The sum of half a number, n, and 15 is 24. What is the value of the number n?

a. The equation sin 3x = 0.99 - x² has two solutions.

b. The solutions are approximately x = -0.667 and x = 0.512, obtained using Newton's method.

a. The equation sin 3x = 0.99 − x² has multiple solutions, but we need to determine how many exist in a specific interval. Let's examine the graph of y = sin 3x and y = 0.99 − x² between x = 0 and x = 1.

By observing the graph, we can see that there are two solutions in the interval [0, 1]. Therefore, the equation has two solutions in this interval.

b. We can use Newton's method to find the solutions. Let f(x) = sin 3x - (0.99 - x²).

First, we need to find the derivative of f(x):

f'(x) = 3cos 3x + 2x

Next, we choose an initial guess for x, let's say x0 = 0.5. Then, we use the following formula to generate the sequence of approximations:

[tex]x_{n+1}[/tex] = [tex]x_n[/tex] - f([tex]x_n[/tex])/f'([tex]x_n[/tex])

We continue this process until we reach a value of [tex]x_{n+1}[/tex] that is close enough to [tex]x_n[/tex].

Starting with x0 = 0.5, we have:

x1 = 0.5 - [sin(30.5) - (0.99 - 0.5²)]/[3cos(30.5) + 20.5] ≈ 0.713

x2 = 0.713 - [sin(30.713) - (0.99 - 0.713²)]/[3cos(30.713) + 20.713] ≈ 0.846

x3 = 0.846 - [sin(30.846) - (0.99 - 0.846²)]/[3cos(30.846) + 20.846] ≈ 0.912

x4 = 0.912 - [sin(30.912) - (0.99 - 0.912²)]/[3cos(30.912) + 20.912] ≈ 0.931

x5 = 0.931 - [sin(30.931) - (0.99 - 0.931²)]/[3cos(30.931) + 20.931] ≈ 0.935

x6 = 0.935 - [sin(30.935) - (0.99 - 0.935²)]/[3cos(30.935) + 20.935] ≈ 0.935

Therefore, the solutions in the interval [0, 1] are approximately x = 0.713 and x = 0.935.

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The question is -

a. How many solutions does the equation sin 3x = 0.99 − x² have?

b. Use Newton's method to find them.

Answer:

To find the inverse of the function y = π/2 + sin(x), we need to first swap the positions of x and y:

x = π/2 + sin(y)

Now, we can solve for y:

sin(y) = x - π/2

y = sin⁻¹(x - π/2)

Therefore, the equation for the inverse of the function y = π/2 + sin(x) is y = sin⁻¹(x - π/2).

Three advanced mathematical topics suitable for an undergraduate college student to write a 4-6 pages report paper are Linear Regression Analysis, Probability Distributions, and Hypothesis Testing.

1. Markov Chains: A Markov chain is a mathematical model that can be used to describe a system that changes over time in a random way. The basic idea is that the future state of the system depends only on its current state, and not on its past history. In this report, you can discuss the basic concepts of Markov chains, including the transition matrix, stationary distribution, and limiting behavior. Some applications of Markov chains can also be explored, such as their use in modeling the stock market or predicting the weather. A good reference for this topic is "Introduction to Probability Models" by Sheldon Ross.

2. Linear Regression: Linear regression is a statistical method for modeling the relationship between two variables, where one variable is considered the dependent variable and the other is considered the independent variable. The goal is to find a linear equation that can be used to predict the value of the dependent variable based on the value of the independent variable. In this report, you can discuss the basic concepts of linear regression, including the formula for the regression line, the coefficient of determination, and the interpretation of regression coefficients. Some applications of linear regression can also be explored, such as its use in predicting housing prices or analyzing trends in data. A good reference for this topic is "Applied Linear Regression" by Sanford Weisberg.

3. Fourier Analysis: Fourier analysis is a mathematical technique for decomposing a function into its component frequencies. The basic idea is that any periodic function can be expressed as a sum of sine and cosine functions of different frequencies, and the relative amplitudes of these functions determine the shape of the original function. In this report, you can discuss the basic concepts of Fourier analysis, including Fourier series, Fourier transforms, and applications in signal processing and image analysis. Some specific examples can also be explored, such as the use of Fourier analysis in music synthesis or the analysis of earthquake signals. A good reference for this topic is "Fourier Analysis and Its Applications" by Gerald B. Folland.

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To find the extreme values of a function subject to a given constraint, Lagrange multipliers can be used. The method involves finding the critical points of the function and the constraint equation, and then solving a system of equations using the Lagrange multiplier. The resulting solutions will give the maximum and minimum values of the function subject to the given constraint.

Suppose we have a function f(x,y,z) and a constraint equation g(x,y,z) = 0. We can set up the Lagrangian function L(x,y,z,λ) = f(x,y,z) - λg(x,y,z) and then find the partial derivatives of L concerning x, y, z, and λ. Setting these partial derivatives to zero and solving the resulting system of equations will give us the critical points and the corresponding values of λ.

Once we have the critical points and values of λ, we can evaluate the function f(x,y,z) at these points to find the maximum and minimum values subject to the given constraint. It is important to note that not all critical points will necessarily correspond to maximum or minimum values, so we must evaluate the function at each point to determine which points give the extreme values.

Overall, Lagrange multipliers provide a powerful method for finding the extreme values of a function subject to a given constraint. The method involves setting up a Lagrangian function, finding the critical points and values of λ, and then evaluating the function at these points to find the maximum and minimum values. This approach can be applied to a wide range of optimization problems in mathematics, physics, and engineering.

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A. The 95% confidence interval is 0.291 < p < 0.435. B. The 80% confidence interval is (0.303, 0.397). C. A 99% confidence interval is 0.613 < p < 0.703.

A. Using the formula:

CI = p ± zsqrt(p(1-p)/n)

where p is the sample proportion, n is the sample size, and z is the critical value from the standard normal distribution. For a 95% confidence level, z is 1.96.

Putting the values:

CI = 131/396 ± 1.96sqrt((131/396)(265/396)/396)

Simplifying:

CI = 0.291 < p < 0.435

Therefore, the 95% confidence interval for the population proportion p is 0.291 to 0.435.

B. For an 80% confidence level, z is 1.282.

Putting the values:

CI = 0.35 ± 1.282sqrt((0.35)(0.65)/367)

Simplifying:

CI = (0.303, 0.397)

Therefore, the 80% confidence interval for the population proportion p is (0.303, 0.397).

C. For a 99% confidence level, z is 2.576.

Putting the values:

CI = 384/600 ± 2.576sqrt((384/600)(216/600)/600)

Simplifying:

CI = 0.613 < p < 0.703

Therefore, the 99% confidence interval for the population proportion p is 0.613 to 0.703. Writing it in tri-inequality form, we get:

0.613 < p < 0.703

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Since there are 4 columns in A, there are no free variables, so the number of parameters in the general solution is equal to the number of non-pivot variables, which is at most 4.

(a) If A is a 4 x 5 matrix, then the number of leading 1's in the reduced row echelon form of A is at most 4. This is because the reduced row echelon form of a matrix has the property that each row has at most one leading 1, and there are only 4 rows in this case.

(b) If A is a 4 x 5 matrix, then the number of parameters in the general solution of Ax = 0 is at most 1. This is because the rank of the matrix A cannot be greater than 4, so there are at most 4 pivot variables in the reduced row echelon form of A. Since there are 5 columns in A, there is one free variable, which corresponds to the number of parameters in the general solution.

(c) If A is a 5 x 4 matrix, then the number of leading 1's in the reduced row echelon form of A is at most 4. This is because the reduced row echelon form of a matrix has the property that each row has at most one leading 1, and there are only 4 columns in this case.

(d) If A is a 5 x 4 matrix, then the number of parameters in the general solution of Ax = 0 is at most 4. This is because the rank of the matrix A cannot be greater than 4, so there are at most 4 pivot variables in the reduced row echelon form of A. Since there are 4 columns in A, there are no free variables, so the number of parameters in the general solution is equal to the number of non-pivot variables, which is at most 4.

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The series for E(X^2) diverges, the variance of X does not exist.

(a) To find the probability mass function of X, we need to calculate the probability of getting the first tail on the kth flip, for each k = 1,2,...

P(X = k) = Pr(Tail on kth flip) * Pr(Head on first k-1 flips)

= (k/(k+1)) * (1/(k+1-1)) * ((k+1)/k)^{k-1}

= (k/(k+1)) * (1/k) * ((k+1)/k)^{k-1}

= 1/(k * (k+1))

Therefore, the probability mass function of X is:

P(X = k) = 1/(k * (k+1)), for k = 1,2,...

(b) To find the mean E(X), we can use the formula:

E(X) = ∑ k * P(X = k), where the summation is over all possible values of X.

E(X) = ∑_{k=1}^∞ k * (1/(k * (k+1)))

= ∑_{k=1}^∞ (1/k - 1/(k+1))

= 1

To find the variance Var(X), we can use the formula:

Var(X) = E(X^2) - (E(X))^2

E(X^2) = ∑ k^2 * P(X = k), where the summation is over all possible values of X.

E(X^2) = ∑_{k=1}^∞ k^2 * (1/(k * (k+1)))

= ∑_{k=1}^∞ (k/(k+1) + 1/(k+1))

= ∑_{k=1}^∞ (1 + 1/k)

  (we split the fraction k/(k+1) into 1 + 1/(k+1))

  = ∞  (diverges)

Since the series for E(X^2) diverges, the variance of X does not exist.

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The value of y when the value of x is 1 would be 60.

In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) by using the data points contained in the table as follows:

Constant of proportionality, k = y/x

Constant of proportionality, k = 60/1

Constant of proportionality, k = 60.

Therefore, the required equation is given by;

y = 60x

y = 60(1)

y = 60.

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The line of reflection is given as follows:

C. y = x.

The coordinates of the original triangle are given as follows:

(-2,5), (0,9) and (3,7).

The coordinates of the reflected triangle are given as follows:

(5,-2), (9,0), (7,3).

We can see that the x-coordinates and the y-coordinates of the vertices were exchanged, hence the reflection rule is given as follows:

(x,y) -> (y,x).

Which represents a reflection over the line y = x, hence the correct option is given by option C.

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The possible outcomes of the experiment is {RR, BB, GG, WW, BB, RW, RB, RG, RW, RG, WB, WG}

The sample space for the experiment gave:

{RR, BB, GG, WW, BB, RW, RB, RG, RW, RG, WB, WG}

where each element of the set represents a different pair of socks, and the first letter represents the colour of the sock on the left foot and the second letter represents the colour of the sock on the right foot.

The possible outcomes of the experiment are the elements of the sample space, which are the different pairs of socks that can be selected. For example, selecting the red socks would be represented by the outcome RR, selecting the blue and white socks would be represented by the outcome BW, and so on.

Recall that

Probability = number of outcomes/total number of outcomes

Then, the probability of selecting a blue pair of socks will be:

P(blue) = number of outcomes with blue socks / total number of outcomes

Since there are only two outcomes with blue socks (BB and WB), then:

P(blue) = 2/12 = 1/6

P(green) = number of outcomes with green socks / total number of outcomes

P(green) = 2/12 = 1/6

P(not red) = number of outcomes without red socks / total number of outcomes

P(not red) = 10/12 = 5/6

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The parameters of the sinusoidal function, y = -3·cos(π·(π - 2)) - 4, obtained from the equation of the function are;

(a) a) 2

b) 4 units down

c) 2 units left

(b) d) Please find attached the graph of the function showing the period created with MS Excel.

A sinusoidal function is a periodic sine or cosine based function.

The specified sinusoidal function can be presented as follows;

y = -3·cos(π·(x - 2)) - 4

The general form of a sinusoidal function is; y = A·cos(B·(x + C)) + D

(a) a) The period of a sinusoidal function is T = 2·π/|B|

A comparison with the general form of a sinusoidal function indicates;

A = 3, B = π, C = -2, D = -4

B = π

Therefore; T = 2·π/π = 2

The period, T = 2

b) The vertical shift of the function, D = -4

c) The horizontal shift of the function, C = -2

(b) d) Please find attached the graph of the function created with MS Excel

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The given function f(x) = x³ + 2x² - 3 has three solutions.

To determine the number and types of solutions for the function:

f(x) = x³ + 2x² - 3,

we need to find the roots of the function. The roots are the values of x where the function equals zero.

The roots of the equation are given as:

x³ + 2x² - 3 = 0

x(x-3)(x+1)=0

From the above expression, x has 3 values for which the function terminates itself to zero. It means the given function has three solutions.

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Out of the given options, the one that is not an advantage of using a sample versus a census is population size. The reason for this is that whether you use a sample or a census, the population size remains the same. However, there are several advantages to using a sample over a census.

Firstly, a sample generates a smaller dataset to analyze, which can save time and resources. Secondly, using a sample can be less expensive than conducting a census, which involves surveying every member of the population. Lastly, using a sample provides ready access to respondents, as it is often easier to reach a smaller group of people than an entire population. However, it is important to note that using a sample also has its limitations, such as the potential for sampling bias and the need to ensure that the sample is representative of the population being studied. Overall, the choice between using a sample or a census depends on the research question, available resources, and the level of accuracy and precision required.

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The data provide convincing evidence that the pediatrician's claim is true

To test the pediatrician's claim that the mean weight of one-year-old boys is greater than 22 pounds, we can use a one-sample t-test.

Null Hypothesis: The true population mean weight of one-year-old boys is 22 pounds or less.

Alternative Hypothesis: The true population mean weight of one-year-old boys is greater than 22 pounds.

We can use a level of significance of 0.05, which corresponds to a 95% confidence level.

The test statistic for this one-sample t-test is:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

Substituting the given values:

t = (23.5 - 22) / (2.8 / sqrt(315)) = 10.15

Using a t-distribution table with 314 degrees of freedom (n-1), we find that the critical value for a one-tailed test at the 0.05 level of significance is 1.646.

Since the calculated t-value (10.15) is greater than the critical value (1.646), we reject the null hypothesis.

Therefore, the data provides convincing evidence that the pediatrician's claim is true, and the mean weight of one-year-old boys is greater than 22 pounds.

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The linear constraints with continuous and integer variables for the following is 100y + x2 ≤ 200.

The linear constraints for the problem are:
x ≥ 100 → y ≤ 100
x ≥ 0
y ≥ 0

y * 100 ≥ x1
(1 - y) * 100 ≥ x1
100y + x2 ≤ 200.

Let x be the investment amount on project 1 (continuous variable) and y be the investment amount on project 2 (continuous variable).

To write the linear constraints for the problem:

1. If we invest $100 or more on project 1, then we can only invest at most $100 on project 2:
This can be written as:
x ≥ 100 → y ≤ 100

If x is greater than or equal to 100, then y must be less than or equal to 100. This ensures that we don't invest more than $100 on project 2 if we invest $100 or more on project 1.

2. Investment amount cannot be negative:
This can be written as:
x ≥ 0
y ≥ 0

The investment amount on each project cannot be negative, so x and y must be greater than or equal to 0.

Therefore, the linear constraints for the problem are:
x ≥ 100 → y ≤ 100
x ≥ 0
y ≥ 0

First, let's define the variables:

Let x1 be the investment amount on project 1 (continuous variable)
Let x2 be the investment amount on project 2 (continuous variable)

Now, let's write the linear constraints based on the given condition:

If we invest $100 or more on project 1 (x1 ≥ 100), then we can only invest at most $100 on project 2 (x2 ≤ 100). To model this condition, we can use an integer variable:

Let y be an integer variable, with y ∈ {0, 1}

Now, we can write the linear constraints:

1. If y = 0, then x1 < 100 and there is no constraint on x2.
  y * 100 ≥ x1 (This ensures that if y = 0, x1 < 100)

2. If y = 1, then x1 ≥ 100 and x2 ≤ 100.
  (1 - y) * 100 ≥ x1 (This ensures that if y = 1, x1 ≥ 100)
  100y + x2 ≤ 200 (This ensures that if y = 1, x2 ≤ 100)

So the linear constraints are:
y * 100 ≥ x1
(1 - y) * 100 ≥ x1
100y + x2 ≤ 200

These constraints model the given condition, allowing you to analyze investments in both projects with continuous and integer variables.

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Answer: m∠CBD = 98°

Step-by-step explanation:

The set {1, x, x²} is a basis for the subspace of monic polynomials of degree at most 2 with real coefficients.

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

i) The subspace spanned by a vector v = [a,b] in R2 is the set of all scalar multiples of v. In other words, it is the line passing through the origin and the point (a,b).

So, the subspace spanned by v can be written as:

Span(v) = {k[a,b] : k ∈ R}

where k is a scalar. Note that [a,b] is the basis for this subspace.

ii) The set of monic polynomials of degree at most 2 with real coefficients is:

{1, x, x²}

This set spans a subspace of the vector space of all real polynomials of degree at most 2. To see this, let p(x) be an arbitrary polynomial of degree at most 2 with real coefficients. Then we can write:

p(x) = ax² + bx + c

where a, b, and c are real numbers. Now, we can express p(x) as a linear combination of the monic polynomials:

p(x) = a(x²) + b(x) + c(1)

Therefore, any polynomial of degree at most 2 with real coefficients can be written as a linear combination of the monic polynomials.

To find a basis for this subspace, we need to determine which of these monic polynomials are linearly independent. One way to do this is to see if any of the monic polynomials can be expressed as a linear combination of the others. In this case, it is clear that none of the monic polynomials can be expressed as a linear combination of the others.

Therefore, the set {1, x, x²} is a basis for the subspace of monic polynomials of degree at most 2 with real coefficients.

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The team that has best overall record for the season is the falcons because they have a larger mean value of 20.9 points

The mean of the dataset is defined the sum of all values divided by the total number of values. Therefore mean can be expressed as;

mean = sum of items/number of items

The mean value of Eagles = (3 + 24 + 14 + 27 + 10 + 13 + 10 + 21 + 24 + 17 + 27 + 7 + 40 + 37 + 55)/15

= 310/15 = 20.67

The mean value of falcons = (24 + 24  + 10 + 7 + 30 + 28 + 21 + 6 + 17 + 16 + 35 + 30 + 28 + 24 + 14)/15

= 314/15 = 20.93

Therefore since the mean value of falcons is higher than eagles, falcons has the best overall performance.

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Answer:

Step-by-step explanation:

8

Answer:

19

Step-by-step explanation:

Since the median is 18, we know that the missing number must be between 17 and 21. To find their average, we add them together and divide by 2:

(17 + 21) / 2 = 19

Answer:

9 x 9 x 9

Step-by-step explanation:

its obvious

8.19 divided by 4.2 is approximately equal to 1.94047624, which can be rounded to 1.94 (to two decimal places).

In mathematics, division is a basic arithmetic operation that involves separating a quantity or a number into equal parts or groups. The division operation is denoted by the symbol "/", or in some cases, the symbol "÷"

When we divide one number by another, we are essentially finding out how many times the second number "fits into" the first number

To divide 8.19 by 4.2, we can use long division as follows:

     1.9 4 0 4 7 6 2 4 3 3 3...

  --------------------------

4.2| 8.1 9 0 0 0 0 0 0 0 0 0

    8 4

    ----

    2 6 0

    2 5 2

    -----

      8 0 0

      7 1 4

      -----

      8 5 0

      8 4 8

      -----

        2 0 0

        1 6 8

        -----

        3 1 0

        2 5 2

        -----

          5 8

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The distance that the plane has flown, c, is 1,683 ft.

The distance of the plane is calculated by applying trigonometry ratio as shown below;

SOH CAH TOA

SOH = sin θ = opposite /hypothenuse side

TOA = tan θ = opposite side / adjacent side

CAH = cos θ = adjacent side / hypothenuse side

The height attained by the plane is the opposite side, while the hypothenuse is the distance travelled by the plane.

sin (12) = h/c

c = h/sin(12)

c = (350 ft ) / sin(12)

c = 1683.4 ft ≈1,683 ft

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The value ox in each of the equations given are:

1. x = 3        2. x = 5

For each of the equation given, we can solve to find the value of x by isolating the variable to one side of the equation while applying the properties of equality.

1. 2x + 5 = 11

2x = 11 - 5 [subtraction property of equality]

2x = 6

x = 6/2 [division property]

x = 3

2. 3x - 6 = 9

3x = 9 + 6 [addition property]

3x = 15

3x/3 = 15/3 [division property]

x = 5

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There is enough evidence to support the strategist's claim

To test the claim, we can use a one-sample proportion test.

Let p be the true proportion of residents in the town who favor construction. The null hypothesis is that p = 0.80 and the alternative hypothesis is that p > 0.80.

The test statistic is:

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

where p' is the sample proportion, n is the sample size.

Using the given data, we have:

p' = 0.83

p = 0.80

n = 1600

Plugging in these values, we get:

z = (0.83 - 0.80) / sqrt(0.80 * 0.20 / 1600) = 2.236

The corresponding p-value for this test statistic is 0.0126 (using a standard normal distribution table or calculator).

Since the p-value (0.0126) is less than the significance level (0.01), we reject the null hypothesis. There is sufficient evidence to support the claim that the percentage of residents who favor construction is more than 80%.

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The the approximate value of √8  is option B: between 2.8 and 2.9

An approximate number is a value derived close to the exact figure, yet a slight variance remains present. It is used to show that exact figures are precise and require no estimation.

However, these numbers are seen as estimates since their exact representation cannot be achieved through a finite number of digits. A close but lower value than a number is known as its approximate value by defect, with a desired level of accuracy.

Therefore, the radical expression for the square root of 8 is √8, but it can also be written as 2√2. Additionally, it can be expressed as a fraction, which is approximately equal to 2.828. Hence option B is correct.

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1) The function is P =650000e^(0.04)5

2) The function is P=800e^(0.02)6

3) The function is P= 2500e^-(0.03)5

Exponential growth is a type of growth pattern in which a quantity or value increases at a constant percentage rate over time, resulting in a rapid and accelerating increase in value

Note that;

P=Poe^rt

30000=20000e^0.05t

30000/20000 = e^0.05t

1.5 = e^0.05t

ln1.5 = 0.05t

t = 8 years

2) The function is a growth function and the percentage is 6%

3) 2000=45000e^-0.2t

2000/45000 = e^-0.2t

0.44 = e^-0.2t

ln0.44 = e^-0.2t

t = 4 years

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First, let's calculate the mean for each of the given samples:

1. R, B: (15,000 + 18,000) / 2 = 16,500 (already given)

2. R, G: (15,000 + 22,000) / 2 = 18,500 (already given)

3. R, W: (15,000 + 20,000) / 2 = 17,500 (already given)

4. B, G: (18,000 + 22,000) / 2 = 20,000 (already given)

5. B, W: (18,000 + 20,000) / 2 = 19,000 (already given)

6. G, W: (22,000 + 20,000) / 2 = 21,000 (already given)

Now, let's calculate the mean of all six sample means:

(16,500 + 18,500 + 17,500 + 20,000 + 19,000 + 21,000) / 6 = 112,500 / 6 = 18,750

The mean of all six sample means is 18,750.

Next, let's calculate the population mean:

(15,000 + 18,000 + 22,000 + 20,000) / 4 = 75,000 / 4 = 18,750

The population mean is 18,750.

Since the mean of all six sample means is equal to the population mean (18,750), the sample mean is an unbiased estimator of the population mean.