Suppose a student carrying a flu virus returns to an isolated college campus of 2000 students. Determine a differential equation governing the number of students x(t) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between students with the flu and students who have not yet contracted it. (Use k > 0 for the constant of proportionality and x for x(t).)

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

Answer:

[tex]P(junior)=\frac{9}{35}[/tex]

[tex]P(freshmen)=\frac{5}{35}=\frac{1}{7}[/tex]

Step-by-step explanation:

In a class there are 13 seniors, 9 juniors, 8 sophomores and 5 freshmen

Total number of students= [tex]13+9+8+5=35[/tex]

one student is selected at random from this class, selected student is Junior

there 9 juniors in the class

[tex]P(junior)= \frac{juniors}{total} =\frac{9}{35}[/tex]

there 5 freshmen in the class

[tex]P(freshmen)= \frac{freshmen}{total} =\frac{5}{35}=\frac{1}{7}[/tex]

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:

[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:

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:

D

Step-by-step explanation:

The correlation coefficient r=-0.84 denotes that there is inverse relationship between x and y. It means that as the x values increase the y values decrease whereas as the x values decreases the y-values increases. Also, r=-0.84 denotes the strong relationship between  x and y because it is close to 1. So, r=-0.84 denotes that there is strong linear relationship between x and y with smaller x values tending to be associated with larger y values.

The correlation coefficient of -0.84 indicates a strong inverse relationship between x and y, with smaller x values generally corresponding to larger y values.

Based on the given correlation coefficient of -0.84, the correct answer is option d. This option states that there is a strong linear relationship between variables x and y, with smaller x values tending to be associated with larger values of the y variable.

A correlation coefficient communicates both the strength and direction of a linear relationship between two variables. In this context, a coefficient of -0.84 indicates a strong relationship (values close to -1 or 1 denote strong relationships), and because the value is negative, it reflects an inverse or negative correlation, meaning y tends to decrease as x increases, and vice versa.

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

Answer:

[tex]\frac{a(450-8a^2)}{4}[/tex]

is the volume in terms of side a

Step-by-step explanation:

Given that you I built a storage shed in the shape of a rectangular box on a square base. The material that I used for the base cost $4 per square foot, the material for the roof cost $2 per square foot, and the material for the sides cost $2.50 per square foot; and I spent $450 altogether on material for the shed.

Let a be the side of square and h be the height

Total cost of materials = cost for floor + cost for roof + cost for sides

= area of floor (4) + lateral area (2.50)+roof area (2)

[tex]=4a^2+4ah+4a^2\\=8a^2+4ah = 450[/tex][tex]h = \frac{450-8a^2}{4a}[/tex]

Now coming to volume

Volume = V = lbh = [tex]a^2 h\\= a^2*\frac{450-8a^2}{4a} \\=\frac{a(450-8a^2)}{4}[/tex]

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:

36

Step-by-step explanation: thats what i got

Answer:the speed of postman from A to B is 6 km/hr

Step-by-step explanation:

Let x represent the speed of the postman when moving from A to B.

From point A, the postman delivered a letter to point B in half hour.

Distance = speed × time

It means that

Distance from A to B = 0.5 × x = 0.5x

In back way he reduced speed by 1km/h and gets back in 36 min. It means that his speed on returning back would be (x - 1)km/h

Converting 36 minutes to hour, it becomes

36/60 = 0.6 hours

Distance from B to A = 0.6(x - 1)

Since distance from A to B = distance from B to A, then

0.5x = 0.6(x - 1) = 0.6x - 0.6

0.6x - 0.5x = 0.6

0.1x = 0.6

x = 0.6/0.1 = 6

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.

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: [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:

a) [tex] P(X \leq 100) = 1- e^{-0.01342*100} =0.7387[/tex]

[tex] P(X \leq 200) = 1- e^{-0.01342*200} =0.9317[/tex]

[tex] P(100\leq X \leq 200) = [1- e^{-0.01342*200}]-[1- e^{-0.01342*100}] =0.1930[/tex]

b) [tex] P(X>223.547) = 1-P(X\leq 223.547) = 1-[1- e^{-0.01342*223.547}]=0.0498[/tex]

c) [tex] m = \frac{ln(0.5)}{-0.01342}=51.65[/tex]

d) [tex] a = \frac{ln(0.05)}{-0.01342}=223.23[/tex]

Step-by-step explanation:

Previous  concepts

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

[tex]P(X=x)=\lambda e^{-\lambda x}[/tex]

Solution to the problem

For this case we have that X is represented by the following distribution:

[tex] X\sim Exp (\lambda=0.01342)[/tex]

Is important to remember that th cumulative distribution for X is given by:

[tex] F(X) =P(X \leq x) = 1-e^{-\lambda x}[/tex]

Part a

For this case we want this probability:

[tex] P(X \leq 100)[/tex]

And using the cumulative distribution function we have this:

[tex] P(X \leq 100) = 1- e^{-0.01342*100} =0.7387[/tex]

[tex] P(X \leq 200) = 1- e^{-0.01342*200} =0.9317[/tex]

[tex] P(100\leq X \leq 200) = [1- e^{-0.01342*200}]-[1- e^{-0.01342*100}] =0.1930[/tex]

Part b

Since we want the probability that the man exceeds the mean by more than 2 deviations

For this case the mean is given by:

[tex] \mu = \frac{1}{\lambda}=\frac{1}{0.01342}= 74.516[/tex]

And by properties the deviation is the same value [tex] \sigma = 74.516[/tex]

So then 2 deviations correspond to 2*74.516=149.03

And the want this probability:

[tex] P(X > 74.516+149.03) = P(X>223.547)[/tex]

And we can find this probability using the complement rule:

[tex] P(X>223.547) = 1-P(X\leq 223.547) = 1-[1- e^{-0.01342*223.547}]=0.0498[/tex]

Part c

For the median we need to find a value of m such that:

[tex] P(X \leq m) = 0.5[/tex]

If we use the cumulative distribution function we got:

[tex] 1-e^{-0.01342 m} =0.5[/tex]

And if we solve for m we got this:

[tex] 0.5 = e^{-0.01342 m}[/tex]

If we apply natural log on both sides we got:

[tex] ln(0.5) = -0.01342 m[/tex]

[tex] m = \frac{ln(0.5)}{-0.01342}=51.65[/tex]

Part d

For this case we have this equation:

[tex] P(X\leq a) = 0.95[/tex]

If we apply the cumulative distribution function we got:

[tex] 1-e^{-0.01342*a} =0.95[/tex]

If w solve for a we can do this:

[tex] 0.05= e^{-0.01342 a}[/tex]

Using natural log on btoh sides we got:

[tex] ln(0.05) = -0.01342 a[/tex]

[tex] a = \frac{ln(0.05)}{-0.01342}=223.23[/tex]

The question involves applying the exponential distribution formula to calculate certain probabilities and expectations involving the distances travelled by kangaroo rats. You need to calculate these by using the formula, P(X≤x) = 1 - e^(-λx), and using that the standard deviation of an exponential distribution is the reciprocal of the parameter. Median and the distance that only 5% will exceed can be calculated by setting the P(X≤a) = 0.50 and 0.95, respectively.

This question surrounds the concepts within probability theory and specifically the exponential distribution.

Firstly, understand that an exponential distribution can be described by the formula: P(X≤x) = 1 - e^(-λx). Given λ = 0.01342, you can solve for Part (a), calculate the probabilities for distances at most 100m, 200m, and between 100 and 200m. Plug the distances into the formula and calculate.

For part (b), you need to know that the standard deviation of an exponential distribution is the reciprocal of the parameter (1/λ). Calculate the mean and standard deviation and use these values to find the required probability.

For Part (c), set P(X≤ a) = 0.50 and solve for 'a', this will give you the median.

For Part (d), set P(X≤ a) = 0.95 and solve for 'a', this will give you the distance that only 5% of animals will exceed.

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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:

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. 0.1216

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

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: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: 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:

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

Final answer:

To answer the student's combinatorics question, there are a total of 8008 potential delegations, 84 of which are all men and 7924 that include at least one woman.

Explanation:

The question involves combinatorics, which is a branch of mathematics. In particular, we are dealing with combinations since the order of selection does not matter for the delegation.

Number of possible delegations:

To find the total number of delegations possible, we must select 6 individuals out of 16 (7 women + 9 men) without regard to order. This is done using the combination formula:

C(n, k) = n! / (k!(n-k)!)

So, C(16, 6) = 16! / (6!(16-6)!) = 8008

Delegations with all men:

To find the number of delegations with all men, we select 6 men out of the 9 available. Using the combination formula again:

C(9, 6) = 9! / (6!(9-6)!) = 84

Delegations with at least one woman:

To find this, we subtract the number of all-male delegations from the total number of delegations:

8008 - 84 = 7924

So, there are 7924 delegations that include at least one woman.

Final answer:

To answer the student's question about the total number of possible delegations, all male delegations, and delegations with at least one woman, combinations are used. There are 8008 total possible delegations, 84 all-male delegations, and 7924 delegations with at least one woman.

Explanation:

To solve this problem, we will use combinatorics, specifically the concept of combinations as the order in which the delegation members are selected does not matter.

a) Total number of delegations possible

The total number of ways to choose 6 members from a board of 16 (7 women + 9 men) is calculated using the combination formula C(n, k) = n!/(k!(n-k)!), where n is the total number of items, and k is the number of items to choose.

Therefore, C(16, 6) = 16!/(6!*(16-6)!) = 8008 different delegations are possible.

b) Delegations with all men

To find the number of all-male delegations, we choose 6 men from a group of 9, which is C(9, 6) = 9!/(6!*(9-6)!) = 84 delegations.

c) Delegations with at least one woman

Instead of calculating this directly, we use the complement principle. We subtract the number of all-male delegations from the total number of delegations. Thus, the number of delegations with at least one woman is 8008 - 84 = 7924 delegations.

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:

13

Step-by-step explanation:

The given expression is (3-2i)(3-2i)

This is of the form (a-b)(a+b)

This is equivalent to [tex]\[a^{2}-b^{2}\][/tex], where a = 3 and b=2i

= [tex]\[3^{2}-(2i)^{2}\][/tex]

= [tex]\[9-4i^{2}\][/tex]

But [tex]\[i^{2} = -1\][/tex]

= [tex]\[9-4*(-1)\][/tex]

= [tex]\[9+4\][/tex]

= [tex]\[13\][/tex]

Hence the value of the expression (3-2i)(3-2i)  is 13.

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:

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:

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)

The radius of the circumscribed circle is [tex]\( 3.2\sqrt{3} \)[/tex] centimeters.
The length of a side of the hexagon is [tex]\( 3.2\sqrt{3} \)[/tex] centimeters.
The perimeter of the hexagon is [tex]\( 19.2\sqrt{3} \)[/tex] centimeters.

Let's solve each part of the problem step by step:

a. **Find the radius of the circumscribed circle:**

The apothem of a regular polygon and the radius of the circumscribed circle are related by the formula:

[tex]\[ \text{Radius} = \frac{\text{Apothem}}{\cos(180^\circ / \text{number of sides})} \][/tex]

For a regular hexagon (6 sides), the formula becomes:

[tex]\[ \text{Radius} = \frac{4.8}{\cos(180^\circ / 6)} \][/tex]

[tex]\[ \text{Radius} = \frac{4.8}{\cos(30^\circ)} \][/tex]

[tex]\[ \text{Radius} = \frac{4.8}{\frac{\sqrt{3}}{2}} \][/tex]

[tex]\[ \text{Radius} = \frac{4.8 \times 2}{\sqrt{3}} \][/tex]

[tex]\[ \text{Radius} = \frac{9.6}{\sqrt{3}} \][/tex]

[tex]\[ \text{Radius} = \frac{9.6\sqrt{3}}{3} \][/tex]

[tex]\[ \text{Radius} = 3.2\sqrt{3} \][/tex]

b. **What is the length of a side of the hexagon?**

The length of a side of a regular hexagon can be found using the formula:

[tex]\[ \text{Side Length} = 2 \times \text{Apothem} \times \tan(180^\circ / \text{number of sides}) \]\\[/tex]

For a regular hexagon with apothem 4.8 cm:

[tex]\[ \text{Side Length} = 2 \times 4.8 \times \tan(30^\circ) \][/tex]

[tex]\[ \text{Side Length} = 2 \times 4.8 \times \frac{1}{\sqrt{3}} \][/tex]

[tex]\[ \text{Side Length} = \frac{9.6}{\sqrt{3}} \][/tex]

[tex]\[ \text{Side Length} = \frac{9.6\sqrt{3}}{3} \][/tex]

[tex]\[ \text{Side Length} = 3.2\sqrt{3} \][/tex]

c. **What is the perimeter of the hexagon?**

Since a regular hexagon has six equal sides, its perimeter is simply:

[tex]\[ \text{Perimeter} = 6 \times \text{Side Length} \][/tex]

[tex]\[ \text{Perimeter} = 6 \times 3.2\sqrt{3} \][/tex]

[tex]\[ \text{Perimeter} = 19.2\sqrt{3} \][/tex]

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)