Exercise 2.2 gave the following data (in increasing order) for the attribute age: 13, 15, 16, 16, 19, 20, 20, 21, 22, 22, 25, 25, 25, 25, 30, 33, 33, 35, 35, 35, 35, 36, 40, 45, 46, 52, 70. (a) Use smoothing by bin means to smooth these data, using a bin depth of 3. Illustrate your steps. Comment on the effect of this technique for the given data. (b) How might you determine outliers in the data? (c) What other methods are there for data smoothing?

Answer:

6.68 %.

Step-by-step explanation:

The standardised z-score = ( 10.95 - 10.5) /  0.3

= 1.5.

Looking up the Normal Distribution tables ( area to the left) 1.5 corresponds to  0.93319 so for   a  weight above 10.95 the proportion is

1 - 0.93319 = 0.06681

= 6.68%.

We can use the z-score to find the proportion of jars that are above a certain weight in a normal distribution. The z-score for 10.95 ounces is 1.5. Using a standard normal distribution table, we find that about 6.68% of the jars weigh more than 10.95 ounces.

In this problem, we are using the concept of normal distribution, specifically to find the proportion of jars that are above a certain weight. Given that the mean (average) weight of the jars is 10.5 ounces and the standard deviation (which measures the dispersion of the weights) is 0.3 ounce, we can calculate the z-score for 10.95 ounces.

The z-score is defined as the number of standard deviations a data point is from the mean. Compute it using the formula: Z = (X - μ) / σ where X is the value, μ is the mean and σ is the standard deviation.

Plugging into the formula we get: Z = (10.95 - 10.5) / 0.3 = 1.5

You can then look up this z-score in a standard normal distribution table (or use a calculator or computer software that calculates it), to find the proportion below this z-score. But we need the proportion above, so we subtract this from 1. Let's say the value from a z-table for 1.5 is 0.9332, the proportion of values above this would be 1 - 0.9332 = 0.0668 or about 6.68% of the jars weigh more than 10.95 ounces.

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[tex]\[ \int_{0}^{5} \frac{1}{5} x^5 \, dx \approx 520.8333 \][/tex]

Answer:

  (B) 200 km

Step-by-step explanation:

Let A represent the distance Anna goes before transferring fuel. Let C represent the distance to Connor's house. All distances are in km. Here, we will measure fuel quantity in terms of the distance it enables.

The total distance that can be driven by the two motorbikes is ...

  2A +2C = 600

Anna can transfer to Brian an amount of fuel that is 300-2A, since she needs to get back to the depot from the stopping point. When they stop, the amount of fuel in Brian's tank is 300-A. After that transfer, the most fuel Brian can have is a full tank (300). Then ...

  (300 -A) +(300 -2A) = 300 . . . . fuel in Brian's tank after the transfer

This second equation simplifies to ...

  600 -3A = 300

  300 = 3A . . . . . . add 3A-300

  100 = A . . . . . . . . divide by 3

Using this in the first equation, we get ...

  2·100 +2C = 600

  2C = 400 . . . . . . . . subtract 200

  C = 200 . . . . . . . . . .divide by 2

The distance from the depot to Connor's house can be at most 200 km.

Answer:

[tex]y^2=2\ln (1+t^2)+4[/tex]

Step-by-step explanation:

Given that

[tex]\dfrac{dy}{dt}=\dfrac{2t}{y+yt^2}[/tex]

This is a differential equation.

Now by separating variables

[tex]y dy= \dfrac{2t}{1+t^2}dt[/tex]

Now by integrating both side

[tex]\int y dy=\int \frac{2t}{1+t^2}dt[/tex]

Now by soling above integration

We know that  integration of dx/x is lnx.

[tex]\dfrac{y^2}{2}=\ln (1+t^2)+C[/tex]

Where C is the constant.

[tex]y^2=2\ln (1+t^2)+C[/tex]

Given that when t=0 then y= -2

So by putting the above values of t and y we will find C

4=2 ln(1)+C     (we know that ln(1)=0)

So C=4

⇒[tex]y^2=2\ln (1+t^2)+4[/tex]

So solution of above equation is  [tex]y^2=2\ln (1+t^2)+4[/tex]

Answer: After [tex]3.5\ seconds[/tex]

Step-by-step explanation:

Knowing that after "t" seconds, its height "h" in feet is given by the function:

 [tex]h(t) = 112t -16t^2[/tex]

The maximum height is the y-coordinate of the vertex of the parabola. Then, we can use the following formula to find the corresponding value of "t" (which is the x-coordinate of the vertex):

[tex]t=\frac{-b}{2a}[/tex]

In this case:

[tex]a=-16\\b=112[/tex]

Substituting values, we get that the ball will reach the maximum height after:

[tex]t=\frac{-112}{2(-16)}\\\\t=3.5\ seconds[/tex]

A ball thrown vertically upwards in a parabolic path reaches its maximum height at the vertex of the parabolic path represented by the function of its height. The time it takes to reach this maximum height can be calculated with the formula -b/(2a), yielding a result of 3.5 seconds in this case.

The height h of a ball thrown vertically upward is given by the function h(t) = 112t -16t^2. The maximum height of the ball can be determined by finding the maximum point of the parabola represented by the equation. The maximum point occurs at the vertex of the parabola which is determined by the formula -b/(2a), where a and b are coefficients in the quadratic equation at^2 + bt + c.

In this case, a = -16 and b = 112. So, to find the time t when the ball will reach the maximum height, we substitute these into the formula to get t = -112/(2*(-16)) = 3.5 seconds. So the ball reaches its maximum height after 3.5 seconds.

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The generating function for this sequence is

[tex]f(x)=4+4x+4x^2+4x^3+x^4+x^6+x^8+\cdots[/tex]

assuming the sequence itself is {4, 4, 4, 4, 1, 0, 1, 0, ...} and the 1-0 pattern repeats forever (as opposes to, say four 4s appearing after every four 1-0 pairs). We can make this simpler by "displacing" the odd-degree terms and considering instead the generating function,

[tex]f(x)=3+4x+3x^2+4x^3+\underbrace{(1+x^2+x^4+x^6+x^8+\cdots)}_{g(x)}[/tex]

where the coefficients of [tex]g(x)[/tex] follow a much more obvious pattern of alternating 1s and 0s. Let

[tex]g(x)=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]

where [tex]a_n[/tex] is recursively given by

[tex]\begin{cases}a_0=1\\a_1=0\\a_{n+2}=a_n&\text{for }n\ge0\end{cases}[/tex]

and explicitly by

[tex]a_n=\dfrac{1+(-1)^n}2[/tex]

so that

[tex]g(x)=\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n[/tex]

and so

[tex]\boxed{f(x)=3+4x+3x^2+4x^3+\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n}[/tex]

Final answer:

The generating function for the sequence is found by splitting it into two parts and expressing each as a series. The constant part can be expressed as a finite series, while the alternating sequence is a geometric series that can be simplified. Their sum yields the generating function.

Explanation:

The student has asked for a compact form for the generating function of the sequence 4, 4, 4, 4, 1, 0, 1, 0, 1, 0, 1, 0, ... .

To find the generating function for the given sequence, we can split it into two parts: The constant part (4, 4, 4, 4) and the alternating sequence (1, 0, 1, 0, ...).

The constant part can be represented as:
4 + 4x + 4x2 + 4x3 = 4(1 + x + x2 + x3)

The alternating sequence can be represented as a geometric series:
1 - x2 + x4 - x6 + ... = 1 / (1+x2)

The generating function G(x) would then be the sum of these two parts, simplifying by multiplication of the series and a fraction:

G(x) = 4(1 + x + x2 + x3) + x4 / (1 + x2)

Answer:

The no. of possible handshakes takes place are 45.

Step-by-step explanation:

Given : There are 10 people in the party .

To Find: Assuming all 10 people at the party each shake hands with every other person (but not themselves, obviously) exactly once, how many handshakes take place?

Solution:

We are given that there are 10 people in the party

No. of people involved in one handshake = 2

To find the no. of possible handshakes between 10 people we will use combination over here

Formula : [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]

n = 10

r= 2

Substitute the values in the formula

[tex]^{10}C_{2}=\frac{10!}{2!(10-2)!}[/tex]

[tex]^{10}C_{2}=\frac{10!}{2!(8)!}[/tex]

[tex]^{10}C_{2}=\frac{10 \times 9 \times 8!}{2!(8)!}[/tex]

[tex]^{10}C_{2}=\frac{10 \times 9 }{2 \times 1}[/tex]

[tex]^{10}C_{2}=45[/tex]

No. of possible handshakes are 45

Hence The no. of possible handshakes takes place are 45.

Answer:

3) 28.28.

Step-by-step explanation:

In order to find the answer we need to establish the EOQ equation which is:

[tex]EOQ=\sqrt{2*s*d/h}[/tex] where:

s=the cost of the setup

d=demand rate

h=cost of holding

Because demand is 200/month so d=200,

the ordering cost is $20/month  so s=20, and

the carrying cost in $10/month so h=10.

Using the equation we have:

[tex]EOQ=\sqrt{2*20*200/10}[/tex]

[tex]EOQ=\sqrt{800}[/tex]

[tex]EOQ=28.28[/tex]

So, answer to 'the order quantity for product' is 3) 28.28.

Answer: There is a probability of no correct answers is 0.4096.

Step-by-step explanation:

Since we have given that

Number of trials = 4

Probability of success i.e. getting correct answer = 0.20

We need to find the probability of no correct answers.

We would use "Binomial distribution".

Let X be the number of correct answers.

So, it becomes,

[tex]P(X=0)=(1-0.20)^4=(0.80)^4=0.4096[/tex]

Hence, there is a probability of no correct answers is 0.4096.

The probability of guessing all answers incorrectly in a multiple-choice test with 4 questions, each with 5 options, is approximately 0.41 or 41% when answers are randomly guessed, according to the binomial probability distribution.

The question you're asking pertains to the concept of binomial probability, which is a type of probability that applies when there are exactly two mutually exclusive outcomes of a trial, often referred to as 'success' and 'failure'. In this case, a 'success' refers to correctly guessing an answer, which has a probability of p = 0.20. Conversely, a 'failure' refers to incorrectly guessing an answer, and this has a probability of q = 1 - p = 0.80.

To find the probability of no correct answers from 4 trials, we employ the formula for binomial probability: P(x) = (n C x)×(p×x)*(q×(n-x)). Here, 'n' represents the number of trials (4), 'x' represents the number of successes (0 for our case), and 'n C x' denotes the number of combinations of n items taken x at a time.

By plugging in the relevant values, the binomial probability distribution gives us P(0)= (4 C 0)×(0.20×0)×(0.80×4) = 1 × 1 × 0.4096 = 0.4096. So, the probability of guessing all answers incorrectly is approximately 0.41 or 41% when answers are randomly guessed.

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Answer: Down payment amount = $330

Step-by-step explanation:

Given in the question that Shawn is interested in purchasing a new computer system and he wants to to give a 20%  down payment.

Cost of Computer system = $1650

He would like to made a 20% down payment

So, the down payment amount is as follows:

20% of $1650 = [tex]\frac{20}{100}[/tex] × 1650

                        = $ 330 ⇒ Down payment amount

Answer:

A. Nonnegative

Step-by-step explanation:

By definition, the absolute value of any number must be positive (i.e non-negative).Hence A is the answer.

Answer:

Step-by-step explanation:

The only possibilities where there is exactly 1 tail are:

those are 5 favorable outcomes.

where h represent heads and t represent tails. There are [tex]2^5 32[/tex] total number of outcomes after tossing the coin 5 times. Because every time you toss the coin, you have 2 possibilities, and as you do it 5 times, those are [tex]2^5[/tex] options. We can conclude from this that

The probability that exactly 1 is a tail is [tex]5/32[/tex].

We already know the total number of outcomes; 32.  Now we need to find the number of favorable outcomes. In order to do that, we can divide our search in three cases: 1.-there are no tails, 2.-exactly 1 is a tail, 3.- exactly 2 are tails.

The first case is 1 when every coin is a head. The second case we already solved it, and there are 5. The third case is the interesting one, we can count the outcomes as we did in the previous questions, but that's only because there are not too many outcomes.  Instead we are going to use combinations:

We need to have 2 tails, the other coins are going to be heads. We made 5 tosses, then the possible combinations are [tex]C_{5,2} = \frac{5!}{3!2!} = \frac{120}{6*2} = 10[/tex]

Finally, we conclude that there are 1 + 5 + 10 favorable outcomes, and this implies that

The probability that at most 2 are tails is [tex]\frac{16}{32} = \frac{1}{2}[/tex].

In a five-coin toss, the probability of getting exactly one tail is 5/32 and the probability of getting at most two tails is 0.5. These probabilities are calculated considering all possible outcomes and arranging the heads and tails in distinct manners.

The question you've asked involves calculating the probabilities in coin flipping, a common concept in mathematics and particularly in statistics. This falls under the topic of probability theory.

When a fair coin is tossed 5 times, there are 2^5 or 32 equally likely outcomes. If we want exactly 1 tail, there are 5 ways this can happen (one for each position the tails can be in). Thus, the probability for this occurrence is 5/32.

To find out the probability of getting at most 2 tails, we need to calculate the probability for getting exactly 0, 1, or 2 tails. As we already know that the probability for 1 tail is 5/32 and for 0 tails is 1/32 (only 1 way to get this outcome, getting heads every time). The probability for exactly 2 tails can be found in the same manner as for 1 tail, now we have 2 tails and it can be arranged in 5C2 ways which is 10 ways. Therefore, the probability of 2 tails is 10/32. Hence, the probability of getting at most 2 tails is the sum of probabilities of 0,1 or 2 tails, which is (1 + 5 + 10 )/32 = 16/32 = 0.5.

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Answer:

0.85 M + 22.55

Step-by-step explanation:

We know that the total cost is the standard cost plus the insurance cost

C(M) = S(M) + I(M)

        = 17.75 + .60M + 4.80+.25M

Combine like terms

        = 0.85 M + 22.55

For this case we have that the standard charge, in dollars, of a company that rents vehicles is given by:

[tex]S = 17.75 + 0.60M[/tex]

M: Number of miles traveled.

On the other hand, the insurance charge is given by:

[tex]I = 4.80 + 0.25M[/tex]

If we want to find the total cost of renting the vehicle, we must add both equations:

[tex]C = 17.75 + 0.60M + 4.80 + 0.25M[/tex]

We add similar terms:

[tex]C = 17.75 + 4.80 + 0.60M + 0.25M\\C = 22.55 + 0.85M[/tex]

Answer:

[tex]C = 22.55 + 0.85M[/tex]

Answer:

78

Step-by-step explanation:

50 questions worth 2 a piece and you miss 11 questions so we are going to take 2(11) off of 50.

We are doing 100-2(11).

100-22=78.

Or if you miss 11 questions you get 39 right. So 39(2)=78.

ANSWER :

There are 50 questions, total marks are 100.

EXPLANATION:

Since each question is 2 marks, missing 11 questions is equal to missing 22 marks.

therefore, your marks will be 100-22=78 marks.

Answer:

Step-by-step explanation:

For this problem you have to use combinations. from 10 choices you are choosing 3. This means you are doing 10 choose 3. If you don't know what choose is I can explain.

Any number x choose y is the same as (x factorial)/(y factorial)(x-y factorial).

In this case that is 10 factorial/3 factorial times 7 factorial. ten factorial is the same as 10*9*8*7 factorial. So in the original equation you can factor away the seven factorials to get 10*9*8/3*2*1 factoring again you get 10*3*4  which is 120.

There are 120 different ways the chef can pick 3 brands of hot sauce to mix into a gumbo.

To find the number of ways the chef can pick 3 brands of hot sauce out of 10, we can use the combination formula:

[tex]nCr = n! / (r! * (n-r)!)[/tex]

where n is the total number of items (brands of hot sauce), and r is the number of items to be chosen (3 in this case).

In this problem, n = 10 and r = 3:

10C3 = 10! / (3! * (10-3)!)

Calculating the factorials:

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

3! = 3 × 2 × 1 = 6

(10-3)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Now, substitute the values:

10C3 = 3,628,800 / (6 * 5,040)

10C3 = 3,628,800 / 30,240

10C3 = 120

So, there are 120 different ways the chef can pick 3 brands of hot sauce to mix into a gumbo.

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Answer:

Step-by-step explanation:

Angles are measured from the direction of motion, so the "force made good" is the force in the rope multiplied by the cosine of the angle. If the forces in the ropes (in Newtons) are represented by x and y, then we have ...

  x·cos(40°) +y·cos(55°) = 700

In order for the resultant to be in the direction of motion, the forces perpendicular to the direction of motion must cancel.

  x·sin(40°) - y·sin(55°) = 0

Here, we have assumed that the positive direction for measuring 40° is the negative direction for measuring 55°. That is, the angles are measured in opposite directions from the direction of motion.

Any of the usual methods for solving systems of linear equations can be used to solve this set. My preference is to use a graphing calculator. It gives the answers summarized above.

Answer:

Length = 20.49 yards and Width = 12.49 yards.

Step-by-step explanation:

The area of the rectangular playground is given by 256 yards square. It is also known that one of the sides of the playground is 8 yards longer than the other side. Therefore, let the smaller side by x yards. Then the longer side will be (x+8) yards. The area of the rectangle is given by:

Area of the rectangle = length * width.

256 = x*(x+8)

x^2 + 8x = 256. Applying the completing the square method gives:

(x)^2 + 2(x)(4) + (4)^2 = 256 + 16

(x+4)^2 = 272. Taking square root on both sides gives:

x+4 = 16.49 or x+4 = -16.49 (to the nearest 2 decimal places).

x = 12.49 or x = -20.49.

Since length cannot be negative, therefore x = 12.49 yards.

Since smaller side = x yards, thus smaller side = 12.49 yards.

Since larger side = (x+8) yards, thus larger side = 12.49+8 = 20.49 yards.

Thus, the length and the width to minimize the perimeter of fencing is 20.49 yards and 12.49 yards respectively!!!

Answer:

even

Step-by-step explanation:

f(-x)=f(x) means f is even

f(-x)=-f(x) means f is odd

If you get neither f(x) or -f(x), you just say it is neither.

f(x)=x^4+7x^2-30

f(-x)=(-x)^4+7(-x)^2-30

f(-x)=x^4+7x^2-30

f(-x)=f(x)

so f is even.

Notes:

(-x)^even=x^even

(-x)^odd=-(x^odd)

Examples (-x)^88=x^88   and    (-x)^85=-(x^85)

Answer: even

Step-by-step explanation:

By definition a function is even if and only if it is fulfilled that:

[tex]f(-x) = f(x)[/tex]

By definition, a function is odd if and only if it is true that:

[tex]f (-x) = -f(x)[/tex]

Then we must prove the parity for the function: [tex]f(x) = x^4 + 7x^2 - 30[/tex]

[tex]f(-x) = (-x)^4 + 7(-x)^2 - 30[/tex]

[tex]f(-x) = x^4 + 7x^2 - 30=f(x)[/tex]

Note that for this case it is true that: [tex]f(-x) = f(x)[/tex]

Finally the function is even

a.  The power series representation for [tex]\( f(x) = \frac{1}{(10 + x)^2} \)[/tex] is:

[tex]f(x) = \sum_{n=0}^{\infty} (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n[/tex]

with a radius of convergence of 10.

b. The power series representation for [tex]\( f(x) = \frac{1}{(10 + x)^3} \)[/tex] is:

[tex]f(x) = \sum_{m=0}^{\infty} (m + 2)(m + 1) \left(-\frac{1}{10}\right)^{m+2} x^m[/tex]

with a radius of convergence of 10.

Question a:

To find a power series representation for the function [tex]\( f(x) = \frac{1}{(10 + x)^2} \)[/tex].

The sum of an infinite geometric series is given by:

[tex]\frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n[/tex]

where [tex]\( |r| < 1 \)[/tex] for convergence.

First, let's consider the function [tex]\( g(x) = \frac{1}{10 + x} \)[/tex]. Its power series can be found by rewriting it in a form similar to the geometric series:

The geometric series with [tex]\( r = -\frac{x}{10} \)[/tex]. Thus, its power series is:

[tex]g(x) = \frac{1}{10} \sum_{n=0}^{\infty} \left(-\frac{x}{10}\right)^n[/tex]

To find the power series for [tex]\( f(x) = \frac{1}{(10 + x)^2} \)[/tex], we can differentiate [tex]\( g(x) \)[/tex] term by term, as the derivative of [tex]\( g(x) \) is \( f(x) \)[/tex]. The derivative of [tex]\( g(x) \)[/tex] is:

[tex]g'(x) = \frac{1}{10} \sum_{n=0}^{\infty} n \left(-\frac{1}{10}\right)^n x^{n-1}[/tex]

Since [tex]\( g'(x) = f(x) \)[/tex], we have:

[tex]f(x) = \frac{1}{10} \sum_{n=0}^{\infty} n \left(-\frac{1}{10}\right)^n x^{n-1}[/tex]

Adjust the index and powers to start the series from [tex]\( n = 0 \)[/tex]. Let's change the index by setting [tex]\( m = n - 1 \)[/tex], so [tex]\( n = m + 1 \)[/tex].

Since the series actually starts from [tex]\( m = 0 \) (equivalent to \( n = 1 \))[/tex], we can rewrite it as:

[tex]$$f(x) = \sum_{m=0}^{\infty} (m + 1) \left(-\frac{1}{10}\right)^{m+1} x^m$$[/tex]

For the radius of convergence, [tex]\( R \)[/tex], we can use the ratio test. The ratio test states that for a series [tex]\( \sum a_n \)[/tex], if [tex]\( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \)[/tex], then the series converges if [tex]\( L < 1 \)[/tex]. The terms of our series are [tex]\( a_n = (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n \)[/tex].

The terms of our series are [tex]\( a_n = (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n \)[/tex]. Applying the ratio test:

[tex]\lim_{n \to \infty} \left| \frac{(n + 2)}{(n + 1)} \cdot \left(-\frac{1}{10}\right) \cdot x \right|[/tex]

As [tex]\( n \)[/tex] approaches infinity, the term [tex]\( \frac{(n + 2)}{(n + 1)} \)[/tex] approaches 1, so the limit simplifies to:

[tex]\lim_{n \to \infty} \left| -\frac{x}{10} \right| = \frac{|x|}{10}[/tex]

For the series to converge, this limit must be less than 1:

[tex]\frac{|x|}{10} < 1[/tex]

[tex]|x| < 10[/tex]

Thus, the radius of convergence, [tex]\( R \)[/tex], is 10.

Power series representation for [tex]\( f(x) = \frac{1}{(10 + x)^2} \)[/tex] is:

[tex]f(x) = \sum_{n=0}^{\infty} (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n[/tex]

Question b:

To find a power series representation for [tex]\( f(x) = \frac{1}{(10 + x)^3} \)[/tex], we can use the result from part (a), where we found a power series for [tex]\( \frac{1}{(10 + x)^2} \)[/tex], and differentiate it once more.

From part (a), we have:

[tex]\frac{1}{(10 + x)^2} = \sum_{n=0}^{\infty} (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n[/tex]

To find the power series for [tex]\( f(x) = \frac{1}{(10 + x)^3} \)[/tex], we differentiate the series for [tex]\( \frac{1}{(10 + x)^2} \)[/tex] term by term.

The derivative of [tex]\( (n + 1) \left(-\frac{1}{10}\right)^{n+1} x^n \)[/tex] with respect to [tex]\( x \)[/tex] is:

[tex](n + 1) n \left(-\frac{1}{10}\right)^{n+1} x^{n-1}[/tex]

power series for [tex]\( f(x) \)[/tex] is:

[tex]f(x) = \sum_{n=0}^{\infty} (n + 1) n \left(-\frac{1}{10}\right)^{n+1} x^{n-1}[/tex]

Change the index by setting [tex]\( m = n - 1 \)[/tex], so [tex]\( n = m + 1 \)[/tex]. Then, our series becomes:

[tex]f(x) = \sum_{m=-1}^{\infty} (m + 2)(m + 1) \left(-\frac{1}{10}\right)^{m+2} x^m[/tex]

Since the series actually starts from [tex]\( m = 0 \)[/tex] (equivalent to [tex]\( n = 1 \))[/tex], we can rewrite it as:

[tex]f(x) = \sum_{m=0}^{\infty} (m + 2)(m + 1) \left(-\frac{1}{10}\right)^{m+2} x^m[/tex]

For the radius of convergence, [tex]\( R \)[/tex], we can use the same approach as in part (a).

Applying the ratio test:

[tex]\lim_{m \to \infty} \left| \frac{(m + 3)}{(m + 1)} \cdot \left(-\frac{1}{10}\right) \cdot x \right|[/tex]

As [tex]\( m \)[/tex] approaches infinity, the term [tex]\( \frac{(m + 3)}{(m + 1)} \)[/tex] approaches 1, so the limit simplifies to:

[tex]\lim_{m \to \infty} \left| -\frac{x}{10} \right| = \frac{|x|}{10}[/tex]

For the series to converge, this limit must be less than 1:

[tex]\frac{|x|}{10} < 1[/tex]

[tex]|x| < 10[/tex]

Power series representation for [tex]\( f(x) = \frac{1}{(10 + x)^3} \)[/tex] is:

[tex]f(x) = \sum_{m=0}^{\infty} (m + 2)(m + 1) \left(-\frac{1}{10}\right)^{m+2} x^m[/tex]

Answer: There are 85 Helena's client own stocks, bonds and mutual funds.

Step-by-step explanation:

Since we have given that

Let A: who own stocks

B : who own bonds

C : who own mutual fund

So, According to question,

n(A) = 288

n(B) = 200

n(C) = 184

n(A∩B) = 123

n(B∩C) = 106

n( A∩C) = 102

n(A∪B∪C) = 426

As we know the formula :

[tex]n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)\\\\426=288+200+184-123-106-102+n(A\cap B\cap C)\\\\426-341=n(A\cap B\cap C)\\\\85=n(A\cap B\cap C)[/tex]

Hence, there are 85 Helena's client own stocks, bonds and mutual funds.

To determine the number of clients who own stocks, bonds, and mutual funds, we can use the principle of inclusion-exclusion. This principle allows us to properly account for overlap in the sets of clients for different investment types.
We were told the following:
- The total number of clients is 426.
- The number of clients who own stocks is 288.
- The number of clients who own bonds is 200.
- The number of clients who own mutual funds is 184.
- The number of clients who own both stocks and bonds is 123.
- The number of clients who own both stocks and mutual funds is 106.
- The number of clients who own both bonds and mutual funds is 102.
Now, when we sum up the number of clients who own stocks, bonds, and mutual funds individually, we're double-counting those clients who have investments in more than one of these. We need to subtract the clients who are counted twice.
So, let's add up all the individuals:
S + B + M = 288 + 200 + 184 = 672
Now, let's subtract the number of clients who were double-counted:
(S + B + M) - (SB + SM + BM) = 672 - (123 + 106 + 102) = 672 - 331 = 341
However, in this calculation, we've subtracted clients who own all three: stocks, bonds, and mutual funds, three times (once for each pair), and then added them back in only once, so we've subtracted them two times too many.
Therefore, we need to correct for this: to find the number of people who own all three, we add the total number of clients (since everyone owns at least one of the three) and then subtract the sum we have just calculated.
Total + All three (overcorrected) = Total clients
426 + All three (overcorrected) = 341
To solve for the overcorrection (the actual number of clients who own all three types), we can now rearrange the equation:
All three (overcorrected) = 341 - 426
All three (overcorrected) = -85
However, since the number of people cannot be negative, this outcome indicates a logical inconsistency. Such an inconsistency generally means there must have been a mistake in either the data provided or the calculations based on that data. Under normal circumstances, you would go back and verify the numbers. But given this answer, it would suggest that the data provided has some inconsistencies, and it is not possible for a negative number of clients to own all three funds.

I'm guessing you intended to say [tex]X[/tex] has mean [tex]\mu_X=E[X]=25[/tex] and standard deviation [tex]\sigma_x=\sqrt{\mathrm{Var}[X]}=6[/tex], and [tex]Y[/tex] has means [tex]\mu_Y=E[Y]=30[/tex] and standard deviation [tex]\sigma_Y=\sqrt{\mathrm{Var}[Y]}=4[/tex].

If [tex]W=X+Y[/tex], then [tex]W[/tex] has mean

[tex]E[W]=E[X+Y]=E[X]+E[Y]=55[/tex]

and variance

[tex]\mathrm{Var}[W]=E[(W-E[W])^2]=E[W^2]-E[W]^2[/tex]

Given that [tex]\mathrm{Var}[X]=36[/tex] and [tex]\mathrm{Var}[Y]=16[/tex], we have

[tex]\mathrm{Var}[X]=E[X^2]-E[X]^2\implies E[X^2]=36+25^2=661[/tex]

[tex]\mathrm{Var}[Y]=E[Y^2]-E[Y]^2\implies E[Y^2]=16+30^2=916[/tex]

Then

[tex]E[W^2]=E[(X+Y)^2]=E[X^2]+2E[XY]+E[Y^2][/tex]

[tex]X[/tex] and [tex]Y[/tex] are independent, so [tex]E[XY]=E[X]E[Y][/tex], and

[tex]E[W^2]=E[X^2]+2E[X]E[Y]+E[Y^2]=661+2\cdot25\cdot30+916=3077[/tex]

so that the variance, and hence standard deviation, are

[tex]\mathrm{Var}[W]=3077-55^2=52[/tex]

[tex]\implies\sqrt{\mathrm{Var}[W]}=\sqrt{52}=\boxed{2\sqrt{13}}[/tex]

# # #

Alternatively, if you've already learned about the variance of linear combinations of random variables, that is

[tex]\mathrm{Var}[aX+bY]=a^2\mathrm{Var}[X]+b^2\mathrm{Var}[Y][/tex]

then the variance of [tex]W[/tex] is simply the sum of the variances of [tex]X[/tex] and [tex]Y[/tex], [tex]\mathrm{Var}[W]=36+16=52[/tex], and so the standard deviation is again [tex]\sqrt{52}[/tex].

The value is W = sqrt(52) = 7.211.

To find the standard deviation of the sum of two independent random variables X and Y, use the formula W = sqrt(X² + Y²), plugging in the given values to calculate the standard deviation of W as 7.211.

The standard deviation of the sum of two independent random variables X and Y is the square root of the sum of their variances:

W = sqrt(X² + Y²)

Substitute the given values to find the standard deviation of W:

Answer: 0.9890

Step-by-step explanation:

Given : Mean : [tex]\mu=50[/tex]

Standard deviation : [tex]\sigma =7[/tex]

We assume the random variable X is normally distributed

The formula to calculate the z-score :-

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

For x=34.

[tex]z=\dfrac{34-50}{7}=-2.2857142\approx-2.29[/tex]

The p-value =[tex]P(z>-2.29)=1-P(z<-2.29)[/tex]

[tex]=1-0.0110107=0.9889893\approx0.9890[/tex]

Hence, [tex]P(X>34)=0.9890[/tex]

Answer:

The product of the y-coordinates of the solutions is equal to zero

Step-by-step explanation:

we have

[tex]x^{2}+y^{2}=9[/tex] -----> equation A

[tex]x+y=3[/tex] ------> equation B

Solve by graphing

Remember that the solutions of the system of equations are the intersection point both graphs

using a graphing tool

The solutions are the points (0,3) and (3,0)

see the attached figure

The y-coordinates of the solutions are 3 and 0

therefore

The product of the y-coordinates of the solutions is equal to

(3)(0)=0

The steps to solve the system of equations involve isolating x in one equation and substituting into the other. Solving yields two solutions for y, y = 0 and y = 3. Their product is 0.

The system of equations given are [tex]x^2+y^2=9[/tex] and x+y=3. From the second equation, we can isolate x as x = 3 - y and substitute into the first equation, yielding: [tex](3 - y)^2 + y^2 = 9[/tex]. This simplifies to [tex]2y^2 - 6y + 9 = 9,[/tex]and then to [tex]2y^2 - 6y = 0[/tex]. If we factor y from this equation, we get y(2y - 6) = 0, giving two possible solutions for y: y = 0, and y = 3. As asked, the product of these y-coordinates is 0 * 3 = 0.

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a. Expected value is defined by

[tex]E[X]=\displaystyle\sum_xx\,p(x)[/tex]

so we get

[tex]E[X]=1\cdot0.05+2\cdot0.10+4\cdot0.30+8\cdot0.45+16\cdot0.10[/tex]

[tex]\boxed{E[X]=6.65}[/tex]

b. Variance is defined by

[tex]V[X]=E[(X-E[X])^2][/tex]

so with the expectation found above, we have

[tex]V[X]=E[(X-6.65)^2][/tex]

[tex]V[X]=\displaystyle\sum_x(x-6.65)^2\,p(x)[/tex]

(by definition of expectation)

[tex]V[X]=(1-6.65)^2\cdot0.05+(2-6.65)^2\cdot0.10+(4-6.65)^2\cdot0.30+(8-6.65)^2\cdot0.45+(16-6.65)^2\cdot0.10[/tex]

[tex]\boxed{V[X]=15.4275}[/tex]

c. Standard deviation is the square root of variance:

[tex]\boxed{\sqrt{V[X]}\approx3.928}[/tex]

d. I assume "shortcut formula" refers to

[tex]V[X]=E[X^2]-E[X]^2[/tex]

which is easily derived from the definition of variance. We have (by def. of expectation)

[tex]E[X^2]=\displaystyle\sum_xx^2\,p(x)[/tex]

[tex]E[X^2]=1^2\cdot0.05+2^2\cdot0.10+4^2\cdot0.30+8^2\cdot0.45+16^2\cdot0.10[/tex]

[tex]E[X^2]=59.65[/tex]

and so the variance is again

[tex]V[X]=59.65-6.65^2[/tex]

[tex]\boxed{V[X]=15.4275}[/tex]

as expected.

To compute E(X), multiply each outcome by its probability and sum them up. Compute V(X) directly from the definition and also using the shortcut formula. Compute the standard deviation of X.

To compute E(X), we need to multiply each outcome x by its corresponding probability p(x) and sum them up. So, E(X) = 1(0.05) + 2(0.10) + 4(0.30) + 8(0.45) + 16(0.10) = 7.6 GB.

To compute V(X) directly from the definition, we need to first compute the squared deviations of each outcome from the expected value, which is 7.6 GB. Then, multiply each squared deviation by its corresponding probability, and sum them up. So, V(X) = (1 - 7.6)^2(0.05) + (2 - 7.6)^2(0.10) + (4 - 7.6)^2(0.30) + (8 - 7.6)^2(0.45) + (16 - 7.6)^2(0.10) ≈ 51.64 GB^2.

The standard deviation of X is the square root of the variance, which is SD(X) ≈ √(51.64) ≈ 7.19 GB.

To compute V(X) using the shortcut formula, we can use the formula: V(X) = E(X^2) - [E(X)]^2. First, we compute E(X^2) by multiplying each outcome squared by its corresponding probability and summing them up. Then, we subtract the square of E(X) to find V(X). This gives us V(X) = (1^2)(0.05) + (2^2)(0.10) + (4^2)(0.30) + (8^2)(0.45) + (16^2)(0.10) - [7.6]^2 ≈ 51.64 GB^2.

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Final answer:

The area under the standard normal distribution curve between z = -2 and z = 2 corresponds to approximately 84.4 %.

Explanation:

To find the indicated area under the curve of the standard normal distribution between z = -2 and z = 2, we refer to a z-table that provides us with the area under the curve to the left of a given z-score.

First, we find the area under the normal curve to the left of z = 2, which typically is around 0.8672.

Since the normal distribution is symmetric about the mean, the area to the left of z = -2 is the same as the area to the right of z = 2, which is 1 - 0.8672 = 0.0228.

The total area between z = -2 and z = 2 is the area to the left of z = 2 minus the area to the left of z = -2, or 0.8672 - 0.0228.

The difference gives us approximately 0.8444, which represents the probability that a value falls within 2 standard deviations of the mean in a standard normal distribution.

Converting this to a percentage, we multiply by 100 to find that about 84.4 % of the area is within 2 standard deviations of the mean.

Answer with explanation:

Equation of Plane having Direction cosines A, B and C passing through points, p, q and r is

⇒A (x-p)+B(y-q)+C(z-r)=0

The plane passes through the points A(-2, 3,-1), B(0, 5, 2) and C(-1, -2, 1).

→A(x+2)+B(y-3)+C(z+1)=0----------(1)

→A(0+2)+B(5-3)+C(2+1)=0

2 A +2 B+3 C=0

→A(-1+2)+B(-2-3)+C(1+1)=0

A -5 B+2 C=0

[tex]\Rightarrow \frac{A}{4+15}=\frac{B}{3-4}=\frac{C}{-10-2}\\\\\Rightarrow \frac{A}{19}=\frac{B}{-1}=\frac{C}{-12}=k\\\\A=19 K,B=-K, C=-12K[/tex]

Substituting the value of A , B and C in equation (1)

⇒19 K(x+2)-K(y-3)-12 K(z+1)=0

⇒19 x +38 -y +3-12 z-12=0

⇒19 x -y -12 z +29=0, is the required equation of Plane in Cartesian form.

⇒(19 i -j -12 k)(xi +y j+z k)+29=0 ,is the required  vector equation of the plane.

Answer:

Part (a) The value of Z is 0.10396. Part (b) The value of Z is 0.410008.

Step-by-step explanation:

Consider the provided information.

Part (a)

In order to find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.5414, simply find 0.5414 in the table and search for the appropriate Z-value.

Now, observing the table it can be concluded that the value of Z is 0.10396.

Part (b)

Consider the number 65.91%

The above number can be written as 0.6591.

Now, find 0.6591 in the table and search for the appropriate Z-value.

By, observing the table it can be concluded that the value of Z is 0.410008.

Using a Z-table, we find that a z-score of approximately 0.1 will give us 0.5414 of observations less than z in a standard normal distribution. Similarly, for 65.91% of observations being greater than z, we subtract this from 1 and find z to be approximately 1.0.

To find a number, z, such that a certain proportion of observations are less than z in a standard normal distribution, we use a Z-table. In the case where observations less than z comprise 0.5414 of the total, we cross reference this probability in the Z-table to find that z is approximately 0.1.

Similarly, when we need to find the number z where 65.91% of all observations from a standard normal distribution are greater than z, we subtract this percentage from 1, as we are interested in the observations to the left of z. Doing this, we get 0.3409. Checking the Z-table shows that the z-score that corresponds with this area under the curve (or probability) is approximately 1.0.

Remember that a standard normal distribution is denoted Z ~ N(0, 1), meaning it has a mean of 0 and a standard deviation of 1. When calculating z-scores, this allows us to see how many standard deviations a certain point is from the mean (µ).

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Answer:

Choice A.

Step-by-step explanation:

Let x = price of a helmet, y = price of a bat, z = price of a ball.

Company A can provide 9 helmets, 6 bats, and 12 balls for $525.

9x + 6y + 12z = 525

Company B can provide 10 helmets, 8 bats, and 10 balls for $600.

10x + 8y + 10z = 600

Company C can provide 8 helmets, 5 bats, and 15 balls for $500.

8x + 5y + 15z = 500

Answer: Choice A.

Answer:

The correct option is A.

Step-by-step explanation:

Let the price of a helmet is x, the price of a bat is y and the price of a ball is z.

It is given that Company A can provide 9 helmets, 6 bats, and 12 balls for $525. The equation for Company A is

[tex]9x+6y+12x=525[/tex]

It is given that Company B can provide 10 helmets, 8 bats, and 10 balls for $600. The equation for Company B is

[tex]10x+8y+10x=600[/tex]

It is given that Company C can provide 8 helmets, 5 bats, and 15 balls for $500. The equation for Company C is

[tex]8x+5y+15x=500[/tex]

The system of equations is

[tex]9x+6y+12x=525[/tex]

[tex]10x+8y+10x=600[/tex]

[tex]8x+5y+15x=500[/tex]

Therefore the correct option is A.

Answer:

Yes.

Step-by-step explanation:

If a graph G doesn't have a circuit, we must have that

[tex]|E(G)|=|V(G)|-1[/tex]

where [tex]|E(G)|[/tex] is the number of edges of the graph and [tex]|V(G)|[/tex] the number of vertices. However, in this case it holds that

[tex]|E(G)|=72>68=|V(G)|.[/tex]