thirty-six students took an exam on which the average was 76 and the standard deviation was 5 . the instructor announces that the distribution is not bell-shaped. what proportion of the students scored within 3 standard deviations of the mean?

Answer:

Step-by-step explanation:

The domain of the set of ordered pairs include the following: {1, 2, 3, 4}.

In Mathematics and Geometry, a domain refers to the set of all real numbers (x-values) for which a particular function (equation) is defined.

In Mathematics and Geometry, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the set of ordered pairs above, we can reasonably and logically deduce the following domain and range:

Domain = {1, 2, 3, 4}.

Range = {3, 4, 5}.

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Therefore, the function f(x) = ax + b is one-to-one, and its inverse function is given by: f1(y) = (y - b)/a.

To determine if the function f(x) = ax + b is one-to-one, we need to show that it passes the horizontal line test. That is, for any horizontal line y = k, the function intersects the line at most once.
To do this, suppose that f(x1) = f(x2), where x1 and x2 are two distinct values in the domain. Then we have:
a x₁ + b = a x2 + b
Subtracting b from both sides gives:
a x₁ = a x₂
Since a ≠ 0, we can divide both sides by a to get:
x₁ = x₂
Therefore, the function f(x) = ax + b is one-to-one, and its inverse function is given by:
f1(y) = (y - b)/a

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Thus, the total cost of the first 64 units is approximately $2730.67.

To find the total cost of the first 64 units, we need to integrate the marginal cost function over the range of production from x=0 to x=64.

The marginal cost function is given by C'(x) = 8√x.

Integrating this function with respect to x, we get:
C(x) = ∫(8√x dx) = 8 * (2/3)x^(3/2) + C

To find the total cost for the first 64 units, we need to evaluate C(x) at x=64 and x=0 and subtract the results:
C(64) - C(0) = (8 * (2/3) * 64^(3/2) + C) - (8 * (2/3) * 0^(3/2) + C)

Simplifying the equation, we get:
C(64) - C(0) = 8 * (2/3) * 64^(3/2)

Now, compute the value:
C(64) - C(0) = 8 * (2/3) * 512 = (16/3) * 512 ≈ 2730.67

So, the total cost of the first 64 units is approximately $2730.67.

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The function notation for the volume of the box is V(a) = (L-2a)(W-2a)(a). However, we need to know the dimensions of the paper in order to determine the volume of the box when the cutout length is 0.8 inches.

To represent the volume of the box (in cubic inches) using function notation, we can use V(a) where "a" represents the cutout length (in inches). To determine the volume of the box when the cutout length is 0.8 inches, we simply substitute 0.8 for "a" in the function V(a). However, we need to know the dimensions of the paper in order to determine the function itself.

Let's assume that the length of the paper is "L" inches and the width is "W" inches. When squares of length "x" are cut out from each corner, the length of the box will be L-2x and the width will be W-2x. The height of the box will be x inches. Therefore, the volume of the box will be V(a) = (L-2a)(W-2a)(a). Since we don't know the dimensions of the paper, we cannot determine the exact value of V(0.8).

So, the volume of the box when the cutout length is 0.8 inches is represented by the function f(0.8) = (L - 1.6)(W - 1.6)(0.8) in cubic inches.

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The theory behind Euler's equation and the boundary conditions for a column with one fixed end and the other end free. Therefore, the answer to the question is d) 1/2, as x = π/2L = (2(1) - 1)π/2L = (2n - 1)π/2L when n = 1/2.

Euler's equation is derived from the Euler-Bernoulli beam theory, which states that a slender column under axial compression will buckle when the compressive stress exceeds a certain critical value. The buckling load is given by the equation P= xt^2π^2Et, where P is the buckling load, x is a dimensionless factor called the slenderness ratio (the ratio of the column length to its cross-sectional dimensions), t is the thickness of the column wall, E is the modulus of elasticity of the column material, and π is the mathematical constant pi.

For a column with both ends pinned, the value of x is given by x = nπ/L, where n is an integer and L is the length of the column. For a column with one end fixed and the other end free, the value of x is given by x = (2n - 1)π/2L, where n is an integer.  In this case, we have a column with one fixed end and the other end free, so we need to use the equation x = (2n - 1)π/2L to find the value of x. Since n can be any integer, we can choose n = 1 to simplify the equation and get x = π/2L.

Substituting this value of x into Euler's equation, we get P = (π/2L)²π²Et = π²Et/4L². This means that the buckling load for a column with one fixed end and the other end free is proportional to the modulus of elasticity and inversely proportional to the square of the length of the column.

Therefore, the answer to the question is d) 1/2, as x = π/2L = (2(1) - 1)π/2L = (2n - 1)π/2L when n = 1/2

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Answer: The area is 83.6

Step-by-step explanation:

7.6 x 11 = 83.6

No, there would not be a collision if the x-coordinate of the second particle is given by x2 = 3 cos(t) instead of x2 = -3 + cos(t).

This is because the x-coordinate of the first particle, x1, has a maximum value of 3 and a minimum value of -3. The x-coordinate of the second particle, x2, also has a maximum value of 3 and a minimum value of -4.

Since the maximum value of x2 is now 3 instead of -3, the two particles can no longer collide.

To confirm this, we can set the x-coordinates of the two particles equal to each other and solve for t. If the resulting values of t are within the interval 0 ≤ t ≤ 2π, then a collision occurs.

However, when we set 3sin(t) = 3cos(t), we get tan(t) = 1, which gives t = π/4 or 5π/4. These values of t are not within the interval 0 ≤ t ≤ 2π, so there is no collision.

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a) The greatest test score is 96.

c) 75% percent of the scores were between 71 and 96

a) The greatest test score is 96.

b) The number that arranges all numbers from large to small in the middle is called the median, and Some numbers. may have more than one of the same the numbers, so the median is not in the middle of the box.

C. 75% ( the upper limit of the box is the upper quartile and the lower quartile is the lower quartile)

d)  Half of the scores were higher than the median. From the box plot, we can see that the median is approximately 84, so half of the scores were higher than 84.

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The required measure of interior angles 1 and 2 are 116° and 62°.

Here,
According to the property of the triangle sum of the remote interior angle is equal to the remote interior triangle.
∠1 + 21 = 137
∠1 = 137 - 21
∠1 = 116

Similarly,
∠1 = ∠2 + 54
116 = ∠2 + 54
∠2 = 116 - 54
∠2 = 62°

Thus, the required measure of interior angles 1 and 2 are 116° and 62°.

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The number of ways you serve up a coffee cup and saucer is 15 ways

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

Colors = red, orange, green, light blue, dark blue, and yellow

So, we have

Colors = 6

To serve up a coffee cup and saucer, we have

n = 6

r = 2

The number of ways is calculated as

Ways = 6C2

Using the combination formula, we have

Ways = 6!/(4! * 2!)

Evaluate

Ways = 15

Hence, the number of ways is 15

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The formula for the volume of a sphere is:

V = (4/3)πr³

where D is the diameter, which is twice the radius (r). So we can calculate the radius as:

r = D/2 = 9.2mm/2 = 4.6mm

Now we can substitute the value of r into the formula for the volume:

V = (4/3)πr³ = (4/3)π(4.6mm)³ = 487.9mm³

Rounding this answer to the nearest tenth gives:

V ≈ 487.9mm³ ≈ 487.8mm³

Therefore, the volume of the sphere is approximately 487.8 cubic millimeters.

Answer:

f(-2)=31

Step-by-step explanation:

f(x)=-x²-8x+19

f(-2)=-(-2)²-8(-2)+19

=-(-2*-2)+16+19

=-4+35

=31

f(-2)=31√

The value of x as required to be determined in the given task content is; 3.

It follows from the task content that the value of x is required to be determined in the given task content.

By observation; the triangles formed by the parallel lines and the common vertex they share are similar triangles.

On this note, the ratio of their corresponding sides are equal and hence; we have that;

15 / (15 + x) = 10 / (10 + 2)

(15 × 12) = 10 (15 + x)

180 - 150 = 10x

30 = 10x

x = 3.

Consequently, it follows that the value of x as required is; 3.

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Answer:

3.8%

Step-by-step explanation:

The correct arrangement of the areas of the figures from the least to the greatest is Y < X < Z.

The area of the figures is calculated as follows;

area of the triangle;

Area = ¹/₂ x base x height

Area = ¹/₂ x 14 m x 22.5 m

Area = 157.5 m²

area of the circle is calculated as follows;

Area = πr²

where;

Area = π x ( 7 m )²

Area = 153.94 m²

The area of the parallelogram is calculated as follows;

Area = base x height

Area = 15.5 m x 10.9 m

Area = 168.95 m²

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The correct graph of amounts of water plot the measurements on a line plot is shown in image.

We have to given that;

The amounts of water plot the measurements on a line plot are,

1/8 oz, 1/8 oz, 1/4 oz, 1./4 oz, 1/4 oz, 1/2 oz, 1/2 oz , 1/2 oz, 3/4 oz, 1 oz

Now, We have;

1/8 oz is repeats two times.

1/4 oz is repeats three times.

1/2 oz is repeats two times.

3/4 oz is only one times

And, 1 oz is repeats one time.

Thus, The correct graph of amounts of water plot the measurements on a line plot is shown in image.

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The wave has:

Period = 3.5 seconds

Frequency = 1/3.5 = 0.29

Amplitude = 42 - 26 = 16 cm

We have,

Period:

The period of a wave refers to the time it takes for one complete cycle of the wave to occur.

T = 1 / f where f represents the frequency.

Frequency:

The frequency of a wave represents the number of complete cycles or oscillations of the wave that occur in a given time period.

f = 1 / T where T represents the period.

Amplitude:

The amplitude of a wave refers to the maximum displacement or distance from the equilibrium position of a particle in the wave.

Now,

The wave has:

Period = 3.5 seconds

Frequency = 1/3.5 = 0.29

Amplitude = 42 - 26 = 16 cm

Thus,

Period = 3.5 seconds

Frequency = 1/3.5 = 0.29

Amplitude = 42 - 26 = 16 cm

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To find the equation of the line passing through the points (-4, -3) and (-4, 6),

we note that the x-coordinate of both points is the same, which means the line is vertical and parallel to the y-axis. In this case, the equation of the line can be written as x = a, where 'a' is the x-coordinate of any point on the line.

Since both points have an x-coordinate of -4, the equation of the line passing through them is x = -4.

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Answer:

x=-4

Step-by-step explanation:

The general equation is y=mx+b where m is the slope and b is the y intercept.

Notice that there are 2 different y coordinates (-3 and 6) for the same x (-4) coordinate!

Slope = rise/run = (y2-y1)/(x2-x1) = (-3-6)/(-4--4) = -9/0 = there's NO slope, you cannot divide by zero!

So the equation is just x=-4.

See attached screenshot.

In the given circle, the measure of arc SM is 106°

From the question, we are to determine the measure of arc SM.

First, we will determine the measure of angle QSM

m ∠QSM = 37° (Angles in the same segment)

Now,

Let the center of the circle be O

Thus,

OS and OM are radii

Therefore,

m ∠OSM = m ∠OMS

m ∠OSM = 37°

But,

Measure of arc SM = m ∠SOM

Now, we will determine the measure of angle SOM

m ∠SOM = 180° - 37° - 37°

m ∠SOM = 106°

Hence, measure of arc SM is 106°

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Therefore, the length of the curve for t ∈ [0, 4] is 4√13.

To find the length of the curve defined by x(t) = 2t and y(t) = 3t - 1 for t ∈ [0, 4], we can use the arc length formula:

L = ∫[a,b] √[x'(t)^2 + y'(t)^2] dt

where x'(t) and y'(t) are the derivatives of x(t) and y(t) with respect to t.

Let's calculate the derivatives first:

x'(t) = d/dt (2t) = 2

y'(t) = d/dt (3t - 1) = 3

Now, we can calculate the integrand:

√[x'(t)^2 + y'(t)^2] = √[(2)^2 + (3)^2] = √[4 + 9] = √13

Substituting the integrand into the arc length formula and integrating with respect to t from 0 to 4:

L = ∫[0,4] √13 dt

= √13 ∫[0,4] dt

= √13 [t] from 0 to 4

= √13 (4 - 0)

= 4√13

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We would employ the technique of comparing proportions from dependent samples to draw conclusions about whether the proportion of students who have known a cancer survivor is representative of all students at the university.

The presented scenario compares the proportion of university students who have knowledge about cancer survivors to the percentage of all university students. The proper procedure for drawing conclusions would be to compare proportions from dependent samples because the same set of students is being polled (dependent samples).

In order to do this research, the study would gather information from a random sample of students and calculate the percentage of those students who knew a cancer survivor. This percentage would be contrasted with the anticipated percentage of all university students who had known a cancer survivor. We may draw conclusions about the total student body at the university by using statistical tests, such as the McNemar's test, to see if the observed proportion in the sample differs significantly from the expected proportion.

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Complete Question: A University of Florida study asks a random sample of students if they have ever known someone that was a cancer survivor. We want to extend the results to all students at the university. In this problem, we want to make inferences about:

a. comparing proportions from dependent samples

b. comparing proportions from 2 independent samples

c. comparing means from 2 independent samples

d. one mean

e. comparing means from dependent samples

f. one proportion

As the size of the house increases, the monthly cost to heat the house with natural gas also increases.Positive, meaning as the size of the house increases, the heating cost also increases.

Based on the data provided, it appears that there is a positive correlation between house size (x) and heating cost (y). As the size of the house increases, the monthly cost to heat the house with natural gas also increases.

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A 1992 study of bull terriers found the following: (i) 50% of the studied bull terriers are white. (ii) 11% of the studied bull terriers are deaf. (iii) 20% of the white bull terriers are deaf. The probability that a randomly chosen bull terrier is white and deaf is 0.1, or 10%.

To find the probability that a randomly chosen bull terrier is white and deaf, we can use the given information from the study:
(i) 50% of the studied bull terriers are white (P(White) = 0.5)
(iii) 20% of the white bull terriers are deaf (P(Deaf|White) = 0.2)
Now, we can apply the conditional probability formula to find the probability of a bull terrier being both white and deaf:
P(White and Deaf) = P(Deaf|White) * P(White)
P(White and Deaf) = 0.2 * 0.5
P(White and Deaf) = 0.1

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The area of the triangle is approximately 27.71 square units.

We can use the formula for the area of a triangle given two sides and the included angle:

Area = 1/2 * |u| * |v| * sin(theta)

where |u| and |v| are the magnitudes of the vectors, and theta is the angle between them.

First, we can find the magnitude of each vector:

Next, we can find the vector difference ~u - ~v:

~u - ~v = (3-6, 3-0, 3-6) = (-3, 3, -3)

Then, we can find the magnitude of ~u - ~v:

|~u - ~v| = √((-3)² + 3² + (-3)²) = 3√(2)

Now, we can find the angle between ~u and ~v using the dot product:

theta [tex]= cos^{-1(2\sqrt(6))}[/tex] ≈ 30.96 degrees

Finally, we can plug in the values to find the area:

Therefor, the area is ≈ 27.71 square units.

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Answer: The value of x is 150.

Option (c) x= 150 is correct

Step-by-step explanation:

Given l1 ║ l2

let ∠1 =30°  

∠1 = ∠2 = 30° [Corresponding angle]

Then , ∠2 + ∠3 = 180° [Linear pair]

30° + x° = 180°

x° = 180° - 30°

∴ x = 150°

You have an SRS of six observations from a Normally distributed population, the correct answer is (c) 2.015

To obtain an 80% confidence interval for the mean µ of a Normally distributed population with a small sample size (n<30), we need to use a t-distribution with n-1 degrees of freedom. In this case, since we have an SRS of six observations, our degrees of freedom are 6-1=5. To determine the critical value for an 80% confidence interval using a t-distribution with 5 degrees of freedom, we can use a t-table or a calculator. Using a t-table, we would find the row corresponding to 5 degrees of freedom and the column for a two-tailed test with an area of 0.10 (80% divided by 2). The intersection of this row and column gives us a critical value of 2.015. Therefore, the correct answer is (c) 2.015. Alternatively, we could use a calculator that has a t-distribution function. In this case, we would enter a confidence level of 0.80, a degree of freedom of 5, and ask the calculator to output the critical value. This would also give us a critical value of 2.015.

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If a and b are mutually exclusive, then P(A ∩ B) = 0. Therefore, we have:

P(~B ∪ ~C) = P(~B) + P(~C) - P(~B ∩ ~C)

= P(B') + P(C') - P(B' ∩ C')

= 1 - P(B) + 1 - P(C) - [1 - P(B ∪ C)]

= 2 - P(B) - P(C) - P(B ∪ C)

= 2 - 0.4 - 0.5 - P(B ∪ C)

= 1.1 - P(B ∪ C)

Also, we know that:

P(A ∪ C) = P(A) + P(C) - P(A ∩ C)

0.6 = 0.2 + 0.5 - P(A ∩ C)

P(A ∩ C) = 0.1

Now, we can use the inclusion-exclusion principle to find P(A ∪ B ∪ C):

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

Since A and B are mutually exclusive, P(A ∩ B) = 0, and we have:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

We can write P(A ∩ B ∩ C) as:

P(A ∩ B ∩ C) = P(A) - P(A ∩ B) + P(B) - P(A ∩ B) + P(C) - P(A ∪ B ∪ C)

Since A and B are mutually exclusive, we have P(A ∩ B) = 0, and we can write:

P(A ∩ B ∩ C) = P(A) + P(B) + P(C) - 2P(A ∪ B ∪ C)

Substituting this into the equation for P(A ∪ B ∪ C), we get:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ C) - P(B ∩ C) + P(A) + P(B) + P(C) - 2P(A ∪ B ∪ C)

= 2P(A) + 2P(B) + 2P(C) - P(A ∩ C) - P(B ∩ C) - 2P(A ∪ B ∪ C)

We can rewrite P(~B ∪ ~C) as :

P(~B ∪ ~C) = P((B ∩ C)')

= 1 - P(B ∩ C)

Substituting this into the equation for P(A ∪ B ∪ C), we get:

P(A ∪ B ∪ C) = 2P(A) + 2P(B) + 2P(C) - P(A ∩ C) - P(B ∩ C) - 2[1.1 - P(~B ∪ ~C)]

= 2P(A) + 2P(B) + 2P(C) - P(A ∩ C) - P(B ∩ C) - 2.2 + 2P(B ∪ C)

= 2P(A) + 4P(B ∪ C) + 2P(C) - P(A

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We can use the error bound formula for the midpoint rule to find the smallest value of N for which the error is less than 10^-9:

Error ≤ K(b-a)^3/(12N^2) where K is the maximum value of the absolute value of the second derivative of f on the interval [a,b]. In this case, we have f(x) = x^(4/3) and we need to integrate from 1 to 2.

First, we find the second derivative of f:

f''(x) = (4/3)(1/3)x^(-2/3)

To find the maximum value of the absolute value of the second derivative on [1,2], we evaluate it at the endpoints and at critical points in the interval. Since the second derivative is decreasing on the interval, its maximum value occurs at the left endpoint, x=1:

|f''(1)| = (4/3)(1/3)(1)^(-2/3) = 1.5874

Next, we need to choose N such that the error bound is less than 10^-9:

K(b-a)^3/(12N^2) ≤ 10^-9

Plugging in the values we have:

(1.5874)(2-1)^3/(12N^2) ≤ 10^-9

Solving for N:

N^2 ≥ (1.5874)(2-1)^3/(12(10^-9))

N^2 ≥ 1.3245×10^9

N ≥ √(1.3245×10^9)

N ≥ 36413.89Since N must be an integer, we round up to get:N = 36414

Therefore the smallest value of N for which Error(SN) 10^-9 is 36414.

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We can use the error bound formula for the midpoint rule to find the smallest value of N for which the error is less than 10^-9:

Error ≤ K(b-a)^3/(12N^2) where K is the maximum value of the absolute value of the second derivative of f on the interval [a,b]. In this case, we have f(x) = x^(4/3) and we need to integrate from 1 to 2.

First, we find the second derivative of f:

f''(x) = (4/3)(1/3)x^(-2/3)

To find the maximum value of the absolute value of the second derivative on [1,2], we evaluate it at the endpoints and at critical points in the interval. Since the second derivative is decreasing on the interval, its maximum value occurs at the left endpoint, x=1:

|f''(1)| = (4/3)(1/3)(1)^(-2/3) = 1.5874

Next, we need to choose N such that the error bound is less than 10^-9:

K(b-a)^3/(12N^2) ≤ 10^-9

Plugging in the values we have:

(1.5874)(2-1)^3/(12N^2) ≤ 10^-9

Solving for N:

N^2 ≥ (1.5874)(2-1)^3/(12(10^-9))

N^2 ≥ 1.3245×10^9

N ≥ √(1.3245×10^9)

N ≥ 36413.89Since N must be an integer, we round up to get:N = 36414

Therefore the smallest value of N for which Error(SN) 10^-9 is 36414.

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Answer: The graph of f(x) = x^2 should be shifted upward by 1 unit to produce the graph of g(x) = x^2 + 1.

Explanation: The function g(x) = x^2 + 1 is obtained by adding 1 to the function f(x) = x^2. This means that for any value of x, the value of g(x) is 1 unit greater than the value of f(x). Graphically, this corresponds to shifting the entire graph of f(x) upward by 1 unit. As a result, the graph of g(x) will be identical to the graph of f(x), but shifted upward by 1 unit.