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Using generating function number of different selections of r hot dogs when there are four types of hot dogs are g(x) = ( 1/(1-x) )⁴.

For a sequence a₀,a₁,...,aₙ,... the corresponding generating function f(x) is the sequence

f(x)=a₀+a₁x+...+aₙxₙ+...= ∑aᵢxᵢ , 0 ≤ i ≤ ∞

Since here we have 4 types of hot dogs, from which we need to select "r" hot dogs, we would use a generating function as follows:

There can be any number of selections from any pile of hot dogs ( repitions is allowed). therefore, from each pile, you could pick up 1 or 2 or 3 or ... hot dogs. The same would go for all types of hot dogs. Therefore ,the generating function for the same would be,

g(x) = (1 + x + x² + x³ + x⁴ + ...)(1 + x + x² + x³ + x⁴ + ...)(1 + x + x² + x³+ x⁴ + ...)(1 + x + x² + x³+ x⁴+ ...)

Therefore, g(x) = (1 + x + x² + x³ + x⁴ + ...)⁴

This is an infinite geometric series, which can be solved to give the sum: g(x) = ( 1/(1-x) )⁴. Therefore, our required the coefficient of xʳ in (1/ (1-x) )⁴.

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The random variable X has a pdf P(X=x) for x = 1,2,3 then,

1) E(x)= 2.2

2) E(3) = 0.9

3) E(x²) = 2.26    

4) E(5x + 3)= 14.

Probability Density Function (pdf) is a statistical expression that defines a probability distribution for a discrete random variable as opposed to a continuouse random variable.

We have given that,  

E(x) = ∑xipi

1) E(x) = ∑xipi

         = x₁p₁ + x₂p₂ + x₃p₃

         = 1(0.1) + 2(0.6) + 3(0.3)

         = 0.1 + 1.2 + 0.9

  E(x) = 2.2

2)E(3) = x₃p₃

          = 3(0.3)

   E(3) = 0.9

3) E(x²) = ∑xi²pi²

           = x₁²p₁² + x₂²p₂² + x₃²p₃²

           = 1(0.1)² + 4(0.6)² + 9(0.3)²

           = 0.01 + 1.44 + 0.81

   E(x²) = 2.26

4) E(5x + 3) = ∑5(xipi) + 3

                  = (5x₁p₁ + 5x₂p₂ + 5x₃p₃) + 3

                  = (5(0.1) + 10(0.6) + 15(0.3)) + 3

                  = 0.5 + 6 + 4.5 + 3

   E(5x + 3) = 14

Therefore the random variable X has a pdf P(X=x) for x = 1,2,3 is

E(x) = ∑xipi then E(x) = 2.2, E(3) = 0.9, E(x²) = 2.26 and E(5x + 3) = 14.

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Answer:                                                                                                                                             Step-by-step explanation:

the value of t such that the area in the left tail of the t -distribution is 0.05, if the sample size is 48. t is 0.76197086

Consider a t distribution with 7 degrees of freedom. Compute P(-1.29 < t < 1.29)

P(-1.29 < t < 1.29) would be the area under the t distribution curve with 7 degrees of freedom between -1.29 and 1.29, that is in the interval (-1.29, 1.29).

This can be done the old style by looking up in a table or by using the technology with a spreadsheet.

In Excel, the function TDIST(x,n,2) with x>0 gives the area outside the interval (-x, x) of the t distribution with n degrees of freedom.

So TDIST(1.29,7,2) gives the area outside (-1.29, 1.29).

If we subtract this value from 1 we get the desired result

Hence  

P(-1.29 < t < 1.29) = 1 - TDIST(1.29,7,2) = 1 - 0.23802914 = 0.76197086

In OpenOffice Calc, the function is the same replacing “,” with “;”  

That is

P(-1.29 < t < 1.29) = 1 - TDIST(1.29;7;2) = 0.76197086

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An investment of $6,400 to increase to $14,000 if it is invested at 5% per year compounded continuously is ≈15.6

Define compound interest?

Given that :

P = $ 6,400

Q = $ 14,000

r = 5%

Let P be the initial investment

r be the rate of interest

t be the time

Q be the increase amount

Q = P [tex]e^{rt} }[/tex]

14,000 = 6,400 ([tex]e^{0.05t}[/tex])

[tex]e^{0.05t}[/tex] = [tex]\frac{14000}{6400}[/tex]

[tex]e^{0.05t}[/tex]  = 2.187

Taking log on both sides , we get

0.05t = ln (2.187)

0.05t = 0.78

       t = [tex]\frac{0.78}{0.05}[/tex]

       t ≈ 15.6

An investment of $6,400 to increase to $14,000 if it is invested at 5% per year compounded continuously is ≈ 15.6

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Answer:

42

Step-by-step explanation:

To simplify the expression -7x3x(-2)x(-5)x(-1), we need to apply the rules of exponentiation and multiplication.

First, we note that the terms -7, -2, -5, and -1 are all negative numbers, which means that they will be multiplied together to give a positive result.

Next, we apply the rule that says that any term raised to the power of 0 is equal to 1. Since -1 raised to the 0 power is 1, this means that the expression -7x3x(-2)x(-5)x(-1) is equivalent to -7x3x(-2)x(-5)x1.

Next, we apply the rule that says that any term raised to the power of 1 is equal to itself. Since 3 raised to the 1 power is 3, and 5 raised to the 1 power is 5, this means that the expression -7x3x(-2)x(-5)x1 is equivalent to -7x3x(-2)x5x1.

Finally, we apply the rule that says that the product of any number and its multiplicative inverse (or reciprocal) is equal to 1. Since the reciprocal of 5 is 1/5, this means that the expression -7x3x(-2)x5x1 is equivalent to -7x3x(-2)x(1/5)x1.

We can now use the rules of multiplication to simplify the expression further. Since any number multiplied by 1 is equal to itself, this means that the expression -7x3x(-2)x(1/5)x1 is equivalent to -7x3x(-2)x(1/5).

Next, we use the rule that says that the product of two negative numbers is a positive number. This means that the expression -7x3x(-2)x(1/5) is equivalent to 7x3x2x(1/5).

Finally, we can apply the rules of exponentiation and multiplication to evaluate the expression. Since any number raised to the power of 1 is equal to itself, this means that the expression 7x3x2x(1/5) is equivalent to 7x3x2. We can then multiply these terms together to get the final result:

7x3x2 = 42

Therefore, the simplified expression is equal to 42.

We cannot conclude that the percentage of people in New York who would vote for the incumbent president is different from 53%.

The correct answer is B) Do not reject H0.

What is a null hypothesis in probability?

The null hypothesis is a typical statistical theory that suggests that no statistical relationship and significance exists in a set of given single observed variables, between two sets of observed data and measured phenomena.

In this situation, the null hypothesis H0 is that the percentage of people in New York who would vote for the incumbent president is equal to 53%. The alternative hypothesis Ha is that the percentage of people in New York who would vote for the incumbent president is different from 53%.

The test statistic for this hypothesis test is z = -1.60. This means that the sample result of 45% is 1.60 standard deviations away from the hypothesized value of 53%.

The p-value is the probability of observing a test statistic as extreme as or more extreme than the one observed, given that the null hypothesis is true. In this case, the p-value is 0.1090, which is greater than the significance level of 0.10.

Since the p-value is greater than the significance level, we do not have sufficient evidence to reject the null hypothesis.

Hence, we cannot conclude that the percentage of people in New York who would vote for the incumbent president is different from 53%.

The correct answer is B) Do not reject H0.

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After solving the given expression, the value obtained for x will be equal to -6,

If a mathematical operation includes at least two words that are connected by an operator and either comprise numbers, variables, or both, it is referred to as an expression.

The operations with reflection coefficients include adding, subtracting, multiplying, and dividing. A mathematical operation like reduction, addition, multiplication, or division is used to include terms in an expression.

As per the given equation in the question,

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

2/3x + 5/6 = 1/2x - 1/6

Substitute like terms together,

2/3x - 1/2x = -1/6 - 5/6

(4x - 3x)/6 = (-1 - 5)/6

x/6 = -6/6

x/6 = -1

x = -6

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The researcher claim is invalid and the p-value based on alpha = 0.05 is   approx. = 0.

In the question ,

it is given that ,

a researches claims that per person yearly consumption (μ) is 52 gallons .

sample size (n) = 50 .

the mean of the consumption (x) = 56.3 .

the standard deviation (σ) is = 3.5 gallons .

the α = 0.05 .

let the null hypothesis ,

H₀ : μ = 52 and

H₁ : μ ≠ 52 .

the test statistics is : z = (x - μ)/(σ/√n)

the critical value ( rejection region ) : {z : |z| ≥ z₀.₀₂₅ = 1.96 }

So , z = (56.3 - 52)/(3.5/√50)

Simplifying further ,

we get ,

z = 8.69

We reject the null hypothesis H₀ , since the absolute value of the test statistics 8.69 is greater than the critical value 1.96 .

Since this is a two tailed test,  p-value must be multiplied by two

2 × P(z > 8.69) ≈  0

Since the p-value is less than α(0.05), So ,we reject the claim of the researcher that the mean yearly consumption of soft drinks per person is 52 gallons

Hence , the claim is not valid .

Therefore , The claim is invalid and p-value is ≈ 0 .

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If the number of solutions to the equation that satisfy the given condition is the number of ways to place all the items into the categories. Then, the answer is 210 .

In the question ,

it is given that ,

number 22 is divided into 22 individual items ,

the three categories in which these items are placed are x₁, x₂ and x₃ .

x₁ + x₂ + x₃ = 22    .....equation(1)

(x₁ - 1) + (x₂ - 1) + (x₃ - 1) = 22 - 3 = 19

let a = x₁ - 1 , b = x₂ - 1  and c = x₃ - 1 .

Hence , a + b + c = 19    ....equation(2)

Now , because the number of solution of equation(1) excluding zero will be equal to the number of solutions of equation(2) including the zero .

So , we subtracted 1 from x₁, x₂ and x₃ respectively .

Now , we have to divide 19 into 3 parts where any part can be zero also or any part is a whole number .

For Example : let us have 19 apples , which must be divided into 3 parts by placing 2 J's among 19 A's .

Now we have reduced the problem to such that we have to find the permutation of 19 A's and 2 J's that is 21 characters in which every permutation is a solution .

So , the  number of ways of arranging 21 characters in which 19 are same(A)  and 2 are same (J)

= 21!/(2! * 19!)

= 21*10 = 210

It means there are 210 non negative integral solutions of a+b+c = 19 .

It means total positive solutions of x₁ + x₂ + x₃ = 22  is 210 .

Therefore , the number of solution to the equation is 210 .  

The given question is incomplete , the complete question is

Start by thinking of the number 22 as divided into 22 individual items and the variables x₁, x₂, and x₃ as three categories into which these items are placed. since each [tex]x_{i}[/tex] is a positive integer, start by placing one item in each category and distribute the remaining items among the categories. The number of solutions to the equation that satisfy the given condition is the number of ways to place all the items into the categories. Thus, the answer is ______  .

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Answer:

Step-by-step explanation:

The odds of selecting 5 cards from a deck of 52 are 52c5. Probability = Maximum outcomes / Possible outcomes for the given circumstance = 52!/47!*5! = 2598960. Now, in poker, the chance of selecting any two numbers from a pool of 13 distinct numbers is 13c2.

Possible poker hands

C(n,r)= C(52,5)

=52!/(5!(52-5!)

=2598960

Hand with pattern

(X,X,Y,Y,Z)

Number of choices for x and y

C(n,r)= C(13,2)

=13!(2!(13-2)!)

=78

Ways to get 2 X's

C(n,r)= C(4,2)= 4!(2!(4-2)!) =6

Similarly, ways to get  2x's is 6

We have 44 choice of occurance for Z

Hands with two pairs =78*44*6*6= 123552

So, Probability of Two pairs is =123552/2598960=0.047539  

P(Two pairs) = 0.0475

How many ways can we choose two pairs in a poker hand?

There are 13 ways to choose the denomination of the first pair, 12 ways to choose the denomination of the second pair, and 4 choices for the remaining card. Thus, there are 13\cdot 12\cdot 4 = 624 ways to choose two pairs in a poker hand.

In poker, a hand is made up of 5 cards from a deck of 52 cards. The different possible hands in poker include a royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, and one pair.

In this problem, we are asked to find the probability of getting a hand with two pairs.

To find the probability of two pairs, we first need to find the total number of possible poker hands. This is done using the combination formula C(n,r) where n is the total number of cards (52 in this case) and r is the number of cards needed to make a hand (5 in this case). This gives us a total of 2598960 possible poker hands.

Next, we need to find the number of possible hands with two pairs. A hand with two pairs has a pattern of X,X,Y,Y,Z where X and Y are the ranks of the pairs and Z is the remaining card. There are 13 ranks in a deck of cards (Ace, 2, 3, ..., 10, Jack, Queen, King) and we need to choose 2 of these ranks to be the ranks of the pairs. This can be done in C(13,2) ways, which is 78.

There are 4 cards of each rank in a deck, so there are 6 ways to choose the 2 cards of rank X and 6 ways to choose the 2 cards of rank Y.

Finally, we have 44 choices for the remaining card (Z) because there are 13 ranks and we have already used up 2 ranks for the pairs.

Therefore, the number of possible hands with two pairs is 78*6*6*44 = 123552. The probability of getting a hand with two pairs is then 123552/2598960 = 0.047539 or approximately 0.0475.

Therefore, the final answer is P(Two pairs) = 0.0475.

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Answer: The probability of receiving a hand with two pairs is  0.04322.

Step-by-step explanation:

How many ways are there in a poker hand to pick two pairs?

There are 13 ways to choose the denomination of the first pair, 12 ways to choose the denomination of the second pair, and 4 choices for the remaining card. Thus, there are 13*12*4 = 624 ways to choose two pairs in a poker hand.

In poker, a hand is made up of 5 cards from a deck of 52 cards. The different possible hands in poker include a royal flush, straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, and one pair.

To choose the denominations of the two pairings, there are C(13,2)=78 possible options. There are C(4,2)=6 methods to select the two cards in each pair for each of these approaches. The remaining card's denomination can be determined in C(4,1)=4 different ways. As a result, there are 78*6*6*4=11232 different ways to deal a hand with two pairs.

A hand of five cards can be dealt in C(52,5)=2598960 different ways. Therefore, the probability of receiving a hand with two pairs is  11232/2598960=0.04322.

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The correct answer is Option E, For values of W repeatedly selected at random from the distribution the mean of the selected values will approach 2.8.

The expected value can be found by using the formula ∑x*p(x)

The random variable W can take on the values of 0, 1, 2, 3, or 4. The expected value of W is 2.8.

Expected value (EV) describes the long-term average level of a random variable based on its probability distribution.

The expected value can be found by using the formula ∑x*p(x)

A randomly selected value of W must be equal to 2.8.

This is not true as the expected value is like the average

B The values of W vary by about 2.8 units from the mean of the distribution.

This is not true, as it is not the difference between an observation to its mean.

с The mean of a random sample of values selected from the distribution will be 2.8 It is false

D A value of randomly selected from the distribution will be less than 2.8 units of the mean. This statement is also false

E For values of W repeatedly selected at random from the distribution the mean of the selected values will approach 2.8.

Thus the correct answer is Option E, For values of W repeatedly selected at random from the distribution the mean of the selected values will approach 2.8.

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The measure of the length (q) of the right-angle triangle will be 2√14 units. Then the correct option is B.

It's a form of a triangle with one 90-degree angle that follows Pythagoras' theorem and can be solved using the trigonometry function.

The right-angle triangles are shown.

In similar triangles ΔQTS and ΔSTR, we have

ST / TR = QT / ST

ST / 4 = 10 / ST

s² = 40

Then the measure of the length (q) is given as.

q² = s² + 4²

q² = 40 + 16

q² = 56

q = 2√14

The measure of the length (q) of the right-angle triangle will be 2√14 units. Then the correct option is B.

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The inequality for the situation is 2x + 4 < 25 and the oldest age of Kim is 10 years old

The inequality is the mathematical statement that shows the relationship between two expressions with inequality sign. The inequality signs are less than, less than or equal, greater than, greater than or equal.

Consider the Kim's age as x

Josh is 4 years older than 2 times Kim's age

Two times Kim's age = 2x

4 years older than 2 times Kim's age = 2x + 4

The age of Josh = 2x + 4

Sum of their ages is less than 25

The inequality is

2x + 4 < 25

2x < 21

x < 21/2

x < 10.5

Therefore, the oldest age of Kim is 10 years old and the inequality is 2x + 4 < 25

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The perimeter of the polygon is 78 cm. (Option D)

The perimeter of a geometric shape refers to the length that encompasses the figure. Hence, the perimeter of any irregular pentagon can be determined by the adding up the lengths of all sides. As the given polygon circumscribes a circle, the sides of the polygon are tangent to the circle. The two tangent theorem states that if two lines are drawn from the same point which lies outside a circle, such that both lines are tangent to the circle, then their lengths are the same. Hence, if two lines that are tangent to a circle meet at a point, then the distance from that point to the points of intersection between the tangents and the circle are equal. Hence, the perimeter of the pentagon is:

P = 8 + 8 + 16 + 16 + 6 + 6 + 9 + 9 = 78 cm

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To correct a number to 3 decimal places, we need to round it to the nearest thousandth. The number 45.7395 is already very close to 45.740, which is the nearest thousandth. Therefore, we can round 45.7395 to 45.740, which is the correct answer to 3 decimal places.

The equation of the line that includes CD, in slope-intercept form, is: B. y = -4/3x + 2.

To write the equations of two lines that are parallel or perpendicular to each other, recall that parallel lines have the same slope (m), while perpendicular lines have slopes that are negative reciprocals.

The slope of y = 3/4x - 11 is 3/4. The negative reciprocal of 3/4 is -4/3. Therefore, the slope of AB that is perpendicular to the graph of y = 3/4x - 11 is, m = -4/3.

Since the CD is parallel to AB, then CD will have the same slope of, m = -4/3.

To write the equation of the line that includes CD, substitute m = -4/3 and (a, b) = (-6, 10) into y - b = m(x - a):

y - 10 = -4/3(x + 6)

Rewrite in slope-intercept form:

y - 10 = -4/3x - 8

y = -4/3x - 8 + 10

y = -4/3x + 2

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Answer:

6y

Step-by-step explanation:

What can you remove form the left side that makes this a false equation?

It's 6y. So place 6y on the right side.

6y + 13 = 6y + 8

Subtract 6y from both sides.

13 = 8

Since 13 = 8 is a false statement, the equation 6y + 13 = 6y + 8 has no solution.

Answer: 6y

Step-by-step explanation:

i did the iready and got it right

its 6y

Answer:

Therefore, Alvin worked 70 hours, Simon worked 55 hours, and Theodore worked 159 hours

Step-by-step explanation:

Let A be the number of hours Alvin worked, S be the number of hours Simon worked, and T be the number of hours Theodore worked.
We know that the total number of hours worked is the sum of the hours each person worked, so we can write the equation:A + S + T = 284
We also know that Alvin worked 15 more hours than Simon, so we can write the equation:A = S + 15
We also know that Theodore worked 6 less than three times as many hours as Simon, so we can write the equation:T = 3S - 6
Substituting the second equation into the first equation, we get:
S + S + 15 + 3S - 6 = 284
Combining like terms, we get:5S + 9 = 284
Subtracting 9 from both sides, we get:5S = 275
Dividing both sides by 5, we get:S = 55
Substituting this value into the equation A = S + 15, we get:
A = 55 + 15A = 70
Substituting this value into the equation T = 3S - 6, we get:T = 3 * 55 - 6T = 165 - 6T = 159

Answer:

depends for what?

Step-by-step explanation:

for shoes, stationary, clothes etc be a bit specific

Therefore , [tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

The probability of winning is actually closer to 1/3 (25/72). Carnival game operators want you to think that a game is fair like I did, so that you will play. If you knew that the probability of winning a game was only about 1/3, most likely you would not waste your money.

Mathematical descriptions of the likelihood that an event will occur or that a claim will be true are called probabilities.

Here,

[tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

Therefore , [tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

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Answer:

x = -7 sqrt(3)

Step-by-step explanation:

Tan(300°)=z/7

Multiply each side by 7

7 Tan(300°)=z

Replacing tan 300 with tan 120 since they are equal

7 Tan(120°)=z

7 ( - sqrt(3)) = z

x = -7 sqrt(3)

Using the confidence interval and null hypothesis we can conclude that the salaries for the finance majors are significantly lower than the salaries of the business analytics majors.

a) The null and alternative hypothesis is given by

H₀:  μ1 − μ2 = 0

H₁:  μ1 − μ2 < 0

b) Test statistic is given by  [tex]\frac{\bar{x}_1-\bar{x}_2-0}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}[/tex]

here:

S₁ = 19000

S₂ = 10000

x₁ = 48237

x₂ = 55217

n₁ = 120

n₂ = 30

Putting the above values we get:

[tex]z=\frac{48237-55217}{\sqrt{\frac{19000^2}{120}+\frac{10000^2}{30}}}[/tex]

or, z = -2.7717744...

or, z ≈ -2.772

c) p-value

= P(Z<-2.772)

= 0.002786

= 0.0028

d) now we will use the p-value to essentially use or reject the the hypothesis.

So the confidence interval of 955 is used for the p-value.

p-value(0.0028) < ∝(0.05)

Hence we will reject the H₀ .

Hence we will conclude that the salaries for the finance majors are significantly lower than the salaries of the business analytics majors.

A confidence interval is a range of estimates for an unknown parameter (CI). The most common confidence level is 95%, but when calculating confidence intervals, other levels, such 90% or 99%, are also occasionally employed.

The confidence level is a measure of how many related CIs over the long run include the actual value of the parameter. For instance, the parameter's true value should be included in 95% of all intervals produced at the 95% confidence level.

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The probability of a negative return if the returns are normally distributed, we can use the mean and standard deviation of the returns and If the distribution is symmetrical but otherwise unknown, it may be more difficult to estimate the probability of a negative return without more information about the shape of the distribution.

To determine how confident we can be that the mean of the given data sample is greater than 40, we would need to perform a statistical test such as a t-test or a z-test to compare the sample mean to the value of 40.

We would need to know the sample size, standard deviation, and any other relevant information about the data to calculate the appropriate test statistic and determine the p-value, which would tell us the probability of observing a sample mean as extreme as the one we obtained if the true population mean was actually 40.

To determine how confident we should be that the portfolio manager will continue to produce positive returns, we would need to analyze the distribution of returns and determine whether it is skewed or symmetrical and whether it follows a normal distribution or a different distribution. If the distribution is skewed or non-normal, it may be more difficult to estimate the probability of positive returns. However, if the distribution is symmetrical and follows a normal distribution, we can use the mean and standard deviation of the returns to calculate the probability of a positive return using the normal distribution.

If the returns of the investment strategy are normally distributed, we can use the mean and standard deviation of the returns to calculate the probability of a negative return using the normal distribution. If the distribution is symmetrical but otherwise unknown, it may be more difficult to estimate the probability of a negative return without more information about the shape of the distribution.

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The question is -

Given the following data sample, how confident can we be that the mean is greater than 40? 64 70 20 58 13 74 84 47 17 2. You are given the following sample of annual returns for the portfolio manager: If you believe that the distribution of returns has been stable over time and will continue to be stable over time; how confident should you be that the portfolio manager will continue to produce positive returns? -7% 7% 19% 23 % -18 % -12% 49% 34% ~6% -20% 3. You are presented with an investment strategy with a mean return of 20% and a standard deviation of 10%_ What is the probability of a negative return if the returns are normally distributed? What if the distribution is symmetrical but otherwise unknown?'

The time that it will take for Kiran to run 6 meters is; 2 seconds

A function table is defined as a visual table that has columns and rows that display the function with regards to the input and output.

Now, the coordinates of the function table are given as;

(0, 0), (3, 1), (6, 2), (9, 3.2), (12, 3.8), (15, 4.6), (18, 6), (21, 6.9), (24, 8.09), (27, 9)

Now, the graph shows that the y-axis represents  the time, t, he took to run various distances, d which is represented on the x-axis.

Now, we want to find the time that it takes Kiran to run 6 meters. From the graph and table attached, it is seen to be 2 seconds.

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The probability that a randomly selected bike is green given that the bike has road bike tires is 0.66

Given :

Nellie is shopping for a new bicycle. She is most interested in color and type of tires.

Probability :

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event .

From the table given :

Total number of 6 road bikes = 4

Total road bikes = 2 + 4 = 6

Probability = 4 / 6

= 2 / 3

= 0.66

Hence the  probability that a randomly selected bike is green given that the bike has road bike tires is 0.66 .

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The lengths of the right-angled triangle are 6 and 8.

A right-angled triangled is a triangle that has one of its angles equal to 90°. The sum of the other two interior angles is also equal to 90°.

The Pythagoras theorem relates the three sides of a right-angled triangle through a simple equation. According to the Pythagoras theorem,

hypotenuse^2 = base^2 + opposite^2

Let base = b, so opposite = b + 2

So,

10^2 = b^2 + (b+2)^2

100 = b^2 + b^2 + 4b + 4

100 = 2b^2 + 4b + 4

50 = b^2 + 2b + 2

b^2 + 2b - 48 = 0

solving this quadratic equation gives, b = 6 and b = -8. Since length cannot be negative, the base is 6 feet and the opposite is 8 feet.

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The acute angle of intersection is  5.2°.

What is acute angle?

Less than 90 degrees is the acute angle measurement. Right angles are 90 degrees in length. Angles that are obtuse are more than 90 degrees. Discover the various sorts of angles and examples of each.

The direction vector normal to the plane is ...

 n = (1, 1, 3)

The direction vector of the line is ...

 m = (1, -3, 1)

Then the angle θ between them can be found from the dot product:

 n•m = |n|·|m|·cos(θ)

 (1·1 +1(-3) +3·1) = 1 -3 +3 = 1 = √(1²+1²+3²)·√(1²+(-3)²+1²)·cos(θ)

 1 = 11·cos(θ)

 θ = arccos(1/11) ≈ 84.8°

This is the angle between the line and the normal to the plane, so the angle between the line and the plane will be the complement of this. Since this angle is not 90°, the line and plane must intersect.

 acute angle = 90° -84.8° = 5.2°

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