The graphs of the quadratic functions f(x) = 6 – 10x2 and g(x) = 8 – (x – 2)2 are provided below. Observe there are TWO lines simultaneously tangent to both graphs.(a) The line simultaneously tangent to both graphs having the LARGEST slope has equation: (Two decimal places of accuracy.)y = ___ x + ___(b) The other line simultaneously tangent to both graphs has equation:(Two decimal places of accuracy.)y = ___ x + ___

Answer:

720

Step-by-step explanation:

If every person has to choose a different character, the first person to choose a character has 6 options, the second has 5, the third has 4, the fourth has 3, and the last person has only two options. Therefore, the total number of ways you can do this if all 5 people dress up as a different character is:

[tex]n=6*5*4*3*2\\n=720[/tex]

There are 720 ways.

Final answer:

There are 720 different ways for 5 people to dress up as the 6 main Avengers characters; this combinatorics problem uses permutations to find the answer.

Explanation:

Combinations for Avengers Characters

The question is about calculations of combinations, which falls under the subject of Mathematics. More specifically, this is a combinatorics problem that can typically be found at a High School level. We want to find out how many different ways 5 people can dress up as any of the 6 main Avengers characters, assuming each person dresses up as a different character. To solve this, we can use the concept of permutations because the order in which we assign the characters to the 5 friends does matter.

In this case, we have 6 characters to choose from, and we want to assign these characters to 5 friends. We are therefore looking for the number of permutations of 6 characters taken 5 at a time, which is calculated using the formula:

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

Here, 'n' is the total number of characters, and 'k' is the number of people to dress up. Therefore, we have:

P(6, 5) = 6! / (6 - 5)! = 6! / 1! = 720 / 1 = 720

There are 720 different ways for the 5 friends to dress up as the Avengers characters.

Answer:

D) An observational study based on available data.

Step-by-step explanation:

This is an observational study based on available data.

If it had been on a sample, he would take a sample of a few pitchers, and not studied the statistics of all major league pitchers during those seasons.

It is not an anecdotal evidence, because an anecdotal evidence is something without study, just an impression.

It is not an experiment, because he just studies(observes, that is why it is an observational study) the data, he does not change anything about the pitchers.

So the correct answer is:

D) An observational study based on available data.

The baseball enthusiast's decision is based on an observational study based on available data.

The baseball enthusiast's decision is based on A) an observational study based on available data. In this case, the enthusiast examined the records of all major league pitchers between 1990 and 1995. This observational study involved collecting and analyzing data that was already available, without manipulating any variables or conducting an experiment.

Answer:

For the company A : [tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]

For the company B: [tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]

So as we can see we have a higher probability for the company B.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Company A

Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(27000,3000)[/tex]  

Where [tex]\mu=27000[/tex] and [tex]\sigma=3000[/tex]

For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:

[tex] z= \frac{x -\mu}{\sigma}[/tex]

And if we use this formula we got:

[tex] P(X>33000)= P(Z> \frac{33000-27000}{3000}) =P(Z>2) = 1-P(Z<2)= 0.0228[/tex]

Company B

Let X the random variable that represent the salaries of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(27000,7000)[/tex]  

Where [tex]\mu=27000[/tex] and [tex]\sigma=7000[/tex]

For this case if we find the probability for [tex] P(X>33000)[/tex] using the z score given by:

[tex] z= \frac{x -\mu}{\sigma}[/tex]

And if we use this formula we got:

[tex] P(X>33000)= P(Z> \frac{33000-27000}{7000}) =P(Z>0.857) = 1-P(Z<0.857)= 0.196[/tex]

The company from which we are more likely to get an offer of $33,000 or more is determined by comparing the probabilities of receiving such an offer from each of the two companies. This probability is determined using a concept called the z-score, with which we can convert any data point into a common scale, which makes comparison easier.

1. Compute the z-scores for obtaining this salary for both companies. The z-score formula is (X - μ) / σ where X is the data point we are interested in, μ stands for the mean amount of the starting salary, and σ is the standard deviation.

   For Company A:
   z = (33000 - 27000) / 3000
   We deduct the average salary for Company A (μ = 27000) from the target salary (X = 33000) and divide it by the standard deviation (σ = 3000).

   For Company B:
   Similarly, the z score is calculated by deducting the average salary for Company B (μ = 27000) the from target salary (X = 33000) and dividing by the standard deviation (σ = 7000).

2. Now we have the z-scores for both companies. The next step is to use the z-score to calculate the probability of getting an offer of $33,000 or more from both companies. We can do this by using the cumulative distribution function (CDF) which gives us the probability that a random variable is less than or equal to a certain value. But since we want the probability that the salary is more than $33,000, we subtract the result from one (1 - CDF(z)).

3. We compare these probabilities. The company with the higher probability value will be the one from which we are more likely to get an offer of $33,000 or more.

4. Based on these calculations, you are more likely to get an offer of $33,000 or more from Company B.

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The mean (average) of the data set is approximately 110.40, the median (the middle value when the data set is arranged in order from least to greatest) is 65, and there is no mode (the most frequently occurring number) in the data set.

The subject of your question is Mathematics, specifically in the field of Statistics. To compute the mean, median, and mode, you would do the following:

For the given data set your mean is approximately 110.40 , the median is 65 and there is no mode as no numbers in the data set are repeated.

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Answer:

  yes

Step-by-step explanation:

You can always separate an equation into two parts and see where those graphs intersect.

Joel's method works well.

_____

Additional comments

Preston should know that the invention of logarithms makes it easy to solve equations like this. x = log₂(14) = log(14)/log(2) ≈ 3.8073549.

As for Joel's method, I prefer to subtract the right side to get the equation ...

  2^x -14 = 0

Then graphing y = 2^x -14, I look for the x-intercept. Most graphing calculators make it easy to find x- and y-intercepts. Not all make it easy to find points of intersection between different curves.

Answer:

Yes, the graph intersects around (3.807,14), so 3.807 is a good estimate of the solution to 2^x=14.

Step-by-step explanation:

Answer:

Check below

Step-by-step explanation:

Vectors are quantities that possess both magnitude and direction. In engineering problems, it is best to think of vectors as arrows, and usually it is best to manipulate vectors using components. In this tutorial, we consider the addition of two vectors using both of these techniques. Consider two vectors [tex]\vec{A}[/tex]and [tex]\vec{B}[/tex] that have lengths A and B, respectively. Vector [tex]\vec{B}[/tex] makes an angle?

1) Vector [tex]\vec{B}[/tex] makes an angle?

Yes, it does. Vector [tex]\vec{B}[/tex] makes an angle with [tex]\vec{A}[/tex], since both have the same origin and different direction.

2) From the direction of A.(Figure 1)In vector notation, the sum is represented by  [tex]\vec{C}=\vec{A}+\vec{B}[/tex]  where [tex]\vec{C}[/tex] is a new vector that is the sum of [tex]\vec{A}[/tex] and [tex]\vec{B}[/tex].  Find C, the length of C, which is the sum of A and B.

C is the resultant vector of this sum of vectors([tex](\vec{C}=\vec{A}+\vec{B})[/tex]

The length of c is found through the law of cosines, after projecting, vector a.

(Check 3rd picture)

2.2) The other technique to add vectors is to write them. As C is the resultant vector then we have

[tex]\vec{A}=\left \langle a_{1},a_{2} \right \rangle \vec{B}=\left \langle b_{1},b_{2} \right \rangle \vec{C}=\left \langle a_{1}+b_{1}, a_{2}+b_{2}\right \rangle[/tex]

In physics, vectors are quantities possessing both magnitude and direction. They can be represented as arrows on a graph, with their length and direction corresponding to the vector's magnitude and direction. Vectors are added geometrically using the head-to-tail method or analytically, where they are broken down into components, and these components are added separately.

This can be visualized as an arrow, where its length corresponds to the magnitude, and its direction is represented by the way the arrow points. Some examples of vectors include displacement, velocity, and force.

When adding vectors, there are two methods to consider: geometric or component wise method. The geometric method involves representing the vectors as arrows on a graph and adding them using the head-to-tail method. The component-wise method involves breaking down the vector into its x and y components, and adding these components separately.

For instance, if you have two vectors A and B, vector A can be broken down into its x and y components: A_x and A_y. Likewise, vector B can be broken down into B_x and B_y. Simply add corresponding components to yield the resultant vector, R_x = (A_x + B_x) and R_y = (A_y + B_y)

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Quota sampling is most commonly used in surveys. It is a non-probability sampling technique where researchers select individuals who meet certain criteria to be included in the sample.

Quota sampling is most commonly used in surveys. It is a non-probability sampling technique where researchers select individuals who meet certain criteria to be included in the sample.

In quota sampling, the researcher sets quotas or targets for each subgroup they want to include in the sample based on certain characteristics. For example, if the researcher wants equal representation of males and females in the sample, they would set quotas for each gender and continue selecting individuals until the quotas are met.

Quota sampling is often used when it is not possible or practical to obtain a random sample, but the researcher still wants to ensure representation of different subgroups within the sample.

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Answer:

The possible number of ways to select distinct (a, b) such that (a + b) is even is 534.

Step-by-step explanation:

The range 1 - 99 has 99 numbers, since 1 and 99 are inclusive.

Of these 50 numbers are odd and 49 are even.

The two distinct numbers a and b must have an even sum and a should be a multiple of 9.

The sum of two numbers is even only when both are odd or both are even.

The possible values that a can assume are,

a = {9, 18, 27, 36, 45, 54, 63, 72, 81, 90 and 99}

Thus, a can assume 6 odd values and 5 even values.

Total number of ways to select a and b such that both are odd and their sum is even is:

[tex]n(Odd\ a\ and\ b)=n(Odd\ value\ of\ a)\times n(Odd\ value\ of\ b)=6\times49=294[/tex]

Total number of ways to select a and b such that both are even and their sum is even is

[tex]n(Even\ a\ and\ b)=n(E\ value\ of\ a)\times n(Even\ value\ of\ b)=5\times48=240[/tex]

Total number of ways to select distinct (a, b) such that (a + b) is even is =

[tex]=n(Odd\ a\ and\ b)+n(Even\ a\ and\ b)=294+240=534[/tex]

Thus, the possible number of ways to select distinct (a, b) such that (a + b) is even is 534.

Answer:

∞

Step-by-step explanation:

Answer: False

Step-by-step explanation:

In Inferential statistics the sample data is analysed rather than analysing the whole population, analysing the whole population may sometimes be impossible, like analysing the population of a whole country. Therefore, for inferential statistics, conclusions about the whole population are drawn by analyzing the corresponding sample data. The sample data are selected from the population and then analysed, the results can then be used to conclude on the whole population.

Final answer:

The z-score for a jug with 128 fl. Oz. of milk and the probability of a jug containing less than 128 fl. Oz. are calculated using the mean and standard deviation provided.

Explanation:

1. Find the z-score corresponding to a jug containing 128 fl. Oz. of milk:

2. Probability that a jug contains less than 128 fl. Oz. of milk:

Answer:

first quadrant

Step-by-step explanation:

To know this we have to take into account that for the breast we have to look at the y-axis and in the cosine we will look at the x-axis

 We have the positive and the negative part of the x-axis, now we notice that it says that the cosine is positive, then the quadrant will be with the positive x,  these quadrants can be the first or the fourth.

Now if we look at sine = 4 is positive, this means that the axis y will also be positive,  this means that it will be between the first and second quadrant.

Now if we look to meet the conditions of the sine and cosine, the angle has to be found in the first quadrant

Answer:

Present Value = $1666666.67

Step-by-step explanation:

Present Value of a Growing Perpuity is calculated using the following formula

PV =D/(r - g)

Where D = Dividend

r = Discount Rate

g = Growth rate

D = $50,000

r = 7%

r = 7/100

r = 0.07

g = 4%

g = 4/100

g = 0.04

PV = D/(r-g)

Becomes

PV = $50,000/(0.07-0.04)

PV = $50,000/0.03

PV = $1,666,666.67

So the Present Value of the perpuity is $1,666,666.67

Answer:

[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]  

The nearest answer for this case would be 3.1%

Step-by-step explanation:

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 now that:

[tex]X \sim Binom(n=20, p=0.71)[/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]

For this case we can begin finding the probability P(X>10). If we find the individual probabilities we got:

[tex]P(X=11)=(20C11)(0.71)^{11} (1-0.71)^{20-11}=0.0563[/tex]

[tex]P(X=12)=(20C12)(0.71)^{12} (1-0.71)^{20-12}=0.1034[/tex]

[tex]P(X=13)=(20C13)(0.71)^{13} (1-0.71)^{20-13}=0.1558[/tex]

[tex]P(X=14)=(20C14)(0.71)^{14} (1-0.71)^{20-14}=0.1907[/tex]

[tex]P(X=15)=(20C15)(0.71)^{15} (1-0.71)^{20-15}=0.1867[/tex]

[tex]P(X=16)=(20C16)(0.71)^{16} (1-0.71)^{20-16}=0.1429[/tex]

[tex]P(X=17)=(20C17)(0.71)^{17} (1-0.71)^{20-17}=0.082[/tex]

[tex]P(X=18)=(20C18)(0.71)^{18} (1-0.71)^{20-18}=0.036[/tex]

[tex]P(X=19)=(20C19)(0.71)^{19} (1-0.71)^{20-19}=0.0087[/tex]

[tex]P(X=20)=(20C20)(0.71)^{20} (1-0.71)^{20-20}=0.00106[/tex]

And if we add the values we got:

[tex] P(X>10)= P(X=11) +.... +P(X=20) = 0.961525[/tex]

And if we use the complement rule the probability that "no more than 10 wear glasses for driving" we can do this:

[tex] P(X\leq 10) = 1- 0.961525= 0.0385[/tex]  

The nearest answer for this case would be 3.1%

Answer:

150

Step-by-step explanation:

You have to divide the correct answers (111) by the total amount of questions (x) in order to find the 74% (0.74)

As per the unitary method, there are 150 questions on the test.

Let's start by breaking down the information we have been given. We know that 74% of the questions on the test were answered correctly, and the number of questions answered correctly is 111.

Step 1: Convert Percentage to Decimal

To make calculations easier, we need to convert the percentage into a decimal. To do this, we divide 74 by 100, which gives us 0.74.

Step 2: Set Up the Equation

Now we can set up an equation to solve for the total number of questions on the test. Let's denote the total number of questions as "x". Since 74% of the questions were answered correctly, we can say that 0.74x questions were answered correctly.

Step 3: Solve for Total Questions

We are given that 111 questions were answered correctly. So, we can set up the equation:

0.74x = 111

Step 4: Solve for "x"

To solve for "x", we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 0.74:

x = 111 / 0.74

x = 150

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Answer:

20

Step-by-step explanation:

At equilibrium quantity demanded Qd equals quantity supply Qs

                              Qd = Qs      (At equilibrium)

     ⇒                   150 - 2P = 50 + 3P

solving for P in the equation,

                             3P + 2P = 150 - 50

                              5P = 100

                                [tex]P = \frac{100}{5}[/tex]

                                P = 20

Answer:

60%

Step-by-step explanation:

The percentage does not change.

Probability at least one of next 3 vehicles fail inspection: approximately 0.784 (to 3 decimal places).

To find the probability that at least one of the next three vehicles fail, we can calculate the probability of the complementary event, i.e., the probability that all three vehicles pass, and then subtract it from 1.

Given that 60% of vehicles pass the inspection, the probability that one vehicle fails the inspection is 1 - 0.60 = 0.40.

Since the vehicles pass or fail independently, the probability that all three vehicles pass is:

[tex]\[0.60 \times 0.60 \times 0.60 = 0.216\][/tex]

Now, the probability that at least one vehicle fails is:

[tex]\[1 - 0.216 = 0.784\][/tex]

So, the probability that at least one of the next three vehicles fail is approximately [tex]\(0.784\)[/tex] (to 3 decimal places).

Answer:

13/4260 tons

Step-by-step explanation:

We have the rate at which they remove tons of dirt per hour. We also know that total that needs to be removed. We can determine the time by dividing  the amount of tons that need to be removed by the rate:

[tex]=(13/6)/\cdot{710}=13/4260[/tex]

will take 13/4260 hours to remove the dirt

Answer:

Present value =  $4,122.4

Accumulated amount = $4,742

Step-by-step explanation:

Data provided in the question:

Amount at the Start of money flow = $1,000

Increase in amount is exponentially at the rate of 5% per year

Time = 4 years

Interest rate = 3.5%  compounded continuously

Now,

Accumulated Value of the money flow = [tex]1000e^{0.05t}[/tex]

The present value of the money flow = [tex]\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t})} \, dt[/tex]

= [tex]1000\int\limits^4_0 {e^{0.015t}} \, dt[/tex]

= [tex]1000\left [\frac{e^{0.015t}}{0.015} \right ]_0^4[/tex]

= [tex]1000\times\left [\frac{e^{0.015(4)}}{0.015} -\frac{e^{0.015(0)}}{0.015} \right][/tex]

= 1000 × [70.7891 - 66.6667]

= $4,122.4

Accumulated interest = [tex]e^{rt}\int\limits^4_0 {1000e^{0.05t}(e^{-0.035t}} \, dt[/tex]

= [tex]e^{0.035\times4}\times4,122.4[/tex]

= $4,742

The present value and interest accumulated would be as follows:

Present Value = $ 4,122.4

Interest Accumulated = $ 4742

Given that,

Principal at the beginning of money flow = $1,000

Exponential interest rate = 5% per year

Time Period = 4 years

So,

The accumulated money flow's worth = [tex]1000e^{0.05t}[/tex]

The current value of the money can be determined by [tex]\int\limits^4_0 1000e^{0.05t}(e^{-0.035t}) {} \, dt[/tex]

On solving, we get

The present value = $ 4,122.4

Interest Accumulated = $4,742

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Answer:

(a) [tex]p(x) = 17-\frac{x}{3,500}[/tex]

(b) $8.50

Step-by-step explanation:

(a) The slope of the demand function, p(x), is determined by:

[tex]m=\frac{11-9}{21,000-28,000}=-\frac{1}{3,500 }[/tex]

Applying the point (21,000; 11) to the general linear equation formula gives us the demand function:

[tex]p(x) - 11 = -\frac{1}{3,500}*(x-21,000)\\p(x) = 17-\frac{x}{3,500}[/tex]

(b) The revenue function, r(x), is given by:

[tex]r(x) =x*p(x) = 17x-\frac{x^2}{3,500}[/tex]

The value of x for which the derivate of the revenue function is zero gives us the attendance for which revenue is maximized:

[tex]\frac{dr(x)}{dx} =0= 17-\frac{2x}{3,500}\\x=29,750[/tex]

At an attendance of 29,750, the price is:

[tex]p = 17-\frac{29,750}{3,500}\\p=\$8.50[/tex]

Tickets should be set at a price of $8.50.

The demand function for the baseball team can be found using the given data points. The revenue function can be derived using the demand function, and its maximum can be determined by finding the critical points. The ticket price should be set at $8.50 to maximize revenue for the baseball team.

(a) To find the demand function, we can use the point-slope form of a linear equation. Let's denote the ticket price as p and the attendance as a. We have two data points: (11, 21000) and (9, 28000). We can use the formula: a - a1 = m(p - p1) where (p1, a1) represents one of the given data points and m represents the slope. Let's use the first data point:

21000 - 28000 = m(11 - 9)

-7000 = 2m

m = -3500

So, the demand function is: a(p) = -3500p + b. To find the constant term b, we can substitute one of the data points into the equation. Let's use the first data point (11, 21000):

21000 = -3500(11) + b

21000 = -38500 + b

b = 59500

Therefore, the demand function is: a(p) = -3500p + 59500.

(b) Revenue is the product of the ticket price and the attendance. Let's denote the revenue as R and the ticket price as p. The revenue function can be written as: R(p) = p * a(p). Substituting the demand function into the revenue function, we have: R(p) = p * (-3500p + 59500). To maximize revenue, we can find the critical points of the function by taking the derivative and setting it equal to zero.

R'(p) = -7000p + 59500 = 0

-7000p = -59500

p = 8.5

The critical point is p = 8.5. To confirm that this gives a maximum, we can take the second derivative and check its sign.

R''(p) = -7000

Since the second derivative is negative, the critical point p = 8.5 corresponds to the maximum revenue. Therefore, the ticket price should be set at $8.50 to maximize revenue.

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Answer:

The only solution is r=2 or r=1

Step-by-step explanation:

Check the attachment

To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, substitute y = tr into the given differential equation and solve for r. The values of r that satisfy the equation are r = 1.

To determine the values of r for which the given differential equation has solutions of the form y = tr for t > 0, we need to substitute y = tr into the given differential equation and solve for r.

First, we find the first and second derivatives of y = tr:

y' = r

y'' = 0

Substituting these derivatives into the differential equation, we get:

t2(0) - 2t(r) + 2(tr) = 0

Simplifying the equation, we have:

(2 - 2r)t = 0

This equation is satisfied when t = 0 or r = 1. Therefore, the values of r for which the given differential equation has solutions of the form y = tr for t > 0 are r = 1.

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Answer:  

120 ways

Step-by-step explanation: If there are three different categories of components, that is, component A, comprising four types; component B comprising, five types and component C comprising, six types. Three types are to be selected, that is, one type from each category. The number of ways one component of each type be selected is:

Component A * Component B * Component C = 4 X 5 X 6 = 120

Answer:

a) left handed people

b) Binomial probability distribution with pdf

[tex]P(X=x)=15Cx0.1^{x} 0.9^{15-x}[/tex]

where x=0,1,2,...,15.

c) Histogram is attached

d) The shape of histogram depicts that distribution is rightly skewed.

e) 1.5

f) 1.35

g) 1.16

Step-by-step explanation:

a)

The random variable in the given scenario is " left handed people"

b)

The scenario represents the binomial probability distribution as the outcome is divided into one of two categories and experiment is repeated fixed number of times i.e. 15 and trails are independent. The pdf of binomial distribution is

[tex]P(X=x)=nCxp^{x} q^{n-x}[/tex]

Here n=15, p=0.1 and q=1-p=0.9.

So, the pdf would be

[tex]P(X=x)=15Cx0.1^{x} 0.9^{n-x}[/tex]

where x=0,1,2,...,15.

c)

Histogram is constructed by first computing probabilities on all x points i.e. x=0, x=1 , .... ,x=15 and then plotting all probabilities with respective x values. Histogram is in attached image.

d)

The tail of histogram is to the right side and thus the histogram depicts that given probability distribution is rightly skewed.

e)

The mean of binomial probability distribution is computed by multiplying number of trails and probability of success.

mean=np=15*0.1=1.5

f)

The variance of binomial probability distribution is computed by multiplying number of trails and probability of success and probability of failure.

variance=npq=15*0.1*0.9=1.35

g)

The standard deviation can be calculated by simply taking square root of variance

S.D=√npq=√1.35=1.16

The proportion of left-handed people follows a binomial distribution

The given parameters are:

[tex]\mathbf{n = 15}[/tex] -- the sample size

[tex]\mathbf{p = 10\%}[/tex] --- the proportion of left-handed people

(a) The random variable

The distribution is about left-handed people.

Hence, the random variable is left-handed people

(b) The probability distribution

If the proportion of left-handed people is 10%, then the proportion of right-handed people is 90%.

So, the probability distribution function is:

[tex]\mathbf{P(x) = ^nC_x p^x (1 - p)^{n -x}}[/tex]

This gives

[tex]\mathbf{P(x) = ^nC_x (10\%)^x (1 - 10\%)^{n -x}}[/tex]

[tex]\mathbf{P(x) = ^nC_x 0.1^x 0.9^{n -x}}[/tex]

Hence, the probability distribution function is [tex]\mathbf{P(x) = ^nC_x 0.1^x 0.9^{n -x}}[/tex]

(c) The histogram

To do this, we calculate P(x) for x = 0 to 15

[tex]\mathbf{P(0) = ^{15}C_0 \times 0.1^0 \times 0.9^{15 -0} = 0.206}[/tex]

[tex]\mathbf{P(1) = ^{15}C_1 \times 0.1^1 \times 0.9^{15 -1} = 0.343}[/tex]

.....

..

[tex]\mathbf{P(15) = ^{15}C_{15} \times 0.1^{15} \times 0.9^{15 -15} = 10^{-15}}[/tex]

See attachment for the histogram

(d) The mean

This is calculated as:

[tex]\mathbf{\bar x = np}[/tex]

So, we have:

[tex]\mathbf{\bar x = 15 \times 10\% }[/tex]

[tex]\mathbf{\bar x= 1.5}[/tex]

Hence, the mean is 1.5

(e) The variance

This is calculated as:

[tex]\mathbf{Var = np(1 - p)}[/tex]

So, we have:

[tex]\mathbf{Var = 15 \times 10\% \times (1 - 10\%)}[/tex]

[tex]\mathbf{Var = 1.35}[/tex]

Hence, the variance is 1.35

(f) The standard deviation

This is calculated as:

[tex]\mathbf{\sigma = \sqrt{Var}}[/tex]

So, we have:

[tex]\mathbf{\sigma = \sqrt{1.35}}[/tex]

[tex]\mathbf{\sigma =1.16}[/tex]

Hence, the standard deviation is 1.16

Read more about distributions at:

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Answer:

66.98% probability that the mean gain for the sample was between 250 and 500.

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 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.

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]\frac{\sigma}{\sqrt{n}}[/tex].

In this problem, we have that:

[tex]\mu = 432, \sigma = 722, n = 40, s = \frac{722}{\sqrt{40}} = 114.16[/tex]

What is the probability that the mean gain for the sample was between 250 and 500?

This is the pvalue of Z when X = 500 subtracted by the pvalue of Z when X = 250.

So

X = 500

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

By the Central Limit Theorem

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

[tex]Z = \frac{500 - 432}{114.16}[/tex]

[tex]Z = 0.6[/tex]

[tex]Z = 0.6[/tex] has a pvalue of 0.7257.

X = 250

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

By the Central Limit Theorem

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

[tex]Z = \frac{250 - 432}{114.16}[/tex]

[tex]Z = -1.59[/tex]

[tex]Z = -1.59[/tex] has a pvalue of 0.0559.

So there is a 0.7257 - 0.0559 = 0.6698 = 66.98% probability that the mean gain for the sample was between 250 and 500.

The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.

To determine the probability that the mean gain for a random sample of 40 years of the Dow Jones Industrial Average is between 250 and 500, we will use the concept of the sampling distribution of the sample mean. The steps involved are:

State the given information:

  - Population mean [tex](\(\mu\))[/tex] = 432

  - Population standard deviation [tex](\(\sigma\))[/tex] = 722

  - Sample size n = 40

Find the standard error of the mean (SEM):

  The standard error of the mean is calculated as:

  [tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{722}{\sqrt{40}} \approx 114.2 \][/tex]

Convert the sample means to z-scores:

  To find the probability that the sample mean [tex](\(\bar{x}\))[/tex] is between 250 and 500, we convert these values to z-scores using the formula:

  [tex]\[ z = \frac{\bar{x} - \mu}{\text{SEM}} \] For \(\bar{x} = 250\): \[ z = \frac{250 - 432}{114.2} \approx \frac{-182}{114.2} \approx -1.59 \] For \(\bar{x} = 500\):[/tex]

  [tex]\[ z = \frac{500 - 432}{114.2} \approx \frac{68}{114.2} \approx 0.60 \][/tex]

Find the probability corresponding to the z-scores:

  Using the standard normal distribution table or a calculator, we find the probabilities corresponding to these z-scores.

   [tex]\(z = -1.59\), \(P(Z \leq -1.59) \approx 0.0559\) \\\(z = 0.60\), \(P(Z \leq 0.60) \approx 0.7257\)[/tex]

Calculate the probability that the mean gain is between 250 and 500:

  \[tex][ P(250 \leq \bar{x} \leq 500) = P(Z \leq 0.60) - P(Z \leq -1.59) \] \[[/tex]

 = 0.7257 - 0.0559 = 0.6698

Conclusion:

The probability that the mean gain for a sample of 40 years is between 250 and 500 is approximately 0.6698, or 66.98%.

Answer:

e. 0.08

Step-by-step explanation:

In the question above, a certain quantity of goods was supplied while a specific quantity of goods was sold per week. In a given week, if the number of proportion sold is X, therefore:

f(x) = {Γ(4+2)/Γ(4)Γ(2) x^3 (1-x), 0≤x≤1 ; 0, elsewhere

and

P(X greater than 0.9) =  [tex]\int\limits^1_ {0.9} \, 20(x^{3} - x^{4}) dx[/tex] = 20*{(y^4/4)[1,0.9] - (y^5/5)[1,0.9]} = 20*{(0.25 - 0.164) - (0.20 - 0.118)}  = 20*{0.086 - 0.0819} = 20*0.0041 = 0.082

Therefore the probability of the proportion sold is approximately 0.082

The probability that at least 90% of the stock will be sold in a given week is approximately 0.05. The correct answer option is b. 0.05

Here's how to calculate it:

1. Given the beta distribution with parameters alpha = 4 and beta = 2, we want to find [tex]\( P(X > 0.9) \)[/tex], where X represents the proportion of stock sold in a week.

2. Since [tex]\( P(X > 0.9) = 1 - P(X \leq 0.9) \)[/tex], we need to find the cumulative distribution function (CDF) of the beta distribution and then subtract it from 1.

3. Using the incomplete beta function formula [tex]\( I_{x}(\alpha, \beta) \)[/tex], we have:

[tex]\[ P(X > 0.9) = 1 - I_{0.9}(4, 2) \][/tex]

4. Calculating [tex]\( I_{0.9}(4, 2) \)[/tex] using software or a calculator gives approximately 0.95.

5. Subtracting this from 1:

[tex]\[ P(X > 0.9) = 1 - 0.95 = 0.05 \][/tex]

So, the correct answer is b. 0.05.

Answer:

Factory A should be operated 6.11 days

Factory B should be operated 6.88 days

The minimum cost for factory A is $6,110

The minimum cost for factory B is $5,504

Step-by-step explanation:

Factory A

Daily production: 16 3-speed and 20 10-speed bikes

Total daily production = 16+20 = 36 speed bikes

Order: 80 3-speed and 140 10-speed bikes

Total order = 80+140 = 220 speed bikes

Number of days = 220/36 = 6.11 days

Cost per day = $1,000

Minimum cost for 6.11 days = $1000 × 6.11 = $6,110

Factory B

Daily production: 12 3-speed and 20 10-speed bikes

Total daily production = 12+20 = 32 speed bikes

Total order = 220 speed bikes

Number of days = 220/36 = 6.88 days

Cost per day = $800

Cost for 6.88 days = $800 × 6.88 = $5,504

The business should operate Factory A for about 2.92 days and Factory B for about 4.17 days to fill the order at a minimum cost of $7148.

This problem can be solved using Linear Programming, a mathematical model used in optimization problems. Suppose we let A be the number of days factory A operates and B be the number of days factory B operates. The cost to operate the factories is given by the equation C = 1000A + 800B.

The constraints are as follows:

By solving these equations, you find that Factory A should be operated for about 2.92 days and Factory B should be operated for 4.17 days, giving a minimum cost of about $7148.

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Answer:

See below

Step-by-step explanation:

Remember some definitions about binary relations. If R⊆S×S then

a) R is not reflexive since (1,1)∉R.

R is irreflexive, since (a,a)∉R for all a=1,2,3,4

R is asymmetric: (2,3)∈R and (3,2)∉R (thus R is not symmetric).

R is antisymmetric, there are no cases to check. R is transitive, there are no cases to check.

b) R is reflexive, checking case by case, (a,a)∈R for all a=1,2,3,4. Hence R is not irreflexive.

R is not asymmetric: (1,2)∈R but (2,1)∈R. R is not symmetric, since (4,1)∈R but (1,4)∉R

R is not antisymmetric: (1,2)∈R and (2,1)∈R but 1≠2.

R is not transitive: (1,2)∈R and (2,4)∈R but (1,4)∉R.

Answer:

z(max)  =  5000

x   =  100

y   =  0

Step-by-step explanation:

Let call

x  floral arrangements  and

y  fruit baskets

Then Objective function is according to profits in each gift

z   =   50*x    +    35*y

Constraints:

1.- Hours available in Assembly department  40  in minutes   is  2400 minutes

20*x  +  15*y  ≤ 2400

2.- Hours available in packaging department 10 in minutes  is  600

6*x  + 2*y  ≤  600

3.- x    and    y must be    x ≥ 0    y  ≥ 0

Then the system is:

z      - 50*x       -   50*y                       =  0        To maximize subject to:

         20*x       +  15*y                        ≤  2400

           6*x        +   2*y                        ≤  600

x ≥ 0    y ≥ 0

Simplex Method:

z          x         y       s₁       s₂         Cte

1         -50     -35     0        0    =     0

0         20       15      1        0    =  2400

0           6         2      0        1    =  600

First iteration: 6 is a pivot   we dvede R3  by 6

z          x         y       s₁       s₂         Cte

1          0        15      0        50/6 =  5000

0         0      -50/6  -1        20/6 = -400                                                                

0         20       15      1        0    =  2400

0          1          2/6     0       1/6   =  100

We have done no negative number in the objective function we stop iteration and

z(max)  =  5000

x   =  100

y   =  0

                                         to add to R1  50*R3 [ 0  50  100/6  0  50/6  5000

                                         to add to R2 20*R3 [ 0   20  40/6   0  20/6  2000

Answer:

[tex]V = \frac{5\pi}{6}[/tex] or 2.62

Step-by-step explanation:

Since our region (on the left) is bounded by x = 1 and x = 0 (where [tex]y = x^2 = 0[/tex], if we take center at x = 2 then our radius will range from 1 to 2 (x=1 to x = 0). We can use the following integration to calculate the volume using shell method

[tex]V = \int\limits^2_1 {2\pi r h} \, dr[/tex]

where r = 2 - x so x = 2 - r and [tex]h = y = x^2 = (2-r)^2[/tex] for [tex]1 \leq r \leq 2[/tex]

[tex]V = \int\limits^2_1 {2\pi r(2-r)^2} \, dr[/tex]

[tex]V = \int\limits^2_1 {2\pi r(4 - 4r + r^2)} \, dr[/tex]

[tex]V = 2\pi \int\limits^2_1 {r^3 - 4r^2 +4r} \, dr[/tex]

[tex]V = 2\pi\left[\frac{r^4}{4} - \frac{4r^3}{3} + 2r^2\right]^2_1[/tex]

[tex]V = 2\pi\left[\left(\frac{2^4}{4} - \frac{4*2^3}{3} + 2*2^2\right) - \left(\frac{1^4}{4} - \frac{4*1^3}{3} + 2*1^2\right)\right][/tex]

[tex]V = 2\pi(4 - 32/3 +8 - 1/4 + 4/3 - 2)[/tex]

[tex]V = \pi(20 - 56/3 - 1/2)[/tex]

[tex]V = \pi\frac{120 - 112 - 3}{6}[/tex]

[tex]V = \frac{5\pi}{6}[/tex] or 2.62

Answer: The answers in order are: Equal, 120°, Equal, 120°, Equal, no

Step-by-step explanation:

The positive sequence components have equal magnitudes and 120° phase displacements in positive sequence.  Their phase sequence is same as that of the system one. Also their phase rotations is same like the system.

The negative sequence components have equal magnitudes with same 120° phase displacements. Their phase sequence is opposite to that of system but their phase rotation is same like the system.

Zero sequence components have equal magnitude with no phase displacement. They behave like negative sequence in terms of phase sequence and phase rotation.

Answer:

Equal, 120°, Equal, 120°, Equal, no

Step-by-step explanation: