3. If a=12, 161-8, and the angle between them is 60°, determine the magnitude and direction of a +b. (4 marks) Include a diagram. 4. A ship has a cruising speed of 25 km/h and a heading of N10°W. There is a current of 6 km/h, travelling N70°W. What is the resultant velocity of the ship? (5 marks)

The value of the test statistic is given as follows:

t = 0.55.

The difference off the sample means is given as follows:

7 - 5.8 = 1.2.

The standard error for each sample is given as follows:

The standard error for the distribution of differences is given as follows:

[tex]\sqrt{2^2 + 0.7^2} = 2.17[/tex]

Hence the test statistic is given as follows:

t = 1.2/2.17

t = 0.55.

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We can rearrange the error bound formula to solve for h: h ≤ sqrt((12 * Error) / (L * M)).

To determine the spacing h required for the computation of an integral using the trapezoidal rule to be correct to five decimal places, we need to consider the error bound of the trapezoidal rule.

The error bound for the trapezoidal rule is given by the formula:

Error ≤ (b - a) * (h^2) * M / 12,

where:

- Error is the maximum error in the approximation,

- (b - a) is the interval of integration,

- h is the spacing between the points of evaluation,

- M is the maximum value of the second derivative of the function over the interval [a, b].

In this case, we want the error to be less than or equal to 0.00001 (five decimal places). Let's assume that (b - a) is denoted as L, and the maximum value of the second derivative of the function is denoted as M.

We can rearrange the error bound formula to solve for h:

h ≤ sqrt((12 * Error) / (L * M)).

Substituting the given values into the formula, we can determine the required spacing h.

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The rules of indices indicates that x = ∛(5⁵)

The rules of indices, which are also known as the laws of exponents, are mathematical rule that govern the manipulation of exponential equations.

The equation in the question is; -7 = 8 - 3·[tex]\sqrt[5]{x^3}[/tex]

The radical term, [tex]\sqrt[5]{x^3}[/tex] can be expressed in index form, using the rules of indices as follows;

[tex]\sqrt[5]{x^3}[/tex] = [tex]x^{\frac{3}{5} }[/tex]

The equation is therefore; -7 = 8 - 3·[tex]\sqrt[5]{x^3}[/tex]  = 8 - 3·[tex]x^{\frac{3}{5} }[/tex]

-7 = 8 - 3·[tex]x^{\frac{3}{5} }[/tex]

3·[tex]x^{\frac{3}{5} }[/tex] = 8 + 7 = 15

3·[tex]x^{\frac{3}{5} }[/tex] = 15

[tex]x^{\frac{3}{5} }[/tex] = 15/3 = 5

[tex]x^{\frac{3}{5} }[/tex] = 5

Raising both sides to the power 5, we get;

[tex]x^{\frac{3}{5} \times 5}[/tex] = x³ = 5⁵

x³ = 5⁵

Finding the cube root of both sides, we get;

∛(x³) = x = ∛(5⁵)

Therefore; x = ∛(5⁵)

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In a normal curve, approximately 68.2% of the observations lie between the mean and one standard deviation above or below the mean. It means that almost 68.2% of the population lies within the standard deviation of 1.

Here, standard deviation is a measure of how much variation or dispersion exists from the average value or mean value in a set of data.In a bell-shaped curve or normal distribution, 68.2% of the data points fall within the first standard deviation away from the mean, while about 95.4% of the data points fall within two standard deviations of the mean, and 99.7% of the data points fall within three standard deviations of the mean.

Therefore, the probability of observations falling within a standard deviation of the mean is very high and it is also known as empirical rule, or 68-95-99.7 rule.

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By using hypothesis test for the given p-value and significance level the correct answer is given by,

option b.  null hypothesis should be rejected because 0.02 is less than 0.05.

To make a decision based on the results of a hypothesis test,

Compare the p-value to the significance level (alpha).

Significance level (alpha) = 0.05

P-value = 0.02

The decision rule is as follows,

If the p-value is less than the significance level (p-value < alpha),

reject the null hypothesis.

If the p-value is greater than or equal to the significance level (p-value ≥ alpha), fail to reject the null hypothesis.

Here, the p-value (0.02) is less than the significance level (0.05).

Here, we reject the null hypothesis.

Therefore, for the p-value and significance level the correct option b.  null hypothesis should be rejected because 0.02 is less than 0.05.

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The slope-intercept equation of the line that passes through (7, 4) and is parallel to the y-axis is x = 7.

If a line is parallel to the y-axis, its slope is undefined because it does not have a change in the x-coordinate. However, we can still write the equation of the line using the given point (7,4).

Since the line is parallel to the y-axis, the x-coordinate of any point on the line will be 7. Therefore, the equation of the line is simply x = 7.

Note that this is a vertical line passing through the point (7,4) and parallel to the y-axis. The equation x = 7 represents all points where the x-coordinate is equal to 7, while the y-coordinate can take any value.

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x can be expressed as a linear combination of the vector [[1]], where the scalar is 2.

(a) To show that the set of vectors P = (1, 2, 1), Q = (1, 0, 2), and R = (1, 1, 0) is a spanning set for R³, we need to demonstrate that any vector in R³ can be expressed as a linear combination of these three vectors.

Let's consider an arbitrary vector v = (a, b, c) in R³. We want to find scalars x, y, and z such that xP + yQ + zR = v.

Setting up the equation, we have:

x(1, 2, 1) + y(1, 0, 2) + z(1, 1, 0) = (a, b, c)

Simplifying the equation, we get:

(x + y + z, 2x + z, x + 2y) = (a, b, c)

Now we can solve the system of equations:

x + y + z = a

2x + z = b

x + 2y = c

Solving these equations, we find:

x = a - b + c

y = (2b - a - c) / 3

z = b - 2a + c

Thus, we have expressed the arbitrary vector v as a linear combination of P, Q, and R:

v = (a, b, c) = (a - b + c)P + ((2b - a - c) / 3)Q + (b - 2a + c)R

Since we have shown that any vector in R³ can be expressed as a linear combination of P, Q, and R, we conclude that P, Q, and R form a spanning set for R³.

(b) Given matrix A = [[2, 1, 0]] and vector x = [[2]], to find 7(ax), we perform the matrix multiplication:

7(ax) = 7(Ax) = 7([[2, 1, 0]])[[2]] = [[2, 1, 0]] [[4]] = [[24 + 12 + 0*0]] = [[10]].

Therefore, 7(ax) is equal to [[10]].

To express x as a linear combination, we can write:

x = [[2]] = 2[[1]].

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a. Gum flavor claim: Null Hypothesis ([tex]H0[/tex]): Gum flavor lasts 39 minutes or less. Alternative Hypothesis ([tex]H1[/tex]): Gum flavor lasts more than 39 minutes. Right-tailed test.  b. Weight specifications of parts: Null Hypothesis ([tex]H0[/tex]): Parts meet weight specifications (19 pounds). Alternative Hypothesis ([tex]H1[/tex]): Parts do not meet weight specifications. Two-tailed test.                    c. Improvement in employee scores: Null Hypothesis ([tex]H0[/tex]): Mean score of employees has not improved ([tex]\mu = 58[/tex]). Alternative Hypothesis ([tex]H1[/tex]): Mean score of employees has improved. Right-tailed test.

a. For the claim that the flavor of gum lasts more than 39 minutes:

Null Hypothesis ([tex]H0[/tex]): The flavor of gum lasts 39 minutes or less.

Alternative Hypothesis ([tex]H1[/tex]): The flavor of gum lasts more than 39 minutes.

This is a right-tailed test as the alternative hypothesis suggests an increase in flavor duration.

b. For the weight specifications of the parts at the automobile manufacturing plant:

Null Hypothesis ([tex]H0[/tex]): The parts meet the weight specifications (weigh precisely 19 pounds).

Alternative Hypothesis ([tex]H1[/tex]): The parts do not meet the weight specifications (do not weigh precisely 19 pounds).

This is a two-tailed test, as the alternative hypothesis suggests a deviation from the specified weight in either direction.

c. For the improvement in employee scores in the last training exercise:

Null Hypothesis ([tex]H0[/tex]): The mean score of the employees has not improved ([tex]\mu = 58[/tex]).

Alternative Hypothesis ([tex]H1[/tex]): The mean score of the employees has improved ([tex]\mu > 58[/tex]).

This is a right-tailed test as the alternative hypothesis suggests an increase in scores.

To test the hypothesis at the 0.01 level of significance, we would compare the test statistic (such as z or t-score) with the critical value corresponding to the chosen significance level. If the test statistic falls in the critical region, we reject the null hypothesis and conclude that there is evidence to show a significant difference.

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The correct answer is: B. The function has no local maximums and has local minimums at approximately x = 5.8 and x = 28.8 (rounded to the nearest tenth).

To determine the location of the local extrema (maxima and minima) of the function f(x) = -0.697x^3 + 16642x^2 - 102407x + 650015, we need to find the critical points where the derivative of the function is equal to zero or does not exist. First, let's find the derivative of f(x) with respect to x: f'(x) = -2.091x^2 + 33284x - 102407. To find the critical points, we set f'(x) = 0 and solve for x: -2.091x^2 + 33284x - 102407 = 0. Using the quadratic formula, we can solve for x: x = (-b ± √(b^2 - 4ac)) / (2a). Plugging in the values a = -2.091, b = 33284, and c = -102407, we can calculate the values of x: x ≈ 5.779 or x ≈ 28.755. These are the potential locations of the local extrema.

To determine whether these points are maxima or minima, we can analyze the concavity of the function. Taking the second derivative, we have: f''(x) = -4.182x + 33284. Setting f''(x) = 0 and solving for x: -4.182x + 33284 = 0; x ≈ 7963.28. Since the second derivative is negative for x < 7963.28, we can conclude that x ≈ 5.779 corresponds to a local maximum, and x ≈ 28.755 corresponds to a local minimum. Therefore, the correct answer is: B. The function has no local maximums and has local minimums at approximately x = 5.8 and x = 28.8 (rounded to the nearest tenth).

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Using the argument principle 22-12 L √₁21-3 2)² 22+1 2 100 gives 22 - 12L(11)(23)²(100) = re^(iθ).

To solve the expression using the argument principle, let's break it down step by step:

Express the given expression in a suitable form for applying the argument principle.

We have the expression:

22 - 12L√(21 - 32)²(22 + 1)²(100)

Simplifying the expression inside the square root:

21 - 32 = -11

Substituting this value back into the expression:

22 - 12L√(-11)²(22 + 1)²(100)

Simplifying further:

22 - 12L√121(23)²(100)

We can simplify the square root:

22 - 12L(11)(23)²(100)

Apply the argument principle.

The argument principle states that if we have a complex number in the form z = r*e^(iθ), the argument of z, denoted as Arg(z), can be calculated as Arg(z) = θ.

In our case, we have the expression:

22 - 12L(11)(23)²(100)

To find the argument, we can write it as:

22 - 12L(11)(23)²(100) = re^(iθ)

Here, r represents the magnitude of the expression, and θ represents the argument.

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The correct answers are:

a. Probability that exactly 11 of those selected would do internet voting is 0.04191

b) No, because the probability of 11 or more is 0.05982 which is not low.

c)The probability that at least one of the selected likely voters would do internet voting is 0.96563

Given that 33% of likely voters would be willing to vote by the internet method instead of the in-person traditional method of voting,

So, the probability of internet voting is P = 0.33 and

the probability of the traditional method is

P = 1 - 0.33

  = 0.67

Now, n = 14 (Sample size)

P(X : 11) = C(14,11) × (0.33)11(0.67)14 - 11

             = 0.04191(rounded to 5 decimal places)

b. No, because the probability of 11 or more is 0.05982 which is not low.

C.Given that 11 of the selected voters would do internet voting.

From (a), we know that P(X : 11) = 0.04191 (rounded to 5 decimal places)

We know that if the probability is less than or equal to 0.05, then it is considered low.

Hence, the probability of 0.04191 is low and hence, 11 is significantly low.

c. Probability that at least one of the selected likely voters would do internet voting is 0.96563 (rounded to 5 decimal places)

Probability that none of the selected voters would do internet voting =

P(X : 0) = C(14,0) × (0.33)0(0.67)14 - 0

            = 0.001374 (rounded to 5 decimal places)

So, the probability that at least one of the selected likely voters would do internet voting is:

P(X ≥ 1) = 1 - P(X : 0)

            = 1 - 0.001374

            = 0.96563 (rounded to 5 decimal places)

Hence, the probability that at least one of the selected likely voters would do internet voting is 0.96563 (rounded to 5 decimal places).

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Let's assume that Alex can complete the task in x hours when working alone.  The combined work rate of Jane and Alex is given by the equation: 1/10 + 1/x = 1/8.

To solve for x, we can multiply through by the least common denominator, which is 40x: 4x + 40 = 5x. Now, we can solve for x by subtracting 4x from both sides: 40 = x. Therefore, it would take Alex working alone approximately 40 hours to sand and refinish the floor.

The solution assumes that the rates of work for Jane and Alex are constant and independent of the time spent working. It also assumes that the work is evenly divided between them when they work together.

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The integral `∫x^2/(x^2-9)^1/2 dx` can be evaluated using the trigonometric substitution `x = 3sec(θ)` as `-9cos(sec^-1 (x/3)) + C`.

Using the given trigonometric substitution, `x = 3sec(θ)`, we need to find the integral `∫x^2/(x^2-9)^1/2 dx`.

Now we will substitute `x` with `3sec(θ)` in the integral `∫x^2/(x^2-9)^1/2 dx`.

So, we get `dx = 3sec(θ)tan(θ) dθ`.

Now we will substitute these values of `x` and `dx` in the integral

`∫x^2/(x^2-9)^1/2 dx`.∫x^2/(x^2-9)^1/2 dx = ∫9tan^2(θ) / (9tan^2(θ)-9)^1/2 * 3sec(θ)tan(θ) dθ= 27 ∫sin^2(θ)dθ / (3sin^2(θ))^1/2

∴ ∫x^2/(x^2-9)^1/2 dx= 27 ∫sin^2(θ)dθ / 3sin(θ)

∴ ∫x^2/(x^2-9)^1/2 dx= 9 ∫sin(θ) dθ= -9cos(θ) + C.

Now we will substitute the value of θ.

θ = sec^-1 (x/3)

∴ cos(θ) = (3/x) (x^2-9)^1/2

∴ ∫x^2/(x^2-9)^1/2 dx = -9cos(sec^-1 (x/3)) + C

We can conclude that the integral `∫x^2/(x^2-9)^1/2 dx` can be evaluated using the trigonometric substitution `x = 3sec(θ)` as `-9cos(sec^-1 (x/3)) + C`.

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The probability of three scenarios was calculated based on the given problem.

(1) P(X > 28) = 0.5,

(2) P(X < 32) = 0.75,

(3) P(30 ≤ X ≤ 36) = 0.375.

Given: The weight of a student textbook orders is uniformly distributed over the interval from 20 to 36 pounds.

(1) Probability that a bag will weigh more than 28 pounds P(X > 28)

P(X > 28) = (36 − 28) / (36 − 20)

= 8 / 16

= 0.5

(0.5 is the probability that a bag will weigh more than 28 pounds.)

(2) Probability that a bag will weigh less than 32 pounds P(X < 32)

P(X < 32) = (32 − 20) / (36 − 20)

= 12 / 16

= 0.75 (0.75 is the probability that a bag will weigh less than 32 pounds.)

(3) Probability that a bag will weigh between 30 and 36 pounds P(30 ≤ X ≤ 36)

P(30 ≤ X ≤ 36) = (36 − 30) / (36 − 20)

= 6 / 16

= 0.375(0.375 is the probability that a bag will weigh between 30 and 36 pounds.)

Conclusion:

In this question, the probability of three scenarios was calculated based on the given problem.

(1) P(X > 28) = 0.5,

(2) P(X < 32) = 0.75,

(3) P(30 ≤ X ≤ 36) = 0.375.

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The estimated IQR of the water bills for ten households in Gombak in September is RM44.80.

The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles. A quartile is a statistical term describing a division of observations into four defined intervals based on the values of the data. The IQR is the range between the first quartile (Q1) and the third quartile (Q3).

IQR= Q3 - Q1

Where, Q3 is the third quartile, Q1 is the first quartile.

IQR for the given data can be calculated as follows

Arrange the data in ascending order.32.10, 45.70, 54.90, 65.50, 79.00, 87.70, 88.90, 98.00, 112.00, 132.60

Find the median of the given data.Q2 = (79.00 + 87.70) / 2Q2 = 83.35

Find the first quartile (Q1).It is the median of the lower half of the data set.Q1 = (54.90 + 65.50) / 2Q1 = 60.20

Find the third quartile (Q3).It is the median of the upper half of the data set.Q3 = (98.00 + 112.00) / 2Q3 = 105.00

Finally, use the formula to calculate the IQR.IQR = Q3 - Q1= 105.00 - 60.20= RM44.80

Thus, the estimated IQR of the water bills for ten households in Gombak in September is RM44.80.

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To set up a Riemann sum that approximates the work to pump all the liquid out of the tank, we need to consider the infinitesimal work done to pump an infinitesimal volume of liquid.

The height of the tank is given by the parabola y = 2x², and the density of the liquid varies according to ρ(y) = 8(y) = 8(5 - y). Let's divide the tank into n subintervals of equal width Δy. Each subinterval corresponds to a vertical slice of the tank. We can choose the y-coordinate of the i-th subinterval as yᵢ = iΔy, where i ranges from 0 to n. The width of each subinterval in the x-direction can be calculated as Δxᵢ = 2√(yᵢΔy). This is because the parabolic shape of the tank is obtained by rotating the parabola y = 2x² around the y-axis. The volume of the i-th subinterval can be approximated as Vᵢ ≈ π(Δxᵢ)²Δy. The infinitesimal work done to pump this volume out is dWᵢ = ρ(yᵢ)ghᵢVᵢ, where g is the acceleration due to gravity and hᵢ is the height from which the liquid is pumped. Therefore, the Riemann sum for the total work can be written as: W ≈ Σ dWᵢ ≈ Σ ρ(yᵢ)ghᵢVᵢ  ≈ Σ 8(5 - yᵢ)g(1 + 2√(yᵢΔy))²πΔy. To obtain the integral to compute the total work, we take the limit as n approaches infinity: W = ∫[0,5] 8(5 - y)g(1 + 2√(y))²π dy.

This integral represents the total work required to pump all the liquid out of the tank. However, the expression provided for the Riemann sum is incomplete, so it cannot be evaluated.

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The domain of y = (x² + 5)√x - 7 is all real numbers x such that x > 0 and x² + 5 ≥ 0. This is because the square root function is only defined for non-negative numbers, and the expression x² + 5 must be greater than or equal to 0 in order for the entire expression to be a real number.

The square root function is only defined for non-negative numbers, so the first requirement for the domain is that x > 0. The second requirement is that x² + 5 ≥ 0. This is because if x² + 5 is less than 0, then the square root of x² + 5 will be an imaginary number, and the entire expression will not be a real number.

Combining these two requirements, we get that the domain of y = (x² + 5)√x - 7 is all real numbers x such that x > 0 and x² + 5 ≥ 0.

Here is a more detailed explanation of the two requirements for the domain:

x > 0: The square root function is only defined for non-negative numbers, so the first requirement for the domain is that x > 0. This means that x cannot be equal to 0, and it cannot be negative.

x² + 5 ≥ 0: This requirement is to ensure that the square root of x² + 5 is a real number. If x² + 5 is less than 0, then the square root of x² + 5 will be an imaginary number, and the entire expression will not be a real number. This requirement can be simplified to x² ≥ -5. This means that x can be any real number, as long as it is not equal to the square root of -5.

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The values of x that satisfy f(x) = f^(-1)(-8) are x = -2 and x = -12.To solve the equation f(x) = f^(-1)(-8), we need to find the value of x that satisfies this equation.

First, let's clarify the notation. f^(-1)(-8) represents the inverse of the function f evaluated at -8.

Given that f(x) = x + 4/(x + 6), we can find the inverse function by swapping x and y and solving for y.

Step 1: Swap x and y in the equation:

x = y + 4/(y + 6)

Step 2: Solve for y:

xy + 6x = y + 4

xy - y = 4 - 6x

y(x - 1) = 4 - 6x

y = (4 - 6x)/(x - 1)

So, the inverse function f^(-1)(x) is (4 - 6x)/(x - 1).

Now, we want to find the value of x that satisfies f(x) = f^(-1)(-8):

f(x) = f^(-1)(-8)

x + 4/(x + 6) = (-8)

To solve this equation, we can multiply both sides by (x + 6) to eliminate the denominator:

(x + 6)(x + 4/(x + 6)) = (-8)(x + 6)

x(x + 6) + 4 = -8x - 48

Expanding and simplifying:

x^2 + 6x + 4 = -8x - 48

x^2 + 14x + 52 = 0

Now, we can solve this quadratic equation. Factoring or using the quadratic formula, we find the solutions:

x = -2 or x = -12

Therefore, the values of x that satisfy f(x) = f^(-1)(-8) are x = -2 and x = -12.

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The set {(1,0,0,1), (0,2,0,2), (1,0,1,0), (0,2,2,0)} is not a basis for ℝ⁴ as it is linearly dependent and does not span the entire space.

To determine whether a set is a basis for ℝ⁴, we need to check two conditions: linear independence and spanning the space.

Linear independence:

We consider the given set of vectors {(1,0,0,1), (0,2,0,2), (1,0,1,0), (0,2,2,0)}. We can create a matrix with these vectors as columns and perform row operations to check for linear independence. If the rank of the matrix equals the number of vectors, the set is linearly independent. However, if the rank is less than the number of vectors, the set is linearly dependent.

Upon performing row operations, we find that the rank of the matrix is 3, which is less than the number of vectors (4). Therefore, the given set is linearly dependent.

Spanning the space:

For a set to be a basis, it must also span the entire space ℝ⁴. Since the given set is linearly dependent, it cannot span ℝ⁴.

In conclusion, the given set {(1,0,0,1), (0,2,0,2), (1,0,1,0), (0,2,2,0)} is not a basis for ℝ⁴ as it fails to satisfy both conditions of linear independence and spanning the space.

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To show that S = X (1) is sufficient for θ, we need to show that the conditional distribution of the sample [tex]X1, X2, ...., Xn[/tex]given S and θ is independent of θ.  [tex]g(x, θ) = 0[/tex]almost surely for all θ. Thus, S = X (1) is a complete statistic for estimating θ.

Now, the joint density of X1, X2, ...., Xn is given by \begin[tex]{align*}L(\theta)=f_{X_{1}}(x_{1};\theta)f_{X_{2}}(x_{2};\theta).....f_{X_{n}}(x_{n};\theta)\\=e^{\sum_{i=1}^{n}\theta-x_{i}}I_{[\theta,\infty)}(x_{i})\end{align*}[/tex]To find the conditional distribution of the sample X1, X2, ...., Xn given S = X (1) and θ, we [tex]\&=\int_{0}^{\infty}g(x,\theta)\frac{d}{dx}(1-e^{-\theta x})dx\\&=-\int_{0}^{\infty}g(x,\theta)\frac{d}{d\theta}e^{-\theta x}dx\\&=\int_{0}^{\infty}g(x,\theta)x e^{-\theta x}dx\end{align*}[/tex]Now, since the above expression is zero for all θ, we must

Differentiating the second integral with respect to θ and using integration by parts, we have\begin[tex]{align*}0=\frac{d}{d\theta}\int_{0}^{\infty}g(x,\theta)x e^{-\theta x}dx&=-\int_{0}^{\infty}g(x,\theta)x^{2} e^{-\theta x}dx\\&\geq 0\end{align*}[/tex]

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Based on this confidence interval, there is sufficient evidence to conclude that a majority of Americans think it's the government's responsibility to promote equality between men and women is True.

The given statement is true. To explain in detail, it was determined from the survey that 82% of the respondents believe it's the government's responsibility to promote equality between men and women, with a margin of error of 2% at a 95% confidence interval.

The margin of error indicates that there is a 95% probability that the actual population parameter lies within two percentage points of the sample estimate. It is a 95% confidence interval, so there is only a 5% chance that the sample data is outside of the confidence interval.

If the 82% falls within the confidence interval of 2%, then it is a statistically significant result.

Therefore, there is sufficient evidence to conclude that a majority of Americans think it is the government's responsibility to promote equality between men and women.

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A) The mean of W is HW = -21.

B)  Rounding to 3 decimal places, we have σW = 46.266.

C)  Rounded to 3 decimal places, P(11X - 5Y > 3) = 0.776.

(a) The mean of W can be calculated as follows:

E(W) = E(11X - 5Y - 3)

= 11E(X) - 5E(Y) - 3   (since X and Y are independent)

= 11(9) - 5(19) - 3

= -21

Therefore, the mean of W is HW = -21.

(b) The variance of W can be calculated as follows:

Var(W) = Var(11X - 5Y - 3)

= 11^2 Var(X) + 5^2 Var(Y)    (since X and Y are independent)

= 11^2 (6.6)^2 + 5^2 (8.5)^2

= 2141.45

The standard deviation of W is therefore:

σW = sqrt(Var(W))

= sqrt(2141.45)

= 46.266

Rounding to 3 decimal places, we have σW = 46.266.

(c) We want to find P(11X - 5Y > 3). Let Z = 11X - 5Y - 3. Then Z is normally distributed with mean μZ = E(Z) = 11μX - 5μY - 3 = -24 and standard deviation σZ = sqrt(Var(Z)) = sqrt(11^2σX^2 + 5^2σY^2) = 31.619.

So we need to find P(Z > 0). We can standardize Z by subtracting the mean and dividing by the standard deviation:

P(Z > 0) = P((Z - μZ)/σZ > -μZ/σZ)

= P(Z* > -0.758)

where Z* is a standard normal random variable. Using a standard normal table or calculator, we find:

P(Z* > -0.758) = 1 - P(Z* < -0.758) = 1 - 0.2236 = 0.7764

Therefore, rounded to 3 decimal places, P(11X - 5Y > 3) = 0.776.

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The doctor claims that adults are more likely than children to have a vitamin D deficiency. This statement can be evaluated by calculating the proportion of adults and children in their respective samples that have a vitamin D deficiency.

In the adult sample, 26 out of 80 have a vitamin D deficiency, which is equal to a proportion of 0.325 or 32.5%. In the children sample, 21 out of 100 have a vitamin D deficiency, which is equal to a proportion of 0.21 or 21%. From these proportions, we can see that the proportion of adults with a vitamin D deficiency is higher than the proportion of children with a vitamin D deficiency.

However, to determine whether this difference is statistically significant, we would need to perform a hypothesis test or calculate a confidence interval. Based on the proportions calculated from the samples, it appears that the doctor's claim is supported by the data. However, it is important to note that the samples may not be representative of the entire population of adults and children, and the results may not be generalizable. In addition, there may be confounding variables that could affect the relationship between age and vitamin D deficiency, such as diet, lifestyle, and health conditions. Further research would be needed to explore these factors and determine whether age is a significant predictor of vitamin D deficiency. Overall, while the results of the samples suggest that adults are more likely than children to have a vitamin D deficiency, it is important to interpret these findings with caution and consider the limitations of the study.

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Based on the four-step hypothesis testing process, the data is not sufficient to conclude that the treatment has a significant effect on blood pressure. The p-value is greater than the specified significance level of 0.05, indicating that the results are not statistically significant.

Step 1: State the hypotheses

The null hypothesis (H0) is that the treatment has no effect on blood pressure, and the alternative hypothesis (H1) is that the treatment has a significant effect.

Step 2: Set the criteria for a decision

Using a two-tailed test and a significance level of α = 0.05, the critical region is split equally between the two tails, with 2.5% in each tail.

Step 3: Compute the test statistic

To compute the test statistic, we use the formula: t = (M - μ) / (s / sqrt(n)), where M is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. Plugging in the values, we get t = (27 - 29) / (sqrt(72/9)) = -2 / 2 = -1.

Step 4: Make a decision

Comparing the test statistic to the critical values, we find that the calculated t-value of -1 does not fall in the critical region. Therefore, we fail to reject the null hypothesis. The p-value associated with the t-value is greater than 0.05, indicating that the results are not statistically significant.

Interpretation:

Based on the analysis, we do not have sufficient evidence to conclude that the treatment has a significant effect on blood pressure. The p-value of the test is greater than the specified significance level of 0.05, suggesting that the observed difference in means could have occurred by chance.

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The interval of convergence is found out to be (-inf, inf) or (-∞, ∞) in interval notation.

To determine the center and radius of convergence for the given power series Σ (-1)^(n)(3x - 5)√(n + 1), n = 1, we can use the ratio test. The ratio test states that for a power series Σ a_n(x - c)^n, the series converges when the limit of the absolute value of the ratio of consecutive terms is less than 1.

Let's apply the ratio test to the given series:

|((-1)^(n+1)(3x - 5)√(n + 2))/((-1)^(n)(3x - 5)√(n + 1))|

= |(-1)(3x - 5)√(n + 2)/√(n + 1)|

= |-3x + 5|√((n + 2)/(n + 1))

To ensure convergence, we want the limit of the above expression to be less than 1 as n approaches infinity. However, we can see that the limit depends on the value of x.

For the series to converge, the term |-3x + 5|√((n + 2)/(n + 1)) must be less than 1.

-3x + 5 < 1  and -3x + 5 > -1

Solving these inequalities, we get:

-3x < -4  and -3x < -6

x > 4/3 and x > 2

Therefore, the series converges when x > 4/3.

The center of convergence is given by the value of x for which the series converges, which is x = 4/3.

The radius of convergence, R, can be determined by finding the distance between the center of convergence and the nearest point where the series diverges. In this case, since the series converges for all values of x greater than 4/3, the radius of convergence is infinite (R = inf).

The interval of convergence is then (-inf, inf) or (-∞, ∞) in interval notation.

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The probability that the number of people receiving calls from telemarketers is exactly 3 is 0.31026. b) The probability that the number of people receiving calls from telemarketers is between 2 and 5 is 0.93556. c) The probability that the number of people receiving calls from telemarketers is more than 3 is 0.52624.

Given that the percentage of U.S adults receiving calls from telemarketers is 43%. Let X denote the number of people receiving calls from telemarketers in a random sample of 7 adults. Because each person in the sample either receives a call from a telemarketer or doesn't, the distribution of X is binomial with n = 7 ,

p = 0.43. a) We are to find the probability that exactly 3 people in the sample receive calls from telemarketers. This is given by P(X = 3)

= (7C3) (0.43)3 (0.57)4

= 0.31026. b) We are to find the probability that the number of people receiving calls from telemarketers is between 2 and 5, inclusive.

This is given by P(2 ≤ X ≤ 5)

= P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)

= (7C2) (0.43)2 (0.57)5 + (7C3) (0.43)3 (0.57)4 + (7C4) (0.43)4 (0.57)3 + (7C5) (0.43)5 (0.57)2

= 0.93556. c) We are to find the probability that more than 3 people in the sample receive calls from telemarketers. This is given by P(X > 3)

= P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)

= (7C4) (0.43)4 (0.57)3 + (7C5) (0.43)5 (0.57)2 + (7C6) (0.43)6 (0.57)1 + (7C7) (0.43)7 (0.57)0

= 0.52624.

Hence, the required probabilities are given by: P(X = 3)

= 0.31026,P(2 ≤ X ≤ 5)

= 0.93556,

P(X > 3) = 0.52624.

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The required probabilities are:(a) Pr(r ≤ 2) = 0.657.(b) Pr(r > 1) = 0.8718.(c) Pr(r < 2) = 0.783.(d) Pr(r ≥ 4) = 0.91854.

Given that r are a binomial random variable with parameters n and p. And the number of successes in a Bernoulli Trials experiment. We need to find the probability of given events.

(a) Pr(r\leq2), n = 3, p = 0.7

So, the binomial probability distribution function is

P (r = k) = (n C k) p^k q^(n-k)

where q = 1-p. Here, n = 3, p = 0.7, q = 0.3.

P (r \leq 2) = P (r = 0) + P (r = 1) + P (r = 2)P (r = k)

= (n C k) p^k q^(n-k)P (r = 0)

= (3 C 0) (0.7)^0 (0.3)^3

= 0.027P (r = 1)

= (3 C 1) (0.7)^1 (0.3)^2

= 0.189P (r = 2)

= (3 C 2) (0.7)^2 (0.3)^1

= 0.441 P (r \leq 2)

= 0.027 + 0.189 + 0.441

= 0.657.

(b) Pr(r>1), n = 4, p = 0.7

So, the binomial probability distribution function is

P (r = k) = (n C k) p^k q^(n-k)

where q = 1-p. Here, n = 4, p = 0.7, q = 0.3.

P (r > 1) = 1 - P (r ≤ 1)

= 1 - [P (r = 0) + P (r = 1)]P (r = 0) = (4 C 0) (0.7)^0 (0.3)^4

= 0.0081P (r = 1)

= (4 C 1) (0.7)^1 (0.3)^3

= 0.1201 P (r > 1)

= 1 - [0.0081 + 0.1201]

= 0.8718.

(c) Pr(r<2), n = 3, p = 0.3

So, the binomial probability distribution function is

P (r = k) = (n C k) p^k q^(n-k)

where q = 1-p. Here, n = 3, p = 0.3, q = 0.7.

P (r < 2) = P (r = 0) + P (r = 1)P (r = k)

= (n C k) p^k q^(n-k)P (r = 0)

= (3 C 0) (0.3)^0 (0.7)^3

= 0.342 P (r = 1)

= (3 C 1) (0.3)^1 (0.7)^2

= 0.441 P (r < 2)

= 0.342 + 0.441

= 0.783

(d) Pr(r\geq4), n = 5, p = 0.9

So, the binomial probability distribution function is

P (r = k) = (n C k) p^k q^(n-k)

where q = 1-p. Here, n = 5, p = 0.9, q = 0.1.

P (r \geq 4) = P (r = 4) + P (r = 5)P (r = k)

= (n C k) p^k q^(n-k)P (r = 4)

= (5 C 4) (0.9)^4 (0.1)^1

= 0.32805 P (r = 5)

= (5 C 5) (0.9)^5 (0.1)^0

= 0.59049 P (r \geq 4)

= 0.32805 + 0.59049

= 0.91854

Therefore, the required probabilities are:(a) Pr(r ≤ 2) = 0.657.(b) Pr(r > 1) = 0.8718.(c) Pr(r < 2) = 0.783.(d) Pr(r ≥ 4) = 0.91854.

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Since the p-value is greater than all the provided significance levels, we accept the null hypothesis.

The hypothesis and null hypothesis for this scenario can be stated as follows:

Hypothesis (H1): People are taking less vacation because they are working from home.

Null Hypothesis (H0): People are not taking less vacation because they are working from home.

To test this hypothesis, we can use a one-sample t-test.

The test will compare the average vacation hours per month from last year (population mean) to the average vacation hours per month from this year (sample mean) to determine if there is a significant difference.

The standard error (SE) can be calculated using the formula:

SE = sample standard deviation / sqrt(sample size)

In this case, the sample standard deviation is 1.5 hours and the sample size is 75, so the standard error is:

SE = 1.5 / √75

  ≈ 0.173

The t-score is calculated using the formula:

t = (sample mean - population mean) / SE

Provided that the sample mean is 6.8 hours, the population mean is 7.3 hours, and the standard error is 0.173, the t-score is:

t = (6.8 - 7.3) / 0.173

 ≈ -2.890

Using the t-distribution table with a significance level of 0.05, the degrees of freedom for this test are n - 1 = 75 - 1 = 74.

The critical t-value at a significance level of 0.05 (two-tailed test) and 74 degrees of freedom is approximately ±1.990.

To determine how likely it is that the number of vacation hours used is not less this year than last year, we need to calculate the p-value associated with the t-score.

The p-value is the probability of obtaining a t-score as extreme as the observed t-score (or more extreme) under the null hypothesis.

Looking up the p-value in the t-distribution table, we obtain:

- p-value > 0.10

- p-value > 0.05

- p-value > 0.025

- p-value < 0.01

- p-value < 0.005

- p-value < 0.001

- p-value < 0.0005

Since the p-value is greater than all the provided significance levels (0.10, 0.05, 0.025, 0.01, 0.005, 0.001, 0.0005), we fail to reject the null hypothesis.

There is not enough evidence to support the claim that people are taking less vacation because they are working from home.

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A die with 6 faces is rolled once. The probability that the number is greater than 3 is 3/6.        

Explanation:When a die with six faces is rolled once, the possible outcomes are 1, 2, 3, 4, 5, or 6. Since the question asks for the probability that the number is greater than 3, we need to consider the outcomes that are greater than 3, which are 4, 5, and 6.There are a total of six possible outcomes, and three of them are greater than 3. Therefore, the probability of rolling a number greater than 3 is 3/6 or 1/2. Simplifying, we can say that the probability is 0.5 or 50%.Option b. 3/6 is the correct answer.    

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