What is the area of the triangle?
2 m
4 m
3 m
square meters

Based on the computed 95% confidence interval, we can conclude that we are 95% confident that the true population mean falls within the range of 12.65 to 25.65.

A confidence interval is a range of values that provides an estimate of the true population parameter. In this case, we are interested in estimating the population mean (μ). The 95% confidence interval, as mentioned, is given as 12.65 ≤ μ ≤ 25.65.

Interpreting this confidence interval, we can say that if we were to repeat the sampling process many times and construct 95% confidence intervals from each sample, approximately 95% of those intervals would contain the true population mean.

The confidence level chosen, 95%, represents the probability that the interval captures the true population mean. It is a measure of the confidence or certainty we have in the estimation. However, it does not guarantee that a specific interval from a particular sample contains the true population mean.

Therefore, based on the computed 95% confidence interval, we can conclude that we are 95% confident that the true population mean falls within the range of 12.65 to 25.65.

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The distance between the two points (4, 1) and (9, 9) is sqrt(89), which is approximately 9.43 units.

To find the distance between the two points (4, 1) and (9, 9), we can use the distance formula.

The distance formula is:
d = sqrt((x2 - x1)² + (y2 - y1)²)

Where (x1, y1) and (x2, y2) are the coordinates of the two points.

Using this formula, we can substitute the values we have:
d = √((9 - 4)² + (9 - 1)²)

Simplifying this equation, we get:
d = √(5² + 8²)
d = √(25 + 64)
d = √(89)

So, the distance between the two points (4, 1) and (9, 9) is sqrt(89), which is approximately 9.43 units.

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The mean mass, per bag of the maize is 98.6 kg.

Given,the masses in kg of 20 bags of maize were:

90,94,96,98,99,402,105,91,102,99,105,94,99,90,94,99,98,96,102 and 105.

The assumed mean of the given data is 96 kg. We need to find the mean mass, per bag of the maize.

First we calculate the deviation of each observation from the assumed mean, i.e., 96 kg.

Deviation = Observation - Assumed mean

We can calculate the deviation of each observation from the assumed mean as follows:

It is observed that one of the observation is much higher than the other observations, i.e., 402.

This indicates that there might be a typing error.

Lets replace 402 with 102 which is close to the values of other observations. Therefore, the corrected data is:

90,94,96,98,99,102,105,91,102,99,105,94,99,90,94,99,98,96,102 and 105.

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The Binomial Distribution is equivalent to the Bernoulli Distribution when the number of experiments/observations equals 1.

In the Bernoulli Distribution, there are only two possible outcomes: success (usually denoted as 1) and failure (usually denoted as 0). It represents a single trial with a fixed probability of success. The Binomial Distribution, on the other hand, represents multiple independent Bernoulli trials with the same fixed probability of success.

The Bernoulli Distribution can be considered as a special case of the Binomial Distribution when there is only one trial or experiment. It is characterized by a single parameter, which is the probability of success in that single trial. Therefore, when the number of experiments/observations equals 1, the Binomial Distribution is equivalent to the Bernoulli Distribution.

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The length of the segment with variable x is 4 units.

A trapezium is a quadrilateral. The midsegment of a trapezoid is the segment connecting the midpoints of the two non-parallel sides.

Therefore, the mid segment of a trapezium is equals to the average of the length of the bases.

Hence,

2x + 1 = 1 / 2 (x + 4x - 2)

2x + 1 = 1 / 2 (5x - 2)

2x + 1 = 1 / 2 (5x - 2)

2x + 1 = 5 / 2 x - 1

2x - 5 / 2x = -1 - 1

- 1 / 2x = -2

cross multiply

-x = - 4

x = 4

Therefore,

length of x = 4 units

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(a) Let X be the length of an eel in the lake. Then X ~ N(84, 18^2). The probability that an eel is a giant (i.e., X > 120) is:

P(X > 120) = P(Z > (120-84)/18) = P(Z > 2) = 0.0228 (using standard normal distribution table)

Therefore, the probability that an eel is a giant is 0.0228 or about 2.28%.

(b) Let Y be the number of giants in a sample of 100 eels. Then Y follows a binomial distribution with parameters n = 100 and p = P(X > 120) = 0.0228. The expected number of giants in a sample of 100 eels is:

E(Y) = np = 100(0.0228) = 2.28

Therefore, we expect about 2.28 giants in a sample of 100 eels.

(c) To find the proportion of eels between 75 cm and 90 cm, we need to standardize these values using the mean and standard deviation of the population:

P(75 < X < 90) = P[(75-84)/18 < (X-84)/18 < (90-84)/18]

= P(-0.5 < Z < 0.33)

= 0.3736 - 0.3085

= 0.0651

Therefore, about 6.51% of eels are between 75 cm and 90 cm.

(d) The distribution of sample means follows a normal distribution with mean μ = 84 and standard error σ/sqrt(n) = 18/sqrt(100) = 1.8 (by Central Limit Theorem). Therefore, the distribution of sample means is N(84, 1.8^2).

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Given statement solution is :-  When θ = 4π/7 and the radius is 5 cm, the arc length is approximately 8.163 cm.

To calculate the arc length of a circle, you can use the formula:

Arc Length = θ * r

where θ is the central angle in radians and r is the radius of the circle.

In this case, the central angle θ is given as 4π/7, and the radius r is 5 cm. Plugging these values into the formula, we can calculate the arc length:

Arc Length = (4π/7) * 5

= (4/7) * π * 5

≈ 8.163 cm (rounded to three decimal places)

Therefore, when θ = 4π/7 and the radius is 5 cm, the arc length is approximately 8.163 cm.

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Therefore, the measures of the two complementary angles are 28° and 62°.

Given expressions for complementary angles are (11y - 5)° and (16y + 14)°.

We know that the sum of complementary angles is 90°.

Therefore, we can set up an equation and solve it as follows:

(11y - 5)° + (16y + 14)° = 90°11y + 16y + 9 = 90 (taking the constant terms on one side)

27y = 81y = 3

Hence, the measures of the two complementary angles are:

11y - 5 = 11(3) - 5

= 28°(16y + 14)

= 16(3) + 14

= 62°

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The geometric average return for this stock over the past 4 years is 2.48%.

To find the geometric average return for this time period, we need to use the formula:

Geometric Average Return =[tex](1 + R1) x (1 + R2) x (1 + R3) x (1 + R4)^(1/4) - 1[/tex]

Where R1, R2, R3, and R4 are the returns for each year.

Using the returns given in the question, we can plug them into the formula:

Geometric Average Return = (1 + 0.06) x (1 + 0.13) x (1 - 0.11) x [tex](1 + 0.17)^(1/4)[/tex] - 1

Simplifying this equation, we get:

Geometric Average Return = 1.0248 - 1

Geometric Average Return = 0.0248 or 2.48%

Therefore, the geometric average return for this stock over the past 4 years is 2.48%.

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Generally speaking, if two variables are unrelated and show no pattern as one increases, their covariance will be a positive or negative number close to zero.

So, the correct answer is C.

Covariance is a measure used to indicate the extent to which two variables change together.

A large positive number would suggest a strong positive relationship, while a large negative number would indicate a strong negative relationship.

However, when the variables are unrelated and display no discernible pattern, the covariance tends to be close to zero, showing that there is little to no relationship between the variables.

Hence the answer of the question is C.

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4 kilometers is the height of the given mountain.

In this case, we can consider the height of the mountain as the length of one side of a right triangle, the distance Ben climbed as the length of another side, and the distance from the base of the mountain to the center as the hypotenuse.

Let's denote the height of the mountain as h. According to the given information, the distance Ben climbed is 5 kilometers, and the distance from the base to the center of the mountain is 3 kilometers.

Using the Pythagorean theorem, we have the equation:

[tex]h^2 = 5^2 - 3^2\\\\h^2 = 25 - 9\\\\h^2 = 16[/tex]

Taking the square root of both sides, we find:

h = √16

h = 4

Therefore, the height of the mountain is 4 kilometers.

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The appropriate statements that make the series statement true are:

The given series is the sum of terms defined by the expression (2n)! / (n!)^2 for n=0 to infinity. This series is known as the central binomial coefficients series, which can also be represented as the coefficients of the binomial expansion of (1+1)^2n.
This series is known as the central binomial coefficients series because each term is equal to the binomial coefficient (2n choose n), which is the number of ways to choose n items out of a set of 2n items. This can be seen from the formula for the binomial coefficient:

(2n choose n) = (2n)! / (n!)^2

So the series is the sum of the binomial coefficients (2n choose n) for n=0 to infinity. These coefficients arise in the binomial expansion of (1+1)^2n, which gives:

(1+1)^2n = sum((2n choose k) * 1^k * 1^(2n-k), k=0 to 2n)

The correct question should be :

Choose all the appropriate statements that make the series statement true for the series from n=0 to infinity: (2n)! / (n!)^2.

Select the appropriate statements from the above options.

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It would be more advantageous for the grandfather to choose option B in this case.

Option B is better for a grandfather who wants to give his grandchild $200 when the child turns 18. The reason why is that it will grow his money faster than option A.

5.5% every year means that the grandfather will be earning interest on his $200 each year, and the amount of interest he earns will also increase over time as the principal balance increases.

This will result in the grandfather having more money to give to his grandchild in the end compared to option A. If the grandfather chooses option A, he will only give the grandchild $360 by the time they turn 18. However, with option B, the grandfather will be able to give the grandchild a larger amount of money.

option B is the better choice because it will result in a larger amount of money for the grandchild. The grandfather will earn 5.5% interest on his $200 each year, which will result in more money to give to his grandchild in the end.

Option A would only provide the grandchild with $360 by the time they turn 18, while option B would provide a larger amount. Therefore, it would be more advantageous for the grandfather to choose option B in this case.

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The gradient of f is ∇f(x, y, z) = (3ye3yz, 3xe3yz, xe3yz).

At point P, ∇f(1, 0, 3) = (9e9, 3e9, e9).

The rate of change of f at P in the direction of the vector u is Duf(1, 0, 3) = 2e9/3.

The gradient of f is the vector of partial derivatives of f with respect to each variable, which is given by ∇f(x, y, z) = (3ye3yz, 3xe3yz, xe3yz).

To evaluate the gradient at point P (1, 0, 3), we substitute x=1, y=0, and z=3 into the gradient formula to get ∇f(1, 0, 3) = (9e9, 3e9, e9).

To find the rate of change of f at point P in the direction of the vector u = <2/3, -1/3, 2/3>, we take the dot product of the gradient at point P and the unit vector u, which is given by

Duf(1, 0, 3) = ∇f(1, 0, 3)·u/|u| = (9e9)(2/3) + (3e9)(-1/3) + (e9)(2/3) / √(4/9 + 1/9 + 4/9) = 2e9/3.

Therefore, the rate of change of f at point P in the direction of u is 2e9/3.

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A proof by contrapositive of the theorem is that Assumed: 15 - 8x + x² < 0 Proven: x < 0 or x > 3. So, the correct answer is C).

Proof by Contrapositive

To prove the theorem by contrapositive, we assume that 15 - 8x + x² ≤ 0 and prove that 0 < x ≤ 3.

Given that 15 - 8x + x² ≤ 0,

Rearranging the terms, we get:

x² - 8x + 15 ≤ 0

Factoring the quadratic expression, we get:

(x - 5)(x - 3) ≤ 0

The roots of the quadratic equation x² - 8x + 15 = 0 are x = 3 and x = 5. Therefore, the quadratic expression is negative between these two roots, and it is positive outside this interval. Hence, the solution to the inequality is x < 3 or x > 5.

Since we know that 0 < x, x cannot be negative. Therefore, x > 5 is not possible. Hence, we have x < 3.

Therefore, we have shown that if 15 - 8x + x² ≤ 0, then 0 < x ≤ 3. This is the contrapositive of the theorem.

Therefore, the correct option is (d): Assumed: 15 - 8x + x² < 0 Proven: x < 0 or x > 3.

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To convert 345 milliliters to liters, the proportion you should use is:1 L = 1000 mL.

The above conversion implies that 1 liter is equal to 1000 milliliters. Therefore, to convert milliliters to liters, you should divide the number of milliliters by 1000. This means that the answer will be in liters. The proportion used is shown below:$$\frac{345 \text{ mL}}{1} \times \frac{1\text{ L}}{1000\text{ mL}}= \frac{345}{1000} \text{ L}= 0.345 \text{ L}$$Therefore, 345 milliliters is equal to 0.345 liters when converted using the above proportion.

The litre is a metric volume unit . Although the litre is not a SI unit, it is classified as one of the "units outside the SI that are accepted for use with the SI," along with units like hours and days. The cubic metre (m3) is the SI unit for volume.

A millilitre is a metric unit of volume that is equal to one thousandth of a litre (also written millilitre or mL). The International System of Units (SI) accepts it as a non-SI unit for usage with its system. In terms of volume, it is exactly equal to one cubic centimetre (cm3, or non-standard, cc).

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The integral can be evaluated as ∫[-1,4] (1-x^2) dx = A1 - A2 = π/4 - 3/2 which is approximately equal to -0.93.

We can evaluate the integral ∫[-1,4] (1-x^2) dx by interpreting it in terms of areas. The integrand 1-x^2 represents a downward facing parabola that intersects the x-axis at x = -1 and x = 1. The limits of integration are -1 and 4, which means we are integrating over the entire region between x = -1 and x = 4.

We can split this region into two parts: the area under the curve from x = -1 to x = 1, and the area under the curve from x = 1 to x = 4. Since the integrand is always positive in the first region and always negative in the second region, we can express the integral as the difference of two areas:

∫[-1,4] (1-x^2) dx = A1 - A2

where A1 is the area under the curve from x = -1 to x = 1, and A2 is the area under the curve from x = 1 to x = 4.

To find A1, we integrate the integrand from x = -1 to x = 1:

A1 = ∫[-1,1] (1-x^2) dx

This represents the area of a quarter circle with radius 1, centered at the origin. Thus,

A1 = π/4

To find A2, we integrate the absolute value of the integrand from x = 1 to x = 4:

A2 = ∫[1,4] |1-x^2| dx

This represents the area of a trapezoid with bases of length 3 and 15/4 and height 1. Thus,

A2 = 3/2

Therefore, the integral can be evaluated as:

∫[-1,4] (1-x^2) dx = A1 - A2 = π/4 - 3/2

which is approximately equal to -0.93.

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It should be noted that the average rate of change of g(x) is 3 times that of f(x) is 3.

The average rate of change of a function is calculated by finding the slope of the secant line that intersects the graph of the function at the interval's endpoints.

The average rate of change of f(x) over 1 sxs4 is:

(f(4) - f(1)) / (4 - 1) = (-36 - 1) / 3 = -12

The average rate of change of g(x) over 1 sxs4 is:

(g(4) - g(1)) / (4 - 1) = (-48 - 15) / 3 = -33

The average rate of change of g(x) is 3 times that of f(x).

(-33) / (-12) = 3

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Let's start by recalling that the negative binomial distribution with parameters m and p has probability mass function:

f(x; m, p) = (x+m-1) choose [tex]x (1-p)^mp^x[/tex]

for x = 0, 1, 2, ...

To find the UMVUE for [tex](1-p)^r[/tex], we need to find an unbiased estimator that depends only on the sample X1, X2, ..., Xn and that has the smallest possible variance among all unbiased estimators.

Since [tex](1-p)^r[/tex] is a function of 1-p, we can use the method of moments to find an estimator for 1-p. Specifically, the first moment of the negative binomial distribution with parameters m and p is:

[tex]E[X] = \frac{m(1-p)}{p}[/tex]

Solving for 1-p, we get: [tex]1-p = \frac{m}{(m+E[X])}[/tex]

Now, let's substitute θ = (1-p) into this expression to get:

θ = (1-p) = [tex]1-p = \frac{m}{(m+E[X])}[/tex]

We can use the above expression to construct an unbiased estimator of θ as follows:

θ_hat = [tex]= \frac{1-m}{(m+X_{bar} )}[/tex],

where X_bar is the sample mean.

Now, let's express [tex](1-p)^r[/tex] in terms of θ:

[tex](1-p)^r = θ^r[/tex]

Using the above estimator for θ, we can construct an unbiased estimator for [tex](1-p)^r[/tex] as follows:

[tex](1-p)^{r_{hat} } = (\frac{1-m}{m+X_{bar} } )^{r}[/tex]

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Answer: The fitted curve is a parabola that passes through the first data point (0,27) and has zeros at x=1, x=2, and x=3.

Step-by-step explanation:

To fit a quadratic polynomial to the given data points using least squares, we need to minimize the sum of the squares of the residuals (the vertical distances between the data points and the fitted curve). The quadratic polynomial can be expressed as:

f(x) = ax^2 + bx + c

where a, b, and c are the coefficients to be determined. We can use the following system of equations to solve for these coefficients:

Σ(y - f(x))^2 = Σ(y - ax^2 - bx - c)^2

where Σ represents the sum over all data points.

Substituting the given data points into the equation above, we obtain:

27 - c = 0

0 - (a + b + c) = 0

0 - (4a + 2b + c) = 0

0 - (9a + 3b + c) = 0

Simplifying these equations, we get:

c = 27

a + b = -27

4a + 2b = -27

9a + 3b = -27

Solving for a and b, we obtain:

a = -3

b = -24

Substituting these values into the equation for f(x), we get:

f(x) = -3x^2 - 24x + 27

Therefor, the fitted curve is a parabola that passes through the first data point (0,27) and has zeros at x=1, x=2, and x=3.

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The one-shot nash equilibrium is (1060, 50).

To find the one-shot Nash equilibrium, we need to find a strategy profile where no player can benefit from unilaterally deviating from their strategy.

Let's consider player 1's strategy. If player 1 chooses 10, player 2 should choose -5 since 10-(-5) = 15, which is greater than 0. If player 1 chooses 1060, player 2 should choose 50 since 1060-50 = 1010, which is greater than 0. If player 1 chooses -5, player 2 should choose 10 since -5-10 = -15, which is less than 0. So, player 1's best strategy is to choose 1060.

Now let's consider player 2's strategy. If player 2 chooses -5, player 1 should choose 10 since 10-(-5) = 15, which is greater than 0. If player 2 chooses 6050, player 1 should choose 1060 since 1060-6050 = -4990, which is less than 0. If player 2 chooses 50, player 1 should choose 1060 since 1060-50 = 1010, which is greater than 0. So, player 2's best strategy is to choose 50.

Therefore, the one-shot Nash equilibrium is (1060, 50).

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The standard form of the equation is x^2 + y^2 = 36

To write the equation of the circle x^2 + y^2 - 36 = 0 in standard form, we need to complete the square for both the x and y terms.

Starting with the given equation:

x^2 + y^2 - 36 = 0

Rearranging the terms:

x^2 + y^2 = 36

To complete the square for the x terms, we need to add (1/2) of the coefficient of x, squared. Since the coefficient of x is 0, there is no x term, and thus no need to complete the square for x.

For the y terms, we add (1/2) of the coefficient of y, squared. The coefficient of y is also 0, so there is no y term to complete the square for y.

The equation remains the same:

x^2 + y^2 = 36

In standard form, the equation for a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

Since there is no x or y term, the center of the circle is at the origin (0, 0), and the radius is the square root of the constant term, which is 6.

Therefore, the standard form of the equation is:

(x - 0)^2 + (y - 0)^2 = 6^2

x^2 + y^2 = 36

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The measurement that is closest to the number of kilometers between these two towns is 392 kilometers.

To determine the distance in kilometers between Mesquite and Houston which is closest to the actual number of kilometers, we can use the following conversion factor;

Approximately 8 kilometers in 5 miles

That is;

1 mile = 8/5 kilometers

And the distance between Mesquite and Houston is 245 miles.

Thus, we can calculate the distance in kilometers as;

245 miles = 245 × (8/5) kilometers

245 miles = 392 kilometers (correct to the nearest whole number)

Therefore, the measurement that is closest to the number of kilometers between these two towns is 392 kilometers.

This is obtained by multiplying 245 miles by the conversion factor 8/5 (approximated to 1.6) in order to obtain kilometers.

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The value of k such that p(−0. 54 ≤ t ≤ k) is 0.3

To find the value of k that satisfies this probability condition, we need to refer to the t-distribution table or use statistical software. The t-distribution table provides critical values for different probabilities and degrees of freedom.

From the given probability, p(−0.54 ≤ t ≤ k) = 0.3, we can interpret this as the area under the t-distribution curve between −0.54 and k is 0.3. We want to find the corresponding value of k.

To solve this problem, we need to locate the area of 0.3 in the t-distribution table. We look for the closest value to 0.3 in the table and find the corresponding t-value. Let's assume the closest value to 0.3 in the table corresponds to t = c.

Therefore, p(−0.54 ≤ t ≤ k) = p(t ≤ k) - p(t ≤ -0.54) = 0.3.

We know that p(t ≤ c) = 0.3, and since we're dealing with a symmetric distribution, we can also say that p(t ≥ -c) = 0.3.

From the t-distribution table, we can find the critical t-value for the probability 0.3 and 14 degrees of freedom. Let's denote this critical value as c.

So, we have p(t ≤ c) = 0.3 and p(t ≥ -c) = 0.3. Combining these, we can write:

p(t ≤ c) + p(t ≥ -c) = 0.3 + 0.3.

Since the t-distribution is symmetric, p(t ≤ c) + p(t ≥ -c) is equivalent to 2 * p(t ≤ c). Therefore, we have:

2 * p(t ≤ c) = 0.6.

Solving for p(t ≤ c), we get:

p(t ≤ c) = 0.6 / 2 = 0.3.

From the t-distribution table, locate the critical t-value that corresponds to a probability of 0.3 and 14 degrees of freedom. Denote this value as c.

Now, we have the condition p(t ≤ c) = 0.3. We need to find the value of k such that p(t ≤ k) = 0.3.

Since the t-distribution is symmetric, we know that p(t ≥ -c) = 0.3. Therefore, the condition p(t ≤ k) = 0.3 is equivalent to p(t ≥ -k) = 0.3.

In other words, we need to find the t-value that corresponds to a probability of 0.3 and 14 degrees of freedom. Denote this value as -k.

Hence, we have p(t ≥ -k) = 0.3. From the t-distribution table, locate the critical t-value that corresponds to a probability of 0.3 and 14 degrees of freedom. Denote this value as -k.

Therefore, k is the positive value of -k. We can now find the corresponding t-value for k from the table.

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The condition that necessary to apply the central limit theorem is random sampling

To apply the Central Limit Theorem (CLT), the following condition is necessary:

Random Sampling: The samples should be selected randomly from the population.

The Central Limit Theorem states that for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution. This holds true under the condition of random sampling.

In your case, since you are selecting random samples of size 50 from a normal population with a mean of $95 and a standard deviation of $14, you satisfy the condition of random sampling required for the application of the Central Limit Theorem.

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Answer: (7,-6)

Step-by-step explanation:

9-x = 2  --> x=7

-8-y = -2  --> y = -6

B = (7,-6)

The matching of vocabulary words with their corresponding examples is as follows:

a. Proportion of the 750 survey participants who will vote for the cause - Statistic

b. List of 750 Yes or No answers to the survey question - Data

c. The proportion of all voters from the district who will vote for the cause - Parameter

d. Answer Yes or No to the survey question - Variable

e. All the voters in the district - Population

f. 750 voters who participated in the survey - Sample

a. The proportion of the 750 survey participants who will vote for the cause corresponds to a statistic. A statistic refers to a numerical summary calculated from a sample of data.

b. The list of 750 Yes or No answers to the survey question represents the data. Data refers to the collection of individual observations, measurements, or responses.

c. The proportion of all voters from the district who will vote for the cause corresponds to a parameter. A parameter refers to a numerical summary calculated from the entire population.

d. The answer Yes or No to the survey question represents a variable. A variable is a characteristic or attribute that can take on different values.

e. All the voters in the district represent the population. A population refers to the entire group of individuals, objects, or events of interest in a statistical study.

f. The 750 voters who participated in the survey correspond to a sample. A sample refers to a subset of the population that is selected and used to represent the entire population for analysis.

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Mr. Hillman can buy a maximum of 29 boxes of colored pencils within his budget.

To calculate how many boxes Mr. Hillman can buy, we need to consider the discounted price, sales tax, and his budget.

First, let's calculate the discounted price of each box. The discount is 30%, so Mr. Hillman will pay 70% of the regular price.

Discounted price = 70% of $1.80

               = 0.70 * $1.80

               = $1.26

Next, we need to add the sales tax of 6% to the discounted price.

Price with sales tax = (1 + 6%) * $1.26

                   = 1.06 * $1.26

                   = $1.3356 (rounded to two decimal places)

Now, we can calculate the maximum number of boxes Mr. Hillman can buy with his $40 budget.

Number of boxes = Budget / Price with sales tax

              = $40 / $1.3356

              ≈ 29.95

Since we cannot buy a fraction of a box, Mr. Hillman can buy a maximum of 29 boxes of colored pencils within his budget.

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In multivariable calculus, the tangent plane is a plane that "touches" a surface at a given point and has the same slope or gradient as the surface at that point.

To find the equation for the tangent plane at the point P0(2, 1, 2) on the surface 2x^2 + 4y^2 +3z^2 = 24, we need to find the gradient vector of the surface at P0, which gives us the normal vector of the plane. Then, we can use the point-normal form of the equation for a plane to find the equation of the tangent plane.

The gradient vector of the surface is given by:

grad(2x^2 + 4y^2 +3z^2) = (4x, 8y, 6z)

At P0(2, 1, 2), the gradient vector is (8, 8, 12), which is the normal vector of the tangent plane.

Using the point-normal form of the equation for a plane, we have:

8(x - 2) + 8(y - 1) + 12(z - 2) = 0

Simplifying, we get:

4x + 4y + 3z = 20

Therefore, the correct equation for the tangent plane is D. 5x + 4y + 3z = 20.

To find the equations for the normal line, we need to use the direction vector of the line, which is the same as the normal vector of the tangent plane. Thus, the direction vector of the line is (8, 8, 12).

The equations for the normal line can be expressed as:

x = 2 + 8t

y = 1 + 8t

z = 2 + 12t

where t is a parameter that can take any real value.

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