If f is a continuous function and if F'(x)=f(x) for all real numbers x, then the integral [1,3] f(2x)dx=

The given distribution is a Geometric distribution since it represents the number of trials needed to achieve the first success with a fixed probability of success for each trial.

What is Geometric distribution?

The Geometric distribution is a probability distribution that models the number of independent and identical Bernoulli trials needed to achieve the first success. In other words, it describes the probability of observing the first success on the kth trial, where k can take on non-negative integer values. The distribution is characterized by a single parameter p, the probability of success in each trial. The pmf of the Geometric distribution is P(X=k) = (1-p)^(k-1)*p for k = 1, 2, 3, ..., where X is the random variable representing the number of trials needed to achieve the first success.

The given distribution is a Geometric distribution, since we are interested in the number of trials (draws) needed to achieve the first success (drawing a jack). The probability of success (drawing a jack) remains the same for each trial, and each trial is independent. Therefore, the distribution of the number of trials needed to achieve the first success follows a Geometric distribution with parameter p, where p is the probability of drawing a jack on each trial.

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If you increase the data of a dot plot by a certain percentage, the shape and spread of the data on the plot may change depending on the nature of the data and the percentage of increase.

If you increase the data by a small percentage, the shape and spread of the data may remain relatively unchanged.

The distribution of the data on the plot may become slightly wider or taller, but the general pattern of the data may remain similar.

This is because small increases in data may not significantly affect the underlying distribution of the data.

If you increase the data by a larger percentage, the shape and spread of the data on the plot may change more significantly.

If the data is skewed or has outliers, a large increase in data may cause the distribution to become more symmetrical and the outliers to become less extreme.

On the other hand, if the data is already symmetric and has a narrow spread, a large increase in data may cause the distribution to become more spread out and potentially skewed.

To changing the shape and spread of the data, increasing the data by a percentage may also affect the center of the distribution.

If the original data was centered around a particular value, such as the mean or median, the center of the distribution may shift slightly as a result of the increased data.

This is because the additional data points may have values that are either above or below the original center of the distribution.

Increasing the data by a percentage can affect the shape, spread, and center of the data on a dot plot.

The extent of these changes depends on the nature of the data and the percentage of increase.

To carefully analyze the resulting dot plot to determine how the additional data has affected the distribution of the data.

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To find the critical value of z for each hypothesis test in Excel, we can use the NORMSINV function.

For a two-tailed test at a 7% level of significance, we would use the formula "=NORMSINV(0.035)" (since we want the area in each tail to be 0.035, or half of the 7% level).

This gives us a critical value of -1.812. For a right-tailed test at a 9% level of significance, we would use the formula "=NORMSINV(0.91)" (since we want the area to the right of the critical value to be 0.09, or 9%).

This gives us a critical value of 1.340. For a left-tailed test at a 3% level of significance, we would use the formula "=NORMSINV(0.03)" (since we want the area to the left of the critical value to be 0.03, or 3%).

This gives us a critical value of -1.880. Remember to round all answers to 3 decimal places.

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The solution to the equation is k = -4.

A fraction is a mathematical value that illustrates the components of a whole.

The given equation is:

(k - 10) / 2 = -7

To dissolve the fraction, we can multiply both sides of the equation by 2:

(k - 10) / 2 × 2 = -7 × 2

Simplifying the left-hand side of the equation, we get:

k - 10 = -14

Adding 10 to both sides of the equation, we get:

k - 10 + 10 = -14 + 10

Simplifying, we get:

k = -4

Therefore, the solution to the equation is k = -4.

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We can order them in increasing x-coordinate as:

(0, 0), (0.585, 6.415), and (5.748, 1.252)

To find the critical points of the predator/prey model, we need to find the values of x and y that make both x' and y' equal to zero.

From the given equations, we have:

x' = 7x - x² - xy = 0

y' = -5y + xy = 0

Factoring x out of the first equation, we get:

So, either x = 0 or 7 - x - y = 0.

If x = 0, then the second equation simplifies to y' = -5y = 0, which has a critical point at y = 0.

If 7 - x - y = 0, then we can solve for y to get:

y = 7 - x

y' = -5y + xy = -5(7 - x) + x(7 - x) = 0

Simplifying, we get:

6x² - 49x + 35 = 0

x = (49 ± sqrt(49² - 4(6)(35))) / (2(6)) ≈ 0.585 or x ≈ 5.748

Substituting these values into y = 7 - x, we get:

y ≈ 6.415 or y ≈ 1.252

(0, 0), (0.585, 6.415), and (5.748, 1.252)

We can order them in increasing x-coordinate as:

(0, 0), (0.585, 6.415), and (5.748, 1.252)

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The length of the hypotenuse of the right triangle with a leg length of 7 in is approximately 10 in.

Applying the 45º-45º-90º Triangle Theorem, the length of the hypotenuse of a right triangle can be found by multiplying the length of a leg by the square root of 2. In this case, with a leg length of 7 in, the length of the hypotenuse can be determined.

The 45º-45º-90º Triangle Theorem states that in a right triangle with two equal legs, the length of the hypotenuse is equal to the length of a leg multiplied by the square root of 2.

In this case, the length of one leg is given as 7 in. To find the length of the hypotenuse, we can multiply the length of the leg by the square root of 2:

Hypotenuse = Leg * sqrt(2)

Substituting the given value, we have:

Hypotenuse = 7 in * sqrt(2)

Using a calculator, the approximate value of the square root of 2 is 1.414. Therefore, we can calculate:

Hypotenuse = 7 in * 1.414 ≈ 9.898 in

Rounding to the nearest inch, the length of the hypotenuse is approximately 10 in.

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Tollens's test shows the presence of aldehydes. A positive Tollens's test appears as a silver mirror while a negative Tollens's test appears as a clear solution.

Tollens's test is a chemical test used to detect the presence of aldehydes. The test involves the reaction of Tollens's reagent (ammoniacal silver nitrate) with an aldehyde, resulting in the reduction of silver ions to metallic silver, which appears as a silver mirror.

A positive Tollens's test, therefore, appears as a silver mirror on the surface of the solution being tested. On the other hand, a negative Tollens's test shows the absence of aldehydes, resulting in a clear solution without any visible silver mirror. The test is commonly used in organic chemistry to distinguish between aldehydes and ketones.

Overall, Tollens's test shows the presence of aldehydes.

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The probability of a student studying for 1 hour is 0.1. The Option A is correct.

A probability refers to how likely something is to happen. To determine probability of a student studying for 1 hour, we must find:

total number of students who studied for 1 hr

total number of students surveyed.

The number of students who studied for 1 hour is 3.The total number of students surveyed is:= 1 + 3 + 2 + 5 + 9 + 7 + 3= 30

The probability of a student studying for 1 hour is:

P(1 hour) = No of who studied for 1 hour / Total students surveyed

P(1 hour) = 3 / 30P(1 hour) = 0.1.

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The coordinates of the center are (-1, 8) and the length of the radius is 4 units.

The equation of a circle can be written in the form:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

Where (h, k) is the center of the circle, and r is the radius.

Starting with the given equation:

[tex]x^2 + y^2 + 2x - 16y + 49 = 0[/tex]

Completing the square for x and y, we get:

[tex](x + 1)^2 - 1 + (y - 8)^2 - 64 + 49 = 0[/tex]

Simplifying, we get:

[tex](x + 1)^2 + (y - 8)^2 = 16[/tex]

So the center of the circle is (-1, 8) and the radius is 4.

Now we can see that the center of the circle is (-1, 8) and the radius is 4. Therefore, the coordinates of the center are (-1, 8) and the length of the radius is 4 units.

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Based on the simple linear regression results for the car's mileage, we can estimate the effect of octane levels on the car's performance.

Using a 71% confidence interval and rounding to 3 decimal places, we can say with some confidence that the impact of octane levels on the car's mileage is between -0.045 and 0.102.

This means that while there is some correlation between octane levels and mileage, the effect is relatively small and falls within a narrow range of values. It's worth noting that there may be other factors at play that influence the car's performance, so this result should be interpreted with caution.

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The eigenvalues of the coefficient matrix A is λ = 5 ± 6i

An eigenvector corresponding to the eigenvalue with a positive imaginary part is v = k(3 + 2i, 1)

The general solution can be expressed as

X(t) = e^(5t) × (C1 × cos(6t) + C2 × sin(6t)) × (3 + 2i, 1)

Given,  dx/dt = 7x + 13y and dy/dt = -2x + 9y.

Eigenvalues of the coefficient matrix A:

The coefficient matrix A is:

[tex]A = \left[\begin{array}{cc}7&13\\-2&9\end{array}\right][/tex]

To get the eigenvalues, we have to solve the characteristic equation.

| (7 - λ) (9 - λ) - (-2)(13) | = 0

63 - 7λ - 9λ +λ² + 26 =0

λ² - 16λ + 89 =0

Solving this equation, we get the eigenvalues λ = 5 ± 6i.

The positive imaginary part eigenvalue λ = 5 + 6i. Let v be the eigenvector now we have to solve the system (A - λI)v = 0.

| (2 - 6i)  13     | |x| = |0|

|  -2      (4 - 6i)| |y| = |0|

After solving this system, we get an eigenvector v = k(3 + 2i, 1), where k is a constant.

The general solution of the given system:

The general solution can be expressed as

X(t) = e^(5t) × (C1 × cos(6t) + C2 × sin(6t)) × (3 + 2i, 1)

Where C1 and C2 are constants.

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a. we are 95% confident that the true proportion of college students who work to pay for tuition and living expenses is between 0.552 and 0.648.

b. we are 99% confident that the true proportion of college students who work to pay for tuition and living expenses is between 0.528 and 0.672.

a.  To find the confidence interval for the population proportion, we need to know the sample size and the proportion of the sample who work to pay for tuition and living expenses. Let's assume that a sample of 500 college students was used in the study and that 60% of them work to pay for tuition and living expenses.

Using a 95% confidence level, we can use the following formula to calculate the confidence interval:

Confidence Interval = Sample Proportion ± Margin of Error

Margin of Error = Z* √( (Sample Proportion * (1 - Sample Proportion)) / Sample Size)

Where Z* is the critical value from the standard normal distribution corresponding to the desired confidence level. For a 95% confidence level, Z* = 1.96.

Plugging in the values we have:

Margin of Error = 1.96 * √((0.6 * 0.4) / 500) = 0.048

Confidence Interval = 0.6 ± 0.048 = (0.552, 0.648)

b. To find the confidence interval for a higher confidence level of 99%, we can use the same formula, but with a different Z* value. For a 99% confidence level, Z* = 2.576.

Margin of Error = 2.576 * √((0.6 * 0.4) / 500) = 0.072

Confidence Interval = 0.6 ± 0.072 = (0.528, 0.672)

c. As the confidence level increases, the margin of error increases as well. This is because a higher confidence level requires a wider interval to capture the true population proportion with greater certainty. This wider interval results in a larger margin of error.

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The research question that could be addressed using a one-way analysis of variance is D) Does mean blood pressure differ for three different age groups?

The one-way analysis of variance is a statistical method used to determine whether there are any significant differences between the means of three or more independent groups.

A one-way analysis of variance (ANOVA) is used to test whether the means of three or more groups are significantly different from each other.

In this case, the research question is asking whether there is a difference in mean blood pressure among three different age groups. ANOVA would be an appropriate statistical test to answer this question.

Let's know each option from the data,

Option A is asking about the relationship between two variables, which could be addressed using correlation or regression analysis.

Option B is asking about proportions, which could be analyzed using chi-square tests.

Option C is asking about the variance of a variable, which could be analyzed using a test for homogeneity of variances.

Therefore,

The research question that could be addressed using a one-way analysis of variance is D) Does mean blood pressure differ for three different age groups?

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The solution for z is 0.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

Assuming the equation is:

1 + iz = 1 - iz

We can start by isolating the term with z on one side:

1 + iz = 1 - iz

2iz = 0

Divide both sides by 2i:

z = 0

Therefore, the solution for z is 0.

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Complete Question:

Solve for z in the following equation: 1-iz = -1 + iz (where i^2 = -1).

Simplify your answer as much as possible.

The expression represents the median benefit for this insurance policy is x = [250 - √(62500 + 4/c)] / 2

To determine the median benefit for this policy, we need to first understand the probability density function (PDF) associated with the policy. The PDF is a mathematical function that describes the probability of a random variable taking on a specific value. In this case, the random variable is the benefit amount provided by the insurance policy.

The cumulative distribution function (CDF) for this PDF is F(x) = cx(250 - x)/2. To find the value of x that corresponds to a cumulative probability of 0.5, we set F(x) = 0.5 and solve for x:

0.5 = cx(250 - x)/2

1 = cx(250 - x)

1/c = x(250 - x)

1/c = -x² + 250x

0 = -x² + 250x - 1/c

We can solve this quadratic equation using the quadratic formula, which gives us:

x = [250 ± √(250² - 4(-1/c)(-1))] / 2

Simplifying this expression gives us:

x = [250 ± √(62500 + 4/c)] / 2

Since we know that x cannot exceed 250, we take the negative root of the expression to get:

x = [250 - √(62500 + 4/c)] / 2

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If your data does not support your hypothesis, you may need to revise it or consider alternative explanations for the relationship between temperature and distance from the heat source.

When writing a hypothesis, it's important to start with a clear statement of the relationship you are exploring. In this case, you are investigating the relationship between temperature and distance from the heat source. Your hypothesis might be something like:

"As the distance from the heat source increases, the temperature will decrease."

To support this hypothesis, you would need to collect data that shows a clear trend in temperature as distance increases. You could create a scatterplot with distance on the x-axis and temperature on the y-axis, and look for a negative correlation between the two variables. You could also calculate a correlation coefficient, which would give you a numerical measure of the strength of the relationship.

If your data supports your hypothesis, you could then make conclusions about the relationship between temperature and distance from the heat source. For example, you could state that your data suggests that temperature decreases as distance from the heat source increases. You could also discuss the practical implications of this relationship, such as the importance of maintaining a safe distance from heat sources to prevent burns or fires.

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Answer:

x = 100

y = 44

Step-by-step explanation:

The interior angles of a quadrilateral add up to 360 degrees, so we know that when all four angles are added together, they need to equal 360.

121 + 111 + x - 31 + 2y - 29 = 360

Additionally, the angles opposite each other equals 180 degrees. This means D + F = 180 degrees, and E + G = 180 degrees.

111 + (x - 31) = 180

Subtract 111 from both sides.

x - 31 = 69

Add 31 to each side

x = 100

Plug in the value for x to check the answer

111 + (100 - 31) = 180

Then, for y, we have the same set up

121 + (2y - 29) = 180

Subtract 121 from both sides

2y - 29 = 59

Add 29 to each side

2y = 88

Divide each side by 2

y = 44

Plug in the value for y to check the answer

121 + (2*44 - 29) = 180

This means that angle F equals 59 degrees and angle G equals 69 degrees.

x = 100

y = 44

The required actual length and width of the pool is 70 feet and 35 feet respectively, and the cost of the pool is $735.

Part A:

Using the scale of 2 inches equals 7 feet, we can set up the following proportions to find the actual dimensions of the pool:

2x = 70

x = 35

For the second part, we can use the same scale to find the actual

2x = 140

x = 70

Part B:

To find the area of the pool, we can multiply the length and width:

Area = 35 feet x 70 feet = 2,450 square feet

Cost = 2,450 square feet x $0.30 per square foot = $735

Therefore, it would cost $735 to buy a cover for the pool that costs $0.30 per square foot.

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4

4 x 4 is 16 to get area. and 4 + 4 + 4 + 4 is 16 for perimeter

The absolute extreme value of the function ln (cos 3x) in interval [-π/12, π/9] is 0 at point x = 0.

Given the function is,

f(x) = ln (cos 3x), where x belongs to the closed interval [-π/12, π/9]

Differentiating the function with respect to 'x' we get,

f'(x) = (1/cos 3x)*(- sin 3x)*3 = -3tan 3x

f''(x) = -3 (sec² 3x)*3 = -9 sec² 3x

Now f'(x) = 0 gives,

-3 tan 3x = 0

tan 3x = 0

3x = ..., -π, 0, π, ....

So, x = ...., -π/3, 0, π/3, .....

So the absolute value of x which lies in [-π/12, π/9] is, x = 0.

At x = 0,

f''(0) = -9 sec² (3*0) = -9 sec²0 = -9 < 0

So at x = 0, the function has maximum value.

Max f(x) = f(0) = ln( cos 3*0) = ln 1 = 0

Hence the maximum value of function is also 0.

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The question is incomplete. Complete question will be -

"Locate and identify the absolute extreme values of the following functions:

ln(cos 3x) on [-π/12, π/9]"

For a continuous function, f and f(0) = -4 and f(10)= 1, the true value for some c in the interval (0,10) is equals the [tex]f'(c) = \frac{ 11 - (-4) }{10 - 0}[/tex] since mean value theorem applies.So, option(c) is right one.

The Mean Value Theorem is an important for determining the maximum and minimum values of a function on an interval. It is states that if a function is continuous on the interval [a, b] and differentiable on the open interval (a, b), then, there exists at least one point c∈(a,b) such that [tex]f'(c) = \frac{ f(b) - f(a)}{b - a}[/tex]. We have a function f is differentiable with f(0) = -4 and f(10) = 11.

We have to determine the true value for some c in the interval (0,10). According to mean value theorem, [tex]f'(c) = \frac{ f(b) - f(a)}{b - a}[/tex]

here, a = 0, b = 10 and f(0) = -4, f(10) = 11 so, we can write as [tex]f'(c) = \frac{ 11 -(-4) }{10 - 0}[/tex]

[tex]= \frac{ 11 + 4}{10 }[/tex] = 1.5

which is equivalent to expression present in option(c) in above figure. Hence, right option is option (c).

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Complete question:

The above figure complete the question.

The position of the mass at any time t is: u(t) = (2/12)cos(22.81t) - 5.71sin(22.81t)

The motion of the mass can be described by the following second-order linear differential equation:

mu''(t) + ku(t) = 0

where m is the mass, k is the spring constant, and u(t) is the position of the mass at time t.

We can find k from the given information about the spring:

k = F/x

where F is the force exerted by the mass and x is the displacement of the spring.

F = mg = 3 lb * 32.2 ft/[tex]s^2[/tex] = 96.6 lbft/[tex]s^2[/tex]

x = 3 in - 1 in = 2 in = 2/12 ft

k = 96.6/(2/12) = 579.6 lb/ft

Substituting these values into the differential equation, we get:

3u''(t) + 579.6u(t) = 0

The characteristic equation is:

3*r^2 + 579.6 = 0

Solving for r, we get:

r = ±sqrt(-1932)/3 = ±22.81i

The general solution is:

u(t) = c1cos(22.81t) + c2sin(22.81t)

To find the constants c1 and c2, we need to use the initial conditions. At t = 0, the mass is at its maximum displacement, so:

u(0) = c1 = 2/12 ft

u'(0) = -22.81*c1 + c2 = -2 ft/s

Solving for c1 and c2, we get:

c1 = 2/12 ft

c2 = -22.81*c1 - 2 ft/s = -5.71 ft

Therefore, the position of the mass at any time t is:

u(t) = (2/12)cos(22.81t) - 5.71sin(22.81t)

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Answer:

Step-by-step explanation:

The length of the other diagonal is 26cm. This can be calculated using the formula for a rhombus, which states that the product of the diagonals is equal to the square of the length of the sides. Therefore, 30cm * 26cm = 17cm * 17cm, so the other diagonal is 26cm.

Franco can read approximately 1.875 chapters of his history book in 1 hour, assuming his reading rate remains constant.

Franco read 3/8 of a chapter in 1/5 of an hour. To find out how many chapters he can read in 1 hour, we need to first determine his rate of reading.

To do this, we can set up a proportion:

3/8 of a chapter = x chapters
1/5 of an hour = 1 hour

Cross-multiplying, we get:

3/8 * 1 hour = x * 1/5

Simplifying:

3/8 = x/5

Cross-multiplying again:

15/8 = x

So Franco can read 15/8 or 1 7/8 chapters in 1 hour.

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For a 3×3 matrix A, with the real eigenvalue is 2 and two complex conjugate eigenvalues, the complex conjugate eigenvalues of matrix A are equal to 1 ± i.

Eigenvalues are defined as a special set of scalar points that is associated with set of linear equations and matrix equations. We have a matrix A of order 3×3. It has real and complex eigenvalues. As we know number of eigenvalues for 3×3 matrix are 3. The real eigenvalue, λ

= 2

Number of complex eigenvalues= 2

Also, the determinant of matrix A, det(A) = 50

Trace of matrix A, tr(A) = 8

We have to determine the complex eigenvalues.

The characteristic polynomial for 3×3 is written as below, f( λ )= det(A − λI3 )= −λ³ + 4λ² - 6 λ + 4.

For eigenvalues, −λ³ + 4λ² - 6 λ + 4 = 0

Now, one of eigenvalue of matrix is 2, λ = 2. Using the synthesis division, for calculating the remaining, follow the steps present in above figure. In the last step of division we get a quadratic equation, -λ² + 2λ - 2 = 0, solve it by quadratic formula, [tex]λ = \frac{-2 ± \sqrt{ 2² - 4 (-2)(-1)}}{2(-1)}[/tex]

[tex]= \frac{-2 ± \sqrt{4 - 8 }}{-2}[/tex]

=> λ = 1 ± i

Hence, required values are 1 ± i.

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To form a triangle, the sum of the lengths of any two line segments must be greater than the length of the third side. This is known as the triangle inequality theorem. For example, if we have line segments with lengths of 5, 7, and 10 units, we can add the first two lengths (5+7=12) and compare it to the length of the third side (10). Since 12 is greater than 10, we can form a triangle with these line segments.

On the other hand, if we have line segments with lengths of 3, 6, and 10 units, we can add the first two lengths (3+6=9) and compare it to the length of the third side (10). Since 9 is not greater than 10, we cannot form a triangle with these line segments.

Therefore, the set of line segments that do not meet the requirements to form a triangle are those where the sum of any two lengths is equal to or less than the length of the third side. It is important to remember the triangle inequality theorem when working with triangles, as it is a fundamental rule that determines if a set of line segments can form a triangle or not.

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The correct option is c) between 32 and 37. The box part of the box plot contains all the values between 32 and 37.

A box-and-whisker plot, also known as a box plot, is a visual representation of a set of data that shows the distribution and variability of the data. The box part of the box plot contains the middle 50% of the data, which is also known as the interquartile range (IQR).

In the given example, the box ranges from 32 to 37, which means that 50% of the data falls within this range. The line that divides the box at 36 represents the median, which is the middle value of the data set.

The whiskers, on the other hand, extend from the box to the minimum and maximum values in the data set that are not considered outliers. In this example, the whiskers range from 27 to 40, which means that the minimum and maximum values in the data set are 27 and 40 respectively.

Therefore, the answer to the question is option C, between 32 and 37, as this is the range that the box covers. The other options do not correctly represent the range of values that fall within the box.

Overall, box-and-whisker plots are useful tools for summarizing and comparing data sets, as they provide a clear and concise visual representation of the distribution and variability of the data.

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The degree of freedom is 3

A frequency table is a statistical utility that displays the spread of data by including each observed datum alongside its recurrence, expressing how often that value occurs in the given set.

To put it another way, it orders information by binding similar values and recording their respective appearances. Frequency tables are very beneficial for determining designs, trends, and exceptions within collections of data, as well as streamlining hefty quantities of information to enhance further scrutiny, such as computing possibilities or executing hypothesis examinations.

They can be conveyed through a tabular format or illustrated with histograms, bar graphs, or pies.

The frequency table is given below:

Interval              Observed frequency           Expected frequency

0-24                             33                                        25

25-49                           16                                         25

50-74                            23                                        25

75-99                            28                                       25

The value of [tex]x^2[/tex] is 3.62

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The discus thrower accelerates the discus to a speed of 8.12 m/s in a time of 0.094 s.

What is acceleration?

Acceleration is the rate at which an object changes its velocity with respect to time. In other words, it is the measure of how quickly the speed or direction of an object changes.

The final angular velocity of the discus is given by:

[tex]$\omega_f = \dfrac{1.26 \cdot 2\pi}{time\ taken}$[/tex]

The linear velocity of the discus is given by:

[tex]$v = \omega \cdot r$[/tex]

The final angular velocity of the discus is given by:

[tex]$\omega_f = \dfrac{1.26 \cdot 2\pi}{t} = 7.89 \text{ rad/s}$[/tex]

The angular acceleration of the discus is given by:

[tex]$\alpha = \dfrac{2 \cdot 1.26 \cdot 2\pi}{t^2} = 84.2 \text{ rad/s}^2$[/tex]

The time taken to reach the final angular velocity is:

[tex]$t = \dfrac{\omega_f}{\alpha} = 0.094 \text{ s}$[/tex]

Substituting the values of [tex]$\omega$[/tex] and [tex]$r$[/tex], we get:

[tex]$v = \omega_f \cdot 1.03$[/tex]

The linear velocity of the discus is given by:

[tex]$v = \omega_f \cdot r = 8.12 \text{ m/s}$[/tex]

Therefore, the discus thrower accelerates the discus to a speed of 8.12 m/s in a time of 0.094 s.

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The expected value of x is 18. We can interpret this as saying that if we were to repeat this experiment many times (tossing a fair coin 10 times), we would expect to see HH appear an average of 18 times in each sequence of 10 tosses.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

We can start by using the definition of expected value, which is the sum of all possible outcomes multiplied by their respective probabilities. In this case, we need to find the probability of getting two heads in a row (HH) in a sequence of 10 coin tosses, and then multiply it by the number of times we expect to see it (the expected value).

Let's start by finding the probability of getting HH in a sequence of two coin tosses. Since the coin is fair, there are two possible outcomes for each toss (H or T), and they are equally likely.

Therefore, the probability of getting HH is:

P(HH) = 1/2 * 1/2 = 1/4

Now, let's consider the sequence of 10 coin tosses. We can count the number of times that HH appears by counting the number of times we get two heads in a row in each possible position of the sequence. For example, if the sequence is:

T H H T H H H H T T

we can see that there are two occurrences of HH, one in the second and third positions, and one in the fifth and sixth positions.

To count the total number of occurrences of HH in a sequence of 10 coin tosses, we need to consider all possible positions of the two heads. There are nine possible positions where the first head can appear, and in each of these positions, there are eight possible positions where the second head can appear (since we don't want to count overlapping occurrences). Therefore, there are a total of 9*8 = 72 possible positions where HH can appear.

Now, we need to find the probability of getting HH in each of these positions. Since the coin tosses are independent, the probability of getting HH in any given position is the same as the probability of getting HH in two tosses (1/4). Therefore, the probability of getting HH in any of the 72 possible positions is:

P(HH) = 1/4

To find the expected value of x, we need to multiply the probability of getting HH in any given position (1/4) by the total number of possible positions (72):

E(x) = P(HH) * 72 = 1/4 * 72 = 18

Therefore, the expected value of x is 18. We can interpret this as saying that if we were to repeat this experiment many times (tossing a fair coin 10 times), we would expect to see HH appear an average of 18 times in each sequence of 10 tosses.

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