4. The t test for two independent samples - One-tailed example using tables
Most engaged couples expect or at least hope that they will have high levels of marital satisfaction. However, because 54% of first marriages end in divorce, social scientists have begun investigating influences on marital satisfaction. [Data source: This data was obtained from the National Center for Health Statistics.]
Suppose a counseling psychologist sets out to look at the role of economic hardship in relationship longevity. He decides to measure marital satisfaction in a group of couples living above the poverty level and a group of couples living below the poverty level. He chooses the Marital Satisfaction Inventory, because it refers to “partner” and “relationship” rather than “spouse” and “marriage,” which makes it useful for research with both traditional and nontraditional couples. Higher scores on the Marital Satisfaction Inventory indicate greater satisfaction. There is one score per couple. Assume that these scores are normally distributed and that the variances of the scores are the same among couples living above the poverty level as among couples living below the poverty level.
The psychologist thinks that couples living above the poverty level will have greater relationship satisfaction than couples living below the poverty level. He identifies the null and alternative hypotheses as:
H₀: μcouples living above the poverty level
μcouples living below the poverty level
H₁: μcouples living above the poverty level
μcouples living below the poverty level
This is a tailed test.
The psychologist collects the data. A group of 39 couples living above the poverty level scored an average of 51.1 with a sample standard deviation of 9 on the Marital Satisfaction Inventory. A group of 31 couples living below the poverty level scored an average of 45.2 with a sample standard deviation of 12. Use the t distribution table. To use the table, you will first need to calculate the degrees of freedom.
The degrees of freedom are .
The t distribution
Proportion in One Tail
0.25 0.10 0.05 0.025 0.01 0.005
Proportion in Two Tails Combined
df 0.50 0.20 0.10 0.05 0.02 0.01
1 1.000 3.078 6.314 12.706 31.821 63.657
2 0.816 1.886 2.920 4.303 6.965 9.925
3 0.765 1.638 2.353 3.182 4.541 5.841
4 0.741 1.533 2.132 2.776 3.747 4.604
5 0.727 1.476 2.015 2.571 3.365 4.032
6 0.718 1.440 1.943 2.447 3.143 3.707
7 0.711 1.415 1.895 2.365 2.998 3.499
8 0.706 1.397 1.860 2.306 2.896 3.355
9 0.703 1.383 1.833 2.262 2.821 3.250
10 0.700 1.372 1.812 2.228 2.764 3.169
11 0.697 1.363 1.796 2.201 2.718 3.106
12 0.695 1.356 1.782 2.179 2.681 3.055
13 0.694 1.350 1.771 2.160 2.650 3.012
14 0.692 1.345 1.761 2.145 2.624 2.977
15 0.691 1.341 1.753 2.131 2.602 2.947
16 0.690 1.337 1.746 2.120 2.583 2.921
17 0.689 1.333 1.740 2.110 2.567 2.898
18 0.688 1.330 1.734 2.101 2.552 2.878
19 0.688 1.328 1.729 2.093 2.539 2.861
20 0.687 1.325 1.725 2.086 2.528 2.845
21 0.686 1.323 1.721 2.080 2.518 2.831
22 0.686 1.321 1.717 2.074 2.508 2.819
23 0.685 1.319 1.714 2.069 2.500 2.807
24 0.685 1.318 1.711 2.064 2.492 2.797
25 0.684 1.316 1.708 2.060 2.485 2.787
26 0.684 1.315 1.706 2.056 2.479 2.779
27 0.684 1.314 1.703 2.052 2.473 2.771
28 0.683 1.313 1.701 2.048 2.467 2.763
29 0.683 1.311 1.699 2.045 2.462 2.756
30 0.683 1.310 1.697 2.042 2.457 2.750
40 0.681 1.303 1.684 2.021 2.423 2.704
60 0.679 1.296 1.671 2.000 2.390 2.660
120 0.677 1.289 1.658 1.980 2.358 2.617
∞ 0.674 1.282 1.645 1.960 2.326 2.576
0.50 0.20 0.10 0.05 0.02 0.01

The functions can be matched with their correct descriptions as follows:

1. Radical function

2. Rational function

3. Absolute function

4. Piecewise function

5. Step function

6. Quadratic function

To identify the functions, you need to know the formats that each of the functions assumes when expressed mathematically. Radical functions involve radicals or roots, such as square roots (√) or cube roots (∛) while rational functions are functions that take the form p(x)/q(x), where p(x) and q(x) are polynomials, and the denominator q(x) cannot be equal to zero.

Absolute functions are functions of the form f(x) = |x-a|, where a is a constant and |x| represents the distance between x and 0 on the number line. Quadratic functions, on the other hand, are functions of the form f(x) = ax^2 + bx + c.

Piecewise functions are functions that have different rules or formulas for different parts of their domain while step functions are functions that "step up" or "step down" at specific values of x. They can be defined using the floor function (⌊x⌋) or ceiling function (⌈x⌉).

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Therefore , the solution of the given problem of function comes out to be  f''(3/2) at x = 3/2.

On the final exam, each subject will feature a variety of problems that cover both real and hypothetical places as well as the development of numerical factors. a diagram showing the relationships between different elements that cooperate to generate the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input.

Here,

To determine the function f(x) = x⁴ - 2x³'s relative minimum,

By taking f(x)'s derivative, we arrive at:

=>  f'(x) = 4x³ - 6x²

To determine the key places, we can set this to zero as follows:

=>  4x³ - 6x² = 0

=>  2x²(2x - 3) = 0

So, x = 0 and x = 3/2 are the turning points.

The second derivative test can be used to determine the type of these key sites. By taking f(x)'s second derivative, we arrive at:

=> f''(x) = 12x² - 12x

We have f''(0) = 0 at x = 0, which indicates that the second derivative test is not convincing. To determine the nature of this crucial issue, we must employ a different technique.

We have a positive value for

=>  f''(3/2) at x = 3/2, which is 9.

This indicates that the function's relative minimum is located at x.

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Answer:

x = -1

y = -1

Step-by-step explanation:

−6x + 5y = 1

6x + 4y = −10

We combine the equations

9y = -9

y = -1

Now we put -1 in for y and solve for x

−6x + 5(-1) = 1

-6x - 5 = 1

-6x = 6

x = -1

Let's Check

−6(-1) + 5(-1) = 1

6 - 5 = 1

1 = 1

So, x = -1 and y = -1 is the correct answer!

Answer:

a

Step-by-step explanation:

Therefore, Troy's actual score is 83.10.

Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an event that is impossible, and 1 represents an event that is certain. The probability of an event can be determined by dividing the number of ways that event can occur by the total number of possible outcomes. Probability is widely used in various fields, including mathematics, statistics, science, engineering, economics, and finance, to model uncertain events and make predictions.

To answer the questions, we need to calculate the z-score for Troy's exam score and then use the z-table to find the probability or percentage associated with that z-score.

First, let's calculate the z-score for Troy's exam score:

[tex]z = (X - M) / SD[/tex]

where X is Troy's score, M is the class average, and SD is the standard deviation.

[tex]z = (83 - 76.80) / 3.10[/tex]

[tex]z = 2.00[/tex]

Next, let's use the z-table to find the probability associated with a z-score of 2.00. We look up the value in the body of the table, which is 0.9772. This means that the probability of getting a z-score of 2.00 or higher (i.e., a score higher than Troy's) is 0.9772.

To find the percentage of students who scored lower than Troy, we can subtract the probability found above from 1, since the total probability for all possible outcomes is 1.0.

[tex]P(X < 83) = 1 - 0.9772[/tex]

[tex]P(X < 83) = 0.0228[/tex]

So, the percentage of students who scored lower than Troy is 2.28%.

Finally, we can use the formula provided to calculate Troy's actual score if we know his z-score, standard deviation, and class average:

[tex]X = (z)(SD) + M[/tex]

[tex]X = (2.00)(3.10) + 76.80[/tex]

[tex]X = 83.10[/tex]

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You must be a minimum of 18 years old to vote. The correct answer is option (b).

we have

x >= 18

The interval is the answer to this inequality-----> [18, ∞)

bigger than or equal to all real numbers

Case A:

To vote, you must be older than 18 years old.

The inequality in this instance that represents the assertion is

v > 18  is not correct

Case B:

You must be a minimum of 18 years old to vote.

The inequality in this instance that represents the assertion is

v >= 18  is correct

Case C:

Before turning 18 years old, you can cast a ballot.

The inequality in this instance that represents the assertion is

V < 18  is not correct

Case D:

When you are 18 years old, you are too old to vote.

You are unable to vote in this instance when

v = 18

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The rest of the question is,

a). You must be older than 18 years old to vote.

b). You must be at least 18 years old to vote.

c). You can vote before you turn 18 years old.

d). You cannot vote when you are 18 years old.

[tex]6\dfrac{1}{6}[/tex] feet tape is left on the roll

[tex]15-8\dfrac{5}{6} =15-\huge \text(8+\dfrac{5}{6} \huge \text)=15-8-\dfrac{5}{6} =7-\dfrac{5}{6} =6\dfrac{1}{6}[/tex]

Answer:

She has 7 whole sheets with a few stickers on the 8th sheet left.

Step-by-step explanation:

First, you need to do a subtraction problem.

This is because you need to have a whole numerical amount as a whole number in order to clearly prove the exact amount of sheets Sara brought.

You take the whole which is 15 sheets and subtract it from the 8 sheets. This is needed since you need a whole numerical amount excluding fractions or decimals.

-This should be 7 whole sheets left.

So now we know that she used 8 whole sheets. But we still need to evaluate the 5/6s.

You can do 5 divided by 6 which is equal to ~0.83. This is almost equivalent to a whole sheet of stickers. This approximation would prove that 8 sheets could possibly be left.
___________________________________________________

This can also be interpreted as this:

You take 15 and subtract it with 8 5/6. This time is different since we're not approximating.
That equals 37/6 which is equivalent to 6 1/6. Thus giving 6 whole sheets left.
___________________________________________________

Hypothetically She has 7 whole sheets with a few stickers on the 8th sheet left.

Answer:

C 163 yd

Step-by-step explanation:

Answer:The answer is  the last one

Step-by-step explanation:

Im goatified

Answer:

x(x+8)(x-6)

Step-by-step explanation:

The GCF is x. First, you factor out x and get x(x² + 2x - 48). Then, you x-factor it and get (x+8)(x-6). Then you add the x on it and get the final answer of x(x+8)(x-6)

Answer:

There are 4^5 ways to answer 5 multiple choice questions.

Step-by-step explanation:

Answer:

132 yd²

Step-by-step explanation:

This shape is a rectangle.

Area of rectangle = L · W

L = 12 yd

W = 11 yd

We Take

12 · 11 = 132 yd²

So, the area of this shape is 132 yd².

The composite function f(g(x)) has a value of -2x^2 - 6.

From the question, we have the following parameters that can be used in our computation:

Fuctions f(x) and g(x)

To find f(g(x)), we first need to evaluate g(x) and then use the result as the input for f(x).

So we have:

g(x) = x^2 + 3

Now we substitute this expression into f(x) and simplify:

f(g(x)) = f(x^2 + 3) = -2(x^2 + 3) = -2x^2 - 6

Therefore, f(g(x)) = -2x^2 - 6.

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The probability is 0.9999045772. Round up to 6 decimal places if necessary.

The probability distribution of a random experiment or event shows the chance of each conceivable outcome. It gives the likelihoods of several potential events. Additionally, see Probability of Events. A measure of the uncertainty of different phenomena is the likelihood. Like when you throw a die, the probability determines the various results. To determine this distribution, any random experiment with ambiguous or unanticipated outcomes could be employed.

First, we determine the likelihood of receiving a particular sort of hand.

The likelihood of receiving this kind of hand is:

P(deal) = 248/2598560

P(dealt) = 1 - P(not dealt)

Then we find probability of not being dealt with one type of hand.

P(not dealt) = 1 - P( dealt)

P(not dealt) = 1 - 248/2598560

P(not dealt) = 25898960-248/2598960

= 2598712/2598960

= 0.9999045772

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The complete question is,

Answer:

[tex]396\pi \: {ft}^{2} [/tex]

Step-by-step explanation:

Given:

A cone

l = 21 ft

r = 12 ft

Find: A (surface) - ?

[tex]a(surface) = \pi {r}^{2} + \pi \times r \times l[/tex]

[tex]a(surface) = \pi \times {12}^{2} + \pi \times 12 \times 21 = 144\pi + 252\pi = 396\pi \: {ft}^{2} [/tex]

Answer:

a = 0

b = -5

r = 3

Step-by-step explanation:

Given that the center of the circle is at (0, -5) and the diameter of the circle is 6 units, we can determine the values of a, b, and r for the standard equation of a circle: (x - a)^2 + (y - b)^2 = r^2.

a: The x-coordinate of the center of the circle, which is 0 in this case.

b: The y-coordinate of the center of the circle, which is -5 in this case.

r: The radius of the circle, which is half of the diameter. So, r = 6/2 = 3 units.

Therefore, the values for a, b, and r are:

a = 0

b = -5

r = 3

Hence, the approximated circle of area 4500[tex]m^{2}[/tex].

A circle is a particular type of ellipse in geometry where the eccentricity is zero and the two focus coincide. The locus of points drawn equally a part from the center is known as a circle. The radius of a circle is measured from the center to the edge. The line that splits a circle into two identical halves is its diameter, which is also twice of radius.A circle is a 2 Dimensional figure . The interior and exterior parts of the are separated by the circles. It re assembles a line segment in several ways. Consider the line segment being twisted till it reaches its ends.

Each graph's y-axis should the following labels:

Circumference  given in Graph A

Area given in Graph B

Radius given in Graph C

Now find the diameter of circle A:

Circle of area A =[tex]r^{2}[/tex] =500

⇒r =[tex]\sqrt{500}[/tex]

⇒r=22.36 m

Circle A has a diameter that is twice its radius, so:

circle A of  diameter is 2 r= 2×22.36 m.=44.72 m.

Circle A of diameter is three times that of circle B,

that is 3×44.72 =134.16

If circle B of radius is equal to half its diameter,

[tex]\frac{134.16}{2}[/tex]=67.08 m is the radius of circle B.

now We find the area of circle B:

area of circle B =[tex]67.08^{2}[/tex]= 4499.73 [tex]m^{2}[/tex].

Hence, the approximated circle of area 4500[tex]m^{2}[/tex].

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A) The annual rate of change between 1994 and 2003 was 0.0617.

B) The value written in percentage form is 6.17%.

C) The value of car in the year 2008 will be $2,500.

The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

All real numbers can be described by the four fundamental operations, sometimes known as "arithmetic operations." In mathematics, the results of operations such as division, multiplication, addition, and subtraction are quotient, product, sum, and difference, respectively.

A) We are given value of car in the year 1994 as $34,000 and in the year 2003, it was $13,000.

So, using subtraction operation, we get

⇒ Amount of depreciation = $34,000 - $13,000

⇒ Amount of depreciation = $21,000

This is over the period of 10 years.

So, annual rate of change is obtained by using the division operation.

⇒ [tex]\frac{21000}{10}[/tex]

⇒ $2100

Now, rate of change is

⇒ [tex]\frac{21000}{10}[/tex]

⇒ 0.0617

B) The value in the percentage form is obtained by using the multiplication operation which is

0.0617 * 100 = 6.17%

C) Since, the value is dropping by same percentage, the value in 2008 will be:

⇒ $13,000 - (2100 * 5)

⇒ $13,000 - $10500

⇒ $2,500

Hence, the required values have been obtained.

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Question: A car was valued at $34,000 in the year 1994. The value depreciated to $13,000 by the year 2003.

A) What was the annual rate of change between 1994 and 2003?

r = Round the rate of decrease to 4 decimal places.

B) What is the correct answer to part A written in percentage form?

r = %.

C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2008 ?

value = $ Round to the nearest 50 dollars

Answer:

(9/5) * (1156/10000)

*IG:whis.sama_ent

Answer:

4 seconds

Step-by-step explanation:

The length of a 30° arc of a circle with a radius of 6 feet is π feet.

What is circle ?

In geometry, a circle is a closed, two-dimensional figure that consists of all the points in a plane that are a fixed distance (called the radius) away from a given point (called the center). The distance across the circle through the center is called the diameter, which is equal to twice the radius.

The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. Therefore, the circumference of this circle is:

C = 2π(6 feet) = 12π feet

Since a full circle is 360°, we can find the length of a 30° arc by multiplying the fraction of the circle represented by the arc by the circumference of the circle. In this case, the fraction of the circle represented by the 30° arc is:

30°÷360° = 1÷12

So the length of the 30° arc is:

(1÷12) × 12π feet = π feet

Therefore, the length of a 30° arc of a circle with a radius of 6 feet is π feet.

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Answer:

  Tuesdays

Step-by-step explanation:

Given men : women ratios of 1:6 and 2:11 on Tuesdays and Thursdays, respectively, you want to know which day has the higher proportion of women.

Ratios are most easily compared of one of the numbers is the same in both ratios. Here, we can double the first ratio to make the number units representing men be 2:

  1 : 6 = 2 : 12 . . . . . ratio on Tuesdays

  2 : 11 . . . . . . . . . . . ratio on Thursdays

There are relatively more women on Tuesdays. (12 > 11)

__

Additional comment

We can also make the number representing men be 1. Dividing the second ratio by 2 gives ...

  2 : 11 = 1 : 5.5

Now, we can see that 5.5 is less than 6, so the proportion of women is higher on Tuesdays. These ratios (6, 5.5) are effectively "unit rates", or women per man.

To be in the top 10% of his class, Troy needs a score of 80.28 or better. With an 83, he placed in the top 10% of his class on this exam.

The percentage is a technique to express a fraction or proportion as a percentage of 100.

We apply the following algorithm to convert Troy's score to a z-score:

z = (X - M) / SD

where X is Troy's score, M represents the class average, and SD represents the standard deviation. Using the values we have:

z = (83 - 76.80) / 3.10 = 2.00

We seek up the equivalent area in the z-table and plot this on a distribution. A z-score of 2.00 equates to a 0.9772 area.

Troy's score should be converted to We use the following formula to calculate a percentile rank:

(number of scores below X / total number of scores) x 100% = percentile rank

(0.9772) x 100% = 97.72% Percentile rank

We may accomplish this by dividing the area to the left of Troy's z-score by the area to the left of the class average's z-score:

Area = 0.9772 - 0.5000 = 0.4772

This means that around 47.72% of the scores fall between Troy's and the class average.

We can solve for X using the z-score formula:

X = (z)(SD) + M = (1.28)(3.10) + 76.80 = 80.28

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Investigate all similar responses

The manager should charge $ 700 per camera and the maximum revenue is $ 73,000.    

Maximum revenue refers to the highest amount of money a business can earn by selling its products or services. To find the maximum revenue, we need to determine the price that would generate the maximum number of sales and then multiply it by the number of cameras sold at that price.

Here we have

Let x be the number of $20 decreases in price, and let y be the number of cameras sold per month.

From the problem statement, we can write:

y = 70 + 5x

The price of a camera after x decreases of $20 each is:

p(x) = 900 - 20x

The revenue earned per month is given by:

R(x) = p(x) × y = (900 - 20x) × (70 + 5x)

Expanding the expression and simplifying, we get:

R(x) = -100x² + 2000x + 63000

To find the value of x that maximizes revenue, we need to find the vertex of the parabola represented by R(x).

The x-coordinate of the vertex is given by:

x = -b / (2a) = -2000 / (-200) = 10

Therefore,

The manager should decrease the price by $200, or $700 per camera, to maximize monthly revenue.

The number of cameras sold per month at this price is:

y = 70 + 5x = 70 + 5(10) = 120

The maximum revenue is:

R(10) = -100(10)² + 2000(10) + 63000 = $73,000  

Therefore,

The manager should charge $ 700 per camera and the maximum revenue is $ 73,000.      

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Answer:

-3

Step-by-step explanation:

-3 is 12 less than its square. Implies that 9 - 12 = -3.

Answer:

[tex]7 \: \: and \: \: 14 \frac{1}{6} [/tex]

Step-by-step explanation:

Assume, the 1st mixed number is x and the 2nd one is y

Let's write 2 equations according to the given information:

[tex]x + y = 21 \frac{1}{6} [/tex]

[tex]x - y = \frac{43}{6} [/tex]

Let's make x the subject from the 1 equation:

[tex]x = 21 \frac{1}{6} - y[/tex]

Replace x in the 2nd equation with its value from the 1st one:

[tex]( 21\frac{1}{6} - y) - y = \frac{43}{3} [/tex]

Eliminate the brackets and collect like-terms:

[tex] - 2y = \frac{43}{6} - 21 \frac{1}{6} [/tex]

Convert the number 21 1/6 to an improper fraction:

[tex] - 2y = \frac{43}{6} - \frac{21 \times 6 + 1}{6} [/tex]

[tex] - 2y = \frac{43}{6} - \frac{127}{6} = \frac{ - 84}{6} [/tex]

Multiply both sides of the equation by 6 to eliminate the fraction:

[tex] - 12y = - 84[/tex]

Divide both sides of the equation by -12 to make y the subject:

[tex]y = 7[/tex]

[tex]x = \frac{127}{6} - \frac{7}{1} = \frac{127}{6} - \frac{42}{6} = \frac{85}{6} = 14 \frac{1}{6} [/tex]

The event you are describing is called a dependent event. In a dependent event, the outcome of one event affects the probability of the outcome of another event.

what is dependent event ?

A dependent event is an event in which the outcome of one event affects the probability of the outcome of another event. In other words, the occurrence or non-occurrence of the first event affects the probability of the second event.

In the given question,

The event you are describing is called a dependent event. In a dependent event, the outcome of one event affects the probability of the outcome of another event. This is in contrast to an independent event, in which the outcome of one event does not affect the probability of the outcome of another event.

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It is important to use the given standard deviation to construct the control chart.

A control chart is a statistical tool used to monitor and control a process over time. It is a graphical representation of process data over time, with the addition of statistical control limits. Control charts are used to detect any unusual variation in a process and to identify when the process is out of control or deviating from its expected performance.

Yes, you need to use the given process standard deviation of 0.27 to construct the three-sigma control chart for the mean weight.

The control limits for the X-Bar Control Chart with a sample size of n = 4 and a Z-value of 3 can be calculated using the following formula:

UCL = X-Bar + Z(σ/√n)

LCL = X-Bar - Z(σ/√n)

where UCL is the upper control limit, LCL is the lower control limit, X-Bar is the sample mean, σ is the process standard deviation, n is the sample size, and Z is the number of standard deviations.

To construct the control chart, you need to calculate the UCL and LCL for each sample means in your data set using the formula above. Then, plot the sample means on the control chart along with the UCL and LCL.

It is important to use the given standard deviation to construct the control chart.

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In the given problem, using Pythagorean theorem, if Ms. Smith wants to put a walkway along each diagonal, she will need a total of  40 feet of walkway.

To find the length of the walkway, we need to find the length of the diagonals of the garden. We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the two sides of the rectangle are 16 feet and 12 feet. So, we can find the length of the two diagonals as follows:

Ms. Smith wants to put a walkway along each diagonal, so she will need a total of 20 + 20 = 40 feet of walkway.

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We may infer after addressing the stated question that As a result, the equation correct answer is (a) $162,000.

A mathematical equation is a formula that links two statements and uses the equals sign (=) to indicate equality. In algebra, an equation is a statement that demonstrates the equality of two mathematical expressions. The equal sign divides the variables 3x + 5 and 14 in the equation 3x + 5 = 14, for instance.

$6.00 x 75 = $450

$450 x 360 = $162,000

As a result, the total amount paid on a $75,000 30-year fixed loan at 6.0% is $162,000.

As a result, the correct answer is (a) $162,000.

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