Lisa put tulips and roses into vases. Each vase has 12 flowers. The red vase has 7 tulips. the blue vase has twice as mnay roses as the red vase. How many roses are in the blue vase?

Answer:

∫∫∫1 dV=4\sqrt{3}π

Step-by-step explanation:

From Exercise we have  

z=6-x^{2}-y^{2}

z=x^{2}+y^{2}

we get

2z=6

z=3

x^{2}+y^{2}=3

We use the polar coordinates, we get

x=r cosθ

y=r sinθ

x^{2}+y^{2}&=r^{2}

r^{2}=3

We get at the limits of the variables that well need for our integral

x^{2}+y^{2}≤z≤3

0≤r ≤\sqrt{3}

0≤θ≤2π

Therefore, we get a triple integral

\int \int \int 1\, dV&=\int \int \left(\int_{x^2+y^2}^{3} 1\, dz\right) dA

=\int \int \left(z|_{x^2+y^2}^{3} \right) dA

=\int \int\ \left(3-(x^2+y^2) \right) dA

=\int \int\ \left(3-r^2 \right) dA

=\int_{0}^{2\pi}\int_{0}^{\sqrt{3}} (3-r^2) dr dθ

=3\int_{0}^{2\pi}\int_{0}^{\sqrt{3}}  1 dr dθ-\int_{0}^{2\pi}\int_{0}^{\sqrt{3}} r^2 dr dθ

=3\int_{0}^{2\pi} r|_{0}^{\sqrt{3}}  dθ-\int_{0}^{2\pi} \frac{r^3}{3}|_{0}^{\sqrt{3}}dθ

=3\sqrt{3}\int_{0}^{2\pi} 1 dθ-\sqrt{3}\int_{0}^{2\pi} 1 dθ

=3\sqrt{3} ·2π-\sqrt{3}·2π

=4\sqrt{3}π

We get

∫∫∫1 dV=4\sqrt{3}π

We find the volume of the region D bounded by the paraboloids z = 6 - x² - y² and z = x² + y² by setting up triple iterated integrals. Six different integrals are presented, and one is evaluated using cylindrical coordinates. The volume is determined to be 9π.

To find the volume of the region D bounded by the paraboloids z = 6 - x² - y² and z = x² + y², we need to set up triple iterated integrals.

The intersection of the two surfaces occurs when 6 - x² - y² = x² + y²,

which simplifies to 6 = 2(x² + y²) or x² + y² = 3, defining a circle of radius √3 in the xy-plane.

Possible Triple Iterated Integrals

Here are six different triple iterated integrals to find the volume of the region D:

Evaluating One of the Integrals

[tex]\int_{0}^{2\pi} \int_{0}^{\sqrt{3}} \int_{r^2}^{6 - r^2} r \, dz \, dr \, d\theta[/tex]

[tex]\int_{r^2}^{6 - r^2}\, dz = \left[ z \right]_{z=r^2}^{z=6-r^2}[/tex]

[tex]= (6-r^2) - (r^2)[/tex]

[tex]= 6-2r^2[/tex]

[tex]int_{0}^{\sqrt{3}} r(6 - 2r^2) dr = \int_{0}^{\sqrt{3}} (6r - 2r^3) dr[/tex]

[tex]= \left[ 3r^2 - \frac{1}{2}r^4 \right]_{r=0}^{r= \sqrt{3}}[/tex]

[tex]= \left[ 3(3) - \frac{1}{2}(9) \right][/tex]

= 9 - 4.5

= 4.5

[tex]\int_{0}^{2\pi} 4.5 \, d\theta = 4.5 \cdot 2\pi = 9\pi[/tex]

So the volume of the region D is 9π.

Answer:

1

Step-by-step explanation:

For a quadratic equation, the roots are expressed by the quadratic formula.

x=(-b+/- Sqrt[b^2-4ac])/2a

In this case a=6, b=-7 and c=k

 So,

x=(7 +/- √[(-7)^2-4(6)(k)]/2(6))

Simplifying gives:

x=(7 +/- √[49-24k])/12

For k=0 the square root simplifies to √[49]=7 which yields roots of 7/6 and 0

For k=1 the square root simplifies to √[49-24]=√[25]=5 which yields roots of 1 and 1/6

For k=2 the square root simplifies to √[49-48]=√[1]=1 which yields roots of 2/3 and 1/2

k= 1 as other roots are fractions

Answer:

k=-5

Step-by-step explanation:

6x^2+7x+k=0 is a quadratic equation.

a= 6; b=7; c=k

Let the roots of the equation be R1 and R2

R1+R2 = -b/a = -7/6 ---------1

R1xR2 = c/a =k/6 or (R1xR2)6=k------------2

From Equation 1:

R1=-7/6-R2

We know 2R1+3R2=-4 Substituting for R1, we get

3 (-7/6-R2)+3R2=-4

R2=-5/3

R1=-7/6-(-5/3)= 1/2

Substituting these values in Eq2,

k= (-5/3 x 1/2) 6

k=-5

Answer:

a) There is a 10.9375% probability that their first child will have green eyes and the second will not.

b) There is a 21.875% probability that exactly one of their two children will have green eyes.

c) There is a 13.74% probability that exactly two will have green eyes.

d) There is a 55.12% probability that at least one will have green eyes.

Step-by-step explanation:

In this problem, the binomial probability distribution is going to be important for itens b,c and d.

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

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

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

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

And p is the probability of X happening.

We have these following probabilities:

0.75 probability of having children with brown eyes, 0.125 probability of having children with blue eyes, and 0.125 probability of having children with green eyes.

(a) What is the probability that their first child will have green eyes and the second will not?

There is a 0.125 probability a child will have green eyes and an 1-0.125 = 0.875 probability a child will not have green eyes.

So

0.125*0.875 = 0.109375

There is a 10.9375% probability that their first child will have green eyes and the second will not.

(b) What is the probability that exactly one of their two children will have green eyes?

Here we use the binomial probability distribution, with [tex]n = 2, p = 0.125[/tex].

We want P(X = 1).

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

[tex]P(X = 1) = C_{2,1}*(0.125)^{1}*(0.875)^{1} = 0.21875[/tex]

There is a 21.875% probability that exactly one of their two children will have green eyes.

(c) If they have six children, what is the probability that exactly two will have green eyes?

Again the binomial probability distribution, with [tex]n = 6, p = 0.125[/tex]

We want P(X = 2)

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

[tex]P(X = 2) = C_{6,2}*(0.125)^{2}*(0.875)^{4} = 0.1374[/tex]

There is a 13.74% probability that exactly two will have green eyes.

(d) If they have six children, what is the probability that at least one will have green eyes?

[tex]n = 6, p = 0.125[/tex]

Either none has green eyes, or at least one has. The sum of the probabilities of these events is decimal 1. So

[tex]P(X = 0) + P(X \geq 1) = 1[/tex]

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

In which

[tex]P(X = 0) = C_{6,0}*(0.125)^{0}*(0.875)^{6} = 0.4488[/tex]

Finally

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.4488 = 0.5512[/tex]

There is a 55.12% probability that at least one will have green eyes.

The probability that their first child will have green eyes and the second will not is 0.109375. The probability that exactly one of their two children will have green eyes is 0.21875. If they have six children, the probability that exactly two will have green eyes is 0.19140625. The probability that at least one of the six children will have green eyes is 0.6499367.

Given that the parents have a probability of 0.125 of having a child with green eyes, the probability of the first child having green eyes is 0.125.

The probability that the second child does not have green eyes is 1 - 0.125 = 0.875.

Therefore, the probability that the first child has green eyes and the second child does not is 0.125 * 0.875 = 0.109375.

There are two possible scenarios: (1) the first child has green eyes but not the second child, or (2) the first child does not have green eyes but the second child does.

The probability of the first scenario is the same as in part (a), which is 0.109375.

The probability of the second scenario is also 0.109375.

The total probability is the sum of the probabilities of the two scenarios, which is 0.109375 + 0.109375 = 0.21875.

This can be calculated using the binomial probability formula.

The probability of having two children with green eyes and four children without green eyes is:

P(2 green, 4 not green) = C(6, 2) * (0.125)^2 * (0.875)^4 = 0.19140625

The probability that none of the six children have green eyes is (1 - 0.125)^6 = 0.3500633.

Therefore, the probability that at least one child will have green eyes is 1 - 0.3500633 = 0.6499367.

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Answer:

The solution is (6.2,6.2)

Step-by-step explanation:

we have

[tex]0.4x+0.6y=6.2[/tex] ----> equation A

For variable x and y equal

[tex]x=y[/tex] ----> equation B

Solve the system by substitution

substitute equation B in equation A

[tex]0.4y+0.6y=6.2[/tex]

solve for y

combine like terms

[tex]y=6.2[/tex]

so

[tex]x=6.2[/tex]

therefore

The solution is (6.2,6.2)

Let p be the true proportion of  registered voters wish to see Mayor Waffleskate defeated.

As per given , we have

[tex]H_0: p\leq0.27\\\\ H_a: p >0.27[/tex]

Sample size : n= 447

Number of of registered voters wish to see Mayor Waffleskate defeated = 157

I.e. sample proportion :  [tex]\hat{p}=\dfrac{157}{447}\approx0.3512[/tex]

Confidence interval for population proportion is given by :-

[tex]\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

, where n= sample size

[tex]\hat{p}[/tex] = sample proportion

z* = critical z-value.

Critical z-value for 98% confidence interval is 2.33.  (By z-table)

Then, the 98% confidence interval for the proportion of registered voters who wish to see Waffleskate defeated will be :

[tex]0.3512\pm2.33\sqrt{\dfrac{0.3512(1-0.3512)}{447}}\\\\=0.3512\pm (2.33)(0.022577656)\\\\=0.3512\pm 0.05260593848\\\\=(0.3512-0.05260593848,\ 0.3512+0.05260593848)\\\\=(0.29859406152,\ 0.40380593848)\approx(0.299,\ 0.404)[/tex]

Since the 0.27 < 0.299 , it means 0.27 does not belong to the above confidence interval.

So , we reject the null hypothesis ([tex]H_0[/tex]).

So ,  98% confidence interval does not support the claim.

Answer:

Reword the question so that is balanced or increase the sample size so that more people respond to the question.

Step-by-step explanation:

Given that the owner of a shopping  mall wishes to expand the number of shops available in the food court. He has a market researcher survey the first 120 customers who come into the food court during weekend morning to determine what types of food the shoppers would like to see added to the food court.

This has sample bias.

Because first 120 customers may not represent the entire population of all customers. There may be bias in selecting the weekend morning.

So this is sampling bias.

Remedy would be

Reword the question so that is balanced or increase the sample size so that more people respond to the question.

Final answer:

The cause of bias is sampling bias. The best remedy is to increase the sample size.

Explanation:

The cause of bias in the given scenario is sampling bias.

The best remedy to this problem would be to increase the sample size so that more people respond to the question. This will help in making the sample more representative of the population and reduce the bias.

Other possible remedies could include asking customers throughout the day on both weekdays and weekends, or rewording the question so that it is balanced.

Answer:

b. 0.1216

Step-by-step explanation:

Given that a sample of  15 from a normal population yields a sample mean of 43 and a sample standard deviation of 4.7.

We have to check the p value for the claim that mean <45

[tex]H_0: \mu =45\\H_a: \mu <45[/tex]

(Left tailed test for population mean)

Sample size n = 15

Sample mean = 45

Sample std dev s = 4.7

Since sample std deviation is being used, we use t test only

Std error of mean = [tex]\frac{s}{\sqrt{n} } \\=1.214[/tex]

Mean difference = 43-45 = -2

t statistic = mean difference/std error

= -1.176

df = n-1 = 14

p value = 0.1216

Answer:

[k*(k+1)*(2*k+1)] / 6

Step-by-step explanation:

We have balls numbered as: 1, 2, 3, ... , k with probabilities as: c, 2*c, 3*c, ... , k*c

Let Y be the discrete random variable defined as: Y = ball number

We know that Expected value of discrete Random Variable is:

E[X] =  Σ₁ⁿ xₐ*f(xₐ)            ,where f(xₐ) is probability of xₐ

then,

E[Y] = 1*c + 2*2*c + 3*3*c + ... + k*k*c

E[Y] = c*(1 + 2*2 + 3*3 + ... + k*k)

E[Y] = c*(1^2 + 2^2 + 3^2 + ... + k^2)

consider c = 1  (because it's constant so you can suppose any you wish)

E[Y] = 1^2 + 2^2 + 3^2 + ... + k^2

using formula of first n squares natural numbers (as attached picture)

E[Y] = [k*(k+1)*(2*k+1)] / 6

Complete Question :

  The question is shown on the first uploaded image

Answer:

The solution is the second uploaded image

Step-by-step explanation:

The step by step explanation is on the third, fourth and fifth uploaded image

Answer:

The point is [tex](-\frac{9}{2},\frac{9\sqrt{3}}{2},3)[/tex]  in rectangular coordinates.

Step-by-step explanation:

To convert from cylindrical to rectangular coordinates we use the relations

[tex]x=r \cdot cos(\theta)\\y=r\cdot sin(\theta)\\z=z[/tex]

To convert the point [tex](9,\frac{2}{3}\pi ,3)[/tex] from cylindrical to rectangular coordinates we use the above relations

Since [tex]r=9[/tex], [tex]\theta=\frac{2}{3} \pi[/tex], and [tex]z=3[/tex],

[tex]x=r \cdot cos(\theta)=9\cdot cos(\frac{2}{3}\pi )=-\frac{9}{2}[/tex]

[tex]y=r\cdot sin(\theta)=9\cdot sin(\frac{2}{3} \pi )=\frac{9\sqrt{3}}{2}[/tex]

[tex]z=z=3[/tex]

Thus, the point is [tex](-\frac{9}{2},\frac{9\sqrt{3}}{2},3)[/tex]  in rectangular coordinates.

Answer: 3, 4,5 and 17, 15,8

give the measures of the legs and hypotenuse of a right triangle.

Step-by-step explanation:

In order for the measures of the legs and hypotenuse given to form a right angle triangle, they must be Pythagorean triples. A Pythagoras triple is a set of numbers that perfectly satisfy the Pythagorean theorem. The Pythagorean theorem is expressed as

Hypotenuse² = opposite side² + adjacent side². We will apply the theorem to each set of numbers given.

1) 3, 4, 5

5² = 3² + 4² = 9 + 16

25 = 25

It is a Pythagorean triple

2) 4, 11, 14

14² = 11² + 4² = 121 + 16

196 = 137

It is a Pythagorean triple

3) 9, 14, 17

17² = 14² + 9² = 196 + 81

289 = 277

It is not a Pythagorean triple

4) 8, 14, 16

16² = 14² + 8² = 196 + 64

256 = 260

It is not a Pythagorean triple

5) 8, 15 , 17

17² = 15² + 8²

289 = 225 + 64

289 = 289

It is a Pythagorean triple

Therefore, 3, 4,5 and 17, 15,8

give the measures of the legs and hypotenuse of a right triangle.

Answer: [tex]x^{2} = 20y[/tex]

Step-by-step explanation:

The directrix given is vertical , so we will use the formula :

[tex](x-h)^{2}=4p(y-k)[/tex]

P is the distance between the focus , that is 5 - 0 = 5

Therefore : p = 5

(h,k) is the mid point between the focus and the directrix , that is

(h,k) = [tex](\frac{x_{1}+x_{2} }{2},\frac{y_{2}+y_{1}}{2})[/tex] = [tex](\frac{0+0}{2} , \frac{5-5}{2})[/tex] = [tex](0,0)[/tex]

Therefore:

h =0

k = 0

substituting into the formula : we have

[tex](x-h)^{2}=4p(y-k)[/tex]

[tex](x-0)^{2}[/tex] = 4(5)([tex]y-0)[/tex]

[tex]x^{2} = 20y[/tex]

Therefore : the equation in vertex form is [tex]x^{2} = 20y[/tex]

Answer:

[tex] W= 34.3 \frac{kg}{s^2} (4^2-0^2)m^2 =548.8 \frac{kg m^2}{s^2} =548.8 J[/tex]

Step-by-step explanation:

Data Given: m = 70 kg , g = 9.8 ms^-2, h =10m.

For this case we can use the following formula:

[tex] W = \int_{x_i}^{x_f} F(x) dx[/tex]

For this case we need to find an expression for the force in terms of the distance. And since on this case the total distance is 10 m long we can write the expression like this:

[tex] F(x) = \frac{ma}{10m}= \frac{mg}{10m} x[/tex]

The only acceleration on this case is the gravity and if we replace the values given we got:

[tex] \frac{70 kg *9.8 m/s^2}{10m} x=68.6 x\frac{kg}{s^2}[/tex]

Now we can find the required work with the following integral:

[tex] W= 68.6 \frac{kg}{s^2} \int_{0}^4 x dx[/tex]

[tex] W= 34.3 \frac{kg}{s^2} x^2 \Big|_0^4[/tex]

[tex] W= 34.3 \frac{kg}{s^2} (4^2-0^2)m^2 =548.8 \frac{kg m^2}{s^2} =548.8 J[/tex]

The amount of work that is required to raise one end of the chain is 548.8 Joules.

Given the following data:

To calculate the amount of work that is required to raise one end of the chain:

We would solve for the magnitude of the force acting on the chain with respect to the distance and this is given by this expression:

[tex]Force = \frac{mgx}{10} \\\\Force = \frac{70 \times 9.8 \times x}{10}[/tex]

Force = 68.6x Newton.

Now, we can calculate the amount of work by using this formula:

[tex]W=\int\limits^{x_2}_{x_1} F({x}) \, dx \\\\W= 68.6 \int\limits^{4}_{0} x \, dx\\\\W= 34.3 x^2 |^4_0\\\\W=34.3 [4^2 -0^]\\\\W=34.3 \times 16[/tex]

W = 548.8 Joules.

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Answer: C.  [tex]0.75^4[/tex]

Step-by-step explanation:

Let x be the binomial variable that denotes the number of makes.

Since each throw is independent from the other throw , so we can say it follows Binomial distribution .

So [tex]X\sim Bin(n=4 , p=0.75)[/tex]

Binomial distribution formula: The probability of getting x success in n trials :

[tex]P(X=x)=^nC_xp^n(1-p)^{n-x}[/tex] , where p = probability of getting success in each trial.

Then, the probability of Michael Beasley making all of his next 4 free throw attempts will be :

[tex]P(X=4)=^4C_4(0.75)^4(1-0.75)^{0}[/tex]

[tex]=(1)(0.75)^4(1)\ \ [\because\ ^nC_n=1]\\\\=(0.75)^4[/tex]

Thus, the probability of Michael Beasley making all of his next 4 free throw attempts is [tex]=0.75^4[/tex]

Hence, the correct answer is C.  [tex]0.75^4[/tex].

Answer:

A. Attachment

Step-by-step explanation:

The powerful survival impulse that leads infants to seek closeness to their caregivers is called Attachment. The infant can count on the caregiver possibly parent(s) for care which gives the infant a solid foundation for dependence and survival.

Final answer:

The instinctual behavior that drives infants to seek closeness with their caregivers is known as A) attachment. It's fostered by reflexes that ensure physical contact and is crucial for an infant's survival, ensuring they receive the necessary care, protection, and opportunity to develop securely.

Explanation:

The powerful survival impulse that leads infants to seek closeness to their caregivers is called A) attachment. This is an intrinsic part of human development and is crucial for the infant's survival. Infants have a set of innate behaviors and reflexes that promote closeness and contact with their caregivers, such as the Moro reflex and the grasping reflex, which help the infant to hold onto the caregiver and thus reduce the risk of falling.

Additionally, behaviors such as crying and the sucking reflex are instinctive methods for infants to express needs and receive care. Furthermore, the rooting reflex is an instinctive behavior that helps the infant find the nipple to feed by touching. John Bowlby's evolutionary theory underscores the importance of attachment by suggesting that the ability to maintain proximity to an attachment figure would have increased the chances of an infant surviving to reproductive age.

Attachments are not just reactions to the provision of food and warmth by the caregivers but are biological imperatives that ensure an infant remains close to those who provide security, learning, and protection, thereby enhancing their chance of survival.

It takes 10.3 years for your bank balance to double from $1,000 to $2,000.

Given that,

The interest rate is 7%.

Principal amount = $1000

Final amount = $2000

Used the formula for the time,

A = P (1 + r)ⁿ

Where, A = Final amount

P = Principal amount

r = interest rate

n = number of years

Substitute all the values,

2000 = 1000 (1 + 0.07)ⁿ

2000/1000 = (1.07)ⁿ

2 = (1.07)ⁿ

Take natural logs on both sides,

ln 2 = n ln (1.07)

0.69 = n × 0.067

n = 0.69/0.067

n = 10.3 years

Therefore, the time for your bank balance to double from $1,000 to $2,000 is 10.3 years.

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It will take approximately 10.1351 years for your bank balance to double from $1,000 to $2,000 at an interest rate of 7%.

To calculate the number of years it will take for your bank balance to double from $1,000 to $2,000 at an interest rate of 7%, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A is the final amount (in this case, $2,000),

P is the initial principal (in this case, $1,000),

r is the annual interest rate (7% or 0.07 as a decimal),

n is the number of times interest is compounded per year (we'll assume it's compounded annually),

and t is the number of years.

We can rearrange the formula to solve for t:

t = (log(A/P) / log(1 + r/n)) / n

Plugging in the values:

A = $2,000

P = $1,000

r = 0.07

n = 1 (since it's compounded annually)

t = (log(2,000/1,000) / log(1 + 0.07/1)) / 1

Simplifying the expression:

t = (log(2) / log(1.07)) / 1

Using a calculator to evaluate the logarithms:

t ≈ (0.3010 / 0.0296) / 1

t ≈ 10.1351 / 1

t ≈ 10.1351

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Answer:

Step-by-step explanation:

Given that two random variables X and Y are independent. Each has a binomial distribution with success probability 0.4 and 2 trials.

When x and y are independent joint probability would be product of individual probabilities

pdf of X

X is Binom (2,0.4)

and Y is Binomi (2,0.4)

Hence joint distribution of XY would be

P(X=x, Y=y) =[tex]2Cx (0.4)^x (0.6)^{2-x} *2Cy (0.4)^y (0.6)^{2-y}[/tex]

for x=0,1,2 and y =0,1,2

b) Joint probability using table

PDF of X is

X        0            1           2

p       0.36    0.48      0.16

and same for Y also

Joint prob would be

X  Y       0            1              2

0      0.1296     0.1728      0.0576

1       0.1728      0.2304     0.0768

2      0.0576     0.0768     0.0256

Joint probability distribution function are used to represent the probability of multiply variables

The joint probability distribution function is [tex]f(x,y) = ^2C_x *0.4^x * 0.6^{2- x} *^2C_y * 0.4^y * 0.6^{2- y}[/tex]

The given parameters are:

[tex]p = 0.4[/tex] --- the probability of success

[tex]n = 2[/tex] ----the number of trials

The joint probability distribution function f(x,y) is calculated as:

[tex]f(x,y) = ^nC_x * p^x * (1 -p)^{n- x} *^nC_y * p^y * (1 -p)^{n- y}\\[/tex]

So, we have:

[tex]f(x,y) = ^2C_x *0.4^x * (1 -0.4)^{2- x} *^2C_y * 0.4^y * (1 -0.4)^{2- y}[/tex]

Evaluate the differences

[tex]f(x,y) = ^2C_x *0.4^x * 0.6^{2- x} *^2C_y * 0.4^y * 0.6^{2- y}[/tex]

The above represents the joint probability distribution function f(x,y)

When x = 0, y = 0;

We have:

[tex]f(0,0) = 0.130[/tex]

When x = 0, y = 1;

We have:

[tex]f(0,1) = 0.173[/tex]

When x = 0, y = 2;

We have:

[tex]f(0,2) = 0.058[/tex]

When x = 1, y = 0;

We have:

[tex]f(1,0) = 0.173[/tex]

When x = 1, y = 1;

We have:

[tex]f(1,1) = 0.230[/tex]

When x = 1, y = 2;

We have:

[tex]f(1,2) = 0.077[/tex]

When x = 2, y = 0;

We have:

[tex]f(2,0) = 0.058[/tex]

When x = 2, y = 1;

We have:

[tex]f(2,1) = 0.077[/tex]

When x = 2, y = 2;

We have:

[tex]f(2,2) = 0.026[/tex]

So, the joint probability as a table is:

X /Y       0            1              2

0      0.1296     0.1728      0.0576

1       0.1728      0.2304     0.0768

2      0.0576     0.0768     0.0256

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Answer:b. 0.22

Step-by-step explanation:

Since the lengths of adult walleye fishes are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = lengths of walleye fishes.

µ = mean length

σ = standard deviation

From the information given,

µ = 44 cm

σ = 4 cm

We want to find the probability or fraction of fishes that are greater than 41 cm in length. It is expressed as

P(x > 41) = 1 - P(x ≤ 41)

For x = 41,

z = (41 - 44)/4 = - 0.75

Looking at the normal distribution table, the probability corresponding to the z score is 0.22

To find the fraction of fish that are greater than 41 cm in length, calculate the z-score with the mean and standard deviation.

To find the fraction of fish that are greater than 41 cm in length, we need to calculate the z-score of 41 cm using the mean and standard deviation. The z-score formula is z = (x - μ) / σ. Plugging in the values, we have z = (41 - 44) / 4 = -0.75. We can then look up the corresponding value on the z-table to find the fraction of fish with a length greater than 41 cm, which is approximately 0.7734. Therefore, the answer is option d, 0.75.

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Answer:

Explanation:

1. Given vector:

2. y-component

The y-component may be determined using the sine ratio, the angle from the x-axys (counterclockwise direction), and the magnitude of the vector.

Answer:

c = 6*e^(-10) -  e^(-5)  ( ≈ -e⁻⁵ = -6.74*10⁻³)

Step-by-step explanation:

for the function

y=c*e^(−2x)+e^(−x)

as a solution of y'+2y=e^(−x)

then for  y(x=−5)=6

6 =c*e^(−2(-5))+e^(−(-5)) = c*e^10 + e^5

6 = c*e^10 + e^5

c = (6 -  e^5)/*e^10 = 6*e^(-10) -  e^(-5)

c = 6*e^(-10) -  e^(-5)  ( ≈ -e⁻⁵ = -6.74*10⁻³)

Answer:

a) Alpha

Step-by-step explanation:

The correct option is alpha because alpha known as type I error is the probability of reject the null hypothesis when null  hypothesis is true. If we take alpha 5%, it means that we are taking 5 out of 100 chance of rejecting the null hypothesis when it is true.

So, statistician controls alpha by establishing the risk of rejecting a null hypothesis when its true.

Step-by-step explanation:

We can not exactly predict the values of mean and median of the data un till and unless we know about the skewness of the data.

Skewness represents the asymmetry or tapering in  the distribution of data  sample. If skewness is

Negative skew: median > mean:

Positive skew: mean > median :

Although this generalization is not always true.

I think the answer is D. hours because its variation does not depend on another variable

Answer:

Option B

Step-by-step explanation:

The data constitutes nominal scale of measurement when the observations can be classified into groups. For example, students are classified into groups on the basis of eye color. The numerical values can also be use in nominal scale for grouping. For example, the students can be categorize into 1,2 and 3  if they have brown, black and green eye color. But they have no numerical significance. Thus, data obtained from a nominal scale can be either numeric or non-numeric.

Answer: 13 outcomes

Step-by-step explanation:

Given:

She wants to get a gift from the following choices;

- one of five books

- one of four ties

- one of four X-box

Since, the three groups of choices are joined with "OR" but not "AND" that means she is getting just one gift from any of the 3 groups.

Total number of gift she needed = 1

Total number of choices = 3 groups with total of 13 options

N = 13P1 = 13!/(13-1)! = 13!/12! =13

N = 13 outcomes.

Answer:

There is a 10.9375% probability that their first child will have green eyes and the second will not.

Step-by-step explanation:

We have these following probabilities:

0.75 probability of having children with brown eyes, 0.125 probability of having children with blue eyes, and 0.125 probability of having children with green eyes.

(a) What is the probability that their first child will have green eyes and the second will not?

There is a 0.125 probability a child will have green eyes and an 1-0.125 = 0.875 probability a child will not have green eyes.

So

0.125*0.875 = 0.109375

There is a 10.9375% probability that their first child will have green eyes and the second will not.

Answer:

[tex](x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 16[/tex]

Step-by-step explanation:

The general equation of a sphere is as follows:

[tex](x - x_{c})^{2} + (y - y_{c})^{2} + (z - z_{c})^{2} = r^{2}[/tex]

In which the center is [tex](x_{c}, y_{c}, z_{c})[/tex], and r is the radius.

In this problem, we have that:

[tex]x_{c} = 5, y_{c} = 4, z_{c} = 1[/tex]

So

[tex](x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = r^{2}[/tex]

Interior contained in the first octant:

The first octant is bounded by:

The xy plane, in which z is 0. The distance from the center of the sphere to the xy plane is 9.

The xz plane, in which y is 0. The distance from the center of the sphere to the xz plane is 4.

The yz plane, in which x is 0. The distance from the center of the sphere to the yz plane is 5.

This means that if the radius is higher than four, the sphere will cross into a different octant.

So the radius for the largest sphere is 4.

The equation is

[tex](x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 4^{2}[/tex]

[tex](x - 5)^{2} + (y - 4)^{2} + (z - 9)^{2} = 16[/tex]

Answer:

a) s = 8.81

b) Q1 = 43.4, Q2 = 49.4, Q3 = 57.35

c) See below

Step-by-step explanation:

(a) Find the standard deviation of the reflectance for these smolts. (Round your answer to two decimal places.)

In order to find the standard deviation, we need the mean first. The mean is defined as

[tex]\bar x=\displaystyle\frac{\displaystyle\sum_{i=1}^{n}x_i}{n}[/tex]

where the [tex]x_i[/tex] are the values of the data collected and n=50 the size of the sample.

So, the mean is

[tex]\bar x=50.882[/tex]

Now, the standard deviation of the sample is defined as  

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

and we have that our standard deviation is

s = 8.81

(b) Find the quartiles of the reflectance for these smolts

To find the quartiles, we must sort the data from lowest to largest:  

33.6,  37.7,  38.3,  38.7,  42,  42,  42.2,  42.4,  42.7,  42.8,  42.9,  43.1,  43.7,  43.8,  44.2,  44.6,  45.5,  45.7,  46.3,  46.6,  47.4,  47.6,  47.7,  47.9,  49.2,  49.6,  50.5,  50.9,  51.3,  53,  53.9,  54.8,  54.9,  55.5,  56,  56,  56.2,  57.1,  57.6,  58.3,  59,  59.5,  60.4,  63.2,  63.4,  63.5,  64.6,  68,  69.1,  69.2

The first quartile is the number between the 12th and the 13th data (so 25% of the data are below it and 75% above it)

So the 1st quartile is

[tex]Q_1=\displaystyle\frac{43.1+43.7}{2}=43.4[/tex]

The 2nd quartile is the median, the point between the 25th and 26th data, it splits the data in two halves.

[tex]Q_2=\displaystyle\frac{49.2+49.6}{2}=49.4[/tex]

The 3rd quartile is the point between the 38th and 39th data (so 75% of the data are below it and 25% above it)

[tex]Q_3=\displaystyle\frac{57.1+57.6}{2}=57.35[/tex]

(c) Do you prefer the standard deviation or the quartiles as a measure of spread for these data? Give reasons for your preference.

In this case, we prefer the quartiles as a measure of spread since the data are very scattered around the mean and there is no a central tendency.

Answer:

not

Step-by-step explanation:

[tex]\left[\begin{array}{ccc}-2&4\\1&3\end{array}\right] *\left[\begin{array}{ccc}-2&1\\3&7\end{array}\right]=[/tex]

First is A and Second is B

Let's find A*B

[tex]\left[\begin{array}{ccc}-2(-2)+4*3&-2*1+4*7\\1(-2)+3*3&1*1+3*7\end{array}\right] =\left[\begin{array}{ccc}16&26\\7&22\end{array}\right][/tex]

b)

[tex]\left[\begin{array}{ccc}-2&1\\3&7\end{array}\right] \left[\begin{array}{ccc}-2&4\\1&3\end{array}\right] =[/tex]

Now let's find B*A

[tex]\left[\begin{array}{ccc}-2(-2)+1*1&-2*4+1*3\\3(-2)+7*1&3*4+7*3\end{array}\right] =\left[\begin{array}{ccc}5&-5\\1&23\end{array}\right][/tex]

c) They are not