Given that Z is a standard normal variable, what is the value k for which P(Z = k) = 0.258 ? O a. 0.84 O b. 0.65 Oc. 0.70 O d. 0.76

If a marble is selected at random from Adrian's Bag of Marbles, then the probability that marble selected from Adrian's bag will not be red is 0.7.

The "Probability" of an "event-A" occurring is defined as the ratio of the number of favorable outcomes for event A to the total number of possible outcomes in a given sample space. It is denoted as P(A).

To find the probability that marble selected will not be red,

we need to find "total-number" of marbles in Adrian's bag and the number of marbles that are not red.

We know that,

⇒ Number of red marbles = 3,

⇒ Number of blue marbles = 7,

So, Total marbles in bag = Number of red marbles + Number of blue marbles,

⇒ 3 + 7 = 10,

The Number of marbles that are not red = Number of blue marbles = 7,

So, probability that marble selected will not be red is :

⇒ Probability (not red) = (Number of marbles that are not red)/(Total number of marbles),

⇒ 7/10,

⇒ 0.7

Therefore, the required probability is 0.7.

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The given question is incomplete, the complete question is

Adrian's Bag of marbles contain 3 Red and 7 Blue Marbles, If a marble is selected at random from Adrian's Bag of Marbles, then What is the probability the Marble selected will NOT be red?

The expected value of investment is 400K.

Let's assume that the expected revenue from the project is 3,200K, which is the midpoint of the given range.

Then, the expected value of the investment can be calculated as follows:

Expected value of investment = Expected revenue - Expected cost

Expected revenue = 3,200K

Expected cost = Expected value of building costs

Expected value of building costs = Mean of building costs = 2,800K

Therefore, the expected value of investment = 3,200K - 2,800K = 400K

Since the expected value of the investment is positive, the project is worth going ahead.

Now, let's investigate if the decision is still valid if the standard deviation varies by +10% from its stated value.

New standard deviation = 220K (10% increase from 200K)

Using the formula from Question 3, we can calculate the probability that the total cost of the project will exceed 2,800K with the new standard deviation:

z = (2,800K - 2,800K) / 220K = 0

P(z > 0) = 0.5

Therefore, the probability that the total cost of the project will exceed 2,800K is still 0.5.

Using the same calculations as above, the expected value of investment with the new standard deviation can be calculated as follows:

Expected value of investment = Expected revenue - Expected cost

Expected revenue = 3,200K

Expected cost = Expected value of building costs

Expected value of building costs = Mean of building costs = 2,800K

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The solution of given expression having scientific notation is 40,000 by using power formula

Scientific notation is a way of writing numbers that are either very large or very small using powers of 10. It is expressed as a number between 1 and 10 multiplied by a power of 10.

An expression is a mathematical statement that contains numbers, variables, and/or operators. It can be a single number, a combination of numbers and operators, or a combination of numbers, variables, and operators.

To evaluate this expression, we can use the rules of scientific notation to multiply the numbers and then simplify the result.

(4.8 x [tex]10^{8}[/tex]) / (1.2 x [tex]10^{4}[/tex]) x (2.2 x [tex]10^{-6}[/tex])

= (4.8 / 1.2) x [tex]10^{8-4-6+6}[/tex] (multiply the numbers and add the exponents)

= 4 x [tex]10^{4}[/tex]

= 40,000

Therefore, the value of the expression is 40,000.

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The value of angle x is 40^o

A given set of angles are said to be supplementary if and only if on addition of the measures of the angles, it forms 180 degrees. This is the measure of an angle on a straight line.

In the given diagram, to find the value of x;

Triangle ABD is an isosceles, thus base angles are equal. So that;

ABD ≅ ADB = 25^o

Thus,

<ABD + x + 115 = 180        (definition of supplementary)

25 + x + 115 = 180

140 + x = 180

x = 180 - 140

 = 40

x = 40^o

Therefore, the value of x is 40^o.

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The length of the spiraling polar curve r = 2[tex]e^{3\theta[/tex] from 0 to 2π is (2/3)√[10]([tex]e^{6\pi[/tex] - 1).

The length of a polar curve can be found using the formula:

L = [tex]\int\limits^a_b[/tex]√[r² + (dr/dθ)²] dθ

where r is the polar function, and a and b are the starting and ending angles of the curve, respectively.

In this case, the polar function is r = 2[tex]e^{3\theta[/tex], and we want to find the length from 0 to 2π. First, we need to find the derivative of r with respect to θ:

dr/dθ = 6[tex]e^{3\theta[/tex]

Plugging in these values into the formula, we get:

L = [tex]\int\limits^{2 \pi}_0[/tex] √[4[tex]e^{6\theta[/tex] + 36[tex]e^{6\theta[/tex]] dθ

L = [tex]\int\limits^{2 \pi}_0[/tex] 2[tex]e^{3\theta[/tex] √[10] dθ

L = 2√[10] [tex]\int\limits^{2 \pi}_0[/tex] [tex]e^{3\theta[/tex] dθ

Using integration by substitution, we can solve this integral:

L = 2√[10] [[tex]e^{3\theta[/tex]/3] (0 to 2π)

L = 2√[10] [([tex]e^{6\pi[/tex] - 1)/3]

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The maximum value is 0 at points (±3, 0) and the minimum value is -12 at points (0, ±3).

To solve this problem using LaGrange Multipliers, let's define the given function and constraint:

f(x, y) = x^2 * y^2 - 4y
g(x, y) = x^2 + y^2 - 9

Now, introduce the LaGrange Multiplier λ (lambda) and find the gradient of both f and g:

∇f(x, y) = (2xy^2, 2x^2y - 4)
∇g(x, y) = (2x, 2y)

The LaGrange Multiplier method states that for optimal points (x, y), ∇f(x, y) = λ∇g(x, y). So, we have the following equations:

2xy^2 = λ(2x)   -> (1)
2x^2y - 4 = λ(2y)   -> (2)

From equation (1), we get:

xy^2 = λ

Substitute λ in equation (2):

2x^2y - 4 = 2y(xy^2)

Now, solve for x and y using the constraint g(x, y) = 0:

x^2 + y^2 = 9

By solving the system of equations, you will find four critical points: (±3, 0) and (0, ±3). Evaluate f(x, y) at these points:

f(±3, 0) = 0
f(0, ±3) = -12

Thus, the maximum value is 0 at points (±3, 0) and the minimum value is -12 at points (0, ±3).

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Let's find the solutions for the given sets O and B. The set O is defined as all odd integers n ∈ Z, while set B contains integers n such that -4 ≤ n ≤ 6.

(a) To find O ∩ B, we need to determine the common elements between sets O and B. The odd numbers in the range -4 to 6 are {-3, -1, 1, 3, 5}. Therefore, O ∩ B = {-3, -1, 1, 3, 5}.

(b) To calculate B - O, we need to remove the elements of O from set B. Set B contains {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}. Removing the odd numbers, we get B - O = {-4, -2, 0, 2, 4, 6}.

(c) To find O in Z, we consider all odd integers n in the set of integers Z. Since Z is an infinite set, O in Z is the set of all odd integers, which can be represented as {..., -5, -3, -1, 1, 3, 5, ...}.

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The Statement asked  based on the Variable is , True.

A variable is a symbol or letter that represents a quantity or value that can vary or change in a given context or problem. Variables can be used to express relationships between quantities or to describe patterns or trends in data. They can be either dependent or independent, depending on the context of the problem. An independent variable is a variable that is changed or controlled by the experimenter or observer, while a dependent variable is a variable that is affected or influenced by the independent variable.

True.

A continuous random variable is one that can take on any value within a certain range or interval, and its value is determined by measurement. Examples of continuous random variables include height, weight, temperature, and time, where the values can be any real number within a certain range.

In contrast, a discrete random variable can only take on certain values, typically integers, and its value is determined by counting. Examples of discrete random variables include the number of heads in a series of coin tosses, the number of cars sold in a day, and the number of defects in a batch of products.

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The general term of the sequence is given by an = (-1)ⁿ⁺¹*2ⁿ/3ⁿ⁻¹, where n is the term number starting from 1.

This formula is obtained by observing that each term is obtained by multiplying the previous term by -2/3 and changing its sign. This pattern can be represented mathematically using exponents, which results in the given formula.

To find the general formula for a sequence, we need to identify the pattern in the given terms. Here, we observe that each term is obtained by multiplying the previous term by -2/3 and changing its sign. This means that the sequence alternates between positive and negative values, and the magnitude of each term is increasing as n increases.

To represent this pattern mathematically, we can use the concept of exponents. Specifically, we can write the numerator of each term as 2ⁿ⁻¹, since the magnitude of each term is increasing by a factor of 2.

Similarly, we can write the denominator of each term as 3ⁿ⁻¹, since the magnitude of each term is decreasing by a factor of 3. Finally, we need to account for the alternating signs of the terms, which is done using the factor.

Putting all these pieces together, we get the formula an = (-1)ⁿ⁺¹*2ⁿ/3ⁿ⁻¹. This formula gives us the nth term of the sequence, assuming that the pattern of the first few terms continues. We can use this formula to find any term of the sequence, without having to compute all the previous terms.

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The probability of C and D being mutually exclusive is not possible because they occur  independently  with a probability of 0.18.

The formula for evaluating the probability of independent events is

[tex]P(A and B) = P(A) * P(B)[/tex]

Then

P(A) and P(B) =  probabilities of events A and B respectively.

For the given case,

we keep two independent events C and D with probabilities

Here,

P(C) = 0.3

P(D) = 0.6

Then, the evaluated  probability of both events occurring together is

[tex]P(C and D) = P(C) * P(D)[/tex]

          [tex]= 0.3 * 0.6[/tex]

          = 0.18

Then, C and D are not mutually exclusive events the reason behind it is  they can occur independently of each other with a probability of 0.18.

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The probability of finding a person with an exact height of 180 cm is close to zero.

The probability of randomly selecting a person who is exactly 180 cm tall is extremely low, since height measurements are typically continuous and not discrete

Height measurements are usually considered as continuous data, meaning they can take on any value within a certain range, and not just specific, discrete values.

In reality, it is highly unlikely to find a person who has an exact height of 180 cm, as height measurements are subject to natural variation and measurement error.

Even with perfect measuring accuracy, the probability of finding a person with an exact height of 180 cm is still extremely low, as human height distribution typically follows a bell-shaped curve or a normal distribution.

The normal distribution of human height is characterized by a range of heights that occur with varying frequencies, and the probability of finding a person with a height that falls exactly at 180 cm is likely to be infinitesimal.

Therefore, based on the principles of continuous data and the natural variation in human height, the probability of randomly selecting a person who is exactly 180 cm tall is essentially negligible.

Therefore, the probability of finding a person who is exactly 180 cm tall is extremely low, close to zero, and can be considered as practically impossible.

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The indefinite integral of [tex]\int\limits x^{(-1/6)} dx[/tex] is [tex]6x^{(5/6) }+ C[/tex] where C is a constant of integration, under the given condition that no limits are provided since its a indefinite integral.

So here we have to proceed by performing a set of calculations that fall under the criteria provided by the principles of indefinite integral
The given indefinite integral is [tex]\int\limits x^{(-1/6)} dx[/tex]
Evaluating the form
[tex]\int\limits x^{(-1/6)} dx[/tex]
[tex]= \int\limits 1/x^{(1/6)} dx[/tex]
[tex]= 6x^{(5/6)} + C[/tex] here C is constant concerning the integration.

Indefinite integral refers to the form expression Which has no limits that projects the family of function that differentiate by a constant.

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The probability that on a randomly selected trip, the number of whales seen is 3 is 22.4%.

To find the probability that on a randomly selected whale watch trip in March, the number of whales seen is 3, we'll use the Poisson distribution with a mean of 4.3. The formula for the Poisson probability is:
P(X=k) = (e^(-λ) * (λ^k)) / k!
Where X is the number of whales seen, k is the desired number of whales (in this case, 3), λ is the mean (4.3), and e is the base of the natural logarithm (approximately 2.71828).
Plugging in the values, we get:
P(X=3) = (e^(-4.3) * (4.3^3)) / 3!
P(X=3) = (0.01353 * 79.507) / 6
P(X=3) ≈ 0.22404
So, the probability of seeing 3 whales on a randomly selected trip is approximately 22.4%.

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There are 1458, 7-digit even numbers that can be formed using all the digits 0, 1, 2, 2, 3, 5, and 5.

To form a 7-digit even number, the last digit must be even. We have two choices for the last digit: 2 or 5. Once we have chosen the last digit, we have 6 digits left to fill the first six positions. We have two 2's, two 5's, one 0, one 1, and one 3 to choose from.

To count the number of 7-digit even numbers, we will use the multiplication principle. There are 2 choices for the last digit and 3 choices for each of the first six digits (since we cannot use the same digit twice). Therefore, the total number of 7-digit even numbers that can be formed is:

2 x 3 x 3 x 3 x 3 x 3 x 3 = 2 x 3^6 = 1458

So there are 1458 7-digit even numbers that can be formed using all the digits 0, 1, 2, 2, 3, 5, and 5.

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The probability that either the die will come up 2 or 3, or the coin will land heads up is 5/6.

To find the probability of either event happening, we can add the probabilities of each individual event happening and then subtract the probability of both events happening together (since that would be counted twice).

The probability of the die coming up 2 or 3 is 2/6, or 1/3, since there are two out of six equally likely outcomes that meet this condition.

The probability of the coin landing heads up is 1/2, since there are two equally likely outcomes (heads or tails).

To find the probability of both events happening together, we can multiply the probabilities of each event: (1/3) * (1/2) = 1/6.

So, the probability of either the die coming up 2 or 3, or the coin landing heads up is:

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

Therefore, the probability is 5/6.

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To ensure no more than 8% of batteries are refunded, Quick Start should guarantee the batteries for 31 months.

(a) To determine the percentage of batteries that fail within 36 months, we will use the Z-score formula:
Z = (X - μ) / σ
Where X is the target value (36 months), μ is the mean (43.8 months), and σ is the standard deviation (8.3 months).
Z = (36 - 43.8) / 8.3 ≈ -0.94
Using a standard normal distribution table, we find that the percentage of batteries failing within 36 months is approximately 17.36%. So, Quick Start can expect to replace about 17.36% of its batteries under the 36-month guarantee.
(b) If Quick Start does not want to make refunds for more than 8% of its batteries, we need to find the corresponding Z-score for 8%. Using a standard normal distribution table, we find a Z-score of approximately -1.41.
Now, we need to find the corresponding number of months (X) for this Z-score:
X = μ + Z × σ
X = 43.8 + (-1.41) × 8.3 ≈ 31.3 months

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For the first part of your question, the equation 2 = 9(2,y) defines a surface in three-dimensional space. This surface is a plane that is perpendicular to the y-axis and intersects the y-axis at the point (0, 2/9, 0). At the point (1, 2, 1), the value of y is 2, so the point lies on the surface. For the second part of your question, the function f(x,y) has four critical points.

For the function f(x,y) = 4.2 + 2x - 10.2y + xy + 10xy + xy - 4xy^2, let's find the critical points. Critical points are the points where the partial derivatives with respect to x and y are both zero.

Step 1: Find the partial derivatives:
fx(x,y) = ∂f/∂x = 2 + y + 10y - 4y^2
fy(x,y) = ∂f/∂y = -10.2 + x + 10x + x - 8xy

Step 2: Set the partial derivatives to zero and solve the system of equations:
2 + y + 10y - 4y^2 = 0
-10.2 + x + 10x + x - 8xy = 0

Step 3: Solve the system of equations for x and y to find the critical points. This may require techniques such as substitution or elimination.

For the second part of your question, the function f(x,y) has four critical points.which are points where the gradient of the function is equal to zero. These points are (2,1), (-6,-2), (12,42), and (21,91). At these points, the partial derivatives of the function with respect to x and y are both zero. The critical points can be used to determine the maximum, minimum, or saddle point of the function.

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a. σ = 5√5

b. The value of K is 30.75 grams.

c. The values of L1 and L2 are 268.25 grams and 331.75 grams, respectively.

(a) We know that the mean weight of a fruit cup is 300 grams, which is

the sum of the mean weights of strawberries and blueberries. Thus, we

have:

160 + μ = 300

Solving for μ, we get:

μ = 140

Next, we can use the formula for the variance of the sum of independent

random variables to find the variance of the weight of a fruit cup:

Var(weight) = Var(strawberries) + Var(blueberries)

The variance of strawberries is given as 10^2 = 100, and the variance of

blueberries is σ^2. We know that the variance of the weight of a fruit cup

is[tex]15^2[/tex]= 225. Thus, we have:

100 + σ^2 = 225

Solving for σ, we get:

σ = 5√5

(b) The middle 96.6% of the fruit cups corresponds to the interval between the 2.17th and 97.83rd percentiles of the distribution of fruit cup weights. We can use the standard normal distribution to find the z-scores corresponding to these percentiles:

z1 = invNorm(0.0217) ≈ -2.05

z2 = invNorm(0.9783) ≈ 2.05

Using the formula for the standard error of the mean, we can find the value of K:

K = z2 × (15 / √n)

We know that the mean weight of a fruit cup is 300 grams, so n = 1. Plugging in the values, we get:

K = 2.05 × (15 / √1) = 30.75

(c) We can use the mean and standard deviation values found in part (a) to find the z-scores corresponding to the 2.17th and 97.83rd percentiles:

z1 = invNorm(0.0217) ≈ -2.05

z2 = invNorm(0.9783) ≈ 2.05

Using the z-scores and the formula for the standard error of the mean, we can find the values of L1 and L2:

L1 = 300 + z1 × (15 / √1) = 268.25

L2 = 300 + z2 × (15 / √1) = 331.75

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the MLE of θ is θ = max(xi)

the MME is consistent.

To find the maximum likelihood estimator (MLE) of θ, we first write the likelihood function as:

[tex]L(\theta|x_1, ..., x_n) = \theta^n / (\pi xi^2)[/tex]

Taking the logarithm of this function, we have:

[tex]l(\theta|x_1, ..., x_n) = n log(\theta) - 2 \Sigma log(xi)[/tex]

To find the maximum, we differentiate with respect to θ and set the derivative equal to zero:

[tex]dl(\theta|x_1, ..., x_n) / d\theta = n/\theta = 0[/tex]

Solving for θ, we obtain the MLE:

[tex]\theta = max(xi)[/tex]

(ii) To find a method of moments estimator (MME) of θ, we first find the population mean:

[tex]E(X) =\int \theta/x f(x; theta) dx[/tex]

[tex]= \theta \int 1/x^2 dx[/tex]

= θ

We set the sample mean equal to the population mean:

[tex]1/n \Sigma xi = \theta[/tex]

Solving for θ, we obtain the MME:

[tex]\theta = (1/n) \Sigma xi[/tex]

To determine whether this estimator is consistent, we use the weak law of large numbers, which states that as n approaches infinity, the sample mean converges in probability to the population mean. Since the MME is based on the sample mean, if the sample mean converges in probability to the population mean, then the MME is consistent.

In this case, since E(X) = θ and the sample mean is an unbiased estimator of E(X), we have:

[tex]\lim_{n \to \infty} P(|(1/n)\Sigma xi - \theta| < \epsilon) = 1[/tex]

for any [tex]\epsilon > 0[/tex].

The MME is consistent.

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Answer: -2/3

Step-by-step explanation:

The distribution of means obtained by taking all possible samples of a specific size from a population and computing the means for each sample is called the sampling distribution of the sample mean.

The sampling distribution of the sample mean is a theoretical probability distribution that describes the possible values of the sample means that can be obtained from the population. The shape of the sampling distribution of the sample mean is approximately normal if the sample size is large enough (typically, greater than 30) and the population is normally distributed, regardless of the shape of the population distribution. This is known as the Central Limit Theorem.

The sampling distribution of the sample mean is important in statistics because it allows us to make inferences about the population based on the characteristics of the sample. Specifically, we can use the sampling distribution of the sample mean to estimate the population mean and to calculate confidence intervals for the population mean. We can also use the sampling distribution of the sample mean to test hypotheses about the population mean.

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To answer this question, we need to use the normal distribution formula and the z-score table.

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years.

First, we need to find the z-score associated with the 4th percentile (since we are looking for the 4% of items with the shortest lifespan).

Using the z-score table, we find that the z-score associated with the 4th percentile is -1.75.

Next, we use the formula:

z = (x - μ) / σ

where z is the z-score, x is the value we are trying to find (the lifespan we want to know), μ is the mean lifespan (3.1 years), and σ is the standard deviation (0.9 years).

Plugging in the values we know:

-1.75 = (x - 3.1) / 0.9

Solving for x:

x - 3.1 = -1.575

x = 1.525

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years (to one decimal place).
To determine the lifespan for the bottom 4% of items, we will use the mean, standard deviation, and z-score. The mean lifespan is 3.1 years, and the standard deviation is 0.9 years. Using a z-score table, we find that the z-score for the 4% percentile is approximately -1.75.

Now, we can use the formula:
Lifespan = Mean + (z-score × Standard Deviation)
Lifespan = 3.1 + (-1.75 × 0.9)

Lifespan ≈ 1.5 years

Therefore, the 4% of items with the shortest lifespan will last less than 1.5 years.

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Answer:

Step-by-step explanation:

Using the rule of 70, we can estimate that it will take approximately 35 years for the population of Sudan to double with an estimated annual growth rate of 2 percent,since 70/2 = 35.

Rate refers to the measure of change in one quantity with respect to another over a given time period. It can be expressed as a ratio or percentage and is commonly used in finance, economics, science, and mathematics.

Percent, denoted by the symbol "%", means "per hundred" and is used to express a fraction or ratio in relation to 100. It is commonly used in business, finance, and statistics to represent changes, growth rates, and other relative measures.

According to the given information:

If a country's population is growing at a constant rate, the time it takes to double its population can be estimated using the rule of 70. The rule of 70 states that you can estimate the number of years it will take for a population to double by dividing 70 by the annual growth rate as a percentage.

In the case of Sudan, with an estimated annual growth rate of 2 percent, it would take approximately 35 years for the population to double. This is calculated by dividing 70 by 2, which equals 35. Therefore, if the growth rate remains constant, Sudan's population is expected to double in approximately 35 years

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There are 43 integers in the range from 92 to 734 that are divisible by 15. The probability that a random integer from 92 to 734 is divisible by 15 is approximately 0.067.

To find the probability that a random integer from 92 to 734 is divisible by 15, we need to first determine the total number of integers in this range. The difference between 734 and 92 is 642, but since we want to include both endpoints, we need to add 1 to this difference. So there are a total of 643 integers in the range from 92 to 734. Next, we need to determine how many of these integers are divisible by 15. To do this, we need to find the smallest multiple of 15 that is greater than or equal to 92, and the largest multiple of 15 that is less than or equal to 734. The smallest multiple of 15 that is greater than or equal to 92 is 105 (which is 7 times 15), and the largest multiple of 15 that is less than or equal to 734 is 720 (which is 48 times 15).

So the integers from 105 to 720 (inclusive) are all divisible by 15. To count the number of integers in this range, we can divide the difference between 720 and 105 by 15, and add 1 to account for the first multiple of 15:
(720 - 105) / 15 + 1 = 43
Therefore, there are 43 integers in the range from 92 to 734 that are divisible by 15.

To find the probability that a random integer from this range is divisible by 15, we can divide the number of integers that are divisible by 15 (43) by the total number of integers in the range (643):
43 / 643 ≈ 0.067
So the probability that a random integer from 92 to 734 is divisible by 15 is approximately 0.067.

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The length AC of the missing sides of the given triangle above would be;

AC= 15.6cm

AB = 9cm

To calculate the length of the missing sides of the triangle, the sin rule of used such as;

a/sinA = b/sinB

a = 18

A = 90°

b =AB= ?

B = 30°

That is:

18/sin90 = b/sin 30°

b = 18×0.5/1

= 9

Using Pythagorean formula;

c² = a² +b²

18² = 9²+b²

b² = 324-81

b² = 243

b = √243

= 15.6cm

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The complete sentence is,

An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions are ''Violated''.

Given that;

To find An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions.

Now, We know that;

Robust inference is inference that is insensitive to (smaller or larger) deviations from the assumptions under which it is derived.

Hence, We get;

An inference procedure is ROBUST if the confidence level or p-value doesn't change much if the assumptions are ''Violated''.

Hence, Option 4 is true.

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[tex]\textit{area of a triangle}\\\\ A=\cfrac{1}{2}bh ~~ \begin{cases} b=base\\ h=height\\[-0.5em] \hrulefill\\ h=\frac{1}{3}\\[1em] A=\frac{5}{36} \end{cases}\implies \cfrac{5}{36}=\cfrac{1}{2}\cdot b\cdot \cfrac{1}{3}\implies \cfrac{5}{36}=\cfrac{b}{6} \\\\\\ 30=36b\implies \cfrac{30}{36}=b\implies \cfrac{5}{6}=b[/tex]

The influence of a data point depends on its location in the predictor space, its leverage, and its contribution to the fit of the model.

a) An outlier is a data point that deviates significantly from the other observations in a dataset, and it can be either due to measurement errors, natural variability, or genuine extreme observations. On the other hand, high leverage data points are observations that have extreme values on one or more predictor variables, and they have the potential to exert a significant influence on the estimation of regression coefficients.

b) Yes, a data point that is flagged as an outlier can also have high leverage. This occurs when an observation has extreme values on both the response variable and one or more predictor variables.

c) Not necessarily. While outliers and high leverage data points can impact the results of a statistical analysis, they may not necessarily be influential. An influential data point is one that significantly affects the estimated coefficients and can change the results of a statistical analysis if removed. The influence of a data point depends on its location in the predictor space, its leverage, and its contribution to the fit of the model.

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The power series are:

1. x^2 - 3x + 3/2(x-1)^2 - 1/2(x-1)^3 + ... and IOC  is (0,2).

2. 2(x-1/2) - 2ln(1-(x-1/2)) + C and IOC is (1/2,3/2).

1. f(x) = x^2 - 3ln(2) + x - 2/(x+2)

Using the power series expansions for ln(1+x), 1/(1+x), and 1/(1-x), we can write:

f(x) = x^2 - 3ln(2) + x - 2/(x+2)

= x^2 - 3ln(1+(x-1)) + x - 2/(x+2)

= x^2 - 3((x-1) - (x-1)^2/2 + (x-1)^3/3 - ...) + x - 2/(x+2)

= x^2 - 3x + 3/2(x-1)^2 - 1/2(x-1)^3 + ...

The series converges for |x-1| < 1, so the interval of convergence is (0,2).

2. g(x) = 2x^2 - x - 1

We can factor g(x) as:

g(x) = 2(x-1/2)(x+1)

Using the power series expansions for 1/(1-x) and ln(1+x), we can write:

g(x) = 2(x-1/2)(x+1)

= 2(x-1/2) - 2(x-1/2)^2 + 2(x-1/2)^3 - ...

= 2(x-1/2) - 2ln(1-(x-1/2)) + C

where C is a constant. The series converges for |x-1/2| < 1, so the interval of convergence is (1/2,3/2).

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