A group of adult males has foot lengths with a mean of 28.48 cm and a standard deviation of 1.41 cm. Use the range rule of thumb for identifying significant values to identify the limits separating values that are significantly low or significantly high. Is the adult male foot length of 31.6 cm significantly low or significantly high? Explain. Significantly low values are cm or lower. (Type an integer or a decimal. Do not round.)

Answer:

10 miles per hour

Step-by-step explanation:

If you run a mile in 6 minutes...

10miles x 6minutes is 60 minutes. An hour.

You can run 10 miles in the hour at that rate. Does that make sense?

Answer:

Step-by-step explanation:

1. b. Alt. Exterior angles

  a. 7x + 8 = 6x + 18

   c.    x + 8 = 18

          x = 10

   d. 6(10) + 18 = 60 + 18 = 78

       7(10) + 8 = 70 + 8 = 78

2. 15x + 10 + 12x + 8 = 180

     same side angles

   27x + 18 = 180

   27x = 162

   x = 6

15(6) + 10 = 90 + 10 = 100 degrees

12(6) + 8 = 72 + 8 = 80 degrees

3. 12x + 2 = 11x + 6

    x + 2 = 6

    x = 4

12(4) + 2 = 48 + 2 = 50 degrees

11(4) + 6 = 44 + 6 = 50 degrees

corresponding angles

Answer:

-1

Step-by-step explanation:

6 (y - 2) = -18

6y - 12 = -18

6y = -18 + 12

6y = -6

y = -1

Answer:

- 14x - 9y = 6

Step-by-step explanation:

- 7x - y = 0 → (1)

7x + 8y = - 6 → (2)

Subtracting (2) from (1) term by term

- 7x - 7x - y - 8y = 0 - (- 6)

- 14x - 9y = 0 + 6

- 14x - 9y = 6

The best approximation for the area of this circle is approximately 50.2 m².Option (c) is the correct answer.

To determine the best approximation for the area of this circle, we need to use the formula for the area of a circle, which is given by A = πr².

Here, we are given the value of π to be approximately equal to 3.14.

Now, we need to determine the radius of the circle.

From the diagram, we can see that the diameter of the circle is 8 meters.

Therefore, the radius is half of this, which is 4 meters.

Substituting the values of π and r into the formula, we get: A = πr²= 3.14 × 4²= 3.14 × 16= 50.24 (to two decimal places)

Therefore, the best approximation for the area of this circle is approximately 50.2 m².Option (c) is the correct answer.

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Mr. Blanco has the right to refuse treatment, even if it means he will die. The doctor should respect Mr. Blanco's decision and engage in open and honest communication with him to understand his concerns.

1. The ethical principles pertinent to this case include autonomy, beneficence, non-maleficence, and respect for patient's wishes. Autonomy refers to Mr. Blanco's right to make decisions about his own healthcare. Beneficence and non-maleficence require the healthcare professionals to act in Mr. Blanco's best interest and do no harm. Respect for patient's wishes entails honoring his decision and preferences.

2. Yes, Mr. Blanco should have the right to refuse treatment even if the consequence may be that he will die. It is his autonomy and right to make decisions about his own body and medical treatment. As long as he is mentally competent, he has the right to make choices about his healthcare, even if they go against medical advice.

3. It is important to consider the possibility that Mr. Blanco may be seriously depressed and not mentally competent to make the decision. In such cases, a thorough evaluation of his mental capacity and emotional state should be conducted by healthcare professionals to ensure that he is able to make an informed decision. If he is found to lack mental competence, appropriate steps should be taken to protect his best interests.

4. Given that Mr. Blanco had previously given his written consent for the surgery but verbally withdrew it later, the doctor should respect his current decision. A patient's consent can be withdrawn at any time, and healthcare professionals should prioritize the patient's current wishes. It is essential to engage in open and honest communication with Mr. Blanco to understand his concerns and explore alternative options if available.

5. The decision may be different if Mr. Blanco were refusing treatment for another condition such as terminal cancer or irreversible heart failure. In such cases, the prognosis and potential outcomes may be different, and the balance between preserving life and respecting autonomy may shift. Each case should be evaluated individually, considering the patient's medical condition, prognosis, quality of life, and their values and preferences. Open discussions with the patient, their family, and the healthcare team are crucial in reaching the most ethically appropriate decision.

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1/4+1/5+1/6=1/x

LCM = 60

15/60+12/60+10/60=1/x

37/60=1/x

37x=60

x=60/37=1.62, round =1.6

1.6 HOURS

The number of hours they would take them all working together is around 1.6 hours.

A ratio is the divide of the two variables and the lower variable known as the denominator should not be zero.

Given that laine takes 4 hours to complete work

In 1 hour Laine will do 1/4 part of the work.

In 1 hour Leslie will do 1/5 part of the work.

In a 1-hour Lance will do 1/6 part of the work.

so let's suppose all done work by x hours then

1/4 + 1/5 + 1/6 = 1/x

x =1.6 so all together done the work by 1.6 hours.

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Answer:

B. 2 12

Step-by-step explanation:

maaf kalo salah ituh jawabbanya menurut guwah

a) Probability of drawing two cards summing to 21: 1/331

b) Probability of drawing two cards summing to 10: 28/1326

c) Probability of drawing a third card to make the sum strictly larger than 21: 2/25

a) To calculate the probability of drawing two cards that sum to 21, we need to determine the number of favorable outcomes and the total number of possible outcomes.

There are four ways to get a sum of 21 with two cards: drawing an Ace and a face card (4 possibilities).

The total number of possible outcomes is given by the combination of choosing two cards from a deck of 52 cards, which is C(52, 2) = 1326.

Therefore, the probability of drawing two cards that sum to 21 is 4/1326, which simplifies to 1/331 or approximately 0.0030.

b) To calculate the probability of drawing two cards that sum to 10, we need to determine the number of favorable outcomes.

There are several combinations that sum to 10: drawing a 4 and a 6, drawing a 5 and a 5, drawing a 6 and a 4, and drawing a face card and a 10.

The total number of possible outcomes remains the same, which is 1326.

Therefore, the probability of drawing two cards that sum to 10 is (4 + 4 + 4 + 16)/1326, which simplifies to 28/1326 or approximately 0.0211.

c) Given that you have already drawn the 10 of clubs and the 4 of hearts, the sum of these two cards is 10 + 4 = 14.

To find the probability that the sum of all three cards is strictly larger than 21, we need to consider the remaining cards in the deck. Since there are 50 cards left in the deck, we calculate the probability of drawing a card that makes the sum exceed 21.

The only way to exceed 21 is to draw an Ace, which is worth 11. There are four Aces in the deck.

Therefore, the probability of drawing a card that makes the sum strictly larger than 21 is 4/50, which simplifies to 2/25 or approximately 0.08.

Complete question:

Assume that in blackjack, an ace is always worth 11, all face cards (jack, queen, king) are worth 10, and all number cards are worth the number they show. given a shuffled deck of cards:

a) What is the probability that you draw two cards and they sum 21

b) What is the probability that you draw two cards and they sum 10

c) Suppose, you have drawn two cards: 10 of clubs and 4 of hearts. You now draw a third card from remaining 50. What is the probability that the sum of all three cards is strictly larger than 21?

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Answer:

-62.8 is the Answer fot the equation, Hope this helped

please give me brainiest i was working on it the whole time.

Answer: 4* 2 3/4* 8 1/2= 93 1/2

The sets A {1.2,3,4,6,12}, A{1,2,3,4,6,12,18,24}, and A{1,2,3,6,12} all yield a poset that is a total ordering.

In order for a partially ordered set (poset) to be a total order, every pair of elements must be comparable, i.e., for every pair of elements x, y in A, either x | y or y | x must be true.

Therefore, we should choose a set A that satisfies this condition.Let us consider each set in turn:(1, 4, 16, 64)This set contains powers of 4, which are not all divisible by each other.

For example, 16 | 64, but 16 does not divide 4.

Hence, this set is not a total order.{1.2,3, 4, 6, 12}In this set, 1 divides every other element, so it satisfies the condition for being a total order. Therefore, this set yields a poset that is a total order.

(1, 2, 3, 4, 6, 12, 18, 24)This set has the same elements as the previous set, plus some more. Hence, it also satisfies the condition for being a total order.

Therefore, this set yields a poset that is a total order.{1 , 2, 3, 6, 12}In this set, 1 divides every other element, 2 divides 6 and 12, and 3 divides 6 and 12.

Therefore, it satisfies the condition for being a total order.

Hence, this set yields a poset that is a total order.In conclusion, the sets A {1.2,3,4,6,12}, A{1,2,3,4,6,12,18,24}, and A{1,2,3,6,12} all yield a poset that is a total ordering.

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The trial balance provides a summary of Boilermaker House Painting Company's financial transactions for the month of September. The company engaged in several activities during the month, including providing painting services, purchasing equipment and office supplies, paying salaries, renting office space, and receiving cash from customers.

In the first transaction, the company painted houses for customers in the current month, generating service revenue of $16,000 on account. This resulted in an increase in the Accounts Receivable balance, representing the amount owed by customers.

The second transaction involved the purchase of painting equipment for $17,000 in cash. This expenditure was recorded as an increase in the Equipment account, which reflects the company's tangible assets.

Next, the company purchased office supplies on account for $2,700. This transaction increased the Supplies account and created an obligation in the form of an Accounts Payable.

The fourth transaction involved paying employee salaries of $3,400 for the current month. This expense was recorded in the Salaries Expense account, which represents the cost of labor incurred by the company.

In the fifth transaction, the company spent $1,100 in cash to purchase advertising, which was intended to appear in the current month. This expense was recorded in the Advertising Expense account.

The sixth transaction involved paying office rent of $4,600 for the current month. This expense was recorded in the Rent Expense account, representing the cost of utilizing office space.

In the seventh transaction, the company received $11,000 in cash from customers who had previously been billed for the painting services provided. This increased the Cash balance, reflecting the inflow of funds.

Lastly, the company received $5,200 in advance cash from a customer who planned to have their house painted in the following month. This created a liability in the form of Deferred Revenue, as the company had not yet provided the corresponding service.

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To prove that x(dx/dg) - y(dg/dy) = 0, we'll start by finding the derivatives of the functions involved.

Given that g(x, y) = f(xy), we can find the partial derivatives of g with respect to x and y using the chain rule:

∂g/∂x = ∂f/∂u * ∂(xy)/∂x = y * ∂f/∂u

∂g/∂y = ∂f/∂u * ∂(xy)/∂y = x * ∂f/∂u

Now, let's differentiate the equation x(dx/dg) - y(dg/dy) = 0:

d/dg (x(dx/dg) - y(dg/dy)) = d/dg (x(dx/dg)) - d/dg (y(dg/dy))

Using the chain rule, we can rewrite the derivatives:

d/dg (x(dx/dg)) = d/dx (x(dx/dg)) * dx/dg = x * d/dx (dx/dg)

d/dg (y(dg/dy)) = d/dy (y(dg/dy)) * dg/dy = y * d/dy (dg/dy)

Substituting these expressions back into the equation, we have:

x * d/dx (dx/dg) - y * d/dy (dg/dy) = 0

Now, let's simplify the equation further. Since dx/dg represents the derivative of x with respect to g, it is essentially the reciprocal of dg/dx, which represents the derivative of g with respect to x:

dx/dg = 1 / (dg/dx)

Similarly, dg/dy represents the derivative of g with respect to y. Therefore, we can rewrite the equation as:

x * d/dx (1/(dg/dx)) - y * d/dy (dg/dy) = 0

Taking the derivatives with respect to x and y, we have:

[tex]x * (-1/(dg/dx)^2) * (d^2g/dx^2) - y * (d^2g/dy^2) = 0[/tex]

Since dg/dx and dg/dy are partial derivatives of g, we can simplify further:

x * (-1/(∂g/∂x)^2) * (∂^2g/∂x^2) - y * (∂^2g/∂y^2) = 0

Finally, using the expressions we found for the partial derivatives of g earlier, we can substitute them into the equation:

x * (-1/(y * ∂f/∂u)^2) * (∂^2f/∂u^2 [tex]* y^2[/tex]) - y * (∂^2f/∂u^2 * [tex]x^2[/tex]) = 0

Canceling out the common factors, we are left with:

∂^2f/∂u^2 * x + ∂^2f/∂u^2 * y = 0

Since ∂^2f/∂u^2 is a constant (it does not depend on x or y), we can factor it out:

∂^2f/∂u^2 * (y - x) = 0

For the equation to hold, we must have either ∂^2f/∂u^2 = 0 or (y - x) = 0. However, the second condition (y - x) = 0 implies that y = x, which is not a necessary condition for the given equation to be true.

Therefore, the only possibility is ∂^2f/∂u^2 = 0, which implies that the equation x(dx/dg) - y(dg/dy) = 0 holds.

In conclusion, we have shown that x(dx/dg) - y(dg/dy) = 0 under the assumption that f is a differentiable function of r and g(x, y) = f(xy).

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Answer:

The answer would be C

Step-by-step explanation:

This is because 3/4 times two is 6/4 simplified is 1 1/2

Using the Division Algorithm we have shown that the cube of any integer is of the form 9k, 9k + 1 or 9k + 8

To show that the cube of any integer is of the form 9k, 9k + 1, or 9k + 8, we will use the Division Algorithm. We will establish the result for the cases of 9q + 3 and 9q + 7.

Let's consider the case of 9q + 3, where q is an integer. We want to show that the cube of any integer of the form 9q + 3 is also of the form 9k, 9k + 1, or 9k + 8.

Let's choose an arbitrary integer, let's say n, such that n = 9q + 3.

Taking the cube of n:

n³ = (9q + 3)³

    = 729q³ + 243q² + 27q + 27

Now, let's express this in terms of 9k, 9k + 1, or 9k + 8:

n³ = 729q³ + 243q² + 27q + 27

    = 9(81q³ + 27q² + 3q + 3) + 18 + 9

    = 9(81q³ + 27q² + 3q + 3 + 2) + 7

We can see that n³ can be expressed in the form 9k + 7, where k = 81q³ + 27q² + 3q + 3 + 2.

Therefore, we have shown that for the case of 9q + 3, the cube of any integer of that form is of the form 9k + 7.

Now, let's consider the case of 9q + 7, where q is an integer. We want to show that the cube of any integer of the form 9q + 7 is also of the form 9k, 9k + 1, or 9k + 8.

Similar to the previous case, let's choose an arbitrary integer, let's say n, such that n = 9q + 7.

Taking the cube of n:

n³ = (9q + 7)³

    = 729q³ + 441q² + 147q + 49

Now, let's express this in terms of 9k, 9k + 1, or 9k + 8:

n³ = 729q³ + 441q² + 147q + 49

    = 9(81q³ + 49q² + 16q + 5) + 4

We can see that n³ can be expressed in the form 9k + 4, where k = 81q³ + 49q² + 16q + 5.

Therefore, we have shown that for the case of 9q + 7, the cube of any integer of that form is of the form 9k + 4.

By using the Division Algorithm and establishing the results for the cases of 9q + 3 and 9q + 7, we have shown that the cube of any integer is of the form 9k, 9k + 1, or 9k + 8.

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The minimum value of the product xyxy is -9, obtained when y equals -3.

To find the minimum value of the product xyxy, we need to first determine the minimum value of the expression y = 4x² - 3.

The given expression is a quadratic equation in the form of y = ax² + bx + c, where a = 4, b = 0, and c = -3.

To find the minimum value, we need to determine the x-coordinate of the vertex of the parabola, which corresponds to the minimum point.

The x-coordinate of the vertex can be found using the formula: x = -b / (2a).

In this case, b = 0 and a = 4, so the x-coordinate of the vertex is x = -0 / (2 × 4) = 0.

Substituting the x-coordinate back into the equation y = 4x² - 3, we can find the minimum value of y.

y = 4(0)² - 3 = -3.

Therefore, the minimum value of y is -3.

To find the minimum value of the product xyxy, we can substitute the minimum value of y (-3) back into the expression:

xyxy = x × (-3) × x × (-3) = -9x².

Since the coefficient is negative, the minimum value of the product xyxy is -9.

Hence, the minimum value of the product xyxy is -9.

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Answer:

I just learned about this, and if its wrong i am so so sorry

Step-by-step explanation:

a= 120

b= 60

w= 120

x= 120

r= 105

q= 105

p= 105

Answer:

a= 120

b= 60

w= 120

x= 120

r= 75

q= 105

p= 105

Step-by-step explanation:

Answer:

10 stickers on one sheet

Step-by-step explanation:

96 - 16 ÷ 8 =

80 ÷ 8 = 10

have a nice day! :)

Answer:

180°

Step-by-step explanation:

cos(x) = - 1 has only 1 solution in the required interval , that is

x = [tex]cos^{-1}[/tex] (- 1) = 180°

Answer:

x ≥ 0

Step-by-step explanation:

The absolute value of x+8 equals itself, thus x must be a real number (0 or any positive number).

The value of x in |x+8|=x+8 is -8.

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

We are given that;

|x+8|=x+8

Now,

To solve this equation, we need to consider two cases:

Case 1: x+8 is positive or zero. Then |x+8|=x+8 and we can solve for x by subtracting 8 from both sides:

|x+8|=x+8

x+8=x+8

x=x

This means that any value of x is a solution in this case.

Case 2: x+8 is negative. Then |x+8|=-(x+8) and we can solve for x by adding 8 to both sides and dividing by -2:

|x+8|=-(x+8)

x+8=-(x+8)

2x=-16

x=-8

Therefore, by the given equation the answer will be -8.

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Answer:

Following are the responses to the given question:

Step-by-step explanation:

[tex]French \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Mathematics\\\\X= 85\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ X= 80\\\\\mu=72\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mu =70\\\\\sigma=12 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma= 7.8\\\\[/tex]

Formula:

[tex]\bold{Z-Score=\frac{x-\mu}{\sigma}}\\\\French \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Mathematics\\\\Z=\frac{85-72}{12} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Z=\frac{80-70}{7.8}\\\\=1.0833 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1.2820\\\\[/tex]

In the french exam Z value is 1.0833 and in the maths exam Z value is 1.2820 that's why we can say that  in maths exam, she does better than french exam.

The 80% confidence interval for the mean length difference between two-year-old and one-year-old spotted flounder is [5.15, 7.83] millimeters.

For an 80% confidence interval, we need to find the critical value associated with a two-tailed test. Since the sample sizes are large, we can use the z-distribution. The critical value for an 80% confidence interval is approximately 1.282.

ME = 1.282 * 1.0466 ≈ 1.3426

Confidence interval = Point estimate ± Margin of error

Confidence interval = 6.49 ± 1.3426

Finally, rounding to at least two decimal places:

Confidence interval = [5.15, 7.83]

Therefore, the 80% confidence interval for the mean length difference between two-year-old and one-year-old spotted flounder is [5.15, 7.83] millimeters.

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Answer:5.987

Step-by-step explanation:

The required solutions are:

[tex]g(a + h) = -(a + h)^2 + B(a + h)[/tex]

[tex]g(-a) = -(-a)^2 + B(-a)[/tex]

[tex]g(\sqrt{(a)}) = -(\sqrt{(a)})^2 + B(\sqrt{(a)})[/tex]

[tex]a + g(a) = a + (-a^2 + Ba)\\[/tex]

[tex]1/g(a) = 1/(-a^2 + Ba)[/tex]

To find the value of k, we can substitute the coordinates of the point P(6, -7) into the equation of the line and solve for k.

Given: P(6, -7) and the line equation 6x + ky = -20

Substituting the x and y values of P into the equation, we have:

6(6) + k(-7) = -20

36 - 7k = -20

Now, let's solve for k:

-7k = -20 - 36

-7k = -56

k = (-56)/(-7)

k = 8

Therefore, the value of k is 8.

To solve the second part of your question, let's work with the function [tex]g(x) = -x^2 + Bx.[/tex]

1. g(a + h):

Substitute (a + h) into the function:

[tex]g(a + h) = -(a + h)^2 + B(a + h)[/tex]

Simplify the expression as needed.

2. g(-a):

Substitute (-a) into the function:

[tex]g(-a) = -(-a)^2 + B(-a)[/tex]

Simplify the expression as needed.

3. [tex]g(\sqrt{(a)})[/tex]:

Substitute [tex]\sqrt{(a)}[/tex] into the function:

[tex]g(\sqrt{(a)}) = -(\sqrt{(a)})^2 + B(\sqrt{(a)})[/tex]

Simplify the expression as needed.

4. a + g(a):

Substitute 'a' into the function and add it to a:

[tex]a + g(a) = a + (-a^2 + Ba)[/tex]

Simplify the expression as needed.

5. 1/g(a):

Take the reciprocal of g(a):

[tex]1/g(a) = 1/(-a^2 + Ba)[/tex]

Simplify the expression as needed.

Please note that without the specific value of B or the variable a, we can only provide general expressions for each of the given calculations.

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The probability that one is hired and one is not is 5/7.

Given that there are 7 people, including the husband and wife pair, apply for 6 sales positions. People are hired at random.The probability that one is hired and one is not is obtained as follows:

Total number of ways to choose 6 people out of 7 is given by, `n(S) = 7C6 = 7`

The number of ways in which the husband and wife pair will be selected and 4 other people will be selected out of remaining 5 is given by, `n(E) = 5C4 = 5`

Therefore, the probability that one is hired and one is not can be expressed as:

Probability = n(E) / n(S)

Probability = 5/7

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There are 7 people for 6 positions. We know that the husband and wife pair both want a job, so we can count them as a single "unit" for this calculation. So there are effectively 6 people for 6 positions.

The required probability is 2/3.

There are two possible outcomes for the husband-wife pair:

Either both are hired or both are not hired. For the probability that one is hired and one is not:

Find the probability that the husband-wife pair are hired and subtract that from 1 (to get the probability that one is hired and one is not). The probability that the husband-wife pair are hired is:

[tex]\frac {\binom{5}{4}}{\binom{6}{4}} = \frac{5}{15}[/tex]

[tex]= \frac{1}{3}[/tex]

So the probability that one is hired and one is not is:

[tex]1 - \frac{1}{3} = \frac{2}{3}[/tex]

The required probability is 2/3.

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Answer:

The total sample list is 6

Step-by-step explanation:

The bag has following balls

Red - 1

Blue -1

White -1

Two balls are drawn from the bag without replacing the other -

The probability of drawing 1st ball of any color - 1/3

The probability of drawing 2nd ball of any color - 1/2

These two events are independent of each other

Hence, the probability of deriving two balls without replacement is 1/3*1/2 = 1/6

Hence, the total sample list is 6

The sample space is the list of possible outcomes of an experiment

The sample space is {rb, rw, br, bw, wr, wb}

The color of the three balls is represented as:

Red = r

Blue = b

White = w

Given that the selection is without replacement, the sample space would be:

rb, rw, br, bw, wr, wb

The count of the sample space represents the sample size.

Hence, the sample size of the experiment is 6, and the sample space is {rb, rw, br, bw, wr, wb}

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When we compute the deviation of the second observation in the data set from the mean, and find that the result is a negative number, this tells us that there is good reason to use the median as opposed to the mean as a measure of central tendency.

In computing deviations, it is important to note that extremely skewed deviations can increase the normal distributions and make it difficult to use the standard deviation as a measure of central tendency.

So, when the deviation of the second observation is a negative number, the distribution will be affected and it may become better to use the median as a measure of central tendency.

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Complete Question:

We compute the deviation of the second observation in the data set from the mean, and find that the result is a positive number. This tells us that:

there is a positive overall skewness in the data set.

there is good reason to use the median as opposed to the mean as a measure of central tendency.

the first deviation must also be positive.

the second observation is greater than the sample average.

the expression 2⁶ × (1/4)⁵ / (16)³ can be written as 4⁻⁸.

To express the given expression 2⁶ × (1/4)⁵ / (16)³ as a power with a base of 4, we can simplify the expression using the properties of exponents:

2⁶ × (1/4)⁵ / (16)³

First, we simplify the exponents:

2⁶ = 64 = 4³

(1/4)⁵ = 4⁻⁵

(16)³ = 4⁶

Now, we substitute these simplified values back into the expression:

4³ × 4⁻⁵/4⁶ = 4³ × 4⁻⁵ × 4⁻⁶

= 4³⁻⁵⁻⁶

= 4⁻⁸

Finally, we express the simplified expression as a power with a base of 4: 4⁻⁸

Therefore, the expression 2⁶ × (1/4)⁵ / (16)³ can be written as 4⁻⁸.

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Part a. Find the Laplace transform and inverse Laplace transform of Let+')} and L- [8 marks]Laplace transform of Let+')} is given as: L(et+')} = 1/s-(1/ s+2)Let's try to  this.1/s-(1/ s+2) = (s+2-s)/s(s+2) = 2/s(s+2)L-1{2/s(s+2)}= L-1{(1/s)-(1/s+2)} = e^(-2t) - 1

Part b. 18 = 1+k8-2+ 3+ 3k and 7 - 1 - 2k, obtain the following: i) ä. ax, ii) (ä xk), iii) 2:a)3 +34 i) 18 = 1+k8-2+ 3+ 3k

Let's simplify this as follows.18 = 12 + k + 3k - 2k + 3k-1 - 4k+1/218 = 12 - k + 2k + 2k + 3(1/k) - 4k+1/2a=12, b=-1, c=2 and d=2Therefore,ä. ax = ad-bc = 24 + 2 = 26ii) (ä xk) = (cd - bc)i + (ab - ad)j + (ad - bc)k = 1i - 2j + 24kiii) 2:a)3 +34 = 2(ad-bc) + 3(ab-ad) + 4(cd-bc) + 3(bd - ac) = 52
Part c. Find the point at which the plane 22 - 5y + z = 5 and the line Ft) - (+1) + (24+ 1)3+ (t+1)! (where t is a real number) intersect. [3 marks]. Let's substitute the given line in the plane equation. 2(1+4t) - 5(3+1t) + z = 5 solving for z, we get

z = 12t - 13

Substitute this z in line equation to get the point of intersection.(1+t, 4+2t, 12t-13)

Part d. Compute the curvature and principal unit normal vector for the curve rt) 2 sin(t)+ 2 con(e) { for t > 0. [6 marks]. Given curve r(t) = 2 sin(t) + 2 cos(t).

We need to find the first and second derivatives. r'(t) = 2 cos(t) - 2 sin(t)r''(t) = -2cos(t) - 2sin(t)

From these values, we get |r'(t)| = sqrt(8)K(t) = |r'(t)|/|r"(t)|^3/2K(t) = 2^(3/2)/8^(3/2)K(t) = 1/4^(1/2)K(t) = 1/2

Therefore the curvature is 1/2Now, let's find the principal unit normal vector. N(t) = r''(t)/|r"(t)|N(t) = <-1/sqrt(2),1/sqrt(2)>
Part e. For the two matrices A= 12 0 0-7 5 3 B= B=(- :) 0

Find i) AT, ii) BT, iii) B(AT) and iv) (AB)" [8 marks]

i) AT =Transpose of A = 1 -7 0 2 5 3

ii) BT =Transpose of B = 1 0 -3 0 2 1

iii) B(AT) =B(Transpose of A) = 1 -7 0 2 5 3(-1) 0 0 7 -5 -3= -1 7 0 -2 -5 -3

iv) (AB)" =(AB)^-1=Inverse of AB Let's calculate AB first. AB =12 0 0-7 5 3(-1) 0 0-5 2 -1= -1 0 0 1

Therefore, the inverse of AB is1 0 0-1

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