(a) When a=0.01 and n=15, 2 Kieft 2 Xright

The component form and magnitude of the vector are;

v = ⟨-5, 3⟩ and ||v|| = √(34)

The difference between the points on the graph can be used to express the vector in component form as follows;

The component of the vectors are the horizontal and the vertical component

The horizontal component is; -(2 - (-3))·i = -5·i·

The vertical component is; ((5 - 2)·j = 3·j

The magnitude of the vector is; ||v|| = √((-3 - 2)² + (5 - 2)²) = √(34)

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The point (5,-2) is reflected over the y = -x. The correct option is (a).

To reflect a point over the line [tex]y = -x[/tex], we need to find the perpendicular distance from the point to the line, and then move the point by twice that distance in the direction perpendicular to the line.

The line [tex]y = -x[/tex] has a slope of -1 and passes through the origin. Therefore, its equation can be written as  

[tex]y=-x[/tex] reflects the point (5,-2)

These steps can be used to reflect a point over a line:

The slope of the line parallel to the reflection line should be determined. This will be the reflection line's slope's reciprocal in the negative direction. Because[tex]y = -x[/tex] in this instance has a slope of 1, the perpendicular line will also have a slope of 1.

A perpendicular line passing through a particular location has an equation; find it. The line's point-slope formula is: [tex]y - y1 = m(x - x1),[/tex] where [tex](x_{1} , y_{1})[/tex] is the provided point and m is the recently discovered slope. When we enter (5, -2) and m = 1, we obtain the result:

[tex]y - (-2) = 1(x - 5)[/tex]

=> [tex]y = x - 3.[/tex]

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Complete Question:

Point (-5,2) is reflected over the y-axis. Where is the new point located?

A. (5,-2)

B. (-5,-2)

C. (5,2)

D. (-5,2)​

The 22nd term of the arithmetic sequence is 59.

We have,

2nd term of Arithmetic sequence= 9

and, 8th term = 24

So, a + d = 9....(1)

a+ 7d = 24...........(2)

Solving equation (1) and (2) we get

7d - d = 24 - 9

6d = 15

d = 5/2

and, a = 9 - 5/2 = 13/2

Now, the 22nd term of the sequence

= a + 21d

= 13/2 + 21(5/2)

= 13/2 + 105/2

= 118 /2

= 59

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Mr. Well has a class of 24 students who were given an assignment to complete. Once they completed the assignment, they were allowed to play math games at the end of class. At the end of class, it was observed that 40% of the class was playing math games. This means that 60% of the class was not playing math games.

To find out what percentage of the class was still taking the test, we subtract 40% (those playing math games) from 100%. Thus, 100% - 40% = 60% of the class was still taking the test.

This information can be useful in determining how much time is needed to complete the test, and how much time can be allotted for math games. It is important to ensure that enough time is given to complete the test, while also allowing for some fun activities to keep the students engaged and motivated.

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To determine the probability that the power supply will be inadequate on any given day, we need to find the probability that the daily consumption of electric power exceeds 12 million kilowatt-hours. We have a gamma distribution with α = 3 and β = 2.

Step 1: Identify the parameters of the gamma distribution.
α = 3 (shape parameter)
β = 2 (scale parameter)

Step 2: Set up the problem.
We want to find the probability P(X > 12), where X is the random variable representing daily power consumption in millions of kilowatt-hours.

Step 3: Calculate the cumulative distribution function (CDF) for the given parameters at X = 12.
We can use a gamma CDF calculator or software to find the CDF. For example, using the R programming language, you can use the "pgamma" function:

pgamma(12, shape = 3, scale = 2)

Step 4: Calculate the probability of power supply being inadequate.
Since we want the probability of X > 12, we can subtract the CDF from 1 to obtain the probability:

P(X > 12) = 1 - CDF(12)

After calculating the CDF with the given parameters, you'll obtain the probability that the power supply will be inadequate on any given day.

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Answer:

9(6+11)=(9/6)*(9/11)

Step-by-step explanation:

The left side of the equation can be simplified as follows:

9(6+11) = 9(17) = 153

On the right side, we use the fact that the product of two fractions is the product of their numerators over the product of their denominators. So:

(9/6)[6/(9/11)] = (9/6) * (611/9) = 11

Therefore, the equation becomes:

153 = 11

which is not true for any value of the missing numbers in the equation. So there is no solution for the missing numbers.

you have a good day too and you're welcome!

Answer:

Step-by-step explanation:

If I found the mean, the answer would be:

3+ 4+4+4+5+6+7+23= 56

56/ 8 = 7

If I found the average value using the median, the answer would be 4.5.

In this set of data, the anomaly is 23 as it is much higher than the other numbers.

The median is more accurate because it find the more ‘central’ number and is not affected as greatly with anomalies whereas the mean is affected greatly with anomalies as it raises the value significantly.

Therefore, the median is better to work out the average in this set of data.

:)

Answer:

Step-by-step explanation:

All the edges of a cube have the same length, and the volume of a cube is the length of an edge taken to the third power. So the length of the edge of a cube with a volume of 125 is 5.

Answer:

true

true

True

true

False

Step-by-step explanation:

Converting mL (milliliters) to L (liters).

5,450 mL is equal to 5.45 L.

We have,

In the metric system, there are different units of measurement for volume, such as milliliters (mL) and liters (L).

One liter is equal to 1000 milliliters.

So, to convert a volume measurement from milliliters to liters, you need to divide the volume in milliliters by 1000.

This is because there are 1000 milliliters in one liter.

So, to convert 5,450 mL to L, you would divide by 1000 as follows:

5,450 mL ÷ 1000

= 5.45 L

Therefore,

5,450 mL is equal to 5.45 L.

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The number of variables recorded are three, and the types of variables are age (continuous), marital status (categorical), and earned income (continuous).

You have recorded three variables for each of the 1463 women in your sample. These variables are:

1. Age - a continuous quantitative variable, as it can take any value within a range.
2. Marital status - a categorical qualitative variable, as it represents distinct categories (e.g., single, married, divorced).
3. Earned income - a continuous quantitative variable, as it can take any value within a range, representing the income earned by each woman.

In total, you have recorded 1 qualitative and 2 quantitative variables for your sample.

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To construct a matrix with the required property (a), (d) & (e) are possible to construct the matrix. (b), (c) are not possible to construct the matrix.

(a) It is possible to construct a matrix with the given properties as follows:

[tex]\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]. The columns of this matrix span the column space, and the vector (-1,0,1) spans the nullspace.

(b) It is not possible to construct a matrix with the given properties because the dimensions of the column space and the nullspace are different. The column space is a subspace of [tex]R^3[/tex], whereas the nullspace is a subspace of[tex]R^1[/tex].

(c) It is not possible to construct a matrix with the given properties because the dimensions of the column space and the left nullspace are different. The column space is a subspace of[tex]R^3[/tex], whereas the left nullspace is a subspace of [tex]R^2[/tex].

(d) It is possible to construct a matrix with the given properties as follows:

[tex]\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]. The rows of this matrix span the row space, and the vector (1,0,3,2) spans the nullspace.

(e) It is possible to construct a matrix with the given properties as follows:

[tex]\left[\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right][/tex]. The rows of this matrix span the row space, and the vector (-3,1) spans the left nullspace.

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We calculated x by multiplying the elementary matrices E1, E2, and E3:

x = E1E2E3 =

(1 0)   (1 5)    (-1/5 0)

(4 1) * (0 1) *  (0    1) =

(-4 1)

(0 -1)

A matrix is a set of numbers that are arranged in rows and columns. Rows and columns are not always present in all matrices since some of them do not follow the same rule.

To express x as a product of elementary matrices, we need to perform elementary row operations on the identity matrix until we obtain x.

Starting with the 2x2 identity matrix:

I = (1 0)

   (0 1)

We can perform the following row operations to obtain x:

1. Add 4 times the first row to the second row:

E1 = (1 0)

    (4 1)

I * E1 = (1 0)

        (4 1)

2. Add 5 times the second row to the first row:

E2 = (1 5)

    (0 1)

(I * E1) * E2 = (1 5)

               (0 1)

3. Multiply the first row by -1/5:

E3 = (-1/5 0)

(0    1)

((I * E1) * E2) * E3 = (-4 1)

(0 -1)

Therefore, we have expressed x as the product of the elementary matrices E1, E2, and E3:

x = E1E2E3 =

(1 0)   (1 5)    (-1/5 0)

(4 1) * (0 1) *  (0    1) =

(-4 1)

(0 -1)

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For the following  equation, L.H.S = R.H.S is proved by solving the left-hand side and equating with it with the right-hand side equation :

[tex]\frac{tan x}{1-cot x} + \frac{cot x }{1-tan x} = 1+ tan x+ cot x[/tex]

L.H.S =  [tex]\frac{tan x}{1- cot x} + \frac{cot x }{1 - tan x}[/tex]

[tex]\frac{- tan^{2}x }{1- tan x} + \frac{cot x }{1 - tan x}[/tex]

[tex]\frac{-tan^{2} x + cot x}{1 - tan x}[/tex]

Multiply [tex]\frac{tan x}{tan x}[/tex] we get,

[tex]\frac{1- tan^{3} x}{tan x (1- tan x)}[/tex]

[tex]\frac{(1 - tan x ) (1 + tan x + tan ^{2}x) }{tan x (1 - tan x )}[/tex]

[tex]\frac{( 1- tan x + tan^{2}x) }{tan x}[/tex]

Divide each term separately,

[tex]\frac{1}{tan x} + \frac{tan x}{tan x} + \frac{tan^{2}x }{tan x}[/tex]

cot x + 1 + tan x

therefore, 1+ tan x + cot x = R.H.S

L.H.S = R.H.S, hence the theory is proved.

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The given net represents a composite solid formed by combining a rectangular prism and a triangular prism. The surface area of the solid is 450 square units, and it is calculated by finding the areas of all the faces and adding them together.

The solid represented by the given net is a composite shape formed by combining three rectangles and two right triangles. The rectangles are arranged adjacent to each other in the same orientation, while the two right triangles are attached to the ends of the rectangles.

Based on the given dimensions, we can visualize that the three rectangles form the top, middle, and bottom sections of a rectangular prism. The two right triangles form the ends of a triangular prism, which is attached to the rectangular prism.

To calculate the surface area of this solid, we need to find the areas of all the faces and then add them together. The rectangular prism has a total of five faces (top, bottom, front, back, and two sides), and the triangular prism has two faces (front and back).

The area of the top and bottom faces of the rectangular prism is the same, which is the product of the length and width of the rectangle. The total area of the top and bottom faces is (6 x 9) + (10 x 9) + (8 x 9) = 162 square units.

The area of the front and back faces of the rectangular prism is the product of the length and height of the rectangle, which is 6 x 8 = 48 square units. The total area of the front and back faces is 2 x 48 = 96 square units.

The area of the two sides of the rectangular prism is the product of the width and height of the rectangle, which is 9 x 8 = 72 square units. The total area of the two sides is 2 x 72 = 144 square units.

The area of the two triangles that make up the front and back faces of the triangular prism is (1/2) x base x height = (1/2) x 6 x 8 = 24 square units. The total area of the front and back faces is 2 x 24 = 48 square units. Adding up all the areas, we get the total surface area of the solid as: 162 + 96 + 144 + 48 = 450 square units.

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The remainder when 2197^631 is divided by 14 is 5.

To find the remainder when 2197^631 is divided by 14, we can use the concept of modular arithmetic. We want to find the remainder when 2197^631 is divided by 14, so we can write:

2197^631 ≡ x (mod 14)

where x is the remainder we are looking for.

To simplify this expression, we can first look at the remainders of the powers of 2197 when divided by 14. We can start with 2197^1, which has a remainder of 5 when divided by 14:

2197^1 ≡ 5 (mod 14)

We can then use this result to find the remainder of 2197^2:

2197^2 = (2197^1)^2 ≡ 5^2 ≡ 11 (mod 14)

Similarly, we can find the remainder of 2197^3:

2197^3 = (2197^2)*2197 ≡ 11*5 ≡ 9 (mod 14)

We can continue this process to find the remainders of higher powers of 2197, but we can also notice a pattern. The remainders seem to repeat after every 6 powers of 2197:

2197^1 ≡ 5 (mod 14)
2197^2 ≡ 11 (mod 14)
2197^3 ≡ 9 (mod 14)
2197^4 ≡ 3 (mod 14)
2197^5 ≡ 1 (mod 14)
2197^6 ≡ 5 (mod 14)

So, we can write:

2197^631 ≡ 2197^(6*105 + 1) ≡ (2197^6)^105 * 2197^1 ≡ 5^105 * 2197 (mod 14)

To simplify further, we can use the fact that 5^2 ≡ 11 (mod 14):

5^105 ≡ (5^2)^52 * 5 ≡ 11^52 * 5 ≡ 9*5 ≡ 11 (mod 14)

So, we have:

2197^631 ≡ 5^105 * 2197 ≡ 11 * 2197 ≡ 5 (mod 14)

Therefore, the remainder when 2197^631 is divided by 14 is 5.

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To find two subsets of H that are fields isomorphic to C under the induced addition and multiplication from H, we need to look for subfields of H that have the same structure as C. One way to do this is to look for subfields that contain a copy of the field of complex numbers C as a subfield.

Here are two possible subsets of H that meet this criteria:

1. The subfield generated by i and j: Let F be the subfield of H generated by i and j. That is, F is the smallest subfield of H that contains i and j. We can check that F is a field: it contains 0, 1, i, j, -i, -j, and all their sums and products. Moreover, F contains a copy of C as a subfield, namely the subfield generated by i. To see that F is isomorphic to C, consider the map phi:F->C defined by phi(a+bi+cj+dk) = a+bi, where a,b,c,d are real numbers. This map is a field isomorphic: it preserves addition, multiplication, and inverses, and it maps i to i.

2. The subfield generated by 1 and i+j: Let G be the subfield of H generated by 1 and i+j. That is, G is the smallest subfield of H that contains 1 and i+j. We can check that G is a field: it contains 0, 1, i+j, -(i+j), and all their sums and products. Moreover, G contains a copy of C as a subfield, namely the subfield generated by 1 and i. To see that G is isomorphic to C, consider the map psi:G->C defined by psi(a+b(i+j)) = a+bi, where a,b are real numbers. This map is a field isomorphism: it preserves addition, multiplication, and inverses, and it maps 1 to 1 and i+j to i.

Note that these two subfields are different from each other and from the original field H, but they have the same structure as C.

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The true statement about the two triangles is ΔDEF is not congruent to ΔTUV because ΔTUV cannot be mapped to ΔDEF by a single rotation, translation, or reflection. (option a).

Triangles are fundamental shapes in geometry that play a crucial role in various mathematical concepts.

When studying triangles, it is essential to understand their properties, such as congruence and similarity, and how they can be transformed through translations, rotations, and reflections. In this context, we can analyze the given statements about two triangles ADEF and ATUV.

The first statement says that ΔDEF is not congruent to ΔTUV because ΔTUV cannot be mapped to ΔDEF by a single rotation, translation, or reflection. Congruent triangles have the same size and shape, and can be transformed into one another through a combination of rotations, translations, and reflections.

Therefore, if ΔTUV cannot be transformed into ΔDEF through a single transformation, then they cannot be congruent. Hence, this statement is true.

Hence the correct option is (a).

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Answer:

15/8

Step-by-step explanation:

18/8 - 3/8 = 15/8

Answer:

15/8

Step-by-step explanation:

[tex]2 \times \frac{1}{4} - \frac{3}{8} [/tex]

First, combine the mixed fraction. 2 (2/1) is equal to 8/4 (multiply by 4/4). This comes out as 9/4.

9/4 - 3/8

Then, we need to multiply the first fraction by 2/2 to get a common denominator

18/8 - 3/8

Since the denominators are the same, we can subtract the numerators, 18 - 3 = 15.

The denominator is kept after doing the subtraction in the numerator. 15/8 is the answer, or 1 7/8 as a mixed fraction

All three reasons (1, 2, and 3) can be valid for using a normal probability model to represent the distribution of sample means.

1, 2, and 3 are correct.We can use a normal probability model to represent the distribution of sample means for the following reasons:

1. The sample is randomly selected. This ensures that each member of the population has an equal chance of being selected, reducing potential biases and allowing the use of a normal probability model.

2. The distribution of the variable in the population is normally distributed. When the population distribution is normal, the distribution of sample means will also be normally distributed, as stated by the Central Limit Theorem.

3. The sample size is large enough to ensure that sample means will be normally distributed. As the sample size increases, the distribution of sample means approaches a normal distribution, even if the original population distribution is not normal. This is also part of the Central Limit Theorem, which typically suggests a sample size of 30 or more.

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Results:

1) Function to describe the number of eggs remaining after the chef makes x omelettes = 108 - 3x

2) The function is a linear function.

To determine the function for the number of eggs remaining after the chef makes x omelettes, we have:

Given:

Chef uses 3 eggs for each omelette, so, number of eggs used to make x omelettes: eggs used = 3x

When the diner opened, there were 9 dozen eggs in the refrigerator, meaning:

initial eggs = 9 x 12 = 108 eggs

To find the number of eggs remaining after x omelettes, we subtract the number of eggs used from the initial number of eggs:

remaining eggs = initial eggs - eggs used

Substituting the values we calculated earlier, we get:

remaining eggs = 108 - 3x

Therefore, the function to describe the number of eggs remaining after the chef makes x omelettes = 108 - 3x.

2. The function above is a linear function because the variable x has a power of 1, which means that the rate of change of remaining eggs is constant.

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The solution to the given linear system is x = -91, y = 66, and z = 28. This was obtained using both Cramer's rule and solution using Gauss-Jordan elimination method is x = -3, y = 4, z = 2.

Using Inverse Method

The augmented matrix is

[1 1 1 5]

[2 3 5 8]

[4 5 2 2]

The determinant of the coefficient matrix is -9, so the system has a unique solution. The inverse of the coefficient matrix is

[-19 3 4]

[14 -2 -3]

[6 -1 -1]

The solution is

x = -19(5) + 3(2) + 4(0) = -91

y = 14(5) - 2(2) - 3(0) = 66

z = 6(5) - (1)(2) - (1)(0) = 28

Using Gauss-Jordan Elimination Method

The augmented matrix is

[1 1 1 5]

[2 3 5 8]

[4 5 2 2]

Using elementary row operations, the matrix can be reduced to

[1 0 0 -3]

[0 1 0 4]

[0 0 1 2]

The solution is

x = -3

y = 4

z = 2

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Using  both and levels of significance and the critical value, we can conclude that the percentage of spam emails sent by the huge firm is different from the percentage in a recent month.

The population proportion of spam emails in a recent month is p = 0.718.

A random sample of emails from a large corporation has a sample proportion of spam emails, p'= 0.645.

We want to test the hypothesis that the population proportion of spam emails in the large corporation is different from p = 0.718.

We will use both 0.05 and 0.01 levels of significance and the critical value method.

To test this hypothesis using the critical value method, we can follow these steps:

The null hypothesis is that the population proportion of spam emails in the large corporation is equal to 0.718:

H0: p = 0.718

The alternative hypothesis is that the population proportion of spam emails in the large corporation is different from 0.718:

Ha: p ≠ 0.718

We will use both 0.05 and 0.01 levels of significance. Since we have a large sample (np > 10 and n(1-p) > 10), we can use the z-test for proportions. The test statistic is calculated as:

z = ( p' - p) / sqrt(p(1-p)/n)

where n is the sample size.

Using a standard normal distribution table, the critical values for a two-tailed test at the 0.05 and 0.01 levels of significance are:

At the 0.05 level: ±1.96

At the 0.01 level: ±2.58

Step 4: Calculate the test statistic and p-value.

Using the formula for the test statistic and the given values, we get:

z = (0.645 - 0.718) / sqrt(0.718(1-0.718)/n)

Since we don't know the population standard deviation, we use the standard error estimated from the sample:

z = (0.645 - 0.718) / sqrt(0.718(1-0.718)/n) = -2.546 / sqrt(0.718(1-0.718)/n)

Using n = 1000 (a reasonable sample size for an email server), we get:

z = -2.546 / sqrt(0.718(1-0.718)/1000) = -9.386

The corresponding p-value for this test statistic is very small (less than 0.0001), indicating strong evidence against the null hypothesis.

At the 0.05 level of significance, the critical value is ±1.96, which does not include the calculated test statistic of -9.386. Therefore, we reject the null hypothesis and conclude that the population proportion of spam emails in the large corporation is different from 0.718.

At the 0.01 level of significance, the critical value is ±2.58, which also does not include the calculated test statistic of -9.386. Therefore, we reject the null hypothesis at this level of significance as well.

In conclusion, we have strong evidence to suggest that the proportion of spam emails in the large corporation is different from the proportion in a recent month (0.718).

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If two functions are f(x) and k(x), then, The correct option is C.

A mapping demonstrates the pairings of the components. It displays the input and output values of a function, much like a flowchart would.  Every element of the domain is associated with exactly one element of the range in a function, which is a unique kind of relation. A mapping demonstrates the pairings of the components.

It displays the input and output values of a function, much like a flowchart would. The two parallel columns of a mapping diagram.

The calculation is as follows:

If two functions are f(x) and k(x),

(f o g)(x) = f[g(x)]

Now according to the picture

We have to find the value of (h o k)(1).

(h o k)(x) = h[k(x)]

             = h[k(1)]

             = h(3) [Since, k(1) = 3]

             = 28 [Since, h(3) = 28]

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Correct Question:

Select the correct answer. Consider functions h and k. What is the value of ?

a) a 95% confidence interval for the mean lifetime of all LED TV is: (2.664, 2.936)

b)Rounding up, we need to add 28 more students to the sample.

c) The critical value for a one-tailed t-test with 95% confidence and n-1 degrees of freedom is t = 1.729.

Substituting the values, we get:

(a) To construct a 95% confidence interval for the mean lifetime of all LED TV, we can use the formula:

CI = X ± z*(s/√n)

where X is the sample mean, s is the sample standard deviation, n is the sample size, z is the critical value from the standard normal distribution corresponding to the desired confidence level.

Given:

Sample mean X = 2.8 years

Sample standard deviation s = 0.45 years

Sample size n is unknown

Confidence level = 95%

Since we do not know the sample size n, we can use the t-distribution instead of the standard normal distribution to find the critical value. With a 95% confidence level and n-1 degrees of freedom, the critical value is t = 2.093.

Substituting the values, we get:

CI = 2.8 ± 2.093*(0.45/√n)

To find the sample size n, we can solve for it by setting the margin of error to half of the width of the confidence interval, which is equal to 2.093*(0.45/√n):

0.5*(2.093*(0.45/√n)) = 0.025

Simplifying and solving for n, we get:

n ≈ 78

Therefore, a 95% confidence interval for the mean lifetime of all LED TV is:

CI = 2.8 ± 2.093*(0.45/√78) = (2.664, 2.936)

(b) To be 90% confident and have a maximum error of estimation of 0.2, we can use the formula:

n = (z*s/E)^2

where E is the maximum error of estimation and z is the critical value from the standard normal distribution corresponding to the desired confidence level.

Given:

Confidence level = 90%

Maximum error of estimation E = 0.2

Sample standard deviation s = 0.45 years

The critical value corresponding to a 90% confidence level is z = 1.645.

Substituting the values, we get:

n = (1.645*0.45/0.2)^2 ≈ 27.95

Rounding up, we need to add 28 more students to the sample.

(c) To test if the mean GPA is more than 2.5 at a 5% level of significance, we can use a one-tailed t-test with the null and alternative hypotheses:

H0: μ ≤ 2.5

Ha: μ > 2.5

where μ is the population mean GPA.

Given:

Sample mean X = 2.8 years

Sample standard deviation s = 0.45 years

Sample size n is unknown

Level of significance = 5%

We do not know the population standard deviation, so we will use a t-distribution with n-1 degrees of freedom. The test statistic is calculated as:

t = (X - μ) / (s/√n)

To reject the null hypothesis at a 5% level of significance, the t-value must be greater than the critical value from the t-distribution with n-1 degrees of freedom and a one-tailed probability of 0.05. Since the alternative hypothesis is one-tailed, we only need to look up the upper tail of the t-distribution.

The critical value for a one-tailed t-test with 95% confidence and n-1 degrees of freedom is t = 1.729.

Substituting the values, we get:

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To isolate f in an equation, we make f the subject of the equation

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

The statement that represents isolating the variable

Take for instance, the equation is

bc + fc = k

To isolate f we make f the subject

So, we have

f = (k - bc)/c

Hence, isolating f means solving for f

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The probability of rolling 1 and 2 is 1/36 of an experiment consists of rolling two fair number cubes.

Hence, the correct option is A

In the sample space diagram of rolling two number cubes, there are 36 equally likely outcomes since there are 6 possible outcomes for the first cube and 6 possible outcomes for the second cube. The outcome of rolling a 1 and 2 is shown in the sample space diagram and there is only one such outcome.

Therefore, the probability of rolling a 1 and 2 is

P(1 and 2) = (number of favorable outcomes) / (total number of outcomes) = 1/36

So the probability of rolling a 1 and 2 is 1/36.

Hence, the correct option is A

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P(X < Y) = 0.1 + 0.2 + 0.15 = 0.45

P(X > Y) = 0.25 + 0.15 = 0.4.

(a) Since the sum of probabilities for each value of X must be equal to 1, we have:

0.1 + a + b = 0.45

c = 0.25

d + e = 0.3

f = 0.15

Also, since the sum of probabilities for each value of Y must be equal to 1, we have:

a + c + d = 0.3

b + e + f = 0.15

Using these equations, we can solve for the unknowns:

a + b = 0.35

a + b + c = 0.7

d + e = 0.3

f = 0.15

From the first two equations, we get:

c = 0.35 - a - b

Substituting this into the equation for Y probabilities, we get:

a + 0.25 + d = 0.3 - 0.35 + a + b + d

0.65 = 2a + b

Using the equation for X probabilities, we get:

a + b = 0.35

d + e = 0.3

Solving for a, b, d, and e, we get:

a = 0.15

b = 0.2

d = 0.15

e = 0.15

Substituting these values back into the equation for Y probabilities, we get:

c = 0.35 - a - b = 0

And for X probabilities, we get:

f = 0.15

Therefore, the values of a, b, c, d, e, and f are:

a = 0.15, b = 0.2, c = 0, d = 0.15, e = 0.15, f = 0.15.

(b) P(X = Y) is the sum of the probabilities along the diagonal of the table. From the table, we can see that P(X = Y) = 0.15.

P(X < Y) is the sum of the probabilities in the upper triangle of the table, and P(X > Y) is the sum of the probabilities in the lower triangle. From the table, we can see that:

P(X < Y) = 0.1 + 0.2 + 0.15 = 0.45

P(X > Y) = 0.25 + 0.15 = 0.4.

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To prove that a² is algebraic, we need to find a polynomial in F[x] that has a² as a root. Let's start by considering the polynomial with coefficients depending on a's:

p(x) = (x - a²)

If we substitute x = a into this polynomial, we get:

p(a) = (a - a²) = a(1 - a)

Since a is a root of ao + a1x +... + anx^n, we know that:

ao + a1a +... + ana^n = 0

Multiplying both sides by a, we get:

a ao + a1a² +... + ana^{n+1} = 0

Substituting a(1 - a) for a², we get:

a ao + a1(a(1 - a)) +... + an(a(1 - a))^n = 0

Simplifying, we get:

a ao + a1a(1 - a) +... + ana^n(1 - a)^n = 0

Multiplying both sides by (1 - a)^n, we get:

a ao(1 - a)^n + a1a(1 - a)^{n+1} +... + ana^n(1 - a)^{2n} = 0

Now, let's group the terms by even powers of a and odd powers of a:

a^0(1 - a)^n ao + a^2(1 - a)^{n+1} a1 +... + a^{2n}(1 - a)^{2n} an = 0

This is a polynomial in F[x] with a² as a root. Therefore, we have proved that a² is algebraic.

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