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Though a time series could be defined more widely, the common simplification is required for analysis is discrete time.

Time series analysis is a specialized method of analyzing a series of data points collected over a period of time. In time series analysis, analysts record data points at consistent intervals over a set period of time, rather than recording data points intermittently or randomly.

What makes time series data different from other data is that analysis can show how a variable changes over time. In other words, time is a key variable that dictates how the data is adjusted over the course of the data points and the final result. It provides an additional source of information and a specified ordering of dependencies between data.

Time series analysis typically requires a large number of data points to ensure consistency and reliability. Using a large data set ensures a representative sample size and allows the analysis to cut out noisy data. Additionally, time series data can be used for forecasting. Predict future data based on historical data.

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The test statistics for one-way Anova follows, the comparing of the T Statistic with the T critical value.

On data that is dispersed normally, the Z test is a parametric process that is applied. The z-test can be applied to proportions, two samples, or one sample to test hypotheses. When the population variance is known, it is possible to determine whether the means of two significant groups differ.

The second is one-way analysis of variance (ANOVA), which checks if three or more samples originate from populations with the same mean using the F-distribution.

The test statistic should be compared to the crucial value. If the test statistic is more extreme in favor of the alternative than the crucial value, the alternative hypothesis should be considered instead of the null hypothesis. If the test statistic is not as extreme as the crucial value, the null hypothesis should not be disregarded.

Therefore,

The test statistics for one-way Anova follows, the comparing of the T Statistic with the T critical value.

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Based on the information provided, the archaeological sample's estimated age is = 0.214136  years

Half life given is 5730

t₆.₅ = Ln2/k

k= ln 2/ 5730

The rate constant ,

k = 0.00012 year⁻¹

Given that 23 % of the 14C remains

then |A1|/|A2|  = 0.23

As we know ,

|A1|/|A2| = e ⁻₍kt₎

Hence, on solving the equation we get ,

ln(0.23)= -1.46967

t = -0.31471/ --1.46967

= 0.214136 years

The amount of time needed for a quantity (of substance) to decrease to half of its initial value is known as the half-life (symbol t12). In nuclear physics, the phrase is frequently used to indicate how rapidly unstable atoms decay radioactively or how long stable atoms last. The phrase is also used more broadly to describe any kind of exponential decay (or, very infrequently, nonexponential decay).

The biological half-life of medications and other compounds in the human body, for instance, is a term used in the medical sciences. Half-opposite life's in exponential growth is time's doubling.

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The work required to stretch the spring is 91.6 Ibs-inch

What is work?

Work is defined as force applied across a distance. Work examples include pushing an object against gravity, driving a car up a hill, and lowering a captive helium balloon. Work is the mechanical expression of energy.

Δx : displacement from equilibrium position

f = k Δ x (force due to spring = -kΔx)

where, k is the spring constant

Given,

f = 20ibs

Δx = 24 -18 Inches = 6 inches

∴ k = 20/6 ibs/inch

Now, PE stored in spring = 1/2 k (Δx)²

Δx : displacement from equilibrium position

Work done = change in PE

= 1/2K (Δx₂)² -1/2k (Δx₁)²

given,

Δx₁ = 21-18 inches = 3 inches

Δx₂ = 26-18 inches = 8 inches

∴work done = 1/2 * 20/6 (8² - 3²)Ibs-inches

=10/6 *5*11

work done = 91.6 ibs-inch

The spring must be stretched with 91.6 Ibs-inch of force.

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Answer:

B. 123

Step-by-step explanation:

⭐What does "divisible" mean?

Thus, we can find which number is divisible by 3 by dividing each number by 3 and seeing if the remainder is 0.

However, that process may be too long.

You can do it if you'd like, but there's a rule you can use:

⭐"If the sum of the digits is divisible by 3, the number is also."

Let's apply this rule:

A.

76

⇒ 7 + 6

⇒ = 13

13 is NOT divisible by 3. Therefore, 76 is NOT divisible by 3.

B.

123

⇒ 1 + 2 + 3

⇒ = 6

6 IS divisible by 3. Therefore, 123 IS divisible by 3.

C.

457

⇒ 4 + 5 + 7

⇒ = 16

16 is NOT divisible by 3. Therefore, 457 is NOT divisible by 3.

B.

9082

⇒ 9 + 0 + 8 + 2

⇒ = 19

19 is NOT divisible by 3. Therefore, 9082 is NOT divisible by 3.

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y=3 is the equation of a horizontal line that passes through the point (9, 3)

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

If a line passes through a point (x₁, y₁) and has a slope of m, then the

equation of the line is given by (y-y₁) = m(x-x₁).  

Because the question states the line is horizontal, the slope is zero, or m=0.

Plugging the information into the slope equation gives:

(y-(3)) = 0(x-(9)).

y-3= 0

y= 3.

Hence, y=3 is the equation of a horizontal line that passes through the point (9, 3).

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q = 40 is the value of q in quadratic equation.

What is quadratic equation explain?

The equation x ax2+bx+c=0 is a quadratic equation, which is a second-order polynomial equation in a single variable. with a ≠ 0 .

                    The fundamental theorem of algebra ensures that it has at least one solution because it is a second-order polynomial equation. The answer could be simple or complicated.

the roots of a quadratic expression

             ax² + bx + c = 0

Since x' and x'' are the roots of this equation, then we can write the Vieta's formula

                 x' + x'' = -b/a

               x' * x'' = c/a

let's replace the 2nd equation of the Vieta's formula by that and make it a Linear System.

                  x' + x'' =  14/1

                 x' * x'' = 6

after solve the equations

               x' = 10

Solving the  equations

       10 + x''=14

           x'' = 4

Since we have a=1

   x^2-14x + q = 0

      Then we can make q as the parameter c.

    q = x' * x'' = 4 * 10 ⇒ 40      

x2−14x+ 40 = 0

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The complete question is -

The difference between the roots of the quadratic equation x2−14x+q=0 is 6. Find q.

Answer:

because the absolute value of -5 is positive 5

The rationalized expression would be [tex]\sqrt{3}+\frac{\sqrt{9} }{\sqrt{x}}[/tex].

What is a rationalized expression?

A rationalized expression is an expression that has been written in a simplified form which eliminates any radicals (square roots, cube roots, etc.) from the denominator. The denominator is made a rational number, meaning it can be expressed as a fraction with an integer numerator and denominator. This is done by multiplying both the numerator and the denominator by the radical of the denominator. This process is called rationalization.

The given expression is

                          [tex]=\frac{\sqrt{9}}{\sqrt{3}+ \sqrt{x} } \\\\=\frac{\sqrt{9} }{\sqrt{3} }+ \frac{\sqrt{9} }{\sqrt{x}}\\\\=\frac{3 }{\sqrt{3} }+\frac{\sqrt{9} }{\sqrt{x}}\\\\=\sqrt{3}+\frac{\sqrt{9} }{\sqrt{x}}[/tex]

Hence, the rationalized expression would be [tex]\sqrt{3}+\frac{\sqrt{9} }{\sqrt{x}}[/tex].

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The scatterplots shown can be matched to the corresponding correlation coefficients , which are as follows :
Scatterplot A : -0.56

Scatterplot B : -0.96

Scatterplot C : 0.92

Scatterplot D : 0.53

What is correlation coefficient?
Correlation coefficient is a numerical measure of the degree of relationship between two variables. It is used to measure how closely related two variables are, and whether one variable is a good predictor of the other. It is measured on a scale of -1 to 1, with -1 representing a strong negative correlation, 0 representing no correlation, and 1 representing a strong positive correlation. A correlation coefficient of 0 suggests that the two variables are completely unrelated, while a coefficient of 1 suggests that the two variables are strongly related. Correlation coefficients can help to identify patterns in data and can be used to predict future values.

Note the following facts :

Scatterplot that shows a trend that rises from left to your right shows a positive correlation between two variables.

Scatterplot that descends from left to left to right shows a negative correlation between two variables.

The farther away the data points are from each other the greater the value the value of correlation coefficient.

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Answer:

After Rajesh took 493 stones, you would have 471 stones left.

Step-by-step explanation:

If you started with 964 stones and Rajesh took 493 stones, then you would have 964 - 493 = 471 stones left.

The number of seconds will it take the ball to hit the ground will be 2.078 seconds.

An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, and a must not be zero.

For example, 3x² + 6x + 8 = 0 here x has the highest term as 2 and the coefficient of x² is not zero.

As per the given,

h(t)=-16t² + 14t + 40

The time when the ball hit the ground h(t) = 0

-16t² + 14t + 40 = 0

-8t² + 7t + 20 = 0

8t² - 7t - 20 = 0

t = (7 ± 26.25)16

t = 2.078 seconds

Since time can never be negative thus, 2.078 seconds will be correct.

Hence "The number of seconds will it take the ball to hit the ground will be 2.078 seconds".

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In the given formula we have variables:

If we have to solve it for P then it should be called 'Principal' or 'Loan amount'.

If we have to solve it for n then it should be called 'Number of payments' or 'Number of installments'.

Since n is the power of a number we need logarithm to solve it for n.

Answer:

"Principal (loan amount)"

"Term of the loan (in months)" or "Number of monthly payments"

Logarithms

Step-by-step explanation:

Monthly Payment Formula

[tex]M=\dfrac{Pr\left(1+r\right)^n}{\left(1+r\right)^n-1}[/tex]

where:

If we solve for P, the name of the formula is "Principal (loan amount)".

If we solve for n, the name of the formula is "Term of the loan (in months)" or "Number of monthly payments".

When solving for n, we need to use logarithms:

[tex]\boxed{\begin{aligned}M&=\frac{Pr\left(1+r\right)^n}{\left(1+r\right)^n-1}\\\\M(\left(1+r\right)^n-1)&=Pr\left(1+r\right)^n\\\\\frac{\left(1+r\right)^n-1}{\left(1+r\right)^n}&=\frac{Pr}{M}\\\\1-\frac{1}{\left(r+1\right)^n}&=\frac{Pr}{M}\\\\\frac{1}{\left(r+1\right)^n}&=1-\frac{Pr}{M}\\\\\left(r+1\right)^{-n}&=1-\frac{Pr}{M}\\\\\ln \left(r+1\right)^{-n}&=\ln \left(1-\frac{Pr}{M}\right)\\\\-n\ln(r+1)&=\ln\left(1-\frac{Pr}{M}\right)\\\\n&=\frac{-\ln\left(1-\frac{Pr}{M}\right)}{\ln(r+1)}\end{aligned}}[/tex]

The scale factor required for the dilation of the living room is 1/3.

A scale factor is a multiplicative fraction that can be used to either increase or decrease (i.e resize) the shape of a given object. It is given by the expression:

scale factor = [tex]\frac{length on the drawing}{length of the original object}[/tex]

Dilation is type of transformation that can be used to resize the dimensions of a given object. So that an image of the object is formed.

Therefore, the required scale factor is;

scale factor = [tex]\frac{length on the drawing}{length of the original object}[/tex]

                   = 7/21 or 9/27

                   = 1/3

scale factor = 1/3

The scale factor of the dilation as required in the question is 1/3.

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The lightbulb should be placed at the co-ordinate F(0,1)

The general equation of a parabola is as y²=4ax for a parabola opening towards the positive x-axis.

Here a represents the point on the x-axis where the focal point lies and through which the directrix passes.

For the given question,

Equation of Parabola: x²=4y

Comparing the given equation to standard parabolic equation for parabola on y-axis( x²= 4ay)

Then,  x²=4y: x²=4ay

thus value of a=1

Hence, the focal point of the parabola will be (0,1)

Since the lightbulb is to be placed on the focal point of the parabolic mirror, the lightbulb would be placed at a unit distance away from the vertex of the parabola

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The amount of candle burn after 6 hours is 4.95 inches.

What is the total amount?

Total refers to an entire or complete amount, and the verb "to total" means to combine two numbers or to destroy anything. In mathematics, you add integers to get the total; the outcome is the total.

Here, we have

Given that the 9-inch candle burns in 6 hours.

We know that we first have to calculate how 1 hour how much candles will burn.

Candle burn in 1 hour is :

= 9 / 10

= 0.9 inch

Now,

To find out how far the candle burnt after 6 hours we need to multiply the part of the candle burned by one hour by 11/2 hours.

= 11/2 × 0.9

= 4.95 inches

Hence, the amount of candle burn after 6 hours is 4.95 inches.

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The mean of a distribution is the average of all the values in the data set. Since the mean is the center of the distribution, any data point that is exactly the same as the mean will have a standard score of 0.

The mean of a distribution is the average of all the values in the data set. It is the central point in the data set, and any data point that is exactly the same as the mean will have a standard score of 0. The standard score is the number of standard deviations that a given data point is away from the mean. Since the mean is the center of the distribution, it is always 0 standard deviations away from itself and thus has a standard score of 0.

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For the given polynomial 5a²b-13ab+7a³-4b, the descending order is 7a³+5a²b-13ab-4b.

Polynomials are sums of terms of the form k⋅xⁿ, where k is any number and n is a positive integer. For example, 3x+2x-5 is a polynomial. Polynomials are introduced. This video defines words such as degree, standard form, monomial, binomial, and trinomial. A polynomial is an expression made up of variables, constants, and exponents that are combined using mathematical operations like addition, subtraction, multiplication, and division (No division operation by a variable).

Here,

The given polynomial is,

=5a²b-13ab+7a³-4b

The descending order,

=7a³+5a²b-13ab-4b

The descending order of polynomial is 7a³+5a²b-13ab-4b for the given polynomial 5a²b-13ab+7a³-4b.

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As per the volume of cylinder, the value of k is 3.06 centimeters.

Volume of cylinder:

According to geometry, the volume of a cylinder is defined as the capacity of the cylinder, which helps to find the amount of material that the cylinder can hold.

Given,

Here the glass pictured to the left can hold a maximum volume of 473 cubic centimeters, which is approximately 16 fluid ounces.

Here we need to find the value of k, in centimeters.

While we looking into the given question we have identified that the value of the following,

Volume of the glass = 473 cubic centimeter

Height of the glass = 16 fluid ounces.

Radius of the glass = k

Then the value of k is calculated by using the volume of cylinder formula,

=> V = πr²h

Here we have to apply the values on the formula, then we get,

=> 473 = π x k² x 16

Apply the value of π = 3.14,

=> 473 = 3.14 x k² x 16

=> k² = 9.41

=> k = 3.06

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The mean x=10 and the median=10 are equal, then the data is a normal distribution and the shape will be symmetrical "bell curve".

In the given question, Colin surveyed 12 teachers at his school to determine how much each person budgets for lunch.

He recorded his results in the table.

A 6-column table with 2 rows

The first row contains entries 10, 5, 8, 10, 12, 6.

The second row contains entries 8, 10, 15, 6, 12, 18.

We have to find the relationship between the mean and median reveal about the shape of the data.

The data elements are given as:

10, 5, 8, 10, 12, 6, 8, 10, 15, 6, 12, 18

Start by sorting the above data elements in ascending order.

5, 6, 6, 8, 8, 10, 10, 10, 12, 12, 15, 18

The mean of the dataset is then calculated as:

X = 5+6+6+8+8+10+10+10+12+12+15+18/12

X = 120/12

X = 10

The median of element is then calculated as:

Median = n/2th

Median = 12/2th

Median = 6th

This means that, the median is of the 6th elements.

So, we have:

Median = 10

When the mean and the median are equal, then

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(a) If the discrimant is > 0, the quadratic equation has 2 real solutions.

(b) If the discrimant is = 0, the quadratic equation has 1 real solution.

(c) If b^2-4ac is < 0, the quadratic equation has 0 real solution, therefore it has 2 imaginary solutions.

The square root's input (i.e., its contents), which is the phrase b^2 -  4ac, is known as the "discriminant" because using its value, you may "discriminate" between (i.e., be able to distinguish between) the different solution types.

To determine how many possible solutions a quadratic equation has, utilize the discriminant formula.

(a) If the discrimant is > 0, the quadratic equation has 2 real solutions.

Example

x^2 - 5x + 2 = 0

where a = 1, b = -5, c= 2

D=b^2 - 4ac = (-5)^2 - 4 ×1 × 2 = 17

∴ It has two real solutions.

(b) If the discrimant is = 0, the quadratic equation has 1 real solution.

Example

x^2 + 2x + 1 = 0

where a = 1, b = 2, c= 1

D=b^2 - 4ac = (2)^2 - 4 ×1 × 1 = 0

∴ It has one real solution.

(c) If b^2-4ac is < 0, the quadratic equation has 0 real solution, therefore it has 2 imaginary solutions.

Example

3x^2 + 2x + 1 = 0

where a = 3, b = 2, c= 1

D=b^2 - 4ac = (2)^2 - 4 ×3 × 1 = -8

∴ It has two imaginary solutions.

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Answer:

49

Step-by-step explanation:

10 weeks = 70 days

0.7 : 1 = x : 70

x = 0.7 * 70

x = 49

Answer:

See below.

Step-by-step explanation:

(a) y=x²-5  NO - the x term is squared.

(b) y=x³     NO - the x term is cubed

(c) y=-9x    YES

(d) y=-x+3  YES

See attached graph.

Answer:

A = NO

B = NO

C = YES

D = YES

Step-by-step explanation:

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be 0 or 1 for each of its variables.

For A: The degree of variable y is 1 and the degree of variable x is 2.

So it is not linear.

For C: The degree of variable y is 1 and the degree of variable x is 1.

So it is linear.

If a graph moves arbitrarily near to a line in one direction (without necessarily crossing it), then the line is a(n) asymptote of the graph.

An object that serves as the limit of another line or curve in mathematics. Asymptotic to an axis, or the asymptote of a curve, refers to a falling curve that approaches but does not reach the horizontal axis.

An asymptote is a value that you approach steadily but never quite achieve. An asymptote is a line that a graph skirts but never touches in mathematics. It can be horizontal, vertical, or slanted. A graph that approaches a line but does not cross it is said to be an asymptote.

An asymptote of a curve in analytical geometry is a line along which the angle between the curve and the line decreases when one or both of the x or y coordinates tend to infinity. A line that is tangent to a curve at a point at infinity is said to be the asymptote of the curve in projective geometry and similar contexts.

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Answer:

153

Step-by-step explanation:

165/3 = 55 sheet of each color

55-4 = 51

51 x 3 = 153

A student is expected to learn approximately 46.535 words in the ninth hour of study.

Generally, To find the number of words a student is expected to learn in the ninth hour of study, we can substitute 9 for t in the learning curve equation:

y = 50 - 50e^-0.3(9)

Simplifying the exponent, we have:

y = 50 - 50e^-2.7

To calculate the value of e^-2.7, we can use a calculator or lookup the value in a table of common exponent values. The value of e^-2.7 is approximately 0.0693.

Substituting this value back into the equation, we have:

y = 50 - 50(0.0693)

Simplifying, we find:

y = 50 - 3.465

y = 46.535

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