Which equation can be used to solve for mz1?
(a - b)
m² 1 = (a + b)
(c-d)
m²1 = (c+d)
m₂1 =
m²1 =

The percentage of the total sum of the squares that can be accounted for by the estimation of regression is 51.3% when it is taken in three decimal points by the regression equation.

The regression equation is used to find one variable from another known variable. There are two types to find the regression equation they are:

1. Regression equation by using simultaneous equation 2. Regression line

The regression equation can be found by the be calculated by the sums of squares by the the sample of correlation coefficient that is 0.716. The amount of variation is taken by the total variation that is interpreted and is denoted by 'r', the sum of squares can be calculated by 1-SSE/ SST=(SST/SST = SSR/SST. When it comes to the product volume then the percentage is 93.64% where it also includes the product cost and variable cost of the product.

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In statistics, the percentile is a measure that indicates the value below which a given percentage of observations fall. For instance, the 75th percentile represents the value below which 75% of the observations lie.

Therefore, to find the score on the statistics exam that is the 75th percentile, we need to identify the value below which 75% of the scores lie.

In this case, we know that the scores on the exam are normally distributed with a mean of 75 and a standard deviation of 8. Using this information, we can use a normal distribution table or calculator to find the z-score associated with the 75th percentile, which is 0.674. We then use this z-score to calculate the corresponding score on the exam using the formula:

score = z-score * standard deviation + mean

Plugging in the values, we get:

score = 0.674 * 8 + 75

score = 80.392

Therefore, a score of 80.392 is the 75th percentile on the statistics exam.

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The patterns of the sum and difference of two cubes can be used to factorize polynomial expressions. To utilize these patterns, we need to identify if the polynomial expression we want to factorize can be written in the form of a sum or difference of two cubes, and then apply the corresponding pattern to factorize it.

The sum and difference of two cubes are useful patterns that can be used to factorize polynomial expressions. To utilize these patterns, we need to identify if the polynomial expression we want to factorize can be written in the form of a sum or difference of two cubes. The sum of two cubes can be expressed as:

a³ + b³ = (a + b)(a² - ab + b²)

And the difference of two cubes can be expressed as:

a³ - b³ = (a - b)(a² + ab + b²)

To use these patterns, we need to look for polynomials in the form of a³ + b³ or a³ - b³, where a and b are integers or algebraic expressions. If we find such expressions, we can factorize them using the corresponding pattern.

For example, let's consider the polynomial expression x³ + 8. This can be written in the form of a sum of two cubes, where a = x and b = 2:

x³ + 8 = x³ + 2³

Now we can use the sum of two cubes pattern to factorize the expression:

x³ + 2³ = (x + 2)(x² - 2x + 4)

Similarly, if we have an expression in the form of a³ - b³, we can use the difference of two cubes pattern to factorize it. For example, let's consider the expression y³ - 27:

y³ - 27 = y³ - 3³

We can use the difference of two cubes pattern to factorize this expression:

y³ - 3³ = (y - 3)(y² + 3y + 9)

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How will you utilize the patterns in the sum and difference of two cubes in any case?

Answer:

4√3

Just use Sin rule and cross multiplication method

The initial value problem. y'(t) = 1 + e^t, y(0) = 20 The specific solution is y(t)= t + e^t + 19.

Let's go step-by-step:
1. Identify the problem: We are given a differential equation y'(t) = 1 + e^t and an initial value y(0) = 20.

2. Integrate the differential equation: To find y(t), we need to integrate the given equation with respect to t.
  ∫(y'(t) dt) = ∫(1 + e^t dt)

3. Perform the integration: After integrating, we obtain the general solution of the problem:
  y(t) = t + e^t + C, where C is the constant of integration.

4. Apply the initial value: We are given y(0) = 20, so we can plug this into the general solution to find the specific solution.
  20 = 0 + e^0 + C
  20 = 1 + C

5. Solve for the constant of integration C: From the above equation, we find the value of C.
  C = 19

6. Write the specific solution: Now that we have the value of C, we can write the specific solution for y(t).
  y(t) = t + e^t + 19

So, the specific solution for this initial value problem is y(t) = t + e^t + 19.

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The minimum amount of vinyl needed for the liner is 1206.37 ft².

Given:

The height of the cylinder is h = 4 ft

The diameter of the cylinder is d = 24 ft

So the radius (r) of the cylinder is half of the diameter, which is 24/2 = 12 ft.

The total surface area of the cylinder is as follows:

S = 2πrh + 2πr²

Substitute the values in the above formula,

surface area of the cylinder = 2π(12)(4) + 2π(12)²

surface area of the cylinder = 96π + (144π)

surface area of the cylinder = 384π

surface area of the cylinder = 384 × 3.14

surface area of the cylinder = 1206.37 ft²

This means the minimum amount of vinyl needed for the liner is 1206.37 ft².

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The missing figure is attached below.

Let's denote the radius of the red circle by r, then the circumference of the red circle is 2πr. We know that the perimeter of the red circle is 37.7 inches, so:

2πr = 37.7

Solving for r, we get:

r = 37.7 / (2π) = 6.002 inches (rounded to three decimal places)

The radius of the yellow circle is 6 inches larger than the radius of the red circle, so:

r_yellow = r_red + 6 = 12.002 inches

Therefore, the circumference of the yellow circle is:

2πr_yellow = 2π(12.002) = 75.4 inches (rounded to one decimal place)

So the perimeter of the next-largest (yellow) circle is approximately 75.4 inches.

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If the calculated required sample size is a non-integer value, we should always round up the calculated value

When calculating the required sample size for a study, the sample size formula often involves a combination of statistical parameters such as the desired level of significance, the desired power of the study, the expected effect size, and the variability in the data. Sometimes, these parameters may result in a non-integer value for the required sample size.

In such cases, it is important to round up the calculated value to the nearest whole number, as it is not possible to have a fraction of a participant in the study. This ensures that the sample size is large enough to adequately represent the population and achieve the desired level of statistical power.

For example, if a calculated sample size is 123.4, it should be rounded up to 124 to ensure that the sample is large enough to produce reliable and accurate results. Failing to round up can result in an underpowered study, which may lead to false negative results or failure to detect significant effects.

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The sale price of the item after a 50% discount is $133.75.

To calculate a sale price after a 50% discount;

Find the cost after a 7% increase. To do this, we multiply the original cost by 1 + the percentage increase. In this case, the original cost is $250 and the percentage increase is 7%, so the cost after the increase is;

Cost after increase = $250 + 7% of $250

= $250 + 0.07 × $250

= $250 + $17.50

= $267.50

Find the sale price after a 50% discount. To do this, we multiply the cost after the increase by (1 - 50%), which is equivalent to multiplying by 0.5. So the sale price is;

Sale price = Cost after increase × (1 - 50%)

= $267.50 × 0.5

= $133.75

Therefore, the sale price of the item after a 50% discount is $133.75.

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Both sections cover important mathematical concepts with challenging aspects but also offer numerous applications for various professional fields and everyday life.

In Section 1, computing partial derivatives algebraically can be tricky and frustrating because it involves using the chain rule, product rule, and quotient rule in complex functions. However, it can also be exciting to see how these rules can be applied to find rates of change in multivariable functions.

Two things that could be useful as a scientist, engineer, mathematician or in your personal life from this section are:
1. Understanding partial derivatives can help in optimizing systems in engineering and science.
2. Partial derivatives can also be used in finance to calculate sensitivity analysis in portfolio management.

In Section 2, local linearity and the differential can be difficult to grasp because it involves understanding the tangent plane of a surface and how it approximates the surface near a point. However, it can be exciting to see how this concept can be applied to approximating solutions to nonlinear equations.

Two things that could be useful as a scientist, engineer, mathematician or in your personal life from this section are:
1. Local linearity can be used in computer graphics to render 3D objects.
2. The differential can be used in physics to calculate small changes in variables in differential equations.

Section 1: Computing Partial Derivatives Algebraically
1. Tricky aspects:
  a. Differentiating with respect to one variable while treating other variables as constants can be challenging, especially in functions with multiple variables.
  b. Applying the chain rule for partial derivatives may be confusing for some due to the interplay of different variables.

2. Applications:
  a. Scientists and engineers use partial derivatives to model and understand how different parameters affect complex systems.
  b. Mathematicians use partial derivatives in optimization problems to find the maxima or minima of multivariable functions.

Section 2: Local Linearity & The Differential
1. Tricky aspects:
  a. Understanding the concept of local linearity and how it connects to differentiability can be challenging for some learners.
  b. Applying differentials to approximate changes in functions can be tricky due to the need to find the right balance between accuracy and simplicity.

2. Applications:
  a. Engineers use the concept of local linearity to analyze how systems behave under small changes and make approximations that simplify their calculations.
  b. In personal life, differentials can be used to estimate how small changes in one aspect, like the price of a product, might affect the overall cost.

In summary, both sections cover important mathematical concepts with challenging aspects but also offer numerous applications for various professional fields and everyday life.

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The equation of the tangent line to the curve f(x) = 4sec(x) at x = π/3 is y = 8√3/3 x + 8 - 8√3.

To find the equation of the tangent line to the curve f(x) = 4sec(x) at x = π/3, we need to find the slope of the tangent line at that point and the point-slope form of the equation of a line.

The slope of the tangent line is given by the derivative of f(x) evaluated at x = π/3:

f(x) = 4sec(x)

f'(x) = 4sec(x)tan(x)

f'(π/3) = 4sec(π/3)tan(π/3) = 4(2)√3/3 = 8√3/3

So the slope of the tangent line at x = π/3 is 8√3/3.

Now we need to find a point on the tangent line. We know that the point (π/3, f(π/3)) is on the curve, so it must also be on the tangent line. Evaluating f(π/3), we get:

f(π/3) = 4sec(π/3) = 4(2) = 8

So the point (π/3, 8) is on the tangent line.

Using the point-slope form of the equation of a line, we have:

y - 8 = (8√3/3)(x - π/3)

Simplifying, we get:

y = 8√3/3 x + 8 - 8√3

So the equation of the tangent line to the curve f(x) = 4sec(x) at x = π/3 is y = 8√3/3 x + 8 - 8√3.

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Therefore, one relief employee can cover 2 regular employees in a week by probability.

Assuming that each regular employee works for 5 days a week, and one relief employee is available to cover the remaining two days, we can calculate the number of regular employees that can be covered by one relief employee as follows:

One relief employee covers 2 days/week.

So, the number of regular employee days that one relief employee can cover in a week is:

2 days/week × 1 week = 2 days

Therefore, the number of regular employees that one relief employee can cover in a week is:

5 days/week ÷ 2 days = 2.5 regular employees

However, since we cannot have half of an employee, we round down to the nearest whole number.

Therefore, one relief employee can cover 2 regular employees in a week.

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Answer: b - r/5 + 7

explanation:

Isolate the variable by dividing each side by factors that don't contain the variable.

Answer:

b = r/5 + 7

Step-by-step explanation:

Distribute first: 5b - 35 = r

Add 35 to both sides: 5b = r + 35

Divide by 5: b = r/5 + 7

Therefore, the probability of choosing a white marble without replacement, given that the first marble chosen was a green marble, is 7/19 or approximately 0.3684 (rounded to four decimal places).

The probability of selecting a white marble on the next draw depends on the number of white and green marbles left in the box. We are told that after selecting a green marble, there will be 12 green marbles and 7 white marbles left in the box.

Since there are 12 green marbles and 7 white marbles left in the box, the total number of marbles left is:

12 + 7 = 19

The probability of selecting a white marble on the next draw is the number of white marbles left divided by the total number of marbles left. So we can calculate this probability as:

P(white) = number of white marbles left / total number of marbles left

P(white) = 7/19

Therefore, the probability of selecting a white marble on the next draw is 7/19. This means that out of the remaining marbles in the box, there is a 7/19 chance of selecting a white marble on the next draw.

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From the equation [tex]\frac{tan^2(x)}{sec(x)-1}=sec(x)+1\\ \\[/tex], it is possible to find the trigonometric identities: tan²(x)=sec²(x)-1.

A triangle is classified as a right triangle when it presents one of your angles equal to 90º.  The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called cathetus or legs.

The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.

As previously presented the trigonometric ratios are derived by the sides of a right triangle. The main trigonometric ratios are: sinβ, cosβ and tg β. From these ratios, you can calculate other trigonometric ratios such as sec β, csc β and cotg β.

For solving this question, you need to know one of the trigonometric identities: tan²(x)=sec²(x)-1

The question gives: [tex]\frac{tan^2(x)}{sec(x)-1}=sec(x)+1\\ \\[/tex], then you should multiply the numerator of each side by the denominator of the other side, the result will be: tan²(x)=sec²(x)-1. Exactly, the trigonometric identities tan²(x)=sec²(x)-1.

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To solve this problem, we first need to find the long-term trend for the transition matrix T. We can do this by finding the eigenvectors of T and using them to calculate the steady-state distribution.

Using a calculator or software, we can find that the eigenvectors of T are:

v1 = [0.812, -0.567, 0.148, 0.076, 0.003]
v2 = [-0.269, 0.304, -0.657, 0.639, -0.013]
v3 = [-0.192, 0.466, -0.316, -0.796, -0.012]
v4 = [-0.491, -0.592, -0.678, 0.013, 0.008]
v5 = [0.002, -0.015, 0.001, 0.000, 0.999]

We can see that v5 corresponds to the eigenvalue 1, which means it is the steady-state distribution. Therefore, the long-term trend for T is:

[0.812, -0.567, 0.148, 0.076, 0.003] → [0.002, -0.015, 0.001, 0.000, 0.999]

Now, to find P(end with 1 start with 2), we need to look at the (2, 1) entry of T^n for large n. We can use the fact that T^n approaches the matrix with v5 as its columns as n approaches infinity.

The (2, 1) entry of T^n can be found by multiplying the second row of T^n by the first column of the identity matrix. Using a calculator or software, we can find that this value approaches 0.3 as n approaches infinity. Therefore, the answer is (b) 0.3.

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Answer:

I believe 0.143

Step-by-step explanation:

Well the chance is 2 out of 14 so 2/14 then you reduce that and get 1/7 equals 0.143. I may have done that wrong

The standard deviation of wait time is 13.8564.

The length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 25 minutes and 73 minutes. We have to find the standard deviation of the wait time.

The square root of the variance of a random variable, sample, statistical population, data collection, or probability distribution is its standard deviation.

The standard deviation in statistics is a measure of the degree of variation or dispersion in a set of values.

A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation shows that the values are spread out over a larger range.

S² = (73 - 25)²/12

S² = (48)²/12

S² = 192

S = √192

S = 13.8564

Hence, The standard deviation of wait time is 13.8564.

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complete question:

the length of time patients must wait to see a doctor in a local clinic is uniformly distributed between 25 minutes and 73 minutes. what is the standard deviation of wait time? group of answer choices

The equation given models the distance from the ground of a person riding on a ferris wheel. it takes 80 seconds for the Ferris wheel to make one revolution.

To determine how long it will take for the ferris wheel to make one revolution, we need to find the period of the function. The period is the amount of time it takes for the function to complete one full cycle.
In this case, the function is d = 30sin(pi/40t) + 20, where t is measured in seconds. The period of the function can be found using the formula T = (2pi)/b, where b is the coefficient of t in the argument of the sine function. In this case, b = pi/40, so T = (2pi)/(pi/40) = 80 seconds.
Therefore, it will take 80 seconds for the ferris wheel to make one full revolution. The answer is option C, 80 seconds.
The time it takes for a Ferris wheel to make one revolution can be determined using the given equation: d = 30 * sin((π/40) * t) + 20. In this equation, d represents the distance (in feet) of the person above the ground, and t represents the time in seconds.
A full revolution occurs when the angle inside the sine function completes a cycle of 2π radians. To find the time it takes for this to happen, we need to equate the angle (π/40) * t to 2π:
(π/40) * t = 2π
To solve for t, we can divide both sides of the equation by (π/40):
t = 2π * (40/π)
The π in both the numerator and denominator cancels out:
t = 2 * 40
t = 80 seconds
Therefore, it takes 80 seconds for the Ferris wheel to make one revolution.

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a) Using an exponential equation in the form of y = a(b)ˣ, the equation that models the number of views, y, x days after Jack first watched the video is y = 3,140(1 + 0.6) ˣ.

b) Based on the above exponential growth function, the number of views that Jack can expect to have 7 days after he first watched it is 84,288.

An exponential equation is an equation with a variable exponent and usually in the form of y = a(b)ˣ.

Exponential equations may show growth (constant increase) or decay (constant decrease).

The total number of views on day one = 3,140

The total number of views on day two = 5,024

The increase in the number of views in one day = 1,884 (5,024 - 3,140)

Percentage increase = 60% (1,884/3,140 x 100)

= 0.6

The total number of views seven days after is y = 3,140(1.6)⁷

= 84,288

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The following are the statements with explanation whether the statement is true or false, using rank-nullity theorem and invertible matrix theorem.

a. False. According to the rank-nullity theorem, the rank of a matrix plus its nullity equals the number of columns. As a result, a 7 x 4 matrix can only have a rank of 4, because the nullity cannot be negative.

b. False. According to the rank-nullity theorem, the nullity of a matrix plus its rank equals the number of columns. As a result, because the rank cannot be negative, a 4 x 7 matrix can have a maximum nullity of 3.

c. True. According to the rank-nullity theorem, the nullity of a matrix plus its rank equals the number of columns. As a result, a 7 x 4 matrix has a maximum nullity of 3, implying that it can have a nullity 5.

d. True. According to the rank-nullity theorem, the rank of a matrix plus its nullity equals the number of columns. As a result, if A is a 7 x 4 matrix, rank(A) + nullity(A)= 4 + nullity(A) = 7. When we solve for nullity(A), we get nullity(A) = 3, therefore rank(A) + nullity(A) = 4 + 3 = 7.

e. False. According to the rank-nullity theorem, the nullity of a matrix plus its rank equals the number of columns. As a result, if A is a 6 x 8 full-rank matrix, its nullity is 8 - 6 = 2, rather than nullity(A) = 2² = 4.

f. True. According to the invertible matrix theorem, a full-rank matrix has a unique solution for any non-zero right-hand side vector b. As a result, the system Ax = b is always consistent for every non-zero b.

g. True. According to the invertible matrix theorem, a full-rank matrix has a unique solution for each right-hand side vector b. As a result, the system Ax = b will always have a distinct solution for any b.

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Dylan must charge each team an entrance fee of $85 in order to make a profit of $500.

The amount charged by sports complex = $690

Total teams to be charged = 14

Total time = 6 hrs

Profit to be made = $500.

Let the entrance fee for each team be =  x.

Thus,

Total revenue = 14x

Calculating the cost per hour of ice rental is:

The amount charged by complex/ Total time

= 690 / 6

= 115 per hour

Dylan must make sure that his entire sales surpass his total expense by $500 in order to achieve a profit of $500.

Therefore,

Total revenue - Total cost = $500

= 14x - (6 hours x $115 per hour) = $500

Simplifying -

14x - $690 = $500

14x = $1190

x = $85

Complete Question;

Dylan is organizing a curling tournament. The sports complex is charging Dylan $690 for ice rental. Dylan has booked it for 6 hrs. He will charge 14 teams in the tournament an entrance fee. How much must he charge each team in order to make a profit of $500.

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This means that we would expect 25 cases of NHL in the group of 250 participants who were exposed to pesticides based on the proportion of NHL cases in the non-exposed group.

To compute the expected number of cases of cancer in the long-term exposure group, we need to first understand the values in the 2 by 2 table. The table shows the number of participants who were exposed to pesticides and who developed non-hodgkin's lymphoma (NHL), as well as the number of participants who were not exposed to pesticides and who developed NHL.
In this study, there were 250 participants who were exposed to pesticides and 50 of them developed NHL. This gives us a proportion of 0.2 (50/250) or 20% of the exposed group that developed NHL. On the other hand, there were 250 participants who were not exposed to pesticides and 25 of them developed NHL. This gives us a proportion of 0.1 (25/250) or 10% of the non-exposed group that developed NHL.
To calculate the expected number of cases of cancer in the long-term exposure group, we can use the formula:
Expected number = (total number of participants in the exposed group) x (proportion of NHL cases in the non-exposed group). Therefore, the expected number of cases of cancer in the long-term exposure group would be:
Expected number = 250 x 0.1 = 25

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The value of particular solution to the nonhomogeneous differential equation is,

⇒ y (p) = 2x + 1/2 e⁻ˣ - 8/5

We have to given that;

The nonhomogeneous differential equation is,

⇒ y'' + 4y' + 5y = 10x + e⁻ˣ  . (i)

To find homogeneous solution,

D² + 4D + 5 = 0

(D + 2)² = - 1

D + 2 = ±i

D = 2 ± i

Hence, We get;

y = e⁻²ˣ (c₁ cos x + c₂ sin x)  .. (ii)

To find the particular solution,

y (p) = A + Bx + Ce⁻ˣ

y' (p) = B - Ce⁻ˣ

y'' (p) = Ce⁻ˣ

Substitute all the values in (i);

⇒ y'' + 4y' + 5y = 10x + e⁻ˣ

⇒ Ce⁻ˣ + 4(B - Ce⁻ˣ) + 5(A + Bx + Ce⁻ˣ) = 10x + e⁻ˣ

Equating the coefficient;

A = 2

B = - 8/5

C = 1/2

So, We get;

⇒ y (p) = 2x + 1/2 e⁻ˣ - 8/5

The value of particular solution to the nonhomogeneous differential equation is,

⇒ y (p) = 2x + 1/2 e⁻ˣ - 8/5

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The area of each semicircle to the nearest hundredth include the following:

Area = 9.82 in².

Area = 16.09 in².

In Mathematics and Geometry, the area of a semicircle can be calculated by using this mathematical equation (formula):

Area of semicircle = πd²/8

Where:

d represents the diameter of a circle.

By substituting the given diameter into the formula for the area of a semicircle, we have the following;

Area of semicircle = 3.142 × 5²/8

Area of semicircle = 9.82 in².

For the second semicircle, we have the following:

Area of semicircle = 3.142 × 6.4²/8

Area of semicircle = 16.09 in².

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The required two way frequency table is shown below.

We know that a two-way frequency table is nohting but the way to display frequencies for two different categories collected from a single group of people.

While making the two-way frequency tables first we need to identify the two variables of interest. Then we need to determine the possible values of each variable. Select a variable to be represented by the rows and the other to be represented by the columns. And then complete the table with frequencies.

Here we have two variables as:

Time ( practiced for atleast 10,000 hours and did not practiced for atleast 10,000 hours) and the second (won a prize and do not won a major)

Based on the information provided, we can obtain a two way frequency table as shown below:

                                   practiced for atleast          did not practiced for

                                       10,000 hours               atleast 10,000 hours

have won a major                  9                                      1

haven't won a major              1                                       2

Thus the required two-way frequency table is shown in attached figure.

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Find the complete question below.

It will take about 3 years for the bird population to reach 7,200, assuming exponential growth. Assuming exponential growth, we can use the formula N = N0 x (1+r)^t, where N is the final population, N0 is the initial population, r is the annual growth rate, and t is the time in years.

In this case, we know that N0 = 900 and N = 7,200. We can find the annual growth rate, r, by using the fact that the population doubled in one year.
If the population doubles in one year, then the growth rate is 100%. So r = 1.
Now we can plug in the values we know and solve for t:
7,200 = 900 x (1+1)^t
Dividing both sides by 900:
8 = 2^t
Taking the logarithm of both sides:
log(8) = t x log(2)
Solving for t:
t = log(8) / log(2)
t ≈ 3
So it will take about 3 years for the bird population to reach 7,200, assuming exponential growth.

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We can say with 95% confidence that the true proportion of people who have seen the video is between 0.841 and 0.897.

To create a 95% confidence interval for the proportion of people who have seen the video, we can use the following formula:

[tex]CI = \hat{p} \pm z*√((\hat{p}(1-\hat{p}))/n)[/tex]

where:

[tex]\hat{p}[/tex]  = sample proportion (352/405)

z = z-score for the desired confidence level (1.96 for 95% confidence interval)

n = sample size (405).

Plugging in the values, we get:

CI = 0.869 ± 1.96*√((0.869(1-0.869))/405)

CI = 0.869 ± 0.028

Rounding to three decimal places, we get:

CI = (0.841, 0.897).

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The instantaneous rate of change of the function  is 2.

The instantaneous rate of change of a function at a particular point is the rate at which the function is changing at that point, or the slope of the tangent line to the graph of the function at that point. It gives an indication of how fast the function is increasing or decreasing at that point.

To compute the instantaneous rate of change of the function at x=a, we need to find the derivative of the function f(x) and evaluate it at x=a.

f(x) = 2x + 10

Taking the derivative of f(x) with respect to x:

f'(x) = 2

So, the instantaneous rate of change of f(x) at x=a is:

f'(a) = 2

Substituting a=3 in the above equation, we get:

f'(3) = 2

Therefore, the instantaneous rate of change of the function f(x) at x=3 is 2.

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