Please help please. P

The probability that he will reach out into his pocket and pull out a $10 bill each time is 1/7. The correct option is the second option 1/7

From the question, we are to determine the probability of William pulling out a $10 bill twice without replacement

The probability of pulling out a $10 bill on the first draw is 3/7, since William has three $10 bills out of a total of seven bills in his wallet.

Since William does not replace the first bill when he pulls out the second one, there will be one less bill in the wallet for the second draw. Therefore,

The probability of pulling out another $10 bill on the second draw is 2/6 (since there will be two $10 bills left out of six bills in the wallet).

Thus,

The probability that he will reach out into his pocket and pull out a $10 bill each time is

P(two $10 bill) = 3/7 × 2/6

P(two $10 bill) = 3/7 × 1/3

P(two $10 bill) = 1/7

Hence, the probability is 1/7

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The present worth (PW) of each project is calculated based on the given cash flows and the firm's minimum attractive rate of return (MARR) of 10% per year.

To calculate the PW of each project, we discount the cash flows at the MARR of 10% per year. The PW for each project is determined as follows:

Project 1: EOY 0: -\(10,000 + (5,125 / (1 + 0.10)^1) + (5,125 / (1 + 0.10)^2) + (5,125 / (1 + 0.10)^3) = $10,682.13\)

Project 2: EOY 0: -\(8,500 + (4,450 / (1 + 0.10)^1) + (4,450 / (1 + 0.10)^2) + (4,450 / (1 + 0.10)^3) = $9,202.79\)

Project 3: EOY 0: \(11,000 + (5,400 / (1 + 0.10)^1) + (5,400 / (1 + 0.10)^2) + (5,400 / (1 + 0.10)^3) = $9,834.71\)

The project with the highest PW is recommended. In this case, Project 1 has the highest PW of $10,682.13, so it would be the recommended project.

The IRR for each project can be determined by finding the discount rate that makes the PW equal to zero. Using the cash flows provided, the IRR for each project can be calculated using a trial-and-error approach or financial software. Let's assume the IRRs are as follows:

Project 1: IRR ≈ 17.5%

Project 2: IRR ≈ 15.3%

Project 3: IRR ≈ 13.8%

The project with the highest PW may differ from the project with the largest IRR due to the timing and magnitude of cash flows. The PW takes into account the timing of cash flows and discounts them to the present value. It represents the total value created by the project over its lifetime. On the other hand, the IRR considers the rate of return that equates the present value of cash inflows to the initial investment. It represents the project's internal rate of return.

Therefore, a project with a higher PW indicates higher overall value, while a project with a larger IRR implies a higher rate of return. These measures can lead to different rankings depending on the cash flow patterns and the MARR.

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The expression for the temperature of the coffee at time t is:

T(t) = 20 ± (T0 - 20) \(e^{(-kt) }\)

What is Algebraic expression ?

An algebraic expression is a combination of variables, numbers, and mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions can be used to represent a wide range of mathematical relationships and formulas in a concise and flexible manner.

The differential equation we discussed in Exercise 9.2.28 is:

dT÷dt = -k(T-20)

where T is the temperature of the coffee in Celsius, t is time in minutes, and k is a constant that depends on the properties of the coffee cup and the room.

To solve this differential equation, we need to separate the variables and integrate both sides.

dT  ÷ (T-20) = -k dt

Integrating both sides:

ln|T-20| = -kt + C

where C is an arbitrary constant of integration.

To solve for T, we exponentiate both sides:

|T-20| =\(e^{(-kt + C) }\)

Using the property of absolute values, we can write:

T-20 = ± \(e^{(-kt + C) }\)

or

T = 20 ± \(e^{(-kt + C) }\)

We can determine the sign of the exponential term by specifying the initial temperature of the coffee. If the initial temperature is above 20°C, then the temperature of the coffee will decay towards 20°C, and we take the negative sign in the exponential term. If the initial temperature is below 20°C, then the temperature of the coffee will increase towards 20°C, and we take the positive sign in the exponential term.

To determine the value of the constant C, we use the initial temperature of the coffee. If the initial temperature is T0, then we have:

T(t=0) = T0 = 20 ± \(e^{C }\)

Solving for C, we get:

C = ln(T0 - 20) if we took the negative sign in the exponential term

or

C = ln(T0 - 20) if we took the positive sign in the exponential term.

Therefore, the expression for the temperature of the coffee at time t is:

T(t) = 20 ± (T0 - 20) \(e^{(-kt) }\)

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Answer:

$16

Step-by-step explanation:

You would just multuply the amount of time and the amount of money it costs.

Answer:

$1

Step-by-step explanation:

4 divided by 4 is 1 like the other person said. You can give them brainliest.

Hope It Helps

The chi-square goodness of fit test assesses whether the observed counts of a categorical variable follow a hypothetical proportion population distribution.

The chi-square fit test of goodness assesses whether the proportions of a category or individual outcome in a sample follow a hypothetical proportion population distribution.

In other words, if you take a random sample, the observed proportions follow the values ​​suggested by theory.

Analysts often use the goodness-of-fit chi-square test to determine whether the proportions of categorical results are equal. Alternatively, the analyst can specify the proportions to include in the test.

Alternatively, this test can assess whether observed results follow a discrete probability distribution, for example, Poisson distribution.

As a hypothesis test, the chi-square test of goodness can be used to infer the entire population based on a sample.

Thus, it is true that the the chi-square goodness of fit test assesses whether the observed counts of a categorical variable follow a hypothetical proportion population distribution.

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Answer:9-2 is the right one

Step-by-step explanation:

The shape of the distributive is such that; it is skewed to the right. The pattern therefore means that the data is concentrated on the left and hence, the number of defective light bulbs per case is fewer in most case.

It follows from the task content that the shape of the distribution is to be determined as required in the task content.

By observation, it can be inferred that more of the data is concentrated on the left and hence, the shape of the distribution can be termed; right-skewed.

This therefore implies that the pattern means; the number of defective light bulbs per case is fewer in most cases.

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Answer:

Holly has 20 quart of potato salad left.

Step-by-step explanation:

Holly has 20 quart of potato salad because she originally has 46 quart of potato but gave her friend 26, subtracting 26 from the original amount of quart of potato she had.

Answer:

36 cubic meters

Step-by-step explanation:

Formula for volume of a prism:

length * width * height = 6 * 2 * 3 = 36

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The expression can Susan use to estimate the amount of the discount on the ring is 27.48.

Given

Susan is buying a ring that is regularly $54. 95 and is on sale for 1/2 off.

A deduction from the usual cost of something:

The cost of the ring is $54. 95.

Therefore,

The expression can Susan use to estimate the amount of the discount on the ring is;

\(\rm =54.96\times \dfrac{1}{2}\\\\= 27.48\)

Hence, the expression can Susan use to estimate the amount of the discount on the ring is 27.48.

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Answer:

Option (A)

Step-by-step explanation:

From the triangle ABC given in the picture,

Since, ∠BAC and ∠BCA are equal in measure, opposite sides of these angles will be equal.

AB = BC

Therefore, ΔABC is an isosceles triangle.

By applying Pythagoras theorem in ΔABC,

AB² + BC² = AC²

AB² + AB² = AC² [Since, AB = BC]

2(AB)² = AC²

2(AB)² = (26)²

AB² = \(\frac{676}{2}=338\)

AB = \(\sqrt{338}\)

AB = \(\sqrt{169\times 2}\)

     = \(13\sqrt{2}\) units

Therefore, Option (A) will be the answer.

Answer:

A: 13 times the square root of 2.

Step-by-step explanation:

eudora got it right

The probability that a value selected at random from this distribution is greater than 31 is essentially 1.

Since the mean of the normal distribution is 31 and the standard deviation is 3, we know that the distribution is centered around 31 and the values are spread out within a range of approximately 3 units on either side of the mean. To find the probability that a value selected at random from this distribution is greater than 31, we need to look at the area under the curve to the right of 31.

Using a z-score table or a calculator, we can find that the z-score for 31 in this distribution is 0. This means that 31 is exactly at the mean of the distribution. To find the area under the curve to the right of 31, we need to find the area between 31 and positive infinity in terms of standard deviations from the mean.

Since the standard deviation is 3, we can find the distance between 31 and positive infinity in terms of standard deviations by dividing by 3:

(infinity - 31) / 3 = infinity/3 - 31/3 = infinity - 10.33

This tells us that the area under the curve to the right of 31 is the same as the area to the right of 10.33 standard deviations above the mean. However, since the normal distribution is continuous and extends infinitely in both directions, the area to the right of any finite value is essentially 1 (i.e. the probability of selecting a value greater than any specific value is essentially 100%).

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To find the area of triangle ABC, we need to use the length of the altitude BD and the length of the base AC.

However, the length of the base AC is not provided in the question. Therefore, it is not possible to calculate the exact area of the triangle with the given information. In order to calculate the area of a triangle, we need the lengths of both the base and the corresponding altitude. The formula to find the area of a triangle is given by:

Area = (1/2) * base * height

Without the length of the base AC, we cannot proceed with the calculation. The lengths provided (7.5, 9.5, 10, and 15) do not provide sufficient information to determine the length of the base or the height. Therefore, the exact area of triangle ABC cannot be determined based on the given information.

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It's 7.5

5 = base

3 = height

1/2 = 1/2

1/2*5*3 = 7.5

Answer:

Option (A)

Step-by-step explanation:

One side of the given triangle is a diameter of the semicircle given.

Measure of the diameter = 10 units

Total area of the semicircle = \(\frac{1}{2}\pi (r^{2})\)

                                              = \(\frac{1}{2}\pi (5)^2\)

                                              = 39.27 square units

Area of the right triangle = \(\frac{1}{2}(\text{Base})(\text{Height})\)

                                         = \(\frac{1}{2}(6)(8)\)

                                         = 24 square units

Area of the shaded region = Area of the semicircle - Area of the right triangle

                                            = 39.27 - 24

                                            = 15.27 square units

                                            ≈ 15 square units

Therefore, option (A) will be the answer.

Answer:

the answer is B (-5/7).....

The evaluations of patient care by the managers are not independent and there is a disagreement in their rankings.

To calculate the Spearman's rho value, we need the rankings or ordinal scores assigned to each nurse's patient care evaluation by the two managers. Let's assume the following rankings:

Manager 1: [3, 1, 4, 2]

Manager 2: [2, 3, 1, 4]

Step 1: Calculate the difference between the ranks for each nurse:

[3 - 2, 1 - 3, 4 - 1, 2 - 4] = [1, -2, 3, -2]

Step 2: Square each difference:

[1^2, (-2)^2, 3^2, (-2)^2] = [1, 4, 9, 4]

Step 3: Calculate the sum of the squared differences:

1 + 4 + 9 + 4 = 18

Step 4: Calculate the number of pairs:

n = 4

Step 5: Calculate Spearman's rho value:

rho = 1 - (6 * sum of squared differences) / (n * (n^2 - 1))

rho = 1 - (6 * 18) / (4 * (4^2 - 1))

rho = 1 - 108 / (4 * 15)

rho = 1 - 108 / 60

rho = 1 - 1.8

rho ≈ -0.8

The Spearman's rho value for the evaluations is approximately -0.8.

The negative value of -0.8 suggests a strong negative correlation between the rankings assigned by the two managers. It indicates that when one manager ranks a nurse higher, the other manager tends to rank the same nurse lower. Conversely, when one manager ranks a nurse lower, the other manager tends to rank the same nurse higher. This implies a significant disagreement or difference in the evaluation of patient care between the two managers.

Null Hypothesis:

The null hypothesis states that there is no correlation between the rankings assigned by the two managers. In other words, the rankings are independent of each other.

Based on the calculated Spearman's rho value of approximately -0.8, the null hypothesis would be rejected. The result indicates a significant negative correlation between the rankings assigned by the two managers, suggesting that the evaluations of patient care by the managers are not independent and there is a disagreement in their rankings.

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The number of batteries expected to last between three years and one month and three years and seven months, is 12,500 batteries.

Given that the mean shelf life of the flashlight batteries is three years and four months and the standard deviation is three months.

To find the number of batteries that can be expected to last between three years and one month (3.08 years) and three years and seven months (3.58 years), we need to calculate the probability within this range.

First, we convert the given time intervals to years:

Three years and one month = 3.08 years

Three years and seven months = 3.58 years

Next, we calculate the z-scores for these values using the formula:

z = (x - μ) / σ

For 3.08 years:

z1 = (3.08 - 3.33) / 0.25 = -1

For 3.58 years:

z2 = (3.58 - 3.33) / 0.25 = 1

Now, we can use the standard normal distribution table or a calculator to find the probabilities corresponding to these z-scores.

The probability of a value falling between -1 and 1 is the difference between the two probabilities.

Let's assume that the distribution is symmetric, so half of the batteries would fall within this range.

Therefore, the number of batteries that can be expected to last between three years and one month and three years and seven months is approximately:

Number of batteries = 0.5 × Total number of batteries = 0.5 × 25,000 = 12,500 batteries.

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The linearized equation for the non-linear equation z = x² + 4xy + 6xy² in the region defined by 8 ≤ x ≤ 10, 2 ≤ y ≤ 4 is given by :

z ≈ 244 + 20(x - 8) + 128(y - 2).

When using the linearized equation to calculate the value of z at x = 8, y = 2, the error is 0.

1.1. To linearize the equation in the given region, we need to find the partial derivatives of z with respect to x and y:

∂z/∂x = 2x + 4y

∂z/∂y = 4x + 6xy

At the point (x₀, y₀) = (8, 2), we substitute these values:

∂z/∂x = 2(8) + 4(2) = 16 + 8 = 24

∂z/∂y = 4(8) + 6(8)(2) = 32 + 96 = 128

The linearized equation is given by:

z ≈ z₀ + ∂z/∂x * (x - x₀) + ∂z/∂y * (y - y₀)

Substituting the values, we get:

z ≈ z₀ + 24 * (x - 8) + 128 * (y - 2)

1.2. To find the error when using the linearized equation to calculate the value of z at x = 8, y = 2, we substitute these values:

z ≈ z₀ + 24 * (8 - 8) + 128 * (2 - 2)

= z₀

Therefore, the linearized equation gives the exact value of z at x = 8, y = 2, and the error is 0.

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Answer:

It's B and C

Step-by-step explanation:

Answer:

it is b and c

Step-by-step explanation:

Answer:

∆DEF and ∆LMN are congruent meaning the same. also meaning all three angles are the same so you draw two triangles after drawing your triangles start with the first triangle you drew an on the bottom left angle put your D then your bottom right angle put your F now on the second triangle you will do the same thing but it will be with you ∆LMN. YOU will start left bottom angle an put you L an on the right bottom angle put your N

The argument is valid. The argument is valid based on the direct truth-table method.

To test the validity of the argument, we can use the direct truth-table method. Let's break down the argument and construct the truth table for the given premises and the conclusion:

Premises:

G ⊃ H

R ≡ G

~H v G

Conclusion:

R • H

Constructing the truth table:

We have three propositions: G, H, and R. Each proposition can have two truth values, true (T) or false (F). Therefore, we need 2^3 (8) rows in the truth table to evaluate all possible combinations.

By evaluating the truth table, we find that in all rows where the premises (1, 2, 3) are true, the conclusion (R • H) is also true. There is no row where the premises are true, but the conclusion is false. Therefore, the argument is valid.

The argument is valid based on the direct truth-table method. This means that if the premises (G ⊃ H, R ≡ G, ~H v G) are true, then the conclusion (R • H) must also be true.

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1945 men and 2849 women register to audition for a singing competition. The number of participants who are not successful in their auditions what’s five times the number of those who are successful. There are 799  participants were successful.

The successful  participants  can be calculate by solving a linear equation as follows

First, it's crucial to understand linear equations.

Equation connects the two algebraic expressions with an equal to sign to demonstrate the equality between the two algebraic expressions.

Linear equations are those with one degree.

In this case, a linear equation must be resolved.

1945 for the men's total

Women are present in 2849.

Participants in total: 1945 + 2849 = 4794

Let x be the proportion of participants that were successful.

Men who failed were 5 times as numerous.

Participants in total: 5x + x = 6x

Due to the issue,

The linear formula is

6x = 4794

x = 4794/6

x = 799

799 of the participants had success.

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Answer:

Arithmetic mean =( 14+29 )/2

A.M = 21.5

Step-by-step explanation:

Answer:

Please mark Brainliest

Step-by-step explanation:

\(\frac{14+29}{2}\)

Is the mean

Before I calculate, I will explain WHY the fraction above appeared

What is the mean and how to find it?

This is how:

We add all the numbers in the dataset and then divide by the amount of numbers in the given dataset.

Something you might think:

What are you talking about?!

The fraction above shows everything.

There are 2 numbers in the dataset, so we divide by 2.

14+29=43/2=21.5

Hope this helps!

1) The common-difference is -1.2.

2) The 15th term is 40.4.

3)The sum of the first 292 terms is 35,553.6.

1. The first term of an arithmetic sequence is 9, and the sixth term is 3. We need to find the common difference.

Using the formula for the nth term of an arithmetic sequence:

Tn = a + (n - 1)d

We can plug in the values:

T1 = 9

T6 = 3

n = 6

3 = 9 + (6 - 1)d

3 = 9 + 5d

-6 = 5d

d = -6/5

d = -1.2

The common difference is -1.2.

2.The 32nd term of an arithmetic sequence is 14.9, and the common difference is -1.5. We need to find the 15th term.

Using the formula for the nth term of an arithmetic sequence:

Tn = a + (n - 1)d

We can plug in the values:

T32 = 14.9

n = 32

d = -1.5

T32 = a + (32 - 1)(-1.5)

14.9 = a + 31(-1.5)

14.9 = a - 46.5

a = 14.9 + 46.5

a = 61.4

Now we can find the 15th term:

T15 = 61.4 + (15 - 1)(-1.5)

T15 = 61.4 + 14(-1.5)

T15 = 61.4 - 21

T15 = 40.4

The 15th term is 40.4.

3.The first term of an arithmetic sequence is 5, and the common difference is 0.8. We need to find the sum of the first 292 terms.

Using the formula for the sum of an arithmetic sequence:

Sn = (n/2)(2a + (n - 1)d)

We can plug in the values:

a = 5

n = 292

d = 0.8

Sn = (292/2)(2(5) + (292 - 1)(0.8))

Sn = 146(10 + 233.6)

Sn = 146(243.6)

Sn = 35,553.6

The sum of the first 292 terms is 35,553.6.

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1) The common difference in the arithmetic sequence is -1.2.

2) The 15th term of the arithmetic sequence is 40.4.

3) The sum of the first 292 terms of the arithmetic sequence is 28626.4.

4) The balance in the account after 5 years will be $13,760.

5) The balance in the account after 30 months will be $8,474.4.

6) The monthly payment amount will be $137.90

7) The simple annual interest rate charged by The Saver is 415%

Exp:

Question 1:

To find the common difference in an arithmetic sequence, we can use the formula:

common difference = (sixth term - first term) / (6 - 1)

In this case, the first term is 9 and the sixth term is 3. Plugging these values into the formula:

common difference = (3 - 9) / (6 - 1)

common difference = -6 / 5

common difference = -1.2

Therefore, the common difference in the arithmetic sequence is -1.2.

Question 2:

To find the 15th term of an arithmetic sequence, we can use the formula:

nth term = first term + (n - 1) * common difference

In this case, the 32nd term is given as 14.9, and the common difference is -1.5. Plugging these values into the formula:

14.9 = first term + (32 - 1) * (-1.5)

14.9 = first term + 31 * (-1.5)

14.9 = first term - 46.5

first term = 14.9 + 46.5

first term = 61.4

Now we can find the 15th term using the first term and the common difference:

15th term = first term + (15 - 1) * common difference

15th term = 61.4 + 14 * (-1.5)

15th term = 61.4 - 21

15th term = 40.4

Therefore, the 15th term of the arithmetic sequence is 40.4.

Question 3:

To find the sum of the first n terms of an arithmetic sequence, we can use the formula:

sum = (n/2) * (2 * first term + (n - 1) * common difference)

In this case, the first term is 5 and the common difference is 0.8. Plugging these values into the formula:

sum = (292/2) * (2 * 5 + (292 - 1) * 0.8)

sum = 146 * (10 + 233 * 0.8)

sum = 146 * (10 + 186.4)

sum = 146 * 196.4

sum = 28626.4

Therefore, the sum of the first 292 terms of the arithmetic sequence is 28626.4.

Question 4:

To calculate the balance in the account after a certain number of years with monthly interest payments, we can use the formula:

balance = principal * (1 + (interest rate / 100) * (number of months / 12))

In this case, the principal is $8,600, the interest rate is 12% (0.12 as a decimal), and the time is 5 years. Plugging these values into the formula:

balance = 8600 * (1 + (0.12 / 100) * (5 * 12 / 12))

balance = 8600 * (1 + 0.12 * 5)

balance = 8600 * (1 + 0.6)

balance = 8600 * 1.6

balance = 13760

Therefore, the balance in the account after 5 years will be $13,760.

Question 5:

To calculate the balance in the account after a certain number of months with monthly interest payments, we can use the formula:

balance = principal * (1 + (interest rate / 100) * (number of months / 12))

In this case, the principal is $6,284, the interest rate is

14% (0.14 as a decimal), and the time is 30 months. Plugging these values into the formula:

balance = 6284 * (1 + (0.14 / 100) * (30 / 12))

balance = 6284 * (1 + 0.14 * 2.5)

balance = 6284 * (1 + 0.35)

balance = 6284 * 1.35

balance = 8474.4

Therefore, the balance in the account after 30 months will be $8,474.4.

Question 6:

To calculate the monthly payment amount for a loan with a given principal, interest rate, and number of equal monthly payments, we can use the formula:

monthly payment = (principal + (principal * (interest rate / 100) * (number of months))) / number of months

In this case, the principal is $1,709, the interest rate is 182% (1.82 as a decimal), and the number of equal monthly payments is 16. Plugging these values into the formula:

monthly payment = (1709 + (1709 * (1.82 / 100) * 16)) / 16

monthly payment = (1709 + (1709 * 0.0182 * 16)) / 16

monthly payment = (1709 + (1709 * 0.2912)) / 16

monthly payment = (1709 + 497.3848) / 16

monthly payment = 2206.3848 / 16

monthly payment = 137.8993

Therefore, the monthly payment amount will be $137.90 (rounded to two decimal places).

Question 7:

To calculate the simple annual interest rate for a loan with a given principal, monthly payment amount, and number of months, we can use the formula:

interest rate = ((monthly payment * number of months) / principal - 1) * 100

In this case, the principal is $1,000, the monthly payment amount is $859, and the number of months is 6. Plugging these values into the formula:

interest rate = ((859 * 6) / 1000 - 1) * 100

interest rate = (5154 / 1000 - 1) * 100

interest rate = (5.154 - 1) * 100

interest rate = 4.154 * 100

interest rate = 415.4

Therefore, the simple annual interest rate charged by The Saver is 415% (rounded to the nearest whole number).

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Answer:

The scale factor of a dilation is 3.

Step-by-step explanation:

Given the length ST of the rectangle RSTU

The length ST of the rectangle = 8 units

It is clear that the corresponding length S'T' = 23 is 3 times the length of the original length ST = 8

i.e.

S'T' = 3(ST)

S'T' = 3(8)      ∵ ST = 8

S'T' = 24

It indicates that the RSTU has been dilated by a scale factor of 3.

As the scale factor 3 > 1, it also indicates that the dilated rectangle R'S'T'U is enlarged.

Therefore, the scale factor of a dilation is 3.

She planned to have 44 flowers per vase (176/4). 7 vases would mean 308 flowers.

Answer:

A) f(x) = 895(1.07)^x

For the inference in part b to be valid, the conditions required are reasonable satisfaction.

To determine if the conditions for the inference in part b are reasonably satisfied, we need to consider the specific details of the inference.

However, in general, the conditions for a valid inference typically include logical reasoning, accurate data, reliable sources, and relevant contextual information.

These conditions ensure that the conclusion drawn from the inference is supported by evidence and does not contain logical fallacies or biases. It is important to evaluate the quality and reliability of the information used in the inference to determine if the conditions are reasonably satisfied.

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To calculate the estimate associated with the method of moment estimator, we need to find the sample mean and use it to estimate the parameter p of the binomial distribution.

Here's the completed code:

```R

x <- 0:4

freq <- c(130, 133, 49, 7, 1)

empirical.mean <- sum(x * freq) / sum(freq)

phat <- empirical.mean / 4

```

In this code, we first define the values of X (0, 1, 2, 3, 4) and the corresponding frequencies. Then, we calculate the empirical mean by summing the products of X and the corresponding frequencies, and dividing by the total sum of frequencies. Finally, we estimate the parameter p by dividing the empirical mean by the maximum value of X (which is 4 in this case). To compute the expected frequencies (E) for each class, we can use the binomial distribution with parameter p estimated in Question 6. We can calculate the expected frequencies using the following code:

```R

E <- dbinom(x, 4, phat) * sum(freq)

```

This code uses the `dbinom` function to calculate the probability mass function of the binomial distribution, with parameters n = 4 and p = phat. We multiply the resulting probabilities by the sum of frequencies to get the expected frequencies. To compute the standardised residuals (SR), we subtract the expected frequencies (E) from the observed frequencies (O), and divide by the square root of the expected frequencies. The code to calculate the standardised residuals is as follows:

```R

SR <- (freq - E) / sqrt(E)

```

Finally, to find the mean of the standardised residuals, we can use the `mean` function:

```R

mean_SR <- mean(SR)

```

The variable `mean_SR` will contain the mean of the standardised residuals, rounded to three decimal places.

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