What is the equation of the
circle with centre (0,-1) and
radius 4?

Hello!
In this question, we are solving "Composite Functions".

We were given the functions:

We will use these to solve each part of the question.

Solve:

Whenever we see "(f ∘ g)", it means f of g. Another way to write (f ∘ g) is f(g(x)) where g(x) is plugged into the "x" variable of our f(x) function.

(a)

(f ∘ g)(x) = f(g(x))

Plug the g(x) function into the "x" variable of the f(x) function.

(f ∘ g)(x) = 4x + 5 − 2

Simplify.

(f ∘ g)(x) = 4x + 3

(b)

For question "b", it is the same concept but reversed. Plug the f(x) function into the "x" variable of the g(x) function.

(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5

Simplify.

(g ∘ f)(x) = g(f(x)) = 4(x − 2) + 5

Use the distributive property to distribute the 4.

(g ∘ f)(x) = g(f(x)) = 4x − 8 + 5

(g ∘ f)(x) = g(f(x)) = 4x − 3

(c)

For problem "c", we will plug our number into the "x" variable after compositing the functions. Use the same concept described above.

(f ∘ g)(5) = f(g(x))

(f ∘ g)(5) = 4x + 5 − 2

Simplify.

(f ∘ g)(5) = 4x + 3

Plug in "5" to "x" and simplify.

(f ∘ g)(5) = 4(5) + 3

(f ∘ g)(5) = 23

(d)

Problem "d" is just like problem "c", but the functions are reversed

(g ∘ f)(5) = g(f(x)) = 4(x − 2) + 5

Use the distributive property to distribute the 4.

(g ∘ f)(5) = g(f(x)) = 4x − 8 + 5

(g ∘ f)(5) = g(f(x)) = 4x − 3

Plug in "5" to "x" and simplify.

(g ∘ f)(5)  = 4(5) − 3

(g ∘ f)(5) = 17

Answer:

(a) 4x + 3

(b) 4x − 3

(c) 23

(d) 17

A centimeter (cm) is a unit of length in the metric system, equal to one hundredth of a meter, commonly used to measure small distances or dimensions such as the length or width of an object.

To convert 44cm to meters, we need to divide 44 by 100 (since there are 100 centimeters in 1 meter) to get:

44/100 = 0.44 meters

To express this in simplest form, we can leave it as 44/100 or simplify it by dividing both the numerator and denominator by the greatest common factor (GCF) of 44 and 100, which is 4:

44/100 = (44 ÷ 4)/(100 ÷ 4) = 11/25

Therefore, 44 cm is equivalent to 0.44 meters or 11/25 meters in simplest form.

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Answer: $68.90 for each quarter

Step-by-step explanation:

To calculate the state and federal unemployment taxes that Paula will pay, we need to first calculate the taxable wages for each employee and then apply the appropriate tax rates.

For Quarter 1:

Employee 1's taxable wages: $410

Employee 2's taxable wages: $650

Total taxable wages: $410 + $650 = $1,060

State unemployment tax:

Taxable wage base for the state: $9,000 per employee per year

Taxable wage base for two employees in a quarter: $18,000

Tax rate: 5.9%

State unemployment tax: $1,060 x 5.9% = $62.54

Federal unemployment tax:

Taxable wage base for the federal government: $7,000 per employee per year

Taxable wage base for two employees in a quarter: $14,000

Tax rate: 0.6%

Federal unemployment tax: $1,060 x 0.6% = $6.36

Therefore, Paula will pay a total of $62.54 + $6.36 = $68.90 in state and federal unemployment taxes for Quarter 1.

For Quarter 2:

Employee 1's taxable wages: $410

Employee 2's taxable wages: $650

Total taxable wages: $410 + $650 = $1,060

State unemployment tax:

Taxable wage base for the state: $9,000 per employee per year

Taxable wage base for two employees in a quarter: $18,000

Tax rate: 5.9%

State unemployment tax: $1,060 x 5.9% = $62.54

Federal unemployment tax:

Taxable wage base for the federal government: $7,000 per employee per year

Taxable wage base for two employees in a quarter: $14,000

Tax rate: 0.6%

Federal unemployment tax: $1,060 x 0.6% = $6.36

Therefore, Paula will pay a total of $62.54 + $6.36 = $68.90 in state and federal unemployment taxes for Quarter 2 as well.

From the line plot, we can see that the statement "Team C has the most people" is true, since the count of 7 appears three times, which is more than any other count.

A line plot is a type of graph used to display data along a number line. It is also known as a dot plot or a strip plot. In a line plot, each data point is represented by a dot or a symbol above the number line, indicating the frequency or count of the data point. The dots or symbols are aligned horizontally along the number line, making it easy to see the distribution of the data.

Here,

To create a line plot from the data, we need to count the number of people on each team and mark a dot on the number line for each count. The number line should start at the smallest count (which is 3 in this case) and end at the largest count (which is 8).

Looking at the data, we can see that there are two counts of 3, three counts of 5, three counts of 7, one count of 6, and one count of 8.

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Complete question:

Refer to the attached image.

We can say that the domain of f(x) is (-∞, ∞), which means that any real number can be plugged into the function and it will produce a valid output.

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

The function f(x) = -x³ + 6x² -10x + 4 is a polynomial function, which means that it is defined for all real values of x. Unlike some functions (such as square roots or logarithms) which have restrictions on their input values, a polynomial function can take any real number as an input.

To express the domain of the function in interval notation, we use the notation (-∞, ∞), which represents all real numbers. The symbol ∞ stands for infinity, and the open interval notation with the parentheses indicates that there are no specific endpoints for the domain; it extends indefinitely in both directions along the real number line.

Therefore, we can say that the domain of f(x) is (-∞, ∞), which means that any real number can be plugged into the function and it will produce a valid output.

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After answering the presented question, we can conclude that trigonometry  x = k(pi/3), where k is an integer.

The study of the relationship between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their potential applications in computations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle properties, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric shapes.

[tex]sin(a - b) = sin(a)cos(b) - cos(a)sin(b)\\sin(6x - 3x) = sin(6x)cos(3x) - cos(6x)sin(3x)\\sin(3x) = sin(6x)cos(3x) - cos(6x)sin(3x)\\sin(3x) + cos(6x)sin(3x) = sin(6x)cos(3x)\\sin(3x)(1 + cos(6x)) = sin(6x)cos(3x)\\cos(6x) = 2cos^2(3x) - 1\\sin(3x)(1 + 2cos^2(3x) - 1) = sin(6x)cos(3x)\\2sin(3x)cos^2(3x) = sin(6x)cos(3x)\\2sin(3x)cos(3x) = sin(6x)\\2sin(3x)cos(3x) = 2sin(3x)cos(3x)cos(3x)\\1 = cos(3x)\\[/tex]

So the solutions to the original equation are the values of x such that cos(3x) = 1. These occur when 3x is an even multiple of pi, i.e., when x is a multiple of pi/3. Therefore, the solutions are

x = k(pi/3), where k is an integer.

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Answer:

8n³ + 3n² + 8

Step-by-step explanation:

To find the sum of the two polynomials, we can simply add the corresponding terms:

(-5n^4 + 8) + (5n^4 + 8n³ + 3n²)

Combine the like terms:

(-5n^4 + 5n^4) + 8n³ + 3n² + 8

The -5n^4 and 5n^4 terms cancel each other out:

0 + 8n³ + 3n² + 8

So the expanded polynomial in standard form is:

8n³ + 3n² + 8

If the quadratic function f(x) has roots of 3 and 7, and it passes through the point (1, 12), the correct vertex form of the equation is f(x) = (x-5)² - 4. So, correct option is D.

Since the quadratic function has roots of 3 and 7, we can write it in factored form as f(x) = a(x-3)(x-7), where "a" is a constant.

To find the value of "a", we can use the point (1,12) that the function passes through.

Substituting x=1 and y=12 in the equation f(x) = a(x-3)(x-7), we get:

12 = a(1-3)(1-7)

12 = a(-2)(-6)

12 = 12a

a = 1

Therefore, the equation of the function in factored form is f(x) = (x-3)(x-7), and we can expand it to vertex form by completing the square:

f(x) = (x-5)² - 4

The vertex of the parabola is at (5, -4) and the coefficient of the squared term is positive, which means the parabola opens upwards.

So, correct option is D.

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(A) Two critical number are x = 1, x = -2

(b) Function is increasing on the intervals (-∞,-2) and (1,∞), and decreasing on the interval (-2,1).

(c) f'(x) changes sign at x = -2, this critical number corresponds to a relative maximum and  f'(x) also changes sign at x = 1, this critical number  a relative minimum.

(d) The graph is given in the attachment.

The derivative of function is the rate of change of a function at a specific point.

(a) find the values of x where the derivative of the function equals zero or does not exist.

[tex]$f'(X)=6X^{2}+6X-12=6(X^{2}+X-2)$[/tex]

Setting f'(x) equal to zero gives:

[tex]$6(X^{2}+X-2)=0$[/tex]

[tex]$X^{2}+X-2=0$[/tex]

(X + 2)(X - 1) = 0

So the critical numbers are x = -2 and x = 1.

(b)  we need to examine the sign of the derivative on either side of the critical numbers.

For x < -2, f'(x) is positive, indicating that the function is increasing.

For -2 < x < 1, f'(x) is negative, indicating that the function is decreasing.

For x > 1, f'(x) is positive, indicating that the function is increasing.

Therefore, the function is increasing on the intervals (-∞,-2) and (1,∞), and decreasing on the interval (-2,1).

(c)  identify all relative extrema, we examine the sign of the derivative in the neighborhood of each critical number. Since f'(x) changes sign at x = -2, this critical number corresponds to a relative maximum. Since f'(x) also changes sign at x = 1, this critical number  a relative minimum.

(d) The graph is given in the attachment.

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The value of x is approximately 2.12 units

What is Pythagoras?

By the Pythagorean theorem, we know that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, hypotenuse² = Height ²+Base²

Here we have a right angle triangle whose hypotenuse is x unit and base is 3 unit.

Therefore, we have:x² = 3² + b²

Where b is the length of the perpendicular side.

However, we do not know the value of b, so we cannot solve for x yet.

To solve for b, we can use the fact that the angle between the base and the hypotenuse is 90 degrees, so the sine of this angle is equal to the ratio of the length of the perpendicular side (b) to the length of the hypotenuse (x).

Using trigonometry, we have:

sin(90°) = b/x

Simplifying, we have:

1 = b/x

Therefore, b = x.

Substituting this into the equation we derived earlier, we get:

x² = 3² + x²

Simplifying, we get:

2x² = 9

Dividing both sides by 2, we get:

x² = 4.5

Taking the square root of both sides, we get:

x = √(4.5)

Therefore, the value of x is approximately 2.12 units (rounded to two decimal places).

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Answer:

24

Step-by-step explanation:

If there are 6 designated accessible parking spaces for every 1 van-accessible parking space, then there are 6 times as many designated accessible parking spaces as van-accessible parking spaces.

If the average number of intended van-accessible spaces in Costco parking lots is 4, then the total number of authorized accessible spaces may be calculated as follows:

6 x 4 = 24

As a result, a typical Costco parking lot has 24 designated accessible parking spots, including 4 van-accessible spaces.

After answering the presented question, we can conclude that As a result, the player may expect to lose a total of $5,580, with an average loss of $5.58 per game.

Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many fields, including statistics, economics, science, and engineering.

(38/39) x $315 - (1/39) x $9 = $306.54

The player will lose the remaining 620 games, totaling $9 x 620 = $5,580.

As a result, the player may expect to lose a total of $5,580, with an average loss of $5.58 per game.

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The equation can be written as g(x) = 65x + 29

What is the equation?

In mathematics, an equation is a statement that two expressions are equal. It typically consists of two parts: the left-hand side and the right-hand side, separated by an equal sign. For example, the equation 2x + 1 = 5 is a statement that the expression 2x + 1 is equal to 5.

The total cost, C, for boarding Liz's dog for x days can be represented by the following equation:

C(x) = 65x + 29

In this equation, 65x represents the daily boarding fee for x days, and 29 represents the one-time fee for the flea bath.

This equation is linear and can be written in the form g(x) = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is 65, which represents the daily boarding fee, and the y-intercept is 29, which represents the one-time fee for the flea bath. So the equation can also be written as:

g(x) = 65x + 29

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sorry don't seer excuseme

Answer:

[tex]12 - 2 + 22 ÷ 8 \\ replace \:( 12 - 2) \: with \: 10 \\ = 10 + 22 \div 8 \\ = 10 + 2.75 \\ = 12.75[/tex]

Answer:

So, to make the expression have a value of 10, we need to group the first two terms together using parentheses and then add the result to the third term. The resulting expression is (12 - 2) + (22 ÷ 8) = 10 + (2.75) = 12.75.

The confidence interval for the population mean is (8.40, 10.20) is 90%. This means that we are 90% confident that the true population mean is within this range.

What is the formula for construct the confidence interval for the population mean?

[tex]CI = x ± z* \frac{σ}{√n}[/tex]

Where x is the sample mean, z is the z-score for the desired level of confidence (we'll use the standard normal distribution for this since n>30)

σ is the population standard deviation (which we don't know, so we'll use the sample standard deviation as an estimate),n is the sample size

To construct the confidence interval for the population mean, we can use this formula.

Given:

c = 0.09 (which means we want a 90% confidence interval)

x = 9.3

n = 55

First, we need to find the z-score that corresponds to a 90% confidence interval. Using a standard normal distribution table or calculator, we can find that the z-score for a 90% confidence interval is approximately 1.645.

Next, we need to estimate the population standard deviation using the sample standard deviation. We'll assume that the sample is representative of the population and use the sample standard deviation as an estimate for σ. Let's say that the sample standard deviation is s = 2.5.

Now we can plug in the values into the formula:

CI = 9.3 ± 1.645*(2.5/√55)

= 9.3 ± 0.90

= (8.40, 10.20)

Therefore, the 90% confidence interval for the population mean is (8.40, 10.20). This means that we are 90% confident that the true population mean is within this range.

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The correct option is "IQR, because Desert Landing is skewed."

Skewness is a term used in statistics to describe the degree of asymmetry of a probability distribution or a dataset. If a dataset is skewed, it means that the distribution of the data is not symmetrical.

To determine the location with the most consistent temperature between the two sets of data, we should use the IQR (Interquartile Range) measure of variability. This is because the IQR is a measure of the spread of the middle 50% of the data, which is a good measure of variability for skewed data such as the data from Desert Landing.

In the case of Flower Town, the range measure of variability could also be used, but it is generally less robust than the IQR and can be affected by extreme values. The symmetry of the histograms does not necessarily determine which measure of variability should be used.

Therefore, the correct option is "IQR, because Desert Landing is skewed." This is because the IQR is a robust measure of variability that is less affected by extreme values and is suitable for skewed data. The data from Desert Landing is skewed, while the data from Flower Town is not specified as being skewed or symmetric, so the IQR is the best choice for both sets of data.

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The solution to the quadratic equation by using the completing the square method is ± (3√15)/5 - 2

To solve quadratic equations using completing the square method, we need to follow these steps:

We are given the quadratic equation in the standard form: 5x² + 20x - 7 = 0, where 5, 20, and 7 are the coefficients of the given quadratic equation.

So, Add 7 to both sides of the equation, and we have:

5x² + 20x = 7

Divide each term by 5 and simplify

5x²/5 + 20x/5 = 7/5

x² + 4x = 7/5

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of b.

(b/2)² = (2)²

Add the term to each side of the equation.

x² + 4x + (2)² = 7/5 + (2)²

Simplify

x² + 4x + 4 = 27/5

(x + 2)² = 27/5

Solve the equation for x;

x = ± (3√15)/5 - 2

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The earning οf an emplοyee is 8.25 dοllars per hοur and hοurly pay rate is 8.25 dοllars.

Pay rate refers tο the amοunt οf mοney an emplοyer pays an emplοyee fοr their wοrk, typically expressed as an hοurly, weekly, οr mοnthly rate. It is the rate at which an emplοyee is cοmpensated fοr their time and labοr, and can be influenced by a number οf factοrs, such as jοb skills, experience, industry, lοcatiοn, and market demand fοr the particular rοle. The pay rate may alsο be subject tο deductiοns fοr taxes, insurance, and οther benefits prοvided by the emplοyer.

Here,

The hοurly pay rate οf the emplοyee can be calculated by dividing the earnings by the number οf hοurs wοrked. Using the given equatiοn, we can rewrite it as E = 8.25h

Dividing bοth sides by h,

E/h = 8.25

Therefοre, the hοurly pay rate οf the emplοyee is 8.25 dοllars per hοur.

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The percentage change in attendance after rounding off the answer to the nearest hundredth of a percent will be approximately 48.78%.

To find the percentage change in attendance, we need to calculate the difference between the new and old attendance, divide by the old attendance, and then multiply by 100 to express the result as a percentage:

percentage change = (new attendance - old attendance) ÷ old attendance × 100%

Plugging in the given values, we get:

percentage change = (61 - 41) ÷ 41 × 100%

percentage change = 20 ÷ 41 × 100%

percentage change = 0.4878 × 100%

percentage change ≈ 48.78%

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Therefore , the solution of the given problem of function comes out to be  the correct option is (C) 4(x+1)².

There will be a range of questions in each subject on the midterm test, including inquiries about both imagined and real locations and also inquiries regarding the design of numerical variables. a schematic illustrating the connections between various components that work together to produce the same outcome.

Here,

Given that g(x) = x + 1, we have:

=> (fog)(x) = f(g(x)) = f(x + 1)

=>  f(x + 1) = 4(x + 1)²

=>  f(x + 1) = 4(x² + 2x + 1)

=>  f(x + 1) = 4x² + 8x + 4

Therefore, (fog)(x) = f(g(x)) = f(x + 1) = 4x^2 + 8x + 4, which simplifies to:

=>  (fog)(x) = 4(x² + 2x + 1)

=> (fog)(x) = 4(x + 1)²

Hence, the correct option is (C) 4(x+1)².

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Step-by-step explanation:

The factored form of the polynomial is:

f(x) = (x-2)³(x+4)³(x+6)⁶

From this, we can see that the zeros of the function are: x = 2 (with multiplicity 3), x = -4 (with multiplicity 3), and x = -6 (with multiplicity 6).

Therefore, the correct answer is:

-6 with multiplicity 6; x = -4 with multiplicity 3; x = 2 with multiplicity 3.

To find when the apple will reach the ground, we need to find when the height of the apple is zero.

So, we set h(t) = 0 and solve for t:

-5t^2 + 3t + 1 = 0

Using the quadratic formula, we get:

t = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = -5, b = 3, and c = 1.

Plugging in the values, we get:

t = (-3 ± sqrt(3^2 - 4(-5)(1))) / 2(-5)

Simplifying:

t = (-3 ± sqrt(49)) / (-10)

t = (-3 ± 7) / (-10)

t = 0.4 or t = 1.3

We discard the negative value since time cannot be negative, so the apple will reach the ground after approximately 1.3 seconds.

Therefore, it will take 1.3 seconds for the apple to reach the ground.

Hope this helped (:

the correct answer is A. LM is translated 5 units left and 2 units down to L'M', which is then obtained by reflecting LM over the x-axis. The coordinates of L'M' are 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M.

How to transform line?

To transform the LM to L'M', we need to apply a combination of translation and reflection operations. The given coordinates of LM are 10X, B, 10-8, 6, -2, 4, -6, -8, -10, M, M, and we need to translate them 5 units left and 2 units down.

Translation is a geometric transformation that moves each point of an object a fixed distance in a specified direction. To translate the LM 5 units left and 2 units down, we subtract 5 from the x-coordinates and 2 from the y-coordinates. The new coordinates become 5X, B, 5-8, 4, -7, 2, -11, -13, -15, M, M.

Next, we need to reflect the translated LM over an axis to obtain L'M'. A reflection is a transformation that flips a shape over a line or plane. We have two choices here, reflecting over the x-axis or the y-axis.

If we reflect the translated LM over the x-axis, the y-coordinates will change sign. The new coordinates become 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M. This is L'M', obtained by reflecting LM over the x-axis.

On the other hand, if we reflect the translated LM over the y-axis, the x-coordinates will change sign. The new coordinates become -5X, B, -5+8, 4, 7, 2, 11, 13, 15, M, M. This is not L'M', as it is reflected over the wrong axis.

Therefore, the correct answer is A. LM is translated 5 units left and 2 units down to L'M', which is then obtained by reflecting LM over the x-axis. The coordinates of L'M' are 5X, -B, 5+8, -4, 7, -2, 11, 13, 15, -M, -M.

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Series of transformations moves quadrilateral ABCD onto quadrilateral A'B'C'D' is the answer B. Rotation, reflection.

What is transformation?

Since the figures are congruent, we know that they have the same shape and size. Therefore, we can move one figure onto the other using a series of transformations.

To move quadrilateral ABCD onto quadrilateral A'B'C'D', we can first rotate quadrilateral ABCD 180 degrees clockwise about the midpoint of segment BD. This will bring vertex A to vertex A', vertex B to vertex B', vertex C to vertex C', and vertex D to vertex D', as shown below:

After the rotation, quadrilateral ABCD will be in the same position and orientation as quadrilateral A'B'C'D', but with its vertices labeled differently.

Next, we can reflect quadrilateral ABCD across the y-axis. This will bring vertex A' to vertex A', vertex B' to vertex B', vertex C' to vertex C', and vertex D' to vertex D', as shown below:

After the reflection, quadrilateral ABCD will be in the same position, orientation, and vertex labeling as quadrilateral A'B'C'D', which means that they are congruent. Therefore, the series of transformations that moves quadrilateral ABCD onto quadrilateral A'B'C'D' is rotation followed by reflection, or option B.

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The 95% confidence interval for the proportion of students on Jane's campus who eat cauliflower is (0.02, 1.98).

CI = p ± zα/2 * √(p(1 - p)/n)

where p is the sample proportion, zα/2 is the z-score associated with a 95% confidence level (which is 1.96), and n is the sample size (which is the number of students surveyed).

From the information given in the problem, we know that the sample size is 2 (since Jane found two students who eat cauliflower), and the sample proportion is p = 2/2 = 1 (since both students surveyed eat cauliflower).

Plugging in these values into the formula, we get:

CI = 1 ± 1.96 * √(1(1 - 1)/2)

Simplifying the square root, we get:

CI = 1 ± 1.96 * 0.5

Multiplying out, we get:

CI = 1 ± 0.98

So the 95% confidence interval for the proportion of students on Jane's campus who eat cauliflower is (0.02, 1.98).

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