Zahra is a company specializing in high-end cosmetics. Its signature nail polish color is Soft Lilac. On a certain day, the company's factory had 5300 bottles of nail polish, and 1400 of those bottles were Soft Lilac. On that day, the inventory manager took a sample of 14 bottles of nail polish in the factory. She found that 3 of those bottles were Soft Lilac. For the inventory manager's sample, find and write with proper notation the population proportion and sample proportion of bottles that were Soft Lilac. Write the proportions as decimals (not percentages) rounded to two decimal places.

Answer:

1/7

Step-by-step explanation:

7/7-3/7=4/7

[tex]\frac{4}{7} /2[/tex]=2/7

4/7+2/7=6/7

7/7-6/7=1/7

So the housewife has 1/7 of the money left

The guy wire is attached to the tower at a height of approximately 15.95 meters.

Length of the guy wire (hypotenuse) = 18.5 m

Angle between the ground and the guy wire = 59º

Using the sine function to find the height of the tower.
sin(angle) = height/hypotenuse

Putting in the known values and solving for the height.
sin(59º) = height/18.5 m
height = sin(59º) × 18.5 m

Calculating the height
height ≈ 15.95 m

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The forecast for Y3 is -3/8.

We can start by writing the MA(1) process as:

Yt = μ + θεt-1 + εt

where μ is the process mean, θ is the MA(1) coefficient, εt is the white noise error term with mean zero and variance σ^2.

From the given information, we know that the process mean is zero, so μ = 0.

The conditional least-squares estimate of θ given the first two observations can be obtained by minimizing the sum of squared errors:

S(θ) = (y1 - θε0)^2 + (y2 - μ - θε1)^2

where ε0 and ε1 are unobserved error terms and y1, y2 are the first two observations.

Substituting the given values, we get:

S(θ) = 1 + θ^2 + (1/4 - θ)^2

Taking the derivative of S(θ) with respect to θ and setting it to zero, we get:

dS(θ)/dθ = 2θ - 2(1/4 - θ) = 0

Solving for θ, we get:

θ = 3/8

Therefore, the conditional least-squares estimate of θ given the first two observations is 3/8.

To find the forecast for Y3, we can use the MA(1) model equation:

Y3 = μ + θε2 + ε3

where ε2 and ε3 are unobserved error terms. Substituting the estimated value of θ and the given value of Y2, we get:

Y3 = (3/8)(-1) + ε3 = -3/8 + ε3

Therefore, the forecast for Y3 is -3/8.

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(a)The linear function that models the monthly cost C as a function of the distance driven d is:

C(d) = 0.25d + 260

(b) we predict that it would cost $625 per month to drive 1,500 miles.

A linear function is simple and easy to interpret, which makes it a useful model for practical purposes.

(a) Let's use the two data points to find the equation of the line that models the monthly cost as a function of the distance driven. The slope of the line is the change in cost over the change in distance, so we have:

slope = (460 - 380) / (800 - 480) = 80 / 320 = 0.25

The y-intercept is the cost when no distance is driven, so we have:

y-intercept = 380 - 0.25 * 480 = 260

(b) To predict the cost of driving 1,500 miles per month, we simply plug in d = 1500 into the linear function we found in part (a):

C(1500) = 0.25(1500) + 260 = $625

Therefore, we predict that it would cost $625 per month to drive 1,500 miles.

(c) The graph of the linear function is a straight line with slope 0.25 and y-intercept 260. The slope represents the rate of change of the cost with respect to the distance driven. In other words, for each additional mile driven, the cost increases by $0.25.

The y-intercept represents the fixed cost of driving the car, which includes expenses such as insurance and maintenance that do not depend on the distance driven.

(d) The y-intercept represents the fixed cost of driving the car, which includes expenses such as insurance and maintenance that do not depend on the distance driven.

(e) A linear function gives a suitable model in this situation because the relationship between the monthly cost and the distance driven is approximately linear over the range of distances we have data for. Additionally, a linear function is simple and easy to interpret, which makes it a useful model for practical purposes.

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one solution of the equation is approximately 1.8925469 radians.

The equation is:

tan(x) + 3 = 0

Subtracting 3 from both sides, we get:

tan(x) = -3

Taking the inverse tangent of both sides, we get:

x = arctan(-3)

However, the tangent function is periodic with period π, which means that there are infinitely many solutions to this equation. In general, the solutions are given by:

x = arctan(-3) + nπ, where n is an arbitrary integer.

Using a calculator to approximate arctan(-3), we get:

arctan(-3) ≈ -1.2490458

Therefore, the general solution to the equation is:

x ≈ -1.2490458 + nπ, where n is an arbitrary integer.

If we substitute n = 1, we get:

x ≈ -1.2490458 + π

Using a calculator to approximate this value, we get:

x ≈ 1.8925469

So one solution of the equation is approximately 1.8925469 radians.

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If Mexico, one of our largest trading partners, goes into a recession, it is likely to decrease its demand for our exports. This would shift the aggregate demand (AD) curve leftward, leading to a decrease in equilibrium GDP and price level in the short run.

In the AD/AS model, a decrease in aggregate demand would cause a leftward shift of the AD curve. As a result, the intersection point of the AD and the short-run aggregate supply (SRAS) curves would move to the left, causing a decrease in equilibrium GDP and price level.

In the long run, however, the economy is likely to adjust to the new equilibrium. The decrease in aggregate demand would cause a decrease in prices, which would shift the SRAS curve rightward. Eventually, the new intersection point of the AD and the SRAS curves would be reached, resulting in a new equilibrium GDP and price level.

Overall, a recession in Mexico would likely have a negative impact on the US economy, leading to a decrease in GDP and price level in the short run. However, the economy would eventually adjust to the new equilibrium in the long run.

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The probability that a randomly selected Atlantic cod has a length of 39.08 cm or more is 0.9981. The probability that a randomly selected Atlantic cod has a length of 59.08 cm or less is 0.9935. The probability that a randomly selected Atlantic cod has a length between 39.08 and 59.08 cm is  0.9914. The probability of a randomly selected Atlantic cod having a length that is at least 62.99 cm is 0.0002. 48% of all Atlantic cod have a length of more than 50.25 cm.

a) The random variable is the length of a sample of Atlantic cod, denoted by X.

b) The probability that a randomly selected Atlantic cod has a length of 39.08 cm or more can be found using the standard normal distribution table or a calculator. We first standardize the value of 39.08 using the formula

z = (x - μ) / σ, where μ is the mean length and σ is the standard deviation.

Therefore, z = [tex](\frac{39.08- 49.9}{ 3.74 } )[/tex]= -2.89.

From the standard normal distribution table, the probability of a z-score less than or equal to -2.89 is 0.0021.

Thus, the probability of a randomly selected Atlantic cod having a length of 39.08 cm or more is 1 - 0.0021 = 0.9981.

c) The probability that a randomly selected Atlantic cod has a length of 59.08 cm or less can be found using the same method as in part (b).

Standardizing the value of 59.08, we get z = [tex](\frac{59.08- 49.9}{ 3.74 } )[/tex]= 2.45.

Using the standard normal distribution table, the probability of a z-score less than or equal to 2.45 is 0.9935. Thus, the probability of a randomly selected Atlantic cod having a length of 59.08 cm or less is 0.9935.

d) The probability that a randomly selected Atlantic cod has a length between 39.08 and 59.08 cm can be found by subtracting the probability in part (b) from the probability in part (c).

Thus, P(39.08 < X < 59.08) = P(X ≤ 59.08) - P(X ≤ 39.08) = 0.9935 - 0.0021 = 0.9914.

e) The probability that a randomly selected Atlantic cod has a length that is at least 62.99 cm can be found using the same method as in parts (b) and (c).

Standardizing the value of 62.99, we get z = [tex](\frac{62.99- 49.9}{ 3.74 } )[/tex] = 3.49.

Using the standard normal distribution table, the probability of a z-score less than or equal to 3.49 is 0.9998.

Thus, the probability of a randomly selected Atlantic cod having a length that is at least 62.99 cm is 1 - 0.9998 = 0.0002.

f) Yes, a length of 62.99 cm is unusually high for a randomly selected Atlantic cod because the probability of having a value of length at least that high is less than or equal to 0.05.

g) To find the length that 48% of all Atlantic cod have more than, we need to find the z-score that corresponds to a cumulative probability of 0.52 (1 - 0.48).

Using the standard normal distribution table, we find that the z-score is approximately 0.10.

Then, we use the formula z = (x - μ) / σ to solve for x, where μ = 49.9 and σ = 3.74.

Thus, x = μ + σz = 49.9 + 3.74(0.10) = 50.25 cm.

Therefore, 48% of all Atlantic cod have a length of more than 50.25 cm. The units for length are in centimeters.

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The circle relation C defined on the set of real numbers is not reflexive and transitive but it is symmetric.

(a) C is not reflexive. To be reflexive, for every real number, xCx must hold true, meaning [tex]x^{2} + x^{2}[/tex]= 1. This is false. For example, let x=0. In this case, [tex]x^{2} + x^{2}[/tex] = 0, which does not equal 1. Therefore, C is not reflexive.

(b) C is symmetric.  If xCy then yCx, for all real numbers x and y. If we see the definition of C, this means that if [tex]x^{2} + y^{2}[/tex] = 1, then [tex]y^{2} + x^{2}[/tex] = 1. This is true due to the commutative property of addition ([tex]x^{2} + y^{2} = y^{2} + x^{2}[/tex] for all real numbers x and y). Thus, C is symmetric.

(c) C is not transitive. To be transitive, if xCy and yCz, then xCz must hold true for all real numbers x, y, and z. This means that if [tex]x^{2} + y^{2}[/tex] = 1 and [tex]y^{2} + z^{2}[/tex] = 1, then [tex]x^{2} + z^{2}[/tex]must equal 1. This is not always true. Let's take an example (x, y, z) = (1, 0, -1). Then [tex]x^{2} + y^{2}[/tex] = 1, [tex]y^{2} + z^{2}[/tex]= 1, but [tex]x^{2} + z^{2}[/tex] = 2, not 1. Thus, C is not transitive.

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The value of z is given as follows:

z = 38.

We have two secants in this problem, and point C is the intersection of the two secants, hence the angle measure of z is half the difference between the angle measure of the largest arc by the angle measure of the smallest arc.

The arc measures are given as follows:

138º and 62º.

Hence the value of z is obtained as follows:

z = 0.5 x (138 - 62)

z = 38.

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Answer: C

Step-by-step explanation: When did the calendar change from 13 months?

The 1752 Calendar Change

Today, Americans are used to a calendar with a "year" based the earth's rotation around the sun, with "months" having no relationship to the cycles of the moon and New Years Day falling on January 1. However, that system was not adopted in England and its colonies until 1752.

Arrange the data in an ascending order and the median is the middle value. If the number of values is an even number, the median will be the average of the two middle numbers.

median: 34

The mode is the element that occurs most in the data set. In this case, 13, 34 occurs 2 times.

mode: 13, 14

The mean of a set of numbers is the sum divided by the number of terms.

mean: 38.2

Subtract the minimum data value from the maximum data value to find the data range. In this case, the data range is

69−13=56.

Range: 56

The table should be completed with the correct key features as follows;

Axis of symmetry (1st graph): x = 1.

Vertex (1st graph): (1, -9).

Minimum (1st graph): -9.

y-intercept (1st graph): (0, -8).

Axis of symmetry (2nd graph): x = 2.

Vertex (2nd graph): (2, 16).

Maximum (2nd graph): 16.

y-intercept (2nd graph): (0, 12).

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the first graph of a quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Based on the second graph of a quadratic function, we can logically deduce that the graph is a downward parabola because the coefficient of x² is negative and the value of "a" is less than zero (0).

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The length of BD is 8x + 4.

The value of angle AED is determined as 100⁰.

The length of BD is calculated as follows;

Based on the property of rectangle;

length BD = length AC

Length AC = AE + EC

Length AC = 3x + 7 + 5x - 3

Length AC = 8x + 4

Length BD = 8x + 4

The value of angle ECB = 40⁰

then, angle EAD = 40⁰ (alternate angles are equal)

angle EDA = 40⁰ (vertical opposite angles )

angle AED = 180 - (40 + 40)  ( sum of angles in a triangle)

angle AED = 100⁰

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Answer: 43

Step-by-step explanation:

Is because the standard error is the standard deviation of the sampling distribution of the difference between the means, and as such, the differences between the sample means would vary across multiple random samples of the same size.

For part b, the standard error for the difference between the sample means can be calculated as:

[tex]se = sqrt((s1^2/n1) + (s2^2/n2))[/tex]

where s1 and s2 are the sample standard deviations for the two groups, and n1 and n2 are the sample sizes.

Substituting the given values, we get:

[tex]se = sqrt((8.43^2/21) + (3.5^2/21)) ≈ 2.15[/tex]

Interpretation: The standard error represents the standard deviation of the sampling distribution of the difference between the sample means. A lower standard error indicates that the sample means are more likely to be representative of their respective populations, and that the difference between the means is more likely to be significant.

The correct answer is (A): If further random samples of these sizes were obtained from these populations, the differences between the sample means would vary. The standard deviation of these values for (x1-x2) would equal about 2.2. This is because the standard error is the standard deviation of the sampling distribution of the difference between the means, and as such, the differences between the sample means would vary across multiple random samples of the same size.

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To determine if the events "a selected student is underweight" (A) and "a selected student is male" (M) are independent, we need to check if the probability of both events occurring together is equal to the product of the probabilities of each event occurring individually.

Step 1: Calculate the probabilities of each event individually.
P(A) = P(Underweight) = (43 + 24) / 300 = 67 / 300
P(M) = P(Male) = (43 + 17) / 300 = 60 / 300

Step 2: Calculate the probability of both events occurring together.
P(A ∩ M) = P(Underweight and Male) = 43 / 300

Step 3: Check if P(A ∩ M) = P(A) * P(M)
(67 / 300) * (60 / 300) ≠ 43 / 300

Since the probabilities are not equal, the events "a selected student is underweight" and "a selected student is male" are not independent.

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All real values of a such that the given matrix is not invertible is -3.

To determine if a matrix is invertible, we can look at its determinant. A matrix is invertible if and only if its determinant is non-zero. Therefore, we need to find the values of a that make the determinant of matrix A equal to zero.

The determinant of matrix A is given by:

|A| = 0 1 a a 1 3 0 a 1 a

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

= -a + 3a + 3 - a

= a + 3

Therefore, the matrix A is not invertible when a = -3.

So the real value of a for which the matrix A is not invertible is -3.

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For all vectors v in the vector space V, the expression 0 - v is equal to the additive inverse of v, or -v.

To prove that for all vectors v in a vector space V, and with 0 as the zero vector, the expression 0 - v is equal to the additive inverse of v.

To prove this, we'll follow these steps,

1. Start with the definition of the zero vector in a vector space V.
2. Show that adding the additive inverse of v to both sides of the equation results in the desired expression.

1. Let V be a vector space and 0 be the zero vector. By definition, the zero vector has the property that for all vectors v in V, we have:
  v + 0 = v

2. To find the expression for 0 - v, we first need to determine the additive inverse of v, denoted by -v. The additive inverse of v has the property:
  v + (-v) = 0

Now, let's consider the expression 0 - v. To find this, we can rewrite it as 0 + (-v). Using the property of the zero vector, we know that:

0 - v = 0 + (-v) = -v

Hence, for all vectors v in the vector space V, the expression 0 - v is equal to the additive inverse of v, or -v.

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Answer:

Informative!

Step-by-step explanation:

The value of x in the equation 2x² + 10x + 12 = 0 is -2 and -3.

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The standard form of a quadratic equation is:

ax² + bx + c = 0

The quadratic formula is given by:

[tex]x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \\\\Given\ the\ equation\ 2x^2+10x+12=0:\\\\a=2;b=10;c=12\\\\x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} \\\\substituting:\\\\x=\frac{-10\pm\sqrt{10^2-4(2)(12)} }{2(2)} \\\\x=-3; and\ x=-2\\[/tex]

The value of x is -2 and -3.

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Answer: 8 2/5 feet (positive)

Step-by-step explanation:

The value of the given expression is -4, which is integer. Therefore, option B is the correct answer.

The given expression is (p²+q)/(-|p|-q).

Here, p=-6 and q=4.

Substitute p=-6 and q=4 in the given expression we get

((-6)²+4)/(-|-6|-4)

= 40/(-10)

= -4

Therefore, option B is the correct answer.

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Most problems involving the Intermediate Value Theorem will require a three-step process:

1. Verify that the function f is continuous on the closed interval [a,b].

2. Find two points, say p and q, in the interval [a,b] such that f(p) and f(q) have opposite signs.

3. Apply the Intermediate Value Theorem, which guarantees the existence of a root of the equation f(x) = 0 in the interval [p,q].

Most problems involving the Intermediate Value Theorem require a three-step process:

Step 1: Verify the conditions for the Intermediate Value Theorem (IVT)

To apply the IVT, ensure that the function is continuous on a closed interval [a, b]. If it's continuous, you can proceed to the next step.

Step 2: Determine the values of the function at the endpoints

Evaluate the function at the given interval's endpoints, f(a) and f(b).

Step 3: Apply the Intermediate Value Theorem

If there is a value 'c' between f(a) and f(b) such that f(a) < c < f(b) (or f(a) > c > f(b)), then by the IVT, there exists a value x in the interval (a, b) such that f(x) = c.

Keep in mind these steps when solving problems involving the Intermediate Value Theorem.

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Lily would need 40.64 centimeters of copper wire for her experiment.

Given data ,

We may use the conversion factor that 1 inch is equivalent to 2.54 centimeters to convert 16 inches to centimeters .

From the unit conversion ,

1 inch = 2.54 inches

Consequently, 16 inches is equivalent to :

40.64 centimeters are equal to 16 inches at 2.54 centimeters per inch.

Hence , Lily would thus want 40.64 centimeters of copper wire for her experiment.

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The break-even point is approximately 0.6 years, or 7.2 months. This means that if Zach plans to keep the loan for at least 7.2 months, purchasing 1 point would be worth it as he would save more in interest than he paid for the point.

To determine whether Zach should purchase 1 point or no points, we need to calculate the cost of each option and compare the total cost of each option over the life of the loan.

Option 1: No points

Loan amount: $200,000

APR: 3.625%

Monthly payment: $1,941.65 (calculated using a loan amortization calculator)

Total interest paid over 10 years: $33,698.03

Option 2: 1 point

Loan amount: $200,000

APR: 3.5% (3.625% - 0.125%)

Cost of 1 point: $2,000 (1% of the loan amount)

Total loan amount: $202,000 ($200,000 + $2,000)

Monthly payment: $1,903.03 (calculated using a loan amortization calculator)

Total interest paid over 10 years: $30,363.06

Comparing the two options, we can see that purchasing 1 point would result in a lower APR and lower monthly payments, which would save Zach money over the life of the loan. However, he would need to pay $2,000 upfront for the cost of the point.

To determine whether the cost of the point is worth the savings in interest, we need to calculate the break-even point. The break-even point is the point at which the savings in interest equal the cost of the point.

Break-even point:

Savings in interest: $33,698.03 - $30,363.06 = $3,334.97

Cost of 1 point: $2,000

Break-even point: $2,000 ÷ $3,334.97 = 0.6

The break-even point is approximately 0.6 years, or 7.2 months. This means that if Zach plans to keep the loan for at least 7.2 months, purchasing 1 point would be worth it as he would save more in interest than he paid for the point. If he plans to pay off the loan earlier than 7.2 months, then he should not purchase the point as he would not have enough time to recoup the cost.

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Answer:

The answer is the fourth option.

Step-by-step explanation:

The reason is that when a number has the line above it means it is continuous which is the meaning of rational numbers.

a) The simple interest for the loan is $927.19.

b) The total amount that Jax will have to pay at the end of 30 months is $9,427.19.

a) To calculate the simple interest for the loan, we can use the formula:

Simple Interest = Principal x Rate x Time

where Principal is the amount borrowed, Rate is the annual interest rate, and Time is the duration of the loan in years.

Since the loan is for 30 months, which is equivalent to 2.5 years, we can substitute the given values:

Simple Interest = 8,500 x 0.0435 x 2.5 = $927.19

b) To determine the total amount that Jax will have to pay at the end of 30 months, we need to add the simple interest to the original amount borrowed. The total amount can be calculated using the formula:

Total Amount = Principal + Simple Interest

Substituting the given values:

Total Amount = 8,500 + 927.19 = $9,427.19

In summary, Jax will have to pay $927.19 in simple interest and a total of $9,427.19 at the end of 30 months to repay the 8,500 loan with an annual simple interest rate of 4.35%.

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This can be modeled with the linear function.

f(x) = 530 - 20x

We can model this with a linear function. We know that the initial volume of the tank is 530 gallons, and we know that she uses 20 gallons per week.

So, if the variable x describes the number of weeks, the volume at week x will be 530 gallons minus 20 gallons times x.

This is written as a linear function:

f(x) = 530 - 20x

That function gives the volume left after x weeks.

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