Solve the initival value problem: y′=7 cos(5x)/(8−3y)y′=7 cos⁡(5x)/(8−3y), y(0)=3y(0)=3. y=y= When solving an ODE, the solution is only valid in some interval. Furthermore, if an initial condition is given, the solution will only be valid in the largest interval in the domain of the solution that is around the xx-value given in the initial condition. In this case, since y(0)=3y(0)=3, then the solution is only valid in the largest interval in the domain of yy around x=0x=0.

Answer:

[tex]P(X=5)=(30C5)(0.2)^5 (1-0.2)^{30-5}=0.172[/tex]

Step-by-step explanation:

Assuming this complete question :"Assume that a procedure yields a binomial distribution with a trial repeated n times. Using the binomial probability formula, what is the probability of x successes given the probability p of success on a single trial? Round your answer to three decimal places.

n=30, x= 5, p=1/5"

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we know that:

[tex]X \sim Binom(n=30, p=0.2)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

And for this case if we find the probability for x=5 we got:

[tex]P(X=5)=(30C5)(0.2)^5 (1-0.2)^{30-5}=0.172[/tex]

Nickel a: 4.874

Nickel b: 4.7

The difference will be:

Higher- lower

4.874 - 4.7

0.174

So the difference is 0.174g

The difference in mass between the two nickels is 0.174 grams.

Given that:

Weight of first nickel, n = 4.7 g

Weight of second nickel, N = 4.7 g

To find the difference in mass between the two nickels, subtract the weight of one nickel from the weight of the other nickel:

Difference in mass = Weight of the second nickel - Weight of the first nickel

Let's calculate the difference:

Weight of the second nickel = 4.874 g

Weight of the first nickel = 4.7 g

Difference in mass = 4.874 g - 4.7 g

Difference in mass = 0.174 g

Hence, the difference in mass between the two nickels is 0.174 grams.

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Answer:

a) 0.259

b) 0.297

c) 0.497

Step-by-step explanation:

To solve this problem, it is important to know the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central limit theorem

The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex]

In this problem, we have that:

[tex]\mu = 11, \sigma = 3[/tex]

a. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 10.8 and 11.2 ​minutes?

Here we have that [tex]n = 25, s = \frac{3}{\sqrt{25}} = 0.6[/tex]

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11.2 - 11}{0.6}[/tex]

[tex]Z = 0.33[/tex]

[tex]Z = 0.33[/tex] has a pvalue of 0.6293.

X = 10.8

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.8 - 11}{0.6}[/tex]

[tex]Z = -0.33[/tex]

[tex]Z = -0.33[/tex] has a pvalue of 0.3707.

0.6293 - 0.3707 = 0.2586

0.259 probability, rounded to three decimal places.

b. If you select a random sample of 25 ​sessions, what is the probability that the sample mean is between 10.5 and 11 ​minutes?

Subtraction of the pvalue of Z when X = 11 subtracted by the pvalue of Z when X = 10.5. So

X = 11

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11 - 11}{0.6}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5.

X = 10.5

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.5 - 11}{0.6}[/tex]

[tex]Z = -0.83[/tex]

[tex]Z = -0.83[/tex] has a pvalue of 0.2033.

0.5 - 0.2033 = 0.2967

0.297, rounded to three decimal places.

c. If you select a random sample of 100 ​sessions, what is the probability that the sample mean is between 10.8 and 11.2 ​minutes?

Here we have that [tex]n = 100, s = \frac{3}{\sqrt{100}} = 0.3[/tex]

This probability is the pvalue of Z when X = 11.2 subtracted by the pvalue of Z when X = 10.8.

X = 11.2

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{11.2 - 11}{0.3}[/tex]

[tex]Z = 0.67[/tex]

[tex]Z = 0.67[/tex] has a pvalue of 0.7486.

X = 10.8

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{10.8 - 11}{0.3}[/tex]

[tex]Z = -0.67[/tex]

[tex]Z = -0.67[/tex] has a pvalue of 0.2514.

0.7486 - 0.2514 = 0.4972

0.497, rounded to three decimal places.

To find the probability that the sample mean is between eight minutes and 8.5 minutes, calculate the z-scores for both values and find the area under the standard normal distribution curve between these z-scores.

To find the probability that the sample mean is between eight minutes and 8.5 minutes, we need to calculate the z-scores for both values and then find the area under the standard normal distribution curve between these two z-scores.

The formula to calculate the z-score is: z = (x - mu) / (sigma / sqrt(n))

where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size.

Using the given information, we can calculate the z-scores as follows:

z1 = (8 - 11) / (3 / sqrt(25))

z2 = (8.5 - 11) / (3 / sqrt(25))

Next, we use a standard normal distribution table or a calculator to find the area between these two z-scores, which represents the probability that the sample mean is between eight minutes and 8.5 minutes.

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Answer:

(a) 0.023835

(b) 0.6281

Step-by-step explanation:

(a) The chance someone from this population will test positive is given by the percentage of people who have the virus multiplied by the change of testing positive (1 - false-negative rate) added to the percentage of people who do not have the virus multiplied by the change of testing positive (false-positive rate)

[tex]P(+) = 0.015*(1-0.002)+(1-0.015)*0.009\\P(+) = 0.023835[/tex]

(b) The probability that someone actually has the virus given that they have tested positive is determined as the probability of having the virus and testing positive divided by the probability of testing positive:

[tex]P(V|+) = \frac{ 0.015*(1-0.002)}{0.023835}\\P(V|+) = 0.6281[/tex]

Answer:

   [tex]\large\boxed{\large\boxed{slope=-2/5}}[/tex]

Explanation:

The problem is: given the points (19,−2) and (−11,10) find the slope of the line that joins them.

The slope of a line is the change in the y-coordinate over the change of the x-coordinate:

Thus:

         [tex]slope=[10-(-2)]/[-11-19]\\\\slope=12/(-30)\\\\slope=-12/30[/tex]

Simplify, dividing both numerator and denominator by 6:

            [tex]slope=-2/5\leftarrow answer[/tex]

Answer:

[tex]V=226.08\ ft^3[/tex]

Step-by-step explanation:

we know that

The volume of a cylinder is equal to

[tex]V=\pi r^{2} h[/tex]

where

r is the radius of the base of the cylinder

h is the height of the cylinder

we have

[tex]r=6\ ft\\h=2\ ft\\\pi=3.14[/tex]

substitute the given values in the formula

[tex]V=(3.14)(6)^{2}(2)\\ V=226.08\ ft^3[/tex]

The explicit formula for the sequence given by an = an-1 - 4 with a1 = 15 is an = 19 - 4n, which is derived using the general formula for an arithmetic sequence.

The question asks to derive the explicit formula for a sequence given by the recursive formula an = an-1 - 4 with the initial term a1 = 15. To find the explicit formula, we recognize this as an arithmetic sequence where the common difference (d) is -4 and the first term (a1) is 15.

The formula for the nth term of an arithmetic sequence is given by an = a1 + (n - 1)d. Substituting a1 = 15 and d = -4 into this formula gives us:

an = 15 + (n - 1)(-4)

Simplifying, we get an = 19 - 4n. This is the explicit formula for the given sequence, allowing us to find any term in the sequence without needing to calculate all the previous terms.

Since we're given the femour's length, we'll have to use the first function.

If we substitute [tex]f=49[/tex] in the expression we have

[tex]g(f)=2.56f+47.24 \implies g(49)=2.56\cdot 49+47.24=125.44+47.24=172.68[/tex]

The height of the woman with a femur length of 49 cm is 172.68 cm.

A function is a relationship between inputs where each input is related to exactly one output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

G(f) = 2.56f + 47.24

f = 49 cm

G(49) = 2.56 x 49 + 47.24

G(49) = 125.44 + 47.24

G(49) = 172.68  cm

Thus,

The height of the woman is 172.68 cm.

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Answer: 87.03

Step-by-step explanation:

The outer parts (2) of the secant line containing α and β refers to the distance travelled where there is no signal. The middle part is where there is signal presence. To get the altitude of the triangle:

Hc= 2(A/c)

To find the area;

A= 1/2(59)(60)

A= 1770

Use Pythagoras theorem to get c:

C= =√(60)^2+(59)^2

=√7081

Hc=2(1770/√7081)

   =3540√7081/7081

Solve for x using Pythagoras theorem:

x= (√44^2-Hc^2) + (√44^2-Hc^2)

where Hc= 3540√7081/7081

      =87.03

By interpreting the problem geometrically and using Pythagoras' theorem, it can be concluded that for approximately 34 miles of the journey from the city 60 miles north of the transmitter to the city 59 miles east of the transmitter will be in range of the radio signal.

This problem can be solved using geometry and the concept of a circle. If we imagine the area the radio transmitter can reach as a circle with the transmitter at the center, any point within a 44-mile radius from the transmitter can pick up its signal. Now, let's analyze the specific scenario proposed.

Firstly, the city 60 miles north is outside the signal range. However, as you drive towards the second city 59 miles east of the transmitter, you'll at some point enter the broadcast range. That's because, at the closest point, you're only about 15 miles away from the transmitter (60 miles - 44 miles), assuming you drive perpendicular to the diameter of the transmission circle.

You need to calculate the intersection of your driving path with the transmission circle. Using Pythagoras' theorem, it can be seen that for about 34 miles of your direct journey from the first city to the second city, you would be in range of the transmitter. The two cities form the hypotenuse of a right triangle, and that hypotenuse intersects the transmission circle creating a segment along which signal will be received.

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Step-by-step explanation:

Let's say S is the number of small boxes and L is the number of large boxes.

S + L = 26

7S + 13L = 254

Solve the system of equations using substitution.

7S + 13(26 − S) = 254

7S + 338 − 13S = 254

84 − 6S = 0

S = 14

L = 26 − S

L = 12

The company shipped 14 small boxes and 12 large boxes.

Answer:14 small boxes and 12 large boxes were shipped.

Step-by-step explanation:

Let x represent the number of small boxes of paper that were shipped.

Let y represent the number of large boxes of paper that were shipped.

A total of 26 boxes of paper were shipped. This means that

x + y = 26

The volume of each small box is 7 cubic feet and the volume of each large box is 13 cubic feet. The total number of boxes shipped have a combined volume of 254 cubic feet. This means that

7x + 13y = 254 - - - - - - - - - - - - 1

Substituting x = 26 - y into equation 1, it becomes

7(26 - y) + 13y = 254

182 - 7y + 13y = 254

- 7y + 13y = 254 - 182

6y = 72

y = 72/6 = 12

x = 26 - y = 26 - 12

x = 14

Answer:

(A) P (D > 0) = 99.38%

(B) P (D > 15) = 10.56%

Step-by-step explanation:

The random variable D = difference, is defined as the difference between the reading test scores after and before the program.

The random variable D follows a normal distribution with mean, [tex]\mu_{D}=10[/tex] and standard deviation, [tex]\sigma_{D}=4[/tex].

(A)

Compute the probability that the children showed any improvement, i.e.

P (D > 0):

[tex]P(D>0)=P(\frac{D-\mu_{D}}{\sigma_{D}} >\frac{0-10}{4} )=P(Z>-2.5)=P(Z<2.5)[/tex]

Use the standard normal random variable to determine the probability.

[tex]P(D>0)=P(Z<2.5)=0.9938[/tex]

The percentage of children showed any improvement is:

0.9938 × 100 = 99.38%

Thus, 99.38% of children showed improvement.

(B)

Compute the probability that the children improved by more than 15 points, i.e. P (D > 15):

[tex]P(D>15)=P(\frac{D-\mu_{D}}{\sigma_{D}} >\frac{15-10}{4} )=P(Z>1.25)=1-P(Z<1.25)[/tex]

Use the standard normal random variable to determine the probability.

[tex]P(D>0)=1-P(Z<1.25)=1-0.8944=0.1056[/tex]

The percentage of children improved by more than 15 points is:

0.1056 × 100 = 10.56%

Thus, 10.56% of children showed improvement by more than 15 points.

50% of children showed some improvement, while 10.56% improved their reading scores by more than 15 points during the summer literacy program.

The library staff is utilizing statistical analysis to assess the effectiveness of their summer literacy program. They have chosen a sample of 30 children out of the many who attended, and provided scores both before and after the program. A mean difference score of 10 and a standard deviation of 4 were determined. This question asks to find out the percentage of students who have improved based on these scores (positive score difference) and those who have improved by more than 15 points.

Firstly, we are assuming that the score differences follow a normal distribution. In a normal distribution, half of the results fall on either side of the mean. Since we are looking for an improvement, we only consider the side above the mean score difference, which is equivalent to 50% of all students.

Secondly, to find the percent of children who improved by more than 15 points, we need to calculate the z-score for the score difference of 15. Z-score is calculated as (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. So, Z = (15 - 10) / 4 = 1.25.

The z-score of 1.25 corresponds to an area of 0.8944 to the left under a standard table of normal distribution. To get the area to the right (which represents the students who improved by >15), we subtract this from 1. So, 1 - 0.8944 = 0.1056 or 10.56% students improved by more than 15 points.

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Answer:

a.[tex]x^2=-1[/tex]

b.a real number x

Step-by-step explanation:

We are given that  statement.

We have to rewrite the given statement using variable or variables.

Statement:Is there a real number whose square is -1.

a.Let x bet the real number

The square of real number x written as [tex]x^2[/tex]

According to question

[tex]x^2=-1[/tex]

Therefore,

Is there a real number x such that [tex]x^2=-1[/tex]

b.Does there exist a real number x such that

[tex]x^2=-1[/tex]

There is no real number whose square is -1. However, in the domain of complex numbers, 'i' is defined as the square root of -1. Complex numbers include both real and imaginary parts.

In the realm of real numbers, there isn't a real number whose square is -1. In the context of complex numbers, however, 'i' is defined to be the square root of -1. In other words, i2 = -1. It's important to note that complex numbers consist of a real part and an imaginary part (where 'i' is the basis for the imaginary part), and are beyond the usual scope of real numbers.

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Answer:

F is assigned the value of 15

[tex]74AF_{16} = 29871_{10}[/tex]

Step-by-step explanation:

Hexadecimal number system is base 16 and it contain the following numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

A has a value of 10

B has a value of 11

C has a value of 12

D has a value of 13

E has a value of 14

F has a value of 15

By completing the expanded notation:

[tex](7*4096) + (4*256) + (A * 16) + (F *1)\\= (7*4096) + (4*256) + (10 * 16) + (15 *1)\\= 28672 + 1024 + 160 + 15\\= 29871[/tex]

In hexadecimal notation, the letter 'F' corresponds to the decimal value 15. Therefore, when converting from hexadecimal to decimal, you would assign the value 15 to 'F'.

In hexadecimal notation, the letters A through F correspond to the decimal values 10 through 15, respectively. When converting from hexadecimal to decimal, you would replace the hexadecimal digit 'F' with its decimal equivalent. Therefore, in the hexadecimal system, the letter 'F' signifies the decimal number 15. To calculate the value of 74AF16 in decimal, you replace 'F' with 15 and compute the expression (7*4096) + (4*256) + (10*16) + (15*1).

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Answer:

240 cm²

Step-by-step explanation:

We are required to determine the surface area of the figure;

To get the area we add the are of all the surfaces;

Area of triangle;

Area = 0.5 × b × h

There are two triangles;

Therefore;

Area of the two triangles;

Area = 0.5 × 6 × 8 × 2

        = 48 cm²

Area of the rectangles;

Area of a rectangle = Length × width

Area of the first rectangle;

 = 6 cm × 8 cm

 = 48 cm²

Area of the second rectangle

  = 8 cm × 8 cm

  = 64 cm²

Area of the third rectangle

 = 10 cm × 8 cm

 = 80 cm²

The total surface area will be;

 Area = 48 cm² + 48 cm² + 64 cm² + 80 cm²

          = 240 cm²

Answer:

a) 48

b) 64

c) 80

d) 96-16h

Step-by-step explanation:

a) s(1)=112 and s(4)=256

average velocity on [1,4] = (256-112)/(4-1) = 48

b) s(1)=112 and s(3)=240

average velocity on [1,3] = (240-112)/(3-1) = 64

c) s(1)=112 and s(2)=192

average velocity on [1,2] = (192-112)/(2-1) = 80

the next one's tricky to type. watch the parentheses carefully:

d) s(1)=112 and s(1+h)= -16(1+h)^2 + 128(1+h)

average velocity on [1,1+h] =

(s(1+h) - s(1))/((1+h)-1) = (-16(1+h)^2 +128(1+h) - (112))/h

= (-16(1+2h+h^2)+128+128h - 112)/h

= ( -16 -32h -16h^2 + 16 + 128h)/h

= ( 96h - 16 h^2)/h

= 96 - 16h

Answer: 12.22

Step-by-step explanation:

Since it is a right angled triangle, we use the trigonometry method of solving triangles for this question.

The given angle is 42° and we recall our trigonometry functions of

Sin Φ = opposite/hypotenuse

Cos Φ= adjacent/hypotenuse

tan Φ = opposite/adjacent

Where

Φ =42°

Opposite of the angle = GH = 11

Adjacent of the angle = HI = ?.

Hence we use the tan Formula.

tan 42 = 11/HI

HI = 11/tan42

HI = 11/0.90

HI = 12.22

Answer:

1) 2x+7

2) -3x+11

3) 0.75x-2

4) -2x+0

5) -1.5x+2

6) -4x+16

Step-by-step explanation:

1) y = mx + c

m = 2 when x=1 , y=9

9 = 2(1)+c

c = 7

y = 2x + 7

2) m = -3

When x=4, y= -1

-1 = -3(4) + c

c = -1+12 = 11

y = -3x + 11

3) m = 0.75

When x= -4, y= -5

-5 = 0.75(-4) + c

-5 = -3 + c

c = -2

y = 0.75x - 2

4) m = (y2-y1)/(x2-x1)

m = (2-(-6))/(-1-3) = 8/-4 = -2

y = -2x + c

When x= -1, y= 2

2 = -2(-1) + c

2 = 2 + c

c = 0

y = -2x + 0

5) m = (-10-(-4))/(8-4)

m = (-10+4)/4 = -6/4 = -1.5

y = -1.5x + c

When x= 4, y= -4

-4 = -1.5(4) + c

-4 = -6 + c

c = 2

y = -1.5x + 2

6) m = (-4-4)/(5-3) = -8/2 = -4

When x= 3, y= 4

4 = -4(3) + c

4 = -12 + c

c = 16

y = -4x + 16

Answer:

1) 2x+7

2) -3x+11

3) 0.75x-2

4) -2x+0

5) -1.5x+2

6) -4x+16

Step-by-step explanation:

Answer:

[tex] Range= Max-Min= 10.2-8.7=1.5 inches[/tex]

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 0.50 inches[/tex]

[tex] s^2 = 0.503^2 = 0.25 in^2[/tex]

D. The statistics are representative because they are taken from a random sample

Step-by-step explanation:

For this case we have the following data:

9.9 8.7 10.1 9.2 9.2 9.9 10.1 9.4 9.1 9.3 10.2

The data was colledted from a random sample of people selected in 1988.

We can order the dataset on increasing way and we got:

8.7  9.1  9.2  9.2  9.3  9.4  9.9  9.9 10.1 10.1  10.2

The range is defined as [tex] Range= Max-Min= 10.2-8.7=1.5 inches[/tex]

The mean is defined as:

[tex]\bar X =\frac{\sum_{i=1}^n X_i}{n}= 9.555 inches[/tex]

The standard deviation can be calculated with the following formula:

[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]

And if we replace we got [tex] s = 0.50 inches[/tex]

The sample variance would be just the deviation squared:

[tex] s^2 = 0.503^2 = 0.25 in^2[/tex]

And since the data comes from a random sample then is representative fo the population data in 1988. So then the best answer for this case would be:

D. The statistics are representative because they are taken from a random sample

Final answer:

The range, variance, and standard deviation must be calculated from the given data, but their representativeness for the current population is not guaranteed, especially due to changes since 1988 and potential sampling issues.

Explanation:

To calculate the range, variance, and standard deviation for the provided sample data of foot lengths, first, we must find the smallest and largest values to determine the range. In this set, the smallest value is 0.1 inches, and the largest is 10.2 inches, so the range is 10.2 - 0.1 = 10.1 inches.

To find the variance and standard deviation, we need the sum of the squared deviations from the mean, divided by the number of observations minus one for the sample variance, and then the square root of the variance for the standard deviation.

The representativeness of these statistics for the current population depends on various factors, including changes in population demographics and sampling methods. A single small sample, especially with an outlying value such as 0.1, may not be indicative of the entire population's foot sizes today.

Hence, the correct answer is A: 'Since the measurements were made in 1988, they may not necessarily be representative of the population today.'

Answer:

68.26% of the data would be between 16.4 and 21.6.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 19, \sigma = 2.6[/tex]

What percentage of the data would be between 16.4 and 21.6?

This is the pvalue of Z when X = 21.6 subtracted by the pvalue of Z when X = 16.4. So

X = 21.6

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{21.6 - 19}{2.6}[/tex]

[tex]Z = 1[/tex]

[tex]Z = 1[/tex] has a pvalue of 0.8413.

X = 16.4

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{16.4 - 19}{2.6}[/tex]

[tex]Z = -1[/tex]

[tex]Z = -1[/tex] has a pvalue of 0.1587

So 0.8413 - 0.1587 = 0.6826 = 68.26% of the data would be between 16.4 and 21.6.

Answer: Percentage = 0.6826 X 100 = 68.26%

Step-by-step explanation: Please find the attached document for the step by step explanation

Answer:

A) 3/7

Step-by-step explanation:

We start by calculating the following probabilities:

P(produced by A) = 0.6

P(produced by A and defective) =  P(A ∩ def) = 0.6*0.01 = 0.006

P(produced by A and not defective) = P(A ∩ not def) = 0.6*0.99 = 0.594

P(produced by B and defective) = P(B ∩ def) = 0.4*0.02 = 0.008

P(produced by B and not defective) = P(B ∩ not def) = 0.4*0.98 = 0.392

The probability that it was produced by A given that it is defective is:

P(A|def) = P(A ∩ def) / P(def) = P(A ∩ def) / (P(A ∩ def)+P(B ∩ def)) = 0.006 / (0.006+0.008) = 6/14 = 3/7

Final answer:

The probability that a defective electronic component was produced by Plant A is 3/7. This was found using conditional probability and the information on production percentages and defect rates from both plants.

Explanation:

The question involves applying the concept of conditional probability to determine the probability that a defective electronic component was produced by Plant A. We need to calculate this using Bayes' theorem with the given probabilities for production and defect rates at both plants.

To answer the question: The probability of a component being defective from either Plant A or Plant B is calculated as follows:

P(Defective) = P(Defective | A)P(A) + P(Defective | B)P(B)
= (0.01)(0.60) + (0.02)(0.40)
= 0.006 + 0.008
= 0.014

Next, the probability that the component was produced at Plant A given that it is defective is:

P(A | Defective) = P(Defective | A)P(A) / P(Defective)
= (0.01)(0.60) / 0.014
= 0.006 / 0.014
= 3/7

Therefore, the correct answer is A. 3/7.

Answer:

50°, 105°

Step-by-step explanation:

a + 130 = 180 ( sum of angles on a straight line)

a = 180 - 130 = 50°

a + b + 45 = 180 ( sum of angles in a Δ)

30 + b + 45 = 180

75 + b = 180

b = 180 - 75 = 105°

Answer: it is verified that:

* y1 and y2 are solutions to the differential equation,

* c1 + c2t^(1/2) is not a solution.

Step-by-step explanation:

Given the differential equation

yy'' + (y')² = 0

To verify that y1 solutions to the DE, differentiate y1 twice and substitute the values of y1'' for y'', y1' for y', and y1 for y into the DE. If it is equal to 0, then it is a solution. Do this for y2 as well.

Now,

y1 = 1

y1' = 0

y'' = 0

So,

y1y1'' + (y1')² = (1)(0) + (0)² = 0

Hence, y1 is a solution.

y2 = t^(1/2)

y2' = (1/2)t^(-1/2)

y2'' = (-1/4)t^(-3/2)

So,

y2y2'' + (y2')² = t^(1/2)×(-1/4)t^(-3/2) + [(1/2)t^(-1/2)]² = (-1/4)t^(-1) + (1/4)t^(-1) = 0

Hence, y2 is a solution.

Now, for some nonzero constants, c1 and c2, suppose c1 + c2t^(1/2) is a solution, then y = c1 + c2t^(1/2) satisfies the differential equation.

Let us differentiate this twice, and verify if it satisfies the differential equation.

y = c1 + c2t^(1/2)

y' = (1/2)c2t^(-1/2)

y'' = (-1/4)c2t(-3/2)

yy'' + (y')² = [c1 + c2t^(1/2)][(-1/4)c2t(-3/2)] + [(1/2)c2t^(-1/2)]²

= (-1/4)c1c2t(-3/2) + (-1/4)(c2)²t(-3/2) + (1/4)(c2)²t^(-1)

= (-1/4)c1c2t(-3/2)

≠ 0

This clearly doesn't satisfy the differential equation, hence, it is not a solution.

Final answer:

The provided solutions y₁(t) = 1 and y₂(t) = [tex]t^{(1/2)[/tex] satisfy the given differential equation, verified through substitution and simplification. However, a linear combination of the form [tex]c_1 + c_2t^{(1/2)[/tex] is not a solution as it does not satisfy the original equation when its derivatives are substituted.

Explanation:

We are given the second-order differential equation yy'' + (y')² = 0, where y = y(t) is a function of t, and we are asked to verify solutions and understand properties of certain types of solutions to this equation.

To verify that y₁(t) = 1 is a solution, we calculate the derivatives: y₁' = 0 and y₁'' = 0. Substituting these into the differential equation yields (1)(0)+(0)² = 0, which holds true, confirming that y₁(t) = 1 is indeed a solution.

Next, to verify y₂(t) = [tex]t^{(1/2)[/tex], we find y₂' = [tex](1/2)t^{(-1/2)[/tex] and [tex]y_2'' = -(1/4)t^{(-3/2)[/tex]. Substituting these values gives [tex](t^{(1/2)})(-(1/4)t^{(-3/2)}) + ((1/2)t^{(-1/2)})^2 = 0[/tex], which simplifies to 0, showing that y₂(t) is also a solution.

For the linear combination of solutions, we consider [tex]y(t) = c_1 + c_2t^{(1/2)[/tex]. Derivatives are [tex]y' = c_2(1/2)t^{(-1/2)[/tex] and [tex]y'' = -c_2(1/4)t^{(-3/2)[/tex]. Substituting into the given differential equation does not yield zero, thus [tex]c_1 + c_2t^{(1/2)[/tex] is not a solution.

The percentage of cans expected to be rejected based on given mean and standard deviation are calculated using the Z score and standard normal distribution table. By adjusting the mean and standard deviation, the rejection rates will change accordingly.

This question is about calculating the expected rejection rate of cans based on different conditions using statistical concepts like mean and standard deviation.

(a) The Z score for 7.9 is (7.9 - 8.05) / 0.05 = -3. We use the standard normal distribution table to find the probability of a can having weight less than 7.9 ounces. That's almost 0.1% (0.001), so about 0.1% of cans are expected to be rejected.

(b) After adjusting the average weight to 8.1 oz, the Z score for 7.9 becomes (7.9 - 8.1) / 0.05 = -4. Again, find the probability in the standard normal distribution table, it is almost 0, so the rejection rate will drastically decrease.

(c) When the standard deviation is reduced to 0.03 but mean remains 8.05, the Z score becomes (7.9 - 8.05) / 0.03 = -5. The rejection rate will be extremely close to 0 as per standard normal distribution table reference.

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The partial fraction are:

a) [tex](x / (x^2 + x - 20))[/tex] = [tex]\dfrac{x}{x^2 + x - 20} = \dfrac{A}{(x - 4)} + \dfrac{B}{(x + 5)}[/tex]

b) [tex]\dfrac{x}{x^2 + x + 2} = \dfrac{A}{(x +1)} + \dfrac{B}{(x + 2)}[/tex]

(a) The partial fraction decomposition of the function [tex](x / (x^2 + x - 20))[/tex] can be written as:

[tex]\dfrac{x}{x^2 + x - 20} = \dfrac{A}{(x - 4)} + \dfrac{B}{(x + 5)}[/tex]

where A and B are constants.

(b) The partial fraction decomposition of the function [tex]\dfrac{x}{x^2 + x - 20}[/tex] can be written as:

[tex]\dfrac{x}{x^2 + x + 2} = \dfrac{A}{(x +1)} + \dfrac{B}{(x + 2)}[/tex]

where A and B are constants.

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The student is asked to perform partial fraction decomposition for two functions. In the first case, the given rational function decomposes to the form: A/(x - 4) + B/(x + 5). In the second case, the decomposition does not exist as the denominator can't be factored using real numbers.

In mathematics, the concept under discussion is the partial fraction decomposition. This is a process used in algebra to break down complex fractions or rational expressions into simpler ones. Given (a) x/(x^2 + x - 20) and (b) x^2/(x^2 + x + 2), you are being asked to perform the decomposition.

For (a), the denominator, x^2 + x - 20, can be factored as (x - 4)(x + 5), so the partial fraction decomposition would have the form: x/(x^2 + x - 20) = A/(x - 4) + B/(x + 5).

For (b), since the denominator x^2 + x + 2 can't be factored using real numbers, the partial fraction decomposition doesn't exist. Here, the answer would be DNE (Does Not Exist).

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Answer:

Elements of AxBb and BxA have been listed in the attached file

Step-by-step explanation:

The concept applied is that of binary operation and generally using the rule of combining more than one operations sign in either communitative or associative property as shown in the attachment.

Answer:

This statistic is misleading because Howard surveys only his department, and not membes of all the departments that the company has.

Step-by-step explanation:

This is a common statistics practice, when we want to study something from a population, we find a sample of this population.

However, the sample has to be representative

For example:

I want to estimate the proportion of New York state residents who are Buffalo Bills fans. So i ask, lets say, 1000 randomly selected Buffalo residents wheter they are Buffalo Bills fans, and expand this to the entire population of New York State residents. This is not representative of all New York State residents, just Buffalo residents.

In this problem, we have that:

Howard wants to know the proportion of employees of a company who use the company's healthcare. He asks only his department. However, a company as multiple departments, which leads to the statistics found in Howard's survey being misleading.

Final answer:

Howard's statistic that 48% of people in his company use the healthcare may be misleading due to potential issues like non-representative sampling, biased survey questions, response bias, and lack of current context.

Explanation:

The question about Howard reporting that 48% of the people in his company use the company's healthcare is misleading may have several underlying reasons. Firstly, the sample size and how the sample was obtained are critical factors that determine the reliability and representation of the poll. A random sample of 600 people is generally a good size, but if the sample is not representative of the entire company's demographics, the statistic could be misleading.

Another potential issue could be the phrasing of the survey question. Questions that are biased or leading can influence the way people respond and thus skew the results. In addition, respondents might have reasons to provide socially desirable answers rather than true ones, or they might misremember their actual usage of healthcare. This is an example of response bias, which can lead to inaccurate data.

Finally, it's vital to consider the context, such as whether the data is recent and if there have been any significant changes in the company's healthcare policy or overall employee demographics since the survey. All these factors can contribute to why a statistic might be considered misleading, as it may not accurately reflect the true situation.

Answer:

We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.

And if the distribution is not bell shaped or symmetric then we can't use it.

Step-by-step explanation:

Previous concepts

The empirical rule, also referred to as the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ). "Broken down, the empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ)".

Solution to the problem

We can conclude that this statement is False. Because the Empirical Rule does not apply to data sets with severely asymmetric distributions, since by definition the use of the rule is satisfied just for symmetric distributions like the normal distribution.

And if the distribution is not bell shaped or symmetric then we can't use it.

The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False. The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.

The statement 'We apply the Empirical Rule when the relative frequency distribution of the sample is not bell-shaped or symmetric' is False.

The Empirical Rule is applied when the distribution of the data is bell-shaped and symmetric.

It states that approximately 68% of the data is within one standard deviation of the mean, 95% is within two standard deviations, and more than 99% is within three standard deviations.

Therefore, the Empirical Rule is not applied when the data is not bell-shaped or symmetric.

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Answer:

Step-by-step explanation:

The area of each rectangular sheet of cardboard made by the factory is is 2 1/2 square feet. Converting

2 1/2 to improper fraction, it becomes 5/2 square feet.

Each sheet of cardboard can be cut into smaller pieces of cardboard measuring 1 1/6 square feet. Converting 1 1/6 to improper fraction, it becomes 7/6 square feet.

Therefore, the number of smaller pieces of cardboard that each sheet of cardboard provides is

5/2 ÷ 7/6 = 5/2 × 6/7 = 30/14

= 2.14 pieces

Answer: k = 200/3

Step-by-step explanation:

If a variable, a varies directly with a variable, c, it means that as a increases, c increases and as a decreases, c decreases.

Also, If a variable, a varies inversely with a variable, b, it means that as a increases, b decreases and as a decreases, c increases.

The variable c varies directly with a and inversely with b. We would introduce a constant of variation, k. Therefore

a = kc/b

If c = 3/20 when a = 2 and b =5, then

2 = (k × 3/20)/5 = 3k/100

Cross multiplying, it becomes

3k = 100 × 2 = 200

k = 200/3

Answer: k=3/8 c=3/40

Step-by-step explanation:

just took the assignment on edg

Answer:

The answer is a. 50%

Explanation:

Beverages (which include energy drinks, fruit drinks, sweetened coffee and tea, soft drinks, energy drinks, alcoholic beverages, etc.) are the major source of added sugars that are being consumed by the population of the United States: it accounts for almost half (around 50%) of added sugars consumed.

Beverages account for about 75% of the added sugars consumed in the U.S.

Beverages account for about 75% of the added sugars consumed in the U.S. This includes soda, fruit drinks, sports drinks, and energy drinks. These beverages are often high in sugar and can contribute to weight gain and other health issues.

It's important to be mindful of our consumption of sugary beverages and choose healthier alternatives like water, unsweetened tea, or low-sugar options.

Therefore, The correct answer is d. 75%.

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