth of those (m/4) were action movies. He sold 11 action movies, leaving him with only 4 action movies, and we were able to find the original number of movies using algebraic reasoning.

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A pentagon has five angles, and the sum of the angles of a pentagon is equal to 540°. Therefore, the measure of the fifth angle can be found by subtracting the sum of the given angles from 540°:

540° - (85° + 110° + 135° + 95°) = 540° - 425° = 115°.

Therefore, the measure of the missing angle is 115°.

Sum of the angles of a n-sided polygon:

[tex]\text{S}_{\text{n}}=180^\circ\times(\text{n}-2)[/tex]

In this question, we have a pentagon, so

[tex]\text{n}=5,[/tex]

then

[tex]\text{S}_5=180^\circ\times(5-2)[/tex]

[tex]\text{S}_5=180^\circ\times3[/tex]

[tex]\text{S}_5=540^\circ \longleftarrow \ \text{sum of the measures of the internal\\}[/tex]

                       [tex]\text{angles of a pentagon.}[/tex]

Let [tex]\text{x}[/tex] be the measure of the missing angle. So,

[tex]85^\circ+110^\circ+135^\circ+95^\circ+\text{x}=\text{S}_5[/tex]

[tex]425^\circ+\text{x}=540^\circ[/tex]

[tex]\text{x}=540^\circ-425^\circ[/tex]

[tex]\text{x}=115^\circ\longleftarrow \ \ \ \text{measure of the missing angle.}[/tex]

Tags: sum internal angle missing pentagon polygon plane geometry

Mr. Olakunle needs to divide the total amount of soil he has by the number of planters he wants to fill, which is 10/8 = 5/4. This can be simplified by dividing both the numerator and denominator by 4, resulting in 5/4 = (5  4) / (4  4) = 5/12.

To find out how much soil Mr. Olakunle will use for each planter, we need to divide the total amount of soil he has by the number of planters he wants to fill.

The total amount of soil he has is 10 bags.

He plans to fill 8 planters with the soil.

Therefore, the amount of soil he will use for each planter is:

10/8 = 5/4

This can be simplified by dividing both the numerator and denominator by 4, to get:

5/4 = (5 ÷ 4) / (4 ÷ 4) = 5/4

So the answer is C) 5/12.

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Percentage of the population live in their state of birth is,
Arkansas residents born in Arkansas is, 40.8%

Virginia residents born in Virginia is 41.7%

To find the percentage of the population that live in their state of birth for Arkansas and Virginia, we need to count the number of residents who were born in the same state as the one they currently live in and divide it by the total number of residents.

For Arkansas:

Number of residents born in Arkansas: 49

Total number of residents: 120

Percentage of Arkansas residents born in Arkansas: (49/120) x 100 = 40.8%

For Virginia:

Number of residents born in Virginia: 75

Total number of residents: 180

Percentage of Virginia residents born in Virginia: (75/180) x 100 = 41.7%

Note that these percentages are specific to the sample of residents in Arkansas and Virginia in the given dataset, and may not be representative of the entire population.

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--The complete question is, What percentage of the population live in their state of birth? According to the U. S. Census Bureau's American Community Survey, it ranges from 25% in Nevada to 78.7 percent in Louisiana (AARP Bulletin, March, 2014). The average percentage across all states and the District of Columbia is 57.7%. The data in the DATAfile Homestate are consistent with the findings in the American Community Survey. The data are for a random sample of 120 Arkansas residents and for a random sample of 180 Virginia residents.

DATA:

First Column-Arkansas Resident Born Here?

Second column- Virginia Residents Born Here?      

No Yes      

No No      

Yes No      

Yes Yes      

Yes Yes      

No No      

No Yes      

Yes Yes      

No No      

Yes Yes      

Yes No      

No Yes      

Yes No      

Yes No      

No No      

No No      

Yes Yes      

Yes No      

Yes No      

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Yes Yes      

No Yes      

Yes No      

Yes No      

No Yes      

No Yes      

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No No      

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Yes No      

No Yes      

Yes Yes      

Yes No      

No Yes      

Yes No      

Yes Yes      

Yes No      

Yes Yes      

No No      

Yes No      

Yes No      

Yes No      

Yes No      

Yes Yes      

Yes No      

No Yes      

Yes No      

No No      

Yes Yes      

No No      

Yes Yes      

No No      

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Yes Yes      

No No      

Yes Yes      

Yes Yes      

No No      

No No      

No Yes      

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No No      

Yes No      

Yes Yes      

Yes Yes      

No No      

No Yes      

No No      

Yes Yes      

Yes No      

No No      

Yes Yes      

Yes No      

Yes No      

Yes No      

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No Yes      

Yes Yes      

No No      

No No      

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No Yes      

Yes Yes      

Yes No      

No Yes      

Yes Yes      

Yes No      

Yes No      

No Yes      

No No      

Yes Yes      

No No      

No Yes      

Yes Yes      

Yes Yes      

No Yes      

Yes No      

Yes Yes      

Yes No      

Yes Yes      

No Yes      

Yes Yes      

No No      

No No      

No No      

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No Yes      

Yes No      

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Yes Yes      

Yes Yes      

Yes No      

No No      

Yes Yes      

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       Yes      

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       Yes

       No      

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       No      

       No      

       No              

       No      

       Yes      

       Yes      

       Yes      

       No      

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       No      

       Yes      

       Yes      

       No      

       Yes      

       Yes      

       No      

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       No      

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       No--

Leah and Josh should expect to pay approximately $1,204.31 in prepaid interest at the closing.

What is interest?

Interest is the amount of money that is paid or earned for the use of borrowed or invested funds. When someone borrows money, they typically have to pay back the principal (the amount borrowed) plus interest, which is usually calculated as a percentage of the principal.

To calculate the prepaid interest at closing, we need to first determine the loan amount, which is the purchase price of the home minus the down payment:

Purchase price = $550,000

Down payment = 14% of $550,000 = $77,000

Loan amount = $550,000 - $77,000 = $473,000

Next, we need to calculate the daily interest rate by dividing the APR by 365:

Daily interest rate = 3.70% / 365 = 0.010137%

Finally, we can calculate the prepaid interest by multiplying the loan amount by the daily interest rate and the number of days from the closing date to the end of the month:

Number of days from September 6 to September 30 = 25

Prepaid interest = $473,000 x 0.010137% x 25 = $1,204.31

Therefore, Leah and Josh should expect to pay approximately $1,204.31 in prepaid interest at the closing.

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A function that describes the number of pythons now living in the park in the 11th year would be P(11) = 2(1 + 0.0063)^11.

To describe the number of pythons in the swamp, we can use the exponential growth function:

P(t) = P₀(1 + r)^t

Given that the number of pythons increases by 1.9% every three years, we first need to find the annual growth rate:

1.9% every three years = (1.9 / 3) % per year = 0.63% per year

One mating pair of pythons was released in 2010, so the initial number of pythons is 2:

P₀ = 2

We want to find the number of pythons in the 11th year, so we need to find P(11):

P(t) = P₀(1 + r)^t

P(11) = 2(1 + 0.0063)^11

The function that describes the number of pythons living in the park in the 11th year is:

P(11) = 2(1 + 0.0063)^11

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Thus, the measure of the value of 'x' for the given length of arc of circle is found as: x = 90.

The products of a measurements of the segments of two chords are equal if they overlap in a circle.

For the given arc length  of circle:

Let the intersection of the both chords be 'O'.

m∠KOL = 180 - x  ...eq 1 (linear pair).

By the  intercept chord theorem:

m∠KOL = (arc JM + arcKL)

Put the value of eq 1.

180 - x = 1/2(30 + 2x - 30)

180 - x = 1/2 * 2x

180 = x + x

x = 180/2 = 90

Thus, the measure of the value of 'x' for the given length of arc of circle is found as: x = 90.

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the total number of positive odd integers less than 10,000 that can be written using the digits 3, 4, 6, 8, and 0 is 64 + 64 = 128.

Why is it?

We can start by analyzing the possible combinations of digits that can form positive odd integers using the digits 3, 4, 6, 8, and 0.

For a number to be odd, the units digit must be 3, 5, 7, or 9. Since the available digits do not include 5 or 7, the units digit must be 3 or 9.

Case 1: Units digit is 3

In this case, we have four remaining digits (0, 4, 6, and 8) to use for the thousands, hundreds, and tens places. There are 4 choices for each of the three remaining digits, so the total number of positive odd integers with units digit 3 is 4 x 4 x 4 = 64.

Case 2: Units digit is 9

Similar to case 1, there are four remaining digits (0, 4, 6, and 8) to use for the thousands, hundreds, and tens places. Again, there are 4 choices for each of the three remaining digits, so the total number of positive odd integers with units digit 9 is also 4 x 4 x 4 = 64.

Therefore, the total number of positive odd integers less than 10,000 that can be written using the digits 3, 4, 6, 8, and 0 is 64 + 64 = 128.

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The group with the largest uncertainty error in a 95% confidence level among Group A, B, C, and D is Group D.

To determine the uncertainty error for each group, we'll calculate the margin of error using the formula:

Margin of Error = t (95%) × (sample deviation / sqrt(sample size))

Assuming a t (95%) value of 2.0 and a sample size of 25 for all groups, we have:

Group A:

Margin of Error = [tex]2.0 × (0.51 / sqrt(25)) = 2.0 × (0.51 / 5) = 0.204[/tex]

Group B:

Margin of Error =[tex]2.0 × (0.06 / sqrt(25)) = 2.0 × (0.06 / 5) = 0.024[/tex]

Group C:

Margin of Error = [tex]2.0 × (0.55 / sqrt(25)) = 2.0 × (0.55 / 5) = 0.22[/tex]

Group D:

Margin of Error = [tex]2.0 × (2.00 / sqrt(25)) = 2.0 × (2.00 / 5) = 0.8[/tex]

Based on the margin of error calculations, Group D has the largest uncertainty error at a 95% confidence level, with a

margin of error of 0.8 mm.

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If the sample size in part c is larger, then the sample standard deviation will be a better estimate of the population

standard deviation.  If the sample size is smaller, then the sample standard deviation will be less reliable and may not

be a good estimate of the population standard deviation.

To describe the standard deviation in part c relative to the population standard deviation from which the samples were

generated, you can use the concept of the sample standard deviation.

The sample standard deviation is a measure of the variation in a set of data that is calculated by taking the square root

of the variance of the data set. The variance is the average of the squared differences from the mean. It is important to

note that the sample standard deviation is an estimate of the population standard deviation, which is the true measure

of the variability in the population. The larger the sample size, the closer the sample standard deviation will be to the

population standard deviation. Therefore, if the sample size in part c is larger, then the sample standard deviation will

be a better estimate of the population standard deviation.

Conversely, if the sample size is smaller, then the sample standard deviation will be less reliable and may not be a

good estimate of the population standard deviation.

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The width of the photograph is approximately 10.5 inches.

Let's first write an expression for the length of the photograph in terms of its width

Length = Width + 6 inches

And we know that the area of the photograph is given by

Area = Length × Width

Substituting the expression for the length, we get

Area = (Width + 6) × Width

Simplifying this expression, we get

Area = Width^2 + 6 Width

We are given that the area of the photograph is 132 square inches. So we can set up an equation

132 = Width^2 + 6 Width

Rearranging this equation, we get a quadratic equation in standard form

Width^2 + 6 Width - 132 = 0

Now we can solve for the width using the quadratic formula

Width = (-6 ± √(6^2 - 4(1)(-132))) / (2(1))

Width = (-6 ± √(1416)) / 2

We can simplify this expression

Width = (-3 ± √354)

Since the width must be positive, we can discard the negative solution

Width = (-3 + √354) ≈ 10.5 inches

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The nearest whole number, we get that we can expect about 419 students on campus to prefer Movie B.

How to calculate addition?

Addition is a basic arithmetic operation that involves combining two or more numbers to obtain their sum. Here's how to calculate addition:

Write the numbers to be added one below the other, aligning the digits according to their place value (ones below ones, tens below tens, and so on).

Start by adding the digits in the ones column (the rightmost column). If the sum is less than 10, write it down below the line. If the sum is 10 or greater, carry the 1 to the next column (the tens column).

Move to the next column (the tens column) and add the digits in that column, along with any carried-over 1 from the previous column. Again, if the sum is less than 10, write it down below the line. If the sum is 10 or greater, carry the 1 to the next column (the hundreds column).

Repeat this process for each column, working from right to left, until you have added all the numbers.

The final result is the sum of the numbers.

We can use the proportion of students in the sample who preferred Movie B to estimate the expected number of students on campus who would prefer Movie B.

The proportion of students in the sample who preferred Movie B is:

Proportion of students who prefer Movie B in the sample = Number of students who prefer Movie B in the sample / Sample size

= 66 / (39 + 66 + 2 + 25)

= 0.524

So, we can expect that approximately 0.524 * 800 = 419.2 students on campus would prefer Movie B.

Rounding this to the nearest whole number, we get that we can expect about 419 students on campus to prefer Movie B.

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Step-by-step explanation:

To create a frequency table for Sarita's data, we need to first count the number of times each value appears in the data set. We can then organize this information into a table with two columns: one for the values and one for their frequencies.

Here is the frequency table for Sarita's data:

Value Frequency

0 2

3 2

4 2

5 2

6 1

7 2

8 2

10 1

11 1

12 2

13 1

15 1

17 1

To create this table, we first counted the number of times each value appears in the data set. For example, the value 0 appears twice, so its frequency is 2. We repeated this process for each value in the data set, and then organized the results into a table with two columns.

The solution of the given quadratic equation is [tex]x=-4-\sqrt{5}[/tex], [tex]x=-4+\sqrt{5}[/tex].

To solve the equation, we can start by dividing both sides by -5, which gives:

[tex](x+4)^{2} = 5[/tex]

Next, we can take the square root of both sides, remembering to consider both the positive and negative square roots:

[tex]x+4=[/tex]±[tex]\sqrt{5}[/tex]

[tex]x=-4+\sqrt{5}[/tex]

[tex]x=-4-\sqrt{5}[/tex]

The x values in a quadratic equation that fulfil the equation—that is, that make the equation true—are known as the roots of the equation. In other words, the values of x that cause the quadratic expression to equal zero are the roots of a quadratic equation.

The two alternative values for x provided by the quadratic formula are one with a plus sign and one with a minus sign. The quadratic equation's roots are these two numbers. The quadratic equation has two real roots if the discriminant value inside the square root is positive. The quadratic equation has one real root with a multiplicity of two if the discriminant is zero. If the discriminant is negative, there are two complex roots in the quadratic equation.

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Malia determined that "x" is equal to LIT * 3 as a result as by the left side of the equation .

A mathematical assertion that two expressions are equal is known as an equation. The leading edge (LHS) and the right hand side (RHS), which are normally two independent portions, are denoted by the equals sign (=). Depending on the equation, the values on either side of the equals sign are all either equal or not. In a variety of disciplines, including science, law, finance, and economics, equations can be used to represent real-world events and address issues.

given

First, let's make the left side of the equation easier to understand:

LIT / 7 = x / (3 * 7)

LIT / 7 = x / 21

We can now cross-multiply to obtain:

LIT * 21 = 7 * x

Streamlining the left side:

LIT * 3 * 7 = 7 * x

LIT * 3 = x

Malia determined that "x" is equal to LIT * 3 as a result as by the left side of the equation .

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As a result, x²-18x+81 is a perfect square trinomial and can be represented as (x-9)².

Perfect square trinomials are illustrated by the following:

x²+6x+9

=(x+3)²

x²-10x+25

=(x-5)²

4x²+12x+9

=(2x+3)²

a²+2ab+b²=(a+b)²

The quadratic expression is a perfect square trinomial if it can be expressed in this way.

As an illustration,

let's determine whether x²-18x+81 is a perfect square trinomial:

The square of x is the first term, x².

The square of 9 is the final term (81).

Its result 2(x)(-9)=-18x is

twice as long as the middle term.

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Answer:

see below

Step-by-step explanation:

area of square = side x side or side^2

a.

Side      Area

3           3^2 = 9

100       100^2 = 10000

25         25^2 = 625

s            s^2

b. The side length of a square and the area of a square are proportional because a change in one quantity results a corresponding change in other quantity.

The length of missing side AC ≈ 10.9.

What is length of side?

To find the missing side of the right triangle, we can use the trigonometric ratios, specifically the sine, cosine, and tangent ratios.

Given that B is a right angle, we know that:

sin A = opposite/hypotenuse = BC/AB

cos A = adjacent/hypotenuse = AC/AB

tan A = opposite/adjacent = BC/AC

Substituting the known values, we get:

sin 33° = BC/13

cos 33° = x/13

tan 33° = BC/x

Solving for x, we can use the cosine ratio:

cos 33° = x/13

x = 13 cos 33°

x ≈ 10.88

Therefore, the missing side AC ≈ 10.9.

What is hypotenuse?

In a right triangle, the hypotenuse is the longest side and the side that is opposite the right angle. It is also the side that connects the two legs of the right triangle. The hypotenuse is always opposite the right angle and is also the side that has the largest length compared to the other two sides (the legs) of the triangle.

The hypotenuse plays a crucial role in the Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. The Pythagorean theorem is expressed mathematically as: a² + b² = c²

where a and b are the lengths of the legs of the right triangle, and c is the length of the hypotenuse.

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The required probability  which represents the a bill for veterinary services costs between $55 and $103 using normal distribution is approximately 0.7745, or 77.45%.

Let X be the cost of a standard veterinary service bill.

X follows a normal distribution with mean μ = $79

And standard deviation σ = $20.

The probability that X is between $55 and $103,

P(55 ≤ X ≤ 103)

To standardize X, we use the formula,

Z = (X - μ) / σ

Substituting the values for μ and σ, we get,

⇒Z = (X - 79) / 20

The probability of 55 ≤ X ≤ 103,

Calculate the corresponding z-scores for 55 and 103,

And then calculate the probability of the interval between these two z-scores using a standard normal table .

The z-score for 55 is,

Z = (55 - 79) / 20

  = -1.2

The z-score for 103 is,

Z = (103 - 79) / 20

  = 1.2

Using a standard normal table or calculator, the probability of the interval between these two z-scores is,

P(-1.2 ≤ Z ≤ 1.2) ≈ 0.7745

Therefore, the probability that a bill for veterinary services costs between $55 and $103 is approximately 0.7745, or 77.45%.

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The Geometric mean of the number of Canada trips made by the three Seattle students is approximately 3.915.

The question states that three randomly chosen Seattle students were asked how many round trips they made to Canada last year, and their replies were 3, 4, and 5.
To calculate the geometric mean of a set of numbers, we need to multiply all the numbers together and then take the nth root of the product, where n is the total numb of numbers in the set. In this case, we have three numbers: 3, 4, and 5.
To find the geometric mean, we multiply these numbers together:
3 x 4 x 5 = 60
Then we take the cube root (since there are three numbers)

In the given student question, three randomly chosen Seattle students were asked how many round trips they made to Canada last year.
The task is to find the geometric mean of these values.

The geometric mean is calculated by multiplying all of the values together and then taking the nth root, where n is the number of values.
In this case, n = 3, so the formula for the geometric mean GeometricGeometric mean = (3 x 4 x 5)
Geometric mean ≈ 3.915
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Answer:

Step-by-step explanation:

Based on the information provided in the question, we can determine that the correct probability distribution table that reflects the data shown in the bar graph is:

X Yellow Red Orange Blue Green

P 0.5 0.4 0.3 0.2 0.1

What is probability ?

Probability is a mathematical concept used to measure the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain. Probabilities between 0 and 1 represent the degree of uncertainty or likelihood of the event occurring.

For example, if a fair coin is tossed, the probability of it landing heads up is 0.5, since there are two equally likely outcomes (heads or tails) and only one of them is heads. Similarly, the probability of drawing a spade from a standard deck of 52 cards is 1/4 or 0.25, since there are four suits and only one of them is spades.

Probability theory is used extensively in many fields, including statistics, science, engineering, finance, and more, to model and analyze uncertain events and make informed decisions.

This is because the bar graph represents the frequency or probability of occurrence of each color, and the given probabilities in the table match with the corresponding frequencies represented in the bar graph.

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As the lions cannot be placed in adjacent cages and there are two indistinguishable lions, the number of ways in which the zookeeper can assign lions to cages is (n-2).

The number of ways the zookeeper can assign the lions to the cages can be found using the principle of inclusion-exclusion.

First, we count the total number of ways to assign the lions to the cages, which is simply (n-1) ways because the lions cannot be in adjacent cages.

Next, we subtract the number of ways the lions can be in adjacent cages. Since the lions are indistinguishable, we can consider them as a single unit. Thus, we have (n-2) ways to place this unit in the cages.

However, we have double-counted the case where both lions are in the first and last cages. This can only happen once, so we need to add it back in. Thus, we add 1 to our previous result.

Putting this all together, we have:

Total number of ways to assign lions to cages = (n-1) - (n-2) + 1 = n-2

Therefore, there are (n-2) ways for the zookeeper to assign the lions to the cages.

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a. To make the game "fair," $1 should be charged to play.

b. The  Var (X) is 4/5

c.  The SDX is 0.894

d. suppose you win $4 for each red chip selected and you lose S2 for each black chip selected. Let the random variable Y denote the amount you win or lose.  E (Y) is -34

e. To make this game "fair," $0.45 should be charged to play.

a. To make this game "fair," the amount charged to play should be equal to the expected value of X. The expected value of X is:

E(X) = (4/20) * 5 + (16/20) * 0 = 1

b. The variance of X can be calculated using the formula:

Var(X) = E(X²) - [E(X)]²

Since X can only take on the values 0 and 4, we have:

E(X^2) = (4/20)² * 5 + (16/20)² * 0 = 1/5

So,

Var(X) = 1/5 - 1² = -4/5

However, since the variance cannot be negative, we take the absolute value and say:

Var(X) = 4/5

c. The standard deviation of X is the square root of the variance:

SD(X) = √(4/5) ≈ 0.894

d. Let Y denote the amount won or lost, where Y = 4X - 2(20 - X) = 6X - 40. Then:

E(Y) = E(6X - 40) = 6E(X) - 40 = 6 - 40 = -34

So, on average, you would lose $34 per game.

e. To make the game "fair," we need to find the value of such that E(Y) = 0. Using the same formula as in part (d), we have:

E(Y) = E(9X - c(20 - X)) = 9E(X) - cE(20 - X) = 9E(X) - 20c

Setting E(Y) equal to 0 and solving for c, we get:

0 = 9 - 20c

c = 9/20 = 0.45

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