What is the difference between the median number of turkey sandwiches sold and the median number of ham sandwiches


sold?

The truth value of Proposition 1, [A ⊃ ~(B · Y)] ≡ ~[B ⊃ (X · ~A)], is true when A and B are true, and X and Y are false.

First, we'll evaluate each part of the proposition:

1. A ⊃ ~(B · Y): Since A is true and B · Y is false (due to Y being false), the statement becomes "true ⊃ ~false", which simplifies to "true ⊃ true". This is true.

2. B ⊃ (X · ~A): Since B is true, X is false, and ~A is false, the statement becomes "true ⊃ (false · false)", which simplifies to "true ⊃ false". This is false.

Now, we'll evaluate the equivalence ([A ⊃ ~(B · Y)] ≡ ~[B ⊃ (X · ~A)]): The statement becomes "true ≡ ~false", which simplifies to "true ≡ true". Therefore, the truth value of Proposition 1 is true.

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The correct answer to this question is a. The regression [tex]r_{2}[/tex] is a measure of the goodness of fit of your regression line. This means that it tells you how well the regression line fits the data and how much of the variation in the dependent variable can be explained by the independent variable. In other words, it is a measure of the strength of the relationship between the two variables being analyzed.

The determinant is a mathematical term used in linear algebra that helps determine the properties of a matrix. It is not directly related to the regression [tex]r_{2}[/tex] value, so option c is incorrect. Option b is also incorrect as the regression [tex]r_{2}[/tex] value does not determine whether or not x causes y. Finally, option d is also incorrect as ess and tss are not related to the goodness of fit of the regression line.

Overall, the regression [tex]r_{2}[/tex] value is an important measure in determining the quality of a regression model and how well it can predict outcomes based on the independent variable. It is calculated by dividing the explained variance by the total variance and is expressed as a percentage. A high [tex]r_{2}[/tex] value indicates a strong relationship between the variables and a good fit of the regression line to the data, while a low [tex]r_{2}[/tex] value indicates a weak relationship and a poor fit.

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How does one interpret a written work? One interprets a written work by explaining the meaning of the text. Therefore, the correct option is B.

By explaining the meaning of the text.

What is the meaning of interpreting a written work?

Interpreting a written work involves understanding the content of a written work. Interpretation enables one to appreciate, analyze, and evaluate the author's content. One can interpret a written work in different ways, including literary analysis, close reading, and critical thinking.

What does evaluating a written work involve?

Evaluating a written work involves analyzing and assessing the author's content. It entails assessing the strength and weaknesses of the content. Evaluation helps to provide an informed critique of the work.

What is the role of personal opinion in interpreting a written work?

Personal opinion plays a role in interpreting a written work since it enables the artist to engage with the text. However, it is crucial to avoid being biased while offering an opinion.

Therefore, one needs to ensure that their opinion is well-informed and supported by the text.

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The value of x, y and z in the given rhombus are -81, 36 and -20 respectively.

Given angles of a rhombus as,

(-x - 8)⁰

107⁰

(3y - 1)⁰

(-4z - 7)°

Here, the figure is given below.

Opposite angles of a rhombus are equal.

So,

3y - 1 = 107

3y = 108

y = 36

Also, adjacent angles are supplementary for rhombus.

-x - 8 + 107 = 180

-x = 81

x = -81

-4z - 7 + 107 = 180

-4z = 80

z = -20

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Assuming that the pattern of the first few terms continues, the formula for the general term an of the sequence is:
an = (-1)^(n+1) / 6^(n-1)

To find a formula for the general term an of this sequence, we need to identify the pattern in the given terms. Looking at the sequence, we can see that each term is either a positive or negative fraction with a denominator that is a power of 6. Specifically, the denominators of the terms are 1, 6, 36, 216, 1296, which are all powers of 6.

Moreover, we can see that the signs of the terms alternate: the first term is positive, the second term is negative, the third term is positive, and so on.

Based on these observations, we can write the formula for the nth term as follows:

an = (-1)^(n+1) / 6^(n-1)

Here, (-1)^(n+1) gives the alternating signs, and 6^(n-1) gives the denominator that is a power of 6.

Therefore, assuming that the pattern of the first few terms continues, the formula for the general term an of the sequence is:

an = (-1)^(n+1) / 6^(n-1)

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The list contains important information that would help library users. They are vital as they offer guidance on how to utilize the resources and services available in the library.

There are several facts on the list that will guide you when you are planning to utilize the library. Here are some of the most crucial ones you should note:1. The library has a computerized catalog that lists all the materials available in the library.2. There is a computer lab in the library where users can access the internet.3. The library has quiet study rooms that can be used by individuals and groups.4. Reference librarians can provide assistance in researching topics.5. Materials can be borrowed for a period of three weeks.The list contains a range of facts about the library's facilities and services, and it is essential to know them as a library user. Users should ensure they adhere to the library's policies and procedures to make the most out of the library's resources and services. Additionally, users should ask librarians for assistance when they need it, as librarians are there to assist them.

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research, a school librarian must be

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Answer:

B) (7, 1)

Step-by-step explanation:

because 1.7=7

By calculations, the width of the walkway should be 5 feet

From the question, we have the following parameters that can be used in our computation:

Dimension = 12 feet by 8 feet

Area of the walkway = 396 feet

The missing diagram is attached

This means that

Area = (12 + 2x) * (8 + 2x)

Recall that

Area of the walkway = 396 feet

So, we have

(12 + 2x) * (8 + 2x) = 396

When solved using a graphing tool, we have

x = 5

Hence, the width of the walkway should be 5 feet

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The probability that x < 30, given a population mean of 54 and a standard deviation of 8, is 0.13%.

To find the probability that x < 30, we can use the properties of a normal distribution. Given a population mean of 54 and a standard deviation of 8, we can calculate the z-score corresponding to the value of 30 using the formula:

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

where x represents the value of interest, μ is the population mean, and σ is the standard deviation.

Substituting the given values, we have:

[tex]\[ z = \frac{30 - 54}{8} = -3 \][/tex]

Next, we consult a standard normal distribution table or use statistical software to find the probability associated with the z-score of -3. The probability of obtaining a value less than 30 can be interpreted as the area under the standard normal curve to the left of the z-score -3.

By referring to the standard normal distribution table or using software, we find that the probability associated with a z-score of -3 is approximately 0.0013. Therefore, the probability that x < 30, given the provided population mean and standard deviation, is approximately 0.0013 or 0.13%.

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Let x, y, and z be the number of vehicles of type A, B, and C, respectively.

The objective is to maximize the shipping capacity in ton-miles per day, which can be expressed as:

capacity = payload capacity * operating speed * operating hours per day

For vehicle A, the capacity is:

10 * 45 * 18 * x = 8100x

For vehicle B, the capacity is:

20 * 40 * 18 * y = 14400y

For vehicle C, the capacity is:

18 * 40 * 21 * z = 15120z

The total cost of purchasing the vehicles cannot exceed the allocated budget of $800,000:

26000x + 36000y + 42000z ≤ 800000

The total number of drivers required cannot exceed the available number of 150 drivers:

x + 2y + 2z ≤ 150

The total number of vehicles cannot exceed 30:

x + y + z ≤ 30

The objective function to be maximized is the total capacity:

Z = 8100x + 14400y + 15120z

Subject to:

26000x + 36000y + 42000z ≤ 800000

x + 2y + 2z ≤ 150

x + y + z ≤ 30

x, y, z ≥ 0 (since the company cannot purchase negative vehicles)

This is a linear programming problem that can be solved using standard techniques, such as the simplex method.

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Amount of rainfall results in the maximum number of mosquitoes is 4 centimeters.

m(r) = -r(r-4)

m(r) = -r² + 4r

let's find the derivative of m(r) with respect to r:

m'(r) = -2r + 4

To find the critical points, we set m'(r) = 0 and solve for r:

-2r + 4 = 0

-2r = -4

r = 2

m''(r) = -2

Evaluating m''(2), we get

m''(2) = -2

the function m(r) has a maximum at r = 2.

Putting the value 2 we get

m(2) = -2² + 4(2)

m(2) = - 4 + 8

m(2) = 4

Therefore, the amount of rainfall that results in the maximum number of mosquitoes is 4 centimeters

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The question is incomplete the complete question is :

The number of mosquitoes in Brooklyn (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by m(r) = -r(r - 4) What amount of rainfall results in the maximum number of mosquitoes?

A truth table is a table that shows the output of a logical expression for all possible combinations of input values.

(a) Truth table for f(x, y, z) = x + y + z:

x y z f(x, y, z)

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1

(b) Canonical DNF for f(x, y, z) using the truth table:

f(x, y, z) = (¬x ∧ ¬y ∧ z) ∨ (¬x ∧ y ∧ ¬z) ∨ (¬x ∧ y ∧ z) ∨ (x ∧ ¬y ∧ ¬z) ∨ (x ∧ ¬y ∧ z) ∨ (x ∧ y ∧ ¬z) ∨ (x ∧ y ∧ z)

(c) Canonical CNF for f(x, y, z) using the truth table:

f(x, y, z) = (x ∨ y ∨ z) ∧ (x ∨ y ∨ ¬z) ∧ (x ∨ ¬y ∨ z) ∧ (x ∨ ¬y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ∧ (¬x ∨ y ∨ ¬z) ∧ (¬x ∨ ¬y ∨ z)

what is combinations?

Combinations refer to the number of ways in which a subset of elements can be selected from a larger set, disregarding the order of the elements. The formula for combinations is:

nCk = n! / (k! * (n - k)!)

where n is the total number of elements in the set, k is the number of elements in the subset, and ! denotes the factorial function (i.e., the product of all positive integers up to and including the given integer).

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The curved surface area (CSA) of a cylinder is given by the formula:

CSA = 2πrh

where r is the radius of the base of the cylinder, h is the height of the cylinder.

In this case, the radius of the base is x cm and the height of the cylinder is y cm. Therefore, the formula for the curved surface area becomes:

CSA = 2πxy

So, the curved surface area of the cylindrical object with radius 'x' cm and height 'y' cm is 2πxy square centimeters.

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To evaluate the surface integral, we first need to find a parameterization of the surface S. The surface integral ∫∫S (x - y)dS, where S is the portion of the plane x + y + z = 1 that lies in the first octant, evaluates to 1/2.

To evaluate the surface integral, we first need to find a parameterization of the surface S. The plane x + y + z = 1 can be parameterized as x = u, y = v, z = 1 - u - v, where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1 - u. The partial derivatives of x and y with respect to u and v are both 1, while the partial derivative of z with respect to u is -1 and the partial derivative of z with respect to v is -1.

Using this parameterization, we can write the surface integral as            ∫∫D (x(u,v) - y(u,v))√(1 + z_u^2 + z_v^2)dudv,

where D is the region in the uv-plane corresponding to the first octant. Simplifying this expression, we get ∫∫D (u - v)√3dudv. Integrating this expression over the region D, we get 1/2, which is the final answer.

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To calculate 3/8 × 2/6 using a grid model, we need to use the following procedure:

First, represent the fraction 3/8 by shading three cells in each of the eight rows.Then, represent the fraction 2/6 by shading two cells in each of the six columns of the grid model.

Next, identify the cells that are shaded blue and textured. There are six cells where the blue shading and the texture overlap.Now count the number of cells that are shaded blue but not textured, there are 18 of them.Now count the number of cells that are textured but not shaded blue, there are 12 of them.

Finally, count the total number of cells that are shaded blue or textured.

There are 24 of them.

Thus, the product 3/8 × 2/6 is equal to the fraction of the total number of cells that are shaded blue or textured. This fraction is equal to 13/24.Therefore, the answer is B. 13/24.

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Light with a frequency of 6.0 x 10¹⁴ Hz travels through a glass block with an index of refraction of 1.5.

When light travels through a medium, such as glass, its speed and direction can be affected due to the change in the refractive index of the medium. The refractive index is a measure of how much the speed of light is reduced when it enters the medium compared to its speed in a vacuum.

In this case, the glass block has an index of refraction of 1.5. The index of refraction is calculated by dividing the speed of light in a vacuum by the speed of light in the medium. Since the speed of light in a vacuum is approximately 3 x 10⁸ meters per second, the speed of light in the glass block can be calculated by dividing the speed of light in a vacuum by the refractive index: 3 x 10⁸ m/s / 1.5 = 2 x 10⁸ m/s.

The frequency of light remains constant as it travels through different media. Therefore, the light with a frequency of 6.0 x 10¹⁴Hz will also have the same frequency while passing through the glass block. However, since the speed of light is reduced in the glass, the wavelength of the light will change. The relationship between frequency, wavelength, and speed of light is given by the equation: speed of light = frequency x wavelength. As the speed of light decreases in the glass, the wavelength will decrease proportionally to maintain the same frequency.

In conclusion, when light with a frequency of 6.0 x 10¹⁴Hz travels in a glass block with an index of refraction of 1.5, its frequency remains unchanged, but its wavelength will decrease proportionally due to the reduction in the speed of light in the glass.

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The solution to the recurrence relation an=-8a_n-1-16a_n-2 with initial conditions a0=2 and a1=-20 is given by:

[tex]an = [(2 + (-20)(4 - 4\sqrt{2} ) / (8 + 8\sqrt{2} )](4 + 4\sqrt{2} )^n + [(2 - c1)](4 - 4\sqrt{2} )^n[/tex]

To solve the recurrence relation an=-8a_n-1-16a_n-2, we can use the characteristic equation method. We assume that the solution has the form an=r^n, where r is a constant to be determined. Substituting this into the recurrence relation, we get:

[tex]r^n = -8r^(n-1) - 16r^(n-2)[/tex]

Dividing both sides by[tex]r^{(n-2),[/tex] we get:

[tex]r^2 = -8r - 16[/tex]

This is the characteristic equation of the recurrence relation. We can solve for r by using the quadratic formula:

r = (-(-8) ± [tex]\sqrt{-8} ^2[/tex] - 4(-16))) / 2

r = (-(-8) ± [tex]\sqrt{128}[/tex] / 2

r = 4 ± 4[tex]\sqrt{2}[/tex]

Therefore, the general solution to the recurrence relation is:

[tex]an = c1(4 + 4\sqrt{2} )^n + c2(4 - 4\sqrt{2} )^n[/tex]

where c1 and c2 are constants determined by the initial conditions. Using the initial conditions a0=2 and a1=-20, we get:

a0 = c1 + c2 = 2

[tex]a1 = c1(4 + 4\sqrt{2} ) - c2(4 - 4\sqrt{2} ) = -20[/tex]

Solving for c1 and c2, we get:

[tex]c1 = (a0 + a1(4 - 4\sqrt{2} ) / (8 + 8\sqrt{2})[/tex]

c2 = (a0 - c1)

Substituting these values of c1 and c2 into the general solution, we get:

[tex]an = [(a0 + a1(4 - 4\sqrt{2} ) / (8 + 8\sqrt{2} ](4 + 4\sqrt{2} )^n + [(a0 - c1)](4 - 4\sqrt{2} )^n[/tex]

Thus, the solution to the recurrence relation an=-8a_n-1-16a_n-2 with initial conditions a0=2 and a1=-20 is given by:

[tex]an = [(2 + (-20)(4 - 4\sqrt{2} ) / (8 + 8\sqrt{2} )](4 + 4\sqrt{2} )^n + [(2 - c1)](4 - 4\sqrt{2} )^n[/tex]

where [tex]c1 = (2 + (-20)(4 - 4\sqrt{2} ) / (8 + 8\sqrt{2} )[/tex]

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To solve the recurrence relation an=-8a_n-1-16a_n-2 with initial conditions a0=2 and a1=-20, we can use the characteristic equation method.

The solution to the recurrence relation is:

an = 2(-4)^n + 3n(-4)^n

We can check this solution by plugging in n=0 and n=1 to see if we get a0=2 and a1=-20, respectively.

When n=0:

a0 = 2(-4)^0 + 3(0)(-4)^0 = 2

When n=1:

a1 = 2(-4)^1 + 3(1)(-4)^1 = -20

Therefore, the solution is correct.
Hi! I'd be happy to help you solve the recurrence relation. Given the relation a_n = -8a_(n-1) - 16a_(n-2) and the initial conditions a_0 = 2 and a_1 = -20, follow these steps:

Step 1: Use the initial conditions to find a_2.
a_2 = -8a_1 - 16a_0
a_2 = -8(-20) - 16(2)
a_2 = 160 - 32
a_2 = 128

Step 2: Use the relation to find a_3.
a_3 = -8a_2 - 16a_1
a_3 = -8(128) - 16(-20)
a_3 = -1024 + 320
a_3 = -704

Step 3: Continue using the relation to find further terms, if needed.

The first few terms of the sequence are: 2, -20, 128, -704, and so on.

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Answer:

-------------------

There are 13 clubs and 13 hearts in the deck.

First, find the number of ways to choose 2 clubs out of 13:

Next, find the number of ways to choose 3 hearts out of 13:

Now, multiply these two results:

If we determine the estimated multiple linear regression equation to predict the overall score given the scores for comfort, amenities, and in-house dining, some steps need to be followed.

Steps are:
Step 1: Collect the data for each variable (Comfort, Amenities, and In-House Dining) along with the corresponding overall scores.
Step 2: Perform a multiple linear regression analysis on the collected data using statistical software or a calculator. This will give you the coefficients (b0, b1, b2, and b3) and the intercept (a) for the linear regression equation.
Step 3: Form the multiple linear regression equation using the coefficients and intercept obtained in Step 2. The equation will have the form:
Overall Score (Y) = a + b1*X1 + b2*X2 + b3*X3

Where:
Y = Overall Score
X1 = Comfort
X2 = Amenities
X3 = In-House Dining
a = Intercept
b1, b2, and b3 = Coefficients for Comfort, Amenities, and In-House Dining, respectively

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The exponential function [tex]f(x) = 14(0.87)^x[/tex] shows exponential decay, and it's graph is given by the image presented at the end of the answer.

The equation of the asymptote is given as follows:

y = 0.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

As the parameter b for this problem has an absolute value less than 1, the function represents exponential decay.

As there is no term adding/subtracting the exponential function, the asymptote is given as follows:

y = 0.

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The equation X/y = w/z satisfies the Dividendo Theorem.

The Dividendo Theorem, also known as the Proportional Division Theorem or the Constant Ratio Theorem, is a principle in mathematics that relates to ratios. According to the theorem, if two ratios are equal, then the ratios of their corresponding parts (dividendo) are also equal.

In the given equation X/y = w/z, we have two ratios on both sides of the equation. To determine if the equation satisfies the Dividendo Theorem, we need to compare the corresponding parts.

In this case, the corresponding parts are X and w, and y and z. If X/y = w/z, then we can conclude that the ratios of their corresponding parts are equal.

To understand why this is true, consider the concept of ratios. A ratio expresses the relationship between two quantities. When two ratios are equal, it means that the relationship between the corresponding quantities in each ratio is the same. In other words, the relative size or proportion of the quantities remains constant.

By applying the Dividendo Theorem to the equation X/y = w/z, we can determine that the ratios of X to y and w to z are equal. This implies that the relative sizes or proportions of X and y are the same as those of w and z.

Therefore, we can confidently say that the equation X/y = w/z satisfies the Dividendo Theorem.

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Answer:

X=3.6

Step-by-step explanation:

To calculate the cost of 4.25 kg of potatoes based on the given information that 2.5 kg costs $1.40. The cost can be determined by finding the ratio of the weights and applying it to the given cost.

Let's set up a proportion to find the cost of 4.25 kg of potatoes. We know that 2.5 kg of potatoes cost $1.40. So, we can write the proportion as follows:

2.5 kg / $1.40 = 4.25 kg / x

To solve for x (the cost of 4.25 kg of potatoes), we cross-multiply:

2.5 kg * x = $1.40 * 4.25 kg

Simplifying the equation:

2.5x = $1.40 * 4.25

Multiplying the numbers:

2.5x = $5.95

Now, divide both sides of the equation by 2.5 to isolate x:

x = $5.95 / 2.5

Evaluating the division:

x = $2.38

Therefore, the cost of 4.25 kg of potatoes is $2.38.

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The boundary of 0 ≤ x ≤ π, 0 ≤ y ≤ sin x is 4. The value of the line integral is 4.

We want to apply Green's theorem to evaluate the integral ∫_C (6y dx + 8y dy), where C is the boundary of the region 0 ≤ x ≤ π, 0 ≤ y ≤ sin x.

Green's theorem states that for a continuously differentiable vector field F = (P, Q) and a piecewise smooth, simple closed curve C that encloses a region D in the plane, the line integral of F around C is equal to the double integral of the curl of F over D, i.e.,

∫_C F · dr = ∬_D ( ∂Q/∂x - ∂P/∂y ) dA,

where dr = (dx, dy) is the differential element of arc length along C, and dA = dxdy is the differential element of area in the xy-plane.

In our case, we have F = (6y, 8y), so that ∂Q/∂x - ∂P/∂y = 8 - 6 = 2. The region D is given by 0 ≤ x ≤ π, 0 ≤ y ≤ sin x, so we have

∫_C F · dr = ∬_D 2 dA = 2 ∫_0^π ∫_0^sin x dy dx.

The inner integral is simply ∫_0^sin x dy = sin x, so that

∫_C F · dr = 2 ∫_0^π sin x dx = 2 [-cos x]_0^π = 4.

Therefore, the value of the line integral is 4.

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Answer:

B

Step-by-step explanation:

B, this is the only one that is linear.

Probability of random variables

a) P[2Y < 1.9] = 0.475.

b) P[Y(n) < 1.9] ≈ 0.999999999999973

a) Since Y follows a Uniform(0, 2) distribution, we know that its density function is f(y) = 1/2 for 0 <= y <= 2. Therefore, we have:

P[2Y < 1.9] = P[Y < 0.95]

= [tex]\int^{0.95}_0 (1/2)dy + \int^{2}_{1.9/2} (1/2)dy[/tex]= (0.5)(0.95-0) + (0.5)(0-0.05/2)

= 0.475

Therefore, P[2Y < 1.9] = 0.475.

b) Since the Y's are independent, we have:

P[min(Y1, Y2, ..., Y100) < 1.9] = 1 - P[Y1 >= 1.9, Y2 >= 1.9, ..., Y100 >= 1.9]

[tex]= 1 - (P[Y > = 1.9])^{100}\\= 1 - ((2-1.9)/2)^{100}\\= 1 - (0.05/2)^{100}\\[/tex]

≈ 0.999999999999973

Therefore, P[Y(n) < 1.9] ≈ 0.999999999999973.

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Answer:

c, 7/11

Step-by-step explanation:

there are 11 marbles total. 7 aren't blue. so p(not blue) = 7/11. Answer C.

Answer:(a) Proportion of attendeees having a room mate and studying for at least five hours a week = (26+14)/100 = 0.4

Proportion of attendeees not having a room mate and studying for at least five hours a week = (20+10)/100 = 0.3

(b) The expected table of students, if there was no association between number of hours spent studying and having a room mate is as below:

No Roommate One Roommate Totals

Three 13.5 16.5 30

Five 20.7 25.3 46

More than five 10.8 13.2 24

Totals 45 55 100

In the above table, 13.5 is derived as 30*45/100; 16.5 is derived as 30*55/100; 20.7 is derived as 46*45/100 and so on.

Since the actual observed data are different, there seems to be some association, but we can't be sure if the association is postitive or negative .

(c) We have the Null Hypothesis, H0: No of hours spent studying and whether they have a roommate have no association.

and the Alternate Hypothesis, H0: No of hours spent studying and whether they have a roommate have an association.

We do the chi-square test in Excel, using the function CHITEST().

The p-value = 0.797

Since the p-value is high, we cannot reject the Null Hypothesis and conclude that there is no association between No of hours spent studying and having a roommate.

Step-by-step explanation:

Proportion with a roommate and study for at least 5 hours per week is 0.34

Proportion without a roommate and study for at least 5 hours per week is 0.36

Proportion with a roommate and study for at least 5 hours per week

= (20 + 14) / 100 = 34 / 100

= 0.34

Proportion without a roommate and study for at least 5 hours per week

= (26 + 10) / 100 = 36 / 100

= 0.36

Among attendees with no roommates, 46 out of 45 (approximately 1.02) proportionally studied for at least 5 hours per week.

Among attendees with one roommate, 40 out of 55 (approximately 0.73) proportionally studied for at least 5 hours per week.

From these calculations, we can infer that a higher proportion of attendees without a roommate studied for at least 5 hours per week compared to those with one roommate. This suggests a potential negative association between having a roommate and studying for at least 5 hours per week.

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