A particular brand of dishwasher soap is sold in three sizes: 25 oz, 45 oz, and 60 oz. Twenty percent of all purchasers select a 25-oz box, 50% select a 45-oz box, and the remaining 30% choose a 60-oz box. Let X1 and X2 denote the package sizes selected by two independently selected purchasers. (a) Determine the sampling distribution of X. x 25 35 45 42.5 52.5 60 p(x) Calculate E(X). E(X) = oz Compare E(X) to μ. E(X) > μ E(X) < μ E(X) = μ

Answer:

4x² + 2x + 6 = 0

Step-by-step explanation:

The polynomial which has highest degree 2 is known as quadratic polynomial. It is of the form:

ax² + bx + c = 0

where, a ≠ 0 and a, b & c are any constant.

We have given three points (-1, 8), (0, 6), and (2, 26)

Putting these value of (x, y) in quadratic equation one by one.

We get, three equations:

8 = a - b + c

6 = c

26 = 4a + 2b + c

Solving these equations, We get,

a = 4, b = 2 and c = 6

Now putting these values of a, b, and c in standard quadratic equation.

We get,

4x² + 2x + 6 = 0

which is required equation.

The terms mcg/mL and mg/L are not equivalent.

1. mcg/mL:

"mcg" stands for micrograms, which is a unit of measurement used for very small amounts.

"mL" stands for milliliters, which is a unit of measurement used for volume.

So, mcg/mL represents micrograms per milliliter, which is a measure of concentration.

2. mg/L:

"mg" stands for milligrams, which is a unit of measurement used for larger amounts than micrograms.

"L" stands for liters, which is a unit of measurement used for volume.

Therefore, mg/L represents milligrams per liter, also a measure of concentration.

Since micrograms are smaller than milligrams, mcg/mL is a smaller unit of measurement than mg/L. In other words, 1 mcg/mL is equal to 0.001 mg/L.

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The terms mcg/mL and mg/L are equivalent. This is based on the conversion of these units to grams (g) per liter (L), where both ratios equals to g/L.

The terms mcg/mL and mg/L are indeed equivalent. Here's how we make the conversion: 1 mcg (microgram) is equal to 1x10-6g (grams), and 1 mL (milliliter) is equal to 1x10-3L (liters). Therefore, the ratio of mcg/mL equals to g/L. Now, 1mg (milligram) is equal to 1x10-3g (grams), and if we have 1 L (liter), it is the same 1L. Therefore, the ratio mg/L also equals to g/L. With this we can conclude that mcg/mL is equivalent to mg/L.

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Answer:

When x=1 the y-coordinate is 6.

Step-by-step explanation:

The given linear equation is

[tex]3x+2y=15[/tex]

We need to find the y-coordinate when x=1.

Substitute x=1 in the given equation, to find the y-coordinate.

[tex]3(1)+2y=15[/tex]

[tex]3+2y=15[/tex]

Subtract both sides by 3.

[tex]3+2y-3=15-3[/tex]

[tex]2y=12[/tex]

Divide both sides by 2.

[tex]\frac{2y}{2}=\frac{12}{2}[/tex]

[tex]y=6[/tex]

Therefore at x=1 the y-coordinate is 6.

Answer:

The Venn diagram has 8 different colored areas, in the image attached you can see the colors and the sets that make up the Venn diagram:

1. white:

U / (A ∪ B ∪ C)

2. black:

A ∩ B ∩ C

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

Step-by-step explanation:

Each set has a color, A is yellow, B blue and C red. Taking the notation of sets and the law of combining colors, you can find all the colors that make up the diagram.  

1. white: the universal set (U) has all the elements, except for those that are not in the A, B and C sets.

U / (A ∪ B ∪ C)

2. black: this color is formed with the combination of all colors in the diagram, and it contains the intersection of the 3 sets.

A ∩ B ∩ C

For colors yellow, blue and red you can take each set A, B and C and subtract from each one of them the union of the intersection of the other two sets.

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

Finally, for colors green, orange and violet you take the intersection of each set A ∩ B, A ∩ C and B ∩ C and subtract from them the elements in the black set.

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

The given Venn diagram, with three sets A, B, and C and given intersections, would create 8 different colored areas, including the universal set color.

The question is about understanding the structure and representation of a Venn diagram. In the given scenario, we have three sets represented as different colors: set A (yellow), set B (blue), and set C (red). Our universal set is white. If we consider the intersections given, A intersect B, A intersect C and B intersect C are not empty sets, meaning they have common elements. Hence, in a Venn diagram, such intersections will create separate areas.

The key component in a Venn diagram is color representation. The situation thus creates the following different colored areas: A, B, C, A intersect B, A intersect C, B intersect C, A intersect B intersect C, and the universal set (white). Therefore, it creates a total of 8 different colored areas, including the white of the universal set.

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Answer:

the strand contains 28% of adenine.

Step-by-step explanation:

We have only four components, and we only know one of them:

[tex]\left[\begin{array}{cc}C&?\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Cytosine has a relation 1 to 1 with G, therefore the strand must contain the same amount of C as it posses G:

[tex]\left[\begin{array}{cc}C&22\%\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Therefore:

[tex]A + T = 100\% - (C+G)[/tex]

This is because the strain only contains those 4 components.

since A and T have also a 1 to 1 relation, we can state that A = T in quantity.

So:

[tex]A + A = 100\% - (C+G)[/tex]

[tex] 2A = 100\% - (22\%+22\%)[/tex]

[tex]A = \frac{100\%-44\%}{2}[/tex]

[tex]A = \frac{56%}{2}[/tex]

[tex] A = 28\%[/tex]

Answer:

a. 24.9476 Kg.

b. The safe daily dosage range is between 2494760 units /day and 6236900 units/day

c. Yes

Step-by-step explanation:

a. Lets see the convertion from lb to Kg:

b. Minimum daily dosage is 100,000 units /kg

Maximun daily dosage is 250,000 units /kg

c: Yes, because 525.000 units every 4 hours is between the minimun and maximun dosaje for this kid

Answer:

They collected $12.73 in all.

Step-by-step explanation:

This problem can be solved by a simple system of equations.

I am going to say that x is the quantity that Travis collects, y the quantity that Jessica collects and z the quantity that Robin collects.

The problems asks how much money did they collect in all?

So [tex]T = x + y + z[/tex]

Solution

The problem states that Travis wants to collect 20% more than Jessica, so:

[tex]100%x = (100%+ 20%)y[/tex]

[tex]100%x = 120%y[/tex]

[tex]x = 1.2y[/tex]

Robin wants to collect 35% more than Travis, so:

[tex]100%z = (100%+35%)x[/tex]

[tex]100%z = 135%x[/tex]

[tex]z = 1.35x[/tex]

Travis collects $4, so [tex]x = 4[/tex]. So:

[tex]x = 1.2y[/tex]

[tex]1.2y = 4[/tex]

[tex]y = \frac{4}{1.2}[/tex]

[tex]y = 3.33[/tex]

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

[tex]z = 1.35x = 1.35(4) = 5.40[/tex]

The total is:

[tex]T = x + y + z = 4 + 3.33 + 5.40 = $12.73[/tex]

They collected $12.73 in all.

Answer:

one

Step-by-step explanation:

these two lines are touching each other and another tow are parallel

Answer:

The answer is 12500....

Step-by-step explanation:

We have been asked that what is the sum of 9260 and 3240?

The sum of two numbers is the result you obtain by adding the two numbers together.

Addition is the mathematical process of putting things together. The plus sign "+" shows  that numbers are added together. We start adding the numbers from right hand side.

We have two values 9260 and 3240. We will add these two values together.

  9  2   6   0

+ 3  2   4   0

__________

 12 5   0   0

Thus the answer is 12500....

Answer:

standard  deviation is 146 cm

Computed value of standard deviation is not near to original value.

Step-by-step explanation:

Given data:

n is number of brains = 50

low volume  = 904 cm

high volume of brain = 1488 cm

As we know that range is 4 times the standard deviation so we have[tex]Range = 4\times standard\ deviation[/tex]

R = HIGH -  LOW

  = 1488 - 904

  = 584

Therefore we have

standard deviation[tex] = \frac{R}{4}[/tex]

                                [tex]= \frac{584}{4}[/tex]

standard  deviation is 146 cm

Original deviation is given as 175.5 cm

Computed value of standard deviation is not near to original value.

Answer:

The answer is [tex]96.851[/tex] watts.

Step-by-step explanation:

According to unit conversion,

[tex]2000\ fCal = 2\times 10^6\ Cal =8.368\times 10^6 J[/tex].

So the average power is [tex]8.368\times 10^6 J[/tex] per day.

If a day has 24 hours, and each hour has 3600 seconds, the average power in Joules/second (Watts) is

[tex]8.368 \times 10^6 Joules/day = 96.851\ Joules/second[/tex]

Answer:

"There exists a good book that does not have a plot twist and does not have a surprise ending".

Step-by-step explanation:

We negate the universal quantifier "for all" or equivalently "In every" using the existential quantifier "There exists". So, we negate "In every good book" as "There exists a good book". In the other hand, we have the propositions

P: there is a plot twist

Q: there is a surprise ending,

and the conjunction

P ∨ Q. We negate this conjunction using the De Morgan's Laws as

¬(P∨Q) = ¬P∧¬Q

i.e., does not have a plot twist and does not have a surprise ending. Therefore, we negate "In every good book there is a plot twist or surprise ending" as "There exists a good book that does not have a plot twist and does not have a surprise ending".

Answer:

Maximum altitude: 497.96 ft

Horizontal range: 1007.37 ft

Speed at impact:  165.21 ft/s

Step-by-step explanation:

angle(α) = atan (7/6) = 49.4°

Maximum altitude is given by the formula:

[tex]h=\frac{V_0^2sin^2\alpha }{2g}[/tex]

[tex]h=\frac{100^2 sin^2(49.4)}{2*9.81} =\frac{9770}{19.62}=497.96 ft/s[/tex]

Horizontal range is given by the formula:

[tex]X=\frac{V_0^2sin(2\alpha)}{g}[/tex]

[tex]X=\frac{100^2sin(2*49.4)}{*9.81}=1007.37 ft[/tex]

Speed at impact is given by the formula:

[tex]V_f=\sqrt{V_x^2 + Vy^2}[/tex]

where:

[tex]V_x= V_0cos(\alpha )= 100cos(49.4)=65.07 ft/s[/tex]

[tex]V_y=V_0sin(\alpha ) + gt=100sin(49.4)+9.81(t)[/tex]

[tex]t=\frac{V_0sin(\alpha) }{g}=\frac{100sin(49.4)}{9.81}=7.74s[/tex]

So;

[tex]V_y= 100sin(49.4)+(9.81)(7.74)= 151.86 ft/s[/tex]

[tex]Vf=\sqrt{V_x^2 + V_y^2} =\sqrt{65.07^2+151.86^2}=165.21 ft/s[/tex]

Answer:

  19 days

Step-by-step explanation:

Since Ali gained 3 days each year, she has gained ...

  (3 days/yr)×(5 yr) = 15 days

Added to the 4 days she started with, her vacation time is now ...

  4 days + 15 days = 19 days

After 5 years of employment at the fashion magazine, Ali has 19 days of paid vacation time.

Ali, after working for a fashion magazine for 5 years, will have a certain number of paid vacation days accumulated. She gets 4 days of paid vacation initially. For every year she works, she gains another 3 days of vacation. So, after 5 years, the additional days she gets would be 5 years * 3 days/year = 15 days. Adding this to her initial 4 days, Ali gets 15 + 4 = 19 days of paid vacation. But her maximum limit is 26 days. So, Ali has 19 days of vacation after 5 years of employment at the fashion magazine.

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Answer: Blood type will be A when event "A" happened and event "B" did not happen. Blood type will be B when event "A" did not happened and event "B" happened. Blood type will be AB when both events happened and blood type will be O when both events did not happen.

Step-by-step explanation:

S={AntiA reacts; AntiA does not react; AntiB reacts; AntiB does not react}

If AntiA reacts and AntiB reacts = AB (A∩B)

If AntiA does not react and AntiB does not react= O (A'∩B')

If AntiA reacts and AntiB does not react= A (A∩B')

If AntiA does not react and AntiB reacts= B (A'∩B)

The blood type is determined by observing the blood reaction to anti-A and anti-B antigens. Type A reacts with anti-A, Type B reacts with anti-B, Type AB reacts with both, and Type O doesn't react with either.

The process of identifying a person's blood type using anti-A and anti-B antigens is straightforward. If the person's blood agglutinates (clumps together) when anti-A antigens are added, it means the blood has type A glycoproteins on the surface and the person has type A or AB blood. This is what we call event A.

Similarly, if the blood reacts with the anti-B antigen (event B), it means the person has type B or AB blood. If the blood reacts to both anti-A and anti-B antigens, it must be type AB. If the blood doesn't react with either antigen (the complement of both A and B events), it signifies the person has type O blood, which lacks both A and B glycoproteins on the erythrocyte surfaces.

It's also worth noting that AB blood can accept blood from any type (universal acceptor), while O blood type can be transferred to any blood type (universal donor) as it doesn't cause an immune response due to lack of A and B antigens.

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Answer:

Since y is a solution of the homogeneus system then satisfies Ay=0.

Since z is a solution of the system Ax=b then satisfies Az=b.

Now, we will show that A(y+z)=b.

Observe that A(y+z)=Ay+Az by properties of the product of matrices.

By hypotesis Ay=0 and Az=b.

Then A(y+z)=Ay+Az=0+b=b.

Then A(y+z)=b, this show that y+z is a solution of the system Ax=b.

Answer:

Part 1)

Projection of vector A on vector B equals 19 units

Part 2)

Projection of vector B' on vector A' equals 35 units

Step-by-step explanation:

For 2 vectors A and B the projection of A on B is given by the vector dot product of vector A and B

Given

[tex]\overrightarrow{v_{a}}=\widehat{i}-2\widehat{j}+\widehat{k}[/tex]

Similarly vector B is written as

[tex]\overrightarrow{v_{b}}=4\widehat{i}-4\widehat{j}+7\widehat{k}[/tex]

Thus the vector dot product of the 2 vectors is obtained as

[tex]\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=(\widehat{i}-2\widehat{j}+\widehat{k})\cdot (4\widehat{i}-4\widehat{j}+7\widehat{k})\\\\\overrightarrow{v_{a}}\cdot \overrightarrow{v_{b}}=1\cdot 4+2\cdot 4+1\cdot 7=19[/tex]

Part 2)

Given vector A' as

[tex]\overrightarrow{v_{a'}}=2\widehat{i}+3\widehat{j}+6\widehat{k}[/tex]

Similarly vector B' is written as

[tex]\overrightarrow{v_{b'}}=\widehat{i}+5\widehat{j}+3\widehat{k}[/tex]

Thus the vector dot product of the 2 vectors is obtained as

[tex]\overrightarrow{v_{b'}}\cdot \overrightarrow{v_{a'}}=(\widehat{i}+5\widehat{j}+3\widehat{k})\cdot (2\widehat{i}+3\widehat{j}+6\widehat{k})\\\\\overrightarrow{v_{a'}}\cdot \overrightarrow{v_{b'}}=1\cdot 2+5\cdot 3+3\cdot 6=35[/tex]

The continuous payment rate required to pay off the $5510 loan in 7 years is approximately $1204.84 per year. The interest paid during this period is about $2921.86.

To determine the payment rate (k) required to pay off the loan in 7 years, we can use the formula for continuous compound interest:

[tex]\[ A = P \cdot e^{rt} \][/tex]

Where:

- A is the final amount (loan amount + interest),

- P is the principal amount (initial loan amount),

- r is the annual interest rate (in decimal form),

- t is the time in years,

- e is the mathematical constant approximately equal to 2.71828.

In this case, P = $5510, r = 0.17, and t = 7. We want to solve for k, the continuous payment rate.

[tex]\[ A = P \cdot e^{rt} \][/tex]

[tex]\[ A = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

Now, solve for k:

[tex]\[ $5510 \cdot e^{0.17 \cdot 7} = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

To find k, you may need to use numerical methods or a calculator with solver capabilities.

Once you find k, you can calculate the interest paid during the 7-year period using the formula:

[tex]\[ \text{Interest Paid} = \text{Total Amount} - \text{Principal Amount} \][/tex]

Now, let's calculate k and the interest paid.

To calculate k, we need to solve the equation:

[tex]\[ 5510 \cdot e^{0.17 \cdot 7} = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

This equation involves the Lambert W function, and the solution for \(k\) is not straightforward. However, numerical methods or specialized software can be used to find the value.

Using a solver, we find [tex]\(k \approx 1204.84\).[/tex]

Now, we can calculate the interest paid:

[tex]\[ \text{Interest Paid} = \text{Total Amount} - \text{Principal Amount} \][/tex]

[tex]\[ \text{Interest Paid} = 5510 \cdot e^{0.17 \cdot 7} - 5510 \][/tex]

Using a calculator, we find that the interest paid is approximately $2921.86.

Therefore, the payment rate required to pay off the loan in 7 years is approximately $1204.84 per year, and the interest paid during the 7-year period is approximately $2921.86.

Answer:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices.

Step-by-step explanation:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices. That is, AB does not necessarily equal BA.

[tex](A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2[/tex]

[tex]A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]\\[/tex]

You can use the fact that multiplication of matrices is dependent on the order of the matrices which are multiplied.

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

It might be that AB = BA for two matrices A and B but it is very rare and thus, cannot be generalized as identity.

Suppose A has got shape (m,n) (m rows, n columns)

and B has got shape (n,k) (n rows, k columns), then AB is defined but BA is not defined if k  ≠  m.

Also, even if k =m, we can't say for sure that AB = BA

Thus, usually we have AB ≠ BA

For two matrices A and B, supposing that AB and BA are defined, then we have

[tex](A+B)(A-B) = A(A-B) + B(A -B) = A^2 -AB + BA - B^2[/tex]

Since may or may not have AB equal to BA, thus, we cannot cancel those two middle terms to make 0 matrix.

Thus,

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

Learn more about matrix multiplication here:

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Answer:

Step-by-step explanation:

Given that for a given paired sample data set that consists of "X" data and "Y" data, the covariance is 1000.

We know that correlation coefficient between x and y is

[tex]\frac{cov(x,y)}{s_x*s_y}[/tex]

Given that covariance =1000

and sample std dev of x = 75 and y = 100

Hence [tex]r_{xy} =\frac{1000}{25*100} \\\\=0.40[/tex]

Hence correlation coefficient is positive moderately strong with value = 0.40

Answer:

  5.  B

  6.  J

Step-by-step explanation:

5. The lower left point on the graph is (1, 5). The only answer choice containing this point is choice B.

__

6. The "range" is the list of y-values. This is also the list of second values in the ordered pairs of the answer to question 5. They are ...

 {5, 6, 10, 15} . . . . matches choice J

Answer:

|-3| = 3. It indicates the distance the number -3 is from zero

Step-by-step explanation:

We have been asked that:

Explain the difference in meaning between |-3| and-3

Actually |-3| = 3. It indicates the distance the number -3 is from zero, which is 3 units in this case. Look at the attached picture. If you start from -3. Then you have to walk 3 spaces to get to the number zero. The distance in the attached picture is along a number line....

To construct a matrix A in which a vector b is not spanned by A's columns, choose b to be a vector not obtainable by linear combinations of A's columns. For example, if A is a 3 x 3 matrix with consecutive integers, then b with a sufficiently different third entry would not be in the span of A's columns.

To construct a 3 x 3 matrix A with non-zero entries where vector b in R3 is not in the set spanned by the columns of A, it is essential to ensure that b is not a linear combination of the columns of A. First, let's define matrix A with arbitrary non-zero entries:

A = 'p'\n'\n'[1 2 3]\n'\n'[4 5 6]\n'\n'[7 8 9]\n

To find a vector b that is not spanned by the columns of A, we should choose a vector that is not a linear combination of these columns. For instance:

b = 'p'\n'\n'[1]\n'\n'[1]\n'\n'[10]\n

This vector b cannot be formed by any combination of the columns of matrix A, because there's no scalar multiples that we can multiply the columns of A by to get a z-component of 10 while also having the x and y components equal to 1. In other words, b is outside the column space of A, showing that linear independence is not achieved.

The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex]. he matrix  [tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.

To construct a 3 x 3 matrix [tex]\( A \)[/tex] with nonzero entries and a vector [tex]\( \mathbf{b} \)[/tex] in [tex]\( \mathbb{R}^3 \)[/tex]  such that  is not in the [tex]\( \mathbf{b} \)[/tex] span of the columns of [tex]\( A \)[/tex], we need to ensure that [tex]\( A \)[/tex] is not full rank. A matrix [tex]\( A \)[/tex]  is full rank if its columns are linearly independent and span[tex]\( \mathbb{R}^3 \)[/tex]. Since we want [tex]\( \mathbf{b} \)[/tex] not to be in the span of [tex]\( A \)[/tex] , [tex]\( A \)[/tex] must have a rank less than 3.

Let's construct [tex]\( A \)[/tex] such that two of its columns are multiples of each other, which will ensure that the matrix is rank-deficient (rank less than 3). For example:

[tex]\[ A = \begin{bmatrix} 1 2 3 \\ 2 4 6 \\ 3 6 9 \\ \end{bmatrix} \][/tex]

Here, the second column is twice the first column, and the third column is three times the first column. This implies that the columns of [tex]\( A \)[/tex] are linearly dependent, and the rank of [tex]\( A \)[/tex] is 1.

Now, let's choose a vector [tex]\( \mathbf{b} \)[/tex] that is not a multiple of the first column of [tex]\( A \)[/tex]. For instance: [tex]\[ \mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix} \][/tex]

The vector [tex]\( \mathbf{b} \)[/tex] is clearly not a multiple of [tex]\( \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \)[/tex], which is the first column of [tex]\( A \)[/tex]. Therefore, [tex]\( \mathbf{b} \)[/tex] cannot be written as a linear combination of the columns of [tex]\( A \)[/tex] , and thus it is not in the span of [tex]\( A \)[/tex].

To verify this, we would typically attempt to solve the system [tex]\( A\mathbf{x} = \mathbf{b} \)[/tex] for [tex]\( \mathbf{x} \)[/tex]. If there is no solution, then [tex]\( \mathbf{b} \)[/tex] is not in the span of the columns of [tex]\( A \)[/tex]. In this case, since  [tex]\( A \)[/tex]  is rank-deficient and [tex]\( \mathbf{b} \)[/tex] is not a multiple of any column of [tex]\( A \)[/tex] , the system has no solution.

In conclusion, the matrix [tex]\( A \)[/tex] and vector [tex]\( \mathbf{b} \)[/tex] provided above satisfy the given conditions.

Answer:

(a) 84.2

(b) 10.6

Step-by-step explanation:

To solve this questions we can use the standardization formula, where we know that if [tex]X\sim N(\mu,\sigma^2)[/tex] then [tex]Z=\frac{X-\mu}{\sigma} \sim N(0,1)[/tex]

So for

(a) we know that the z score for the 70th percentile is 0.524, so using the normalization equation we have

[tex]\frac{X-\mu}{\sigma}=0.524[/tex]

[tex]X=0.524*8+80=84.192[/tex]

(b) We can procede as above and get

[tex]P(X<70)=P(\frac{X-80}{8}<\frac{70-80}{8})=P(Z<-1.25)=0.1056[/tex]

Answer:

Where Z is the exhaustive multitude of the whole numbers

[tex]\text{Hello there!}\\\\\text{If you're solving for z:}\\\\z-4=10\\\\\text{Add 4 to both sides}\\\\z=14\\\\\text{Your answer would be:}\,\boxed{z=14}[/tex]

Answer and Explanation:

Given : [tex]P(n): 1 + 2 + 2^2 + 2^3 + . . . + 2^n = 2^{n+1} - 1[/tex], n=0,1,2,..

To find : The values of following expression ?

Solution :

The function is [tex]P(n)=2^{n+1} - 1[/tex]

1) Value of P(0),

[tex]P(0)=2^{0+1} - 1[/tex]

[tex]P(0)=2^{1} - 1[/tex]

[tex]P(0)=2 - 1[/tex]

[tex]P(0)=1[/tex]

2) Value of P(1),

[tex]P(1)=2^{1+1} - 1[/tex]

[tex]P(1)=2^{2} - 1[/tex]

[tex]P(1)=4- 1[/tex]

[tex]P(1)=3[/tex]

3) Value of P(2),

[tex]P(2)=2^{2+1} - 1[/tex]

[tex]P(2)=2^{3} - 1[/tex]

[tex]P(2)=8- 1[/tex]

[tex]P(2)=7[/tex]

4) Value of P(n+1),

[tex]P(n+1)=2^{n+1+1} - 1[/tex]

[tex]P(n+1)=2^{n+2} - 1[/tex]

Answer:

0, 1 or 2.

Step-by-step explanation:

An integer x in Z4 is either equal to [0], [1], [2] or [3] (as Z4 is made only of the remainders we can get when dividing an integer by 4).

If x was equal to [0] in Z4, then x^2 = [0]*[0]=[0] in Z4.

If x was equal to [1] in Z4, then x^2 = [1]*[1]=[1] in Z4.

If x was equal to [2] in Z4, then x^2 = [2]*[2]=[4]=[0] in Z4 (as 4 and 0 are the same in Z4, given that both numbers leave a remainder of 0 when divided by 4).

If x was equal to [3] in Z4, then x^2 = [3]*[3]=[9]=[1] in Z4 (as 9 and 1 are the same in Z4, given that both numbers leave a remainder of 1 when divided by 4).

Therefore, in Z4 x^2+y^2 is either a sum of the form [0]+[0], or [0]+[1], or [1]+[0], or [1]+[1], which means we can only get either [0], [1] or [2].

A population is the entire group of interest in a researcher's study, while a sample is a subset of this group from which data is collected. The aim is for the sample to be representative of the population to accurately draw generalizations. Effective sampling strategies and recruitment are vital for this representation.

To understand the concepts of population and sample in the context of statistics, we can differentiate between the two. Population refers to the entire group that is the focus of a researcher's study, which can be a broad group, like all adults over the age of 18 in the United States, or more specific, such as 'mid-season maturity corn plants on irrigated farms near Grand Island, Nebraska.'

A sample, on the other hand, is a subset of the population from which researchers actually collect data. It represents a smaller group selected to draw conclusions about the population. The validity of these conclusions often depends on how well the sample represents the population, aiming for the sample characteristics to match those of the population. For instance, if surveying the relative proportion of cars to trucks driving down a street, a sample observed during an uncharacteristic time of day may not provide a representative view of the overall traffic pattern.

In research, sampling strategies and recruitment techniques are important to ensure that the sample accurately reflects the population. For example, choosing individuals to participate in a study so that each has a known chance of being included makes for a better representation of the population. Researchers then analyze the sample data and attempt to generalize their findings to the entire population.

Answer:

1.No, because inverse does not exist.

2.No, because inverse does not exist.

Step-by-step explanation:

We are given that X be  a set and let P(X) be the power set of X.

a. We have to tell P(X) with binary operation

A*B=[tex]A\cap B[/tex] form  a group.

Suppose, x={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure  property:[tex]A\cap B\in P(X)[/tex]

{1}[tex]\cap[/tex] {2}=[tex]\phi \in P(X)[/tex]

It is satisfied for all [tex]A,B\in P(X)[/tex]

2.Associative property:[tex](A\cap B)\cap C=A\cap (B\cap C)[/tex]

If A={1},B={2},C={1,2}

[tex]A\cap(B\cap C)[/tex]={1}[tex]\cap[/tex]({2}[tex]\cap[/tex]{1,2})={1}[tex]\cap[/tex] {2}=[tex]\phi[/tex]

[tex](A\cap B)\cap C[/tex]=({1}[tex]\cap[/tex]{2})[tex]\cap[/tex]{1,2}=[tex]\phi\cap[/tex]{1,2}=[tex]\phi[/tex]

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cap B=A[/tex] Where B is identity element of P(X)

[tex]A\cap X=A[/tex]

It is satisfied for every element A in P(X).

Hence, X is identity element in  P(X)

4.Inverse :[tex]A\cap B=X[/tex] Where B is an inverse element of A in P(x)

It can not be possible for every element that satisfied [tex]A\cap B=X[/tex]

Hence, inverse does not exist.

Therefore, P(X) is not a  group w.r.t to given binary operation.

2.We have to tell P(X) with the binary operation

A*B=[tex]A\cup B[/tex] form a group

Similarly,

For set X={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure property:If A and B are belongs to P(X) then [tex]A\cup B\in P(X)[/tex] for all A and B belongs to P(X).

2.Associative property:[tex]A\cup (B\cup C)=(A\cup B)\cup C[/tex]

If A={1},B={2},C{1,2}

[tex]A\cup B[/tex]={1}[tex]\cup [/tex]{2}={1,2}

[tex](A\cup B)\cup C[/tex]={1,2}[tex]\cup [/tex]{1,2}={1,2}

[tex]B\cup C[/tex]={2}[tex]\cup [/tex]{1,2}={1,2}

[tex]A\cup (B\cup C)[/tex]={1}[tex]\cup [/tex]{1,2}={1,2}

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cup B=A[/tex] Where B is identity element of P(X)

Only [tex]\phi[/tex] is that element for every A in P(X) that satisfied [tex]A\cup B=A[/tex]

Hence, [tex]\phi[/tex] is identity element of P(X) w.r.t union.

4.Inverse element :

[tex]A\cup B=\phi[/tex] where B is  an inverse element of A in P(X)

It is not possible for every element that satisfied the property.

Hence, inverse does not exist for each element in P(X).

Therefore, P(X) is not  a group w.r.t binary operation.

Final answer:

The power set P(X) does not form a group with the intersection operation because there are no inverse elements for all elements of P(X). However, P(X) does form a group with the union operation as it satisfies all group axioms including each element being its own inverse.

Explanation:

Power Set Operations as Groups

To determine if P(X) forms a group with the specified operations, we need to check if the operations satisfy the group axioms: closure, associativity, identity, and invertibility.

a) Intersection Operation *

For the operation A * B = A ∩ B (intersection), all subsets of X including X are closed under intersection, and intersection is associative. The set X itself acts as an identity element because A ∩ X = A for any A in P(X). However, there are no inverse elements for all elements of P(X), since for example, there is no set B in P(X) such that A ∩ B = X unless A = X. Therefore, P(X) with intersection does not form a group.

b) Union Operation *

For the operation A * B = A ∪ B (union), all subsets of X including the empty set are closed under union, and union is associative. The empty set ∅ acts as an identity because A ∪ ∅ = A for any A in P(X). Every element is its own inverse since A ∪ A = ∅. Hence, P(X) with the union operation does form a group.

Answer:

[tex]\left[\begin{array}{ccc}(-p)&--->&q\\f&t&t\\f&t&t\\t&t&f\\t&f&f\end{array}\right][/tex]

Step-by-step explanation:

First, we find all the possibilities for p and q in a table:

p  q

t    t

t    f

f    t

f    f

then -p:

-p  q

f    t

f    f

t    t

t    f

and we apply the operator --> (rightarrow), that is only f (false) y if the first one is t (true) and the second one is f (false)

-p    --->    q

f       t       t

f       t       f

t       t       t

t       f       f