What is the y-intercept of the following linear equation?
2x +9y = 18
(9,0)
(0,2)
(9,2)
(0, 18)

(a) The 90% confidence interval for the proportion of students who plan on voting is (0.386 - 0.069, 0.386 + 0.069) = (0.317, 0.455). The smaller number is 0.317, so that is entered in the first box.
(b) The correct interpretation for the answer in part (a) is A: We can be 90% confident that the proportion of students who will vote in the upcoming election lies in the interval (0.317, 0.455).

(a) To find the 90% confidence interval, follow these steps:

1. Determine the sample proportion (p') by dividing the number of students who plan on voting by the total number of students surveyed:
  p' = 97 / (97 + 154) = 97 / 251 ≈ 0.3865

2. Find the critical value (z) for a 90% confidence interval using a z-table. For 90%, the z-value is 1.645.

3. Calculate the standard error (SE) using the formula:
  SE = √(p'(1 - p') / n) = √(0.3865(1 - 0.3865) / 251) ≈ 0.0302

4. Find the margin of error (ME) using the formula:
  ME = z × SE = 1.645 × 0.0302 ≈ 0.0497

5. Calculate the lower and upper limits of the confidence interval:
  Lower Limit = p' - ME = 0.3865 - 0.0497 ≈ 0.3368
  Upper Limit = p' + ME = 0.3865 + 0.0497 ≈ 0.4362

Confidence interval: (0.3368, 0.4362)

(b) The correct interpretation for the 90% confidence interval is:

A. We can be 90% confident that the proportion of students who will vote in the upcoming election lies in the interval (0.3368, 0.4362). This means that if we were to repeat this survey many times and construct a 90% confidence interval each time, we would expect 90% of those intervals to contain the true proportion of students who plan on voting in the election. It does not mean that there is a 90% chance that the true proportion lies in this interval, nor does it provide any information about the actual number of students who will vote.

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At least 289 people must be observed to construct a 99% confidence interval with a margin of error of no more than ±2 seconds.

To construct a 99% confidence interval for the mean number of seconds people spend brushing their teeth with a margin of error of no more than ±2 seconds and a standard deviation of 21 seconds, you'll need to determine the minimum sample size required.

Step 1: Identify the z-score for a 99% confidence level. The z-score is 2.576 (you can find this using a standard normal distribution table or online calculator).

Step 2: Use the margin of error (E) formula:
E = z * (σ / √n)

where E is the margin of error, z is the z-score, σ is the standard deviation, and n is the sample size.

Step 3: Plug in the values and solve for n:
2 = 2.576 * (21 / √n)

Step 4: Rearrange the equation to solve for n:
n = (2.576 * 21 / 2)^2

Step 5: Calculate the value of n:
n ≈ 288.39

Since you cannot have a fraction of a person, you must round up to the nearest whole number. Therefore, at least 289 people must be observed to construct a 99% confidence interval with a margin of error of no more than ±2 seconds.

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Original dimensions: 9 x 9 ft. New dimensions: 27 x 5 ft.

Let x be the original side length. The new dimensions are (3x) and (x-4). The new area is 135 sq ft:

[tex]3x(x-4) = 135[/tex]

[tex]3x^2 - 12x = 135[/tex]

[tex]3x^2 - 12x - 135 = 0[/tex]

Factor the quadratic equation:

(x - 9)(3x + 15) = 0

x = 9 (original side length)

3x = 27 (new length)

x-4 = 5 (new width)

Original dimensions: 9 x 9 ft. New dimensions: 27 x 5 ft.

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Answer:

  D.  y = 3sin(1/2x)

Step-by-step explanation:

You want to know the equation that corresponds to the graph.

The graph shows one cycle of a sine function. It has a peak value of 3, indicating a vertical scaling by a factor of 3. All of the answer choices have this scale factor:

  y = 3sin( )

That one cycle extends from x = -2π to x = 2π, a total of 2π -(-2π) = 4π units. The period of the parent sine function y=sin(x) is 2π, so this represents a horizontal expansion by a factor of 2. That scaling is represented in the function's equation as a divisor of x:

  y = 3sin(x/2)

  [tex]\boxed{y=\sin{\left(\dfrac{1}{2}x\right)}}[/tex]

The population proportion of males is 0.2, or 20%. This means that in the population of one man and four women, 20% of the individuals are male.

The population is defined as the entire group of individuals that we are interested in studying. In this exercise, the population consists of one man and four women, for a total of five individuals.

The attribute of interest is "male", which means we are looking to determine the proportion of individuals in the population who are male. In this case, there is only one male in the population (Noah) and a total of five individuals, so the proportion of males in the population can be calculated as:

p = number of males / total population size

In this case, the number of males is 1, and the total population size is 5, so the proportion of males in the population is:

p = 1/5

p = 0.2

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If car weighs "1650-kg" and truck weighs "3-tons", then weight of car as percentage of weight of truck is 55%.

In order to express the weight of car as a percentage of weight of truck, we need to convert the weights to the same unit of measurement.

We know that,

⇒ Weight of the car = 1650 kg,

⇒ Weight of the truck = 3 tons,

Since 1 ton is equal to 1000 kg, we convert the weight of truck from "ton" to "Kg" by multiplying it by 1000,

So, Weight of truck in kg = 3×1000,

Weight of truck in kg = 3000 kg,

So, Percentage = (Weight of car)/(Weight of truck) × 100,

Substituting the values,

⇒ Percentage = (1650/3000) × 100,

⇒ Percentage = 0.55 × 100,

⇒ Percentage = 55%

Therefore, the weight of the car is 55% of the weight of the truck.

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If  y = 5 √5x/ 5x - 3. dy/dx = 25/(2√(5x))

To find dy/dx for y = 5√(5x)/(5x-3), we need to use the chain rule.

Let u = 5x, then y = 5√(5x)/(5x-3) = 5√u/(u-3)

Using the power rule, we can find du/dx = 5

Using the chain rule, we get:

dy/dx = dy/du * du/dx

To find dy/du, we need to use the power rule and the constant multiple rule:

dy/du = (5/2) * u(-1/2)

Substituting back for u, we get:

dy/du = (5/2) * (5x)(-1/2)

Now, substituting all the values we found, we get:

dy/dx = dy/du * du/dx = (5/2) * (5x)(-1/2) * 5

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Answer:

103

Step-by-step explanation:

Given: 54+c², c=7

We can substitute 7 for c into the equation first:

54+7²

The question is asking to evaluate the exponent first, which is the ² part of the expression.

Broken down, we have 54 and 7², so let's focus on 7² first.

7² says that you have to multiply 7 x 7, or in words, multiply 7 by itself.

7 x 7 =49, so c²=49.

Back to the entire equation, we now have:

54+49

=103

Hope this helps :)

The general indefinite integral is F(x) = (11/3)x³ + 7x + C.

To find the general indefinite integral of ∫(5x²+6+6x²+1)dx, you can first simplify the integrand as 11x² + 7. Now, apply the Fundamental Theorem of Calculus and linearity property to find the antiderivatives of each term.

1. Simplify the integrand: 5x² + 6 + 6x² + 1 = 11x² + 7

2. Find the antiderivative of 11x²: ∫(11x²)dx = (11/3)x³ + C₁

3. Find the antiderivative of 7: ∫(7)dx = 7x + C₂

4. Add the antiderivatives and combine constants: F(x) = (11/3)x³ + 7x + (C₁ + C₂)

5. Use a single constant, C: F(x) = (11/3)x³ + 7x + C

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The intervals on which f(x) is concave down are (-∞,∞).

To determine the intervals on which the function f(x) = √(2x+3) is concave up or concave down, we need to find its second derivative.

First, we find the first derivative of f(x):

f'(x) = (1/2)(2x+3)^(-1/2) * 2

f'(x) = (1/√(2x+3)) * 2

f'(x) = 2/√(2x+3)

Next, we find the second derivative of f(x):

f''(x) = d/dx (2/√(2x+3))

f''(x) = -4(2x+3)^(-3/2)

f''(x) = -4/(2x+3)^(3/2)

Now, to determine where the function is concave up or down, we need to find where the second derivative is positive (concave up) or negative (concave down).

f''(x) > 0 if -4/(2x+3)^(3/2) > 0

-4 > 0

This inequality is never true for any value of x, so f(x) is never concave up.

f''(x) < 0 if -4/(2x+3)^(3/2) < 0

-4 < 0

This inequality is always true for any value of x, so f(x) is always concave down.

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The random samples of the given scores of each block represents ,

Inter quartile range of each block is Option D. Block 1 IQR = 20 and Block 2 IQR= 5.

Outliers present in each block is Option B. Block One= 25, Block Two= 100.

Each data display represents Option B. Block 1 is skewed right, Block 2 is skewed left.

Mean and standard deviation in Block 1  is Option A.  Mean = 76.5, standard deviation = 21.6.

Score of two blocks,

Block 1

25, 60, 70, 75, 80, 85, 85, 90, 95, 100.

Block 2

70, 70, 75, 75, 75, 75, 80, 80, 85, 100.

To get Interquartile range 'IQR' of each block,

First find the quartiles.

Median of Block 1 = 80

Median of Block 2 = 75

First quartile Q₁ and third quartile Q₃ of each block,

Split the data into two halves at the median and find the median of each half.

Block 1

Lower half is,

25, 60, 70, 75, 80

Q₁ = median of the lower half

    = 70

Upper half is,

85, 85, 90, 95, 100.

Q₃ = median of the upper half

     = 90

IQR = Q₃ - Q₁

      = 90 - 70

      = 20

Block 2

Lower half is,

70, 70, 75, 75, 75

Q₁ = median of the lower half

     = 75

Upper half is,

75, 80, 80, 85, 100.

Q₃ = median of the upper half

     = 80

IQR = Q₃ - Q₁

      = 80 - 75

       = 5

Option D. Block 1 IQR = 20 and Block 2 IQR= 5

Outliers in each block,

First find lower and upper bounds.

Any data point outside the bounds is considered an outlier.

The lower bound is Q₁ - 1.5(IQR),

and the upper bound is Q₃ + 1.5(IQR).

Block 1,

Q₁ = 70

Q₃ = 90

IQR = 20

Lower bound

= Q₁ - 1.5(IQR)

= 70 - 1.5(20)

= 70 - 30

= 40

Upper bound

=Q₃ + 1.5(IQR)

= 90 + 1.5(20)

= 120

The data point 25 is less than the lower bound,

so it is an outlier in Block 1.

Block 2,

Q₁ = 75

Q₃ = 80

IQR = 5

Lower bound

= 75 - 1.5(5)

= 67.5

Upper bound

= 80 + 1.5(5)

= 87.5

100 is more than the upper bounds in Block 2,

so it is an outliers of Block 2.

Option B. Block One= 25, Block Two= 100

The data displays as symmetric skewed left or skewed right,

Examine the shape of the histograms.

Block 1 has a histogram that is skewed right.

With more scores on the higher end of the range.

Block 2 has a histogram that is slightly skewed left.

With more scores on the lower end of the range.

Option B. Block 1 is skewed right, Block 2 is skewed left.

Mean and standard deviation of Block 1,

Use the formulas,

Mean = sum of scores / number of scores

Standard deviation = √ [(sum of (scores - mean)^2) / (n - 1)]

Block 1

Mean

= (25 + 60 + 70 + 75 + 80 + 85 + 85 + 90 + 95 + 100) / 10

= 76.5

Standard deviation

= √[((25-76.5)^2 + (60-76.5)^2 + ... + (100-76.5)^2) / (10 -1 )]

=√2652.25 + 272.25+ 42.25 +2.25 + 12.25 + 72.25 + 72.25 + 182.25 + 342.25 + 552.25 /9

= √4202.5/9

= 21.6

Option A.  Mean = 76.5, standard deviation = 21.6.

Therefore, for the given scores answer of the following questions are,

Inter quartile range is Option D. Block 1 IQR = 20 and Block 2 IQR= 5.

Outliers in each block is Option B. Block One= 25, Block Two= 100.

Data display is Option B. Block 1 is skewed right, Block 2 is skewed left.

In Block 1 Option A.  Mean = 76.5, standard deviation = 21.6.

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the probability that the sample mean is less than 7 is approximately zero

Since the stress scores follow a uniform distribution from 1 to 5, the mean of the distribution is:

mu = (1 + 5) / 2 = 3

The standard deviation of the distribution is:

sigma = (5 - 1) / sqrt(12) = sqrt(2.33)

The sample size is n = 55.

The sample mean follows a normal distribution with mean mu = 3 and standard deviation sigma/sqrt(n) = sqrt(2.33)/sqrt(55) = 0.313.

To find the probability that the sample mean is less than 7, we standardize the variable:

z = (7 - 3) / 0.313 = 12.78

This is an extremely large value of z, indicating a probability that is essentially zero. Therefore, the probability that the sample mean is less than 7 is approximately zero

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Answer:

140J

Step-by-step explanation:

work done = Mass x acceleration x distance

mass = 7kg

acceleration = 10m/s^2

distance= 2m

work = 7 x 10 x 2

= 140j

Answer:

The answer is 140J or 140Nm

Step-by-step explanation:

Work done=mgh

W=7×10×2

W=140J or 140Nm

The probability of the algorithm being incorrect when the photo is about fashion depends on the accuracy of the algorithm in predicting fashion-related content. This probability can be calculated based on the given information.

(a) The probability of the algorithm being incorrect, given that the photo is about fashion, can be denoted as P(incorrect | fashion). This probability depends on the accuracy of the algorithm in correctly identifying fashion-related content. If the algorithm has a high accuracy in predicting fashion-related content, then P(incorrect | fashion) would be low. On the other hand, if the algorithm has a low accuracy in predicting fashion-related content, then P(incorrect | fashion) would be high.

(b) Using the answers from part (a) and 3.29(b), we can compute P(mach learn is pred fashion | truth is fashion) + P(mach learn is pred not | truth is fashion), where P(mach learn is pred fashion | truth is fashion) is the probability that the algorithm predicts fashion correctly when the truth is indeed about fashion, and P(mach learn is pred not | truth is fashion) is the probability that the algorithm predicts fashion incorrectly when the truth is about fashion.

(c) The sum of P(mach learn is pred fashion | truth is fashion) + P(mach learn is pred not | truth is fashion) is 1 because these two probabilities together represent all possible outcomes when the truth is about fashion. The algorithm can either predict fashion correctly (P(mach learn is pred fashion | truth is fashion)) or predict fashion incorrectly (P(mach learn is pred not | truth is fashion)), but there is no other possibility. Therefore, the sum of these two probabilities is equal to 1, as it accounts for all possible outcomes.

Therefore, the sum of P(mach learn is pred fashion | truth is fashion) + P(mach learn is pred not | truth is fashion) is 1.

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A) The value of the integral is 2

2) The value of the integral is 1/12.

(a) To evaluate ∫ ∫R f(x,y) dA where R = [0, 1] x [0,1] and f(x,y) = 4 - 2y, we integrate the function over the region R.

Step 1: Integrate with respect to y
∫(4 - 2y) dy from 0 to 1

Step 2: Integrate the result with respect to x
∫(4 - 2y) dx from 0 to 1

The result is 2. The average value of f(x,y) on R is (2)/(1*1) = 2.

(b) To evaluate ∫ ∫D(x² + 2y) dA where D is bounded by y = x, y = x³, and x > 0:

Step 1: Determine the limits of integration
x: 0 to 1 (x > 0)
y: x³ to x (bounded by y = x and y = x³)

Step 2: Integrate with respect to y
∫(x² + 2y) dy from x³ to x

Step 3: Integrate the result with respect to x
∫(result) dx from 0 to 1

The result is 1/12.

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In the regression model without an intercept, yi = B1xi + Er for i=1,2,...,n, the total sum of squares (SST) need not be equal to the residual sum of squares (SSR) + explained sum of squares (SE) due to the absence of the intercept term.

SST represents the total variation in the dependent variable (yi), which is typically decomposed into explained variation (SE) and unexplained variation (SSR) in the presence of an intercept. However, when the model doesn't include an intercept, this decomposition does not hold true.

The reason is that without an intercept, the regression line is forced to go through the origin (0,0), which can result in a poor fit of the data. Consequently, the explained sum of squares (SE) may not accurately capture the variability explained by the model, and the residual sum of squares (SSR) might not account for the remaining unexplained variation.

Therefore, in a regression model without an intercept, the relationship SST = SSR + SE may not hold true, as the decomposition of total variability into explained and unexplained components is disrupted by the lack of an intercept term.

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The estimate of at most 25% of workers taking the bus does not seem to be a valid estimate based on the given sample data.

To determine whether the estimate of at most 25% of workers taking the bus is valid, we can perform a hypothesis test using the given sample data.

Null Hypothesis: The true proportion of workers who take the bus to work is equal to or less than 0.25.

Alternative Hypothesis: The true proportion of workers who take the bus to work is greater than 0.25.

We can use a one-tailed z-test for proportions to test this hypothesis, with a significance level of alpha = 0.05. The test statistic is calculated as follows:

z = (p - P) / sqrt(P*(1-P)/n)

where:

p = sample proportion of workers who take the bus = 28/90 = 0.3111

P = hypothesized proportion of workers who take the bus = 0.25

n = sample size = 90

Substituting the values, we get:

z = (0.3111 - 0.25) / sqrt(0.25*(1-0.25)/90) = 1.87

The critical value for a one-tailed test with alpha = 0.05 and a right-tailed alternative hypothesis is 1.645 (found using a standard normal distribution table). Since our test statistic of 1.87 is greater than the critical value of 1.645, we reject the null hypothesis and conclude that the true proportion of workers who take the bus to work is likely greater than 0.25. Therefore, the estimate of at most 25% of workers taking the bus does not seem to be a valid estimate based on the given sample data.

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Answer:

5 units to the left

Step-by-step explanation:

Look at point A.  To get to A' you need to move 5 units to the left.

Helping in the name of Jesus.

The unknown angle measure based on the information is 35⁰.

An angle is a geometric figure formed by two rays (or lines) that share a common endpoint called the vertex. The rays are typically represented as line segments with the vertex at their endpoint, and the angle is measured in degrees or radians based on the amount of rotation between the two rays.

Angles are commonly denoted using the vertex as the center point and a letter at each end of the rays, for example, angle ABC, where B is the vertex and A and C are the endpoints of the two rays forming the angle.

Since the angles on a triangle are 80°, x, and 65°, the value of the unknown angle will be:

= 180 - (65 + 80)

= 35°

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The angles on a triangle are 80°, x, and 65°. What is the value of the unknown angle?

The area inside one loop of the lemniscate r2 = 10 sin 2θ is 5 square units.

To find the area inside one loop of the lemniscate r2 = 10 sin 2θ, we need to use the formula for the area enclosed by a polar curve:

A = (1/2) ∫[a,b] r(θ)2 dθ

Since we want the area inside one loop, we can choose the limits of integration to be from 0 to π/2, which covers one complete loop of the lemniscate. Plugging in the given polar equation, we get:

A = (1/2) ∫[0,π/2] (10 sin 2θ) dθ

Using a double angle identity, we can simplify the integrand to:

A = (1/2) ∫[0,π/2] (10 sin θ) cos θ dθ

Integrating this expression gives:

A = (1/2) [(-10/2) cos2θ] [0,π/2]

A = (1/2) [(-10/2) (1)] = -5

However, since we are looking for a positive area, we need to take the absolute value of this result, which gives:

A = 5 square units

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Normalcdf and Invnorm are two statistical functions commonly used in probability and statistics.

Normalcdf is a function that calculates the cumulative probability of a standard normal distribution. It takes two arguments, the lower and upper limits, and returns the probability that a random variable falls between those limits.

This function is particularly useful for finding the area under the normal curve, which can be interpreted as the probability of a random variable taking on a certain value.

Invnorm, on the other hand, is a function that calculates the inverse normal distribution. It takes a probability value as an argument and returns the z-score (standard deviation) corresponding to that probability.

This function is useful for finding the value of a random variable that corresponds to a certain probability in a normal distribution.

Together, these functions can be used to solve a variety of statistical problems, such as calculating confidence intervals, finding probabilities of events, and determining the significance of test results.

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The p-value is less than the significance level of 0.05, we reject the null hypothesis.

Using the given data, we can calculate the t-test statistic as follows:

t = (117 - 138) / √[(29²/37) + (34²/35)] = -2.21

where 117 is the sample mean for the day after Thanksgiving, 138 is the sample mean for the day after Christmas, 29 is the standard deviation for the day after Thanksgiving, 34 is the standard deviation for the day after Christmas, 37 is the sample size for the day after Thanksgiving, and 35 is the sample size for the day after Christmas.

Using a t-table or a statistical software, we can find the p-value associated with this t-test statistic. At the 0.05 level of significance, the p-value is 0.016. This means that if the null hypothesis were true, we would observe a t-test statistic as extreme or more extreme than -2.21 only 1.6% of the time.

This means that we have evidence to suggest that the mean amount of money spent by shoppers on the day after Thanksgiving is less than the mean amount of money spent by shoppers on the day after Christmas.

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Answer:

Step-by-step explanation:

Answer:

The answer is x³√5x

Step-by-step explanation:

√5x⁷=x³√5x

The expression that shows "3 less than a number" is B. n - 3.

Mathematical operations must be carried out in a specific order according to a set of rules known as the order of operations. The following is the order of events:

Operation inside brackets should come first. Expressions containing exponents should be evaluated next.

Division and Multiplication: Carry out division and multiplication from left to right. Addition and subtraction: Move from left to right when adding and subtracting. It is crucial to adhere to the order of operations since it guarantees accurate and consistent evaluation of mathematical expressions.

When we break the given phrase in to mathematical expression we have:

n - 3.

Hence, the expression that shows "3 less than a number" is B. n - 3.

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The center of the curve is c. Hyperbola; center (-5, -2). The correct option is:c.

The given equation is 9y² - 4x² - 36y - 82 = 0.

To identify the curve and find the center, we need to check the signs of the coefficients of x² and y². Here, x² has a negative coefficient and y² has a positive coefficient, which suggests that the given equation represents a hyperbola.

To find the center of the hyperbola, we need to rewrite the equation in the standard form, which is of the form:

[(y - k)² / a²] - [(x - h)² / b²] = 1

where (h, k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the foci.

Rewriting the given equation in the standard form, we get:

9(y² + 4y) - 4(x² + 20x) = 328

Dividing both sides by 36, we get:

[(y + 2)² / 4²] - [(x + 5)² / (3/2)²] = 1

Comparing this equation with the standard form, we get:

Center = (-5, -2)

Therefore, the correct option is:

c. Hyperbola; center (-5, -2)

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The value is y" + xy = 5x^4/2 - 8x^7/(3·2) - 8x^3/3 + 14x^6/(3·5) - 20x^9/(3·5·7

(A) To find the radius of convergence of the power series, we can use the ratio test. The ratio of consecutive terms of the series is:

| x^3/(3·2) · (3·2·5)/(6·5·8) · (6·5·11)/(9·8·14) · ... | = | x^3/6 | · | (5/8) · (11/14) · (17/20) · ... |

The second factor is a product of terms of the form (4n + 1)/(4n + 4), which simplifies to (1 + 1/(4n + 4)). Thus, the product can be written as:

| (5/8) · (11/14) · (17/20) · ... | = | (1 + 1/4) · (1 + 1/8) · (1 + 1/12) · ... | = ∏(1 + 1/(4n + 4))

Using the ratio test, the series converges absolutely if the limit of the ratio of consecutive terms is less than 1:

| x^3/6 | · ∏(1 + 1/(4n + 4)) < 1

The limit of the product is known to be the Wallis product, which is equal to π/2. Thus, we have:

| x^3/6 | · π/2 < 1

Solving for |x|, we get:

|x| < (2/π)^(1/3)

Therefore, the radius of convergence is:

R = (2/π)^(1/3)

(B) To show that the function y(x) satisfies the differential equation y" + xy = 0, we need to show that its second derivative and the product xy(x) satisfy the equation. Differentiating the power series term by term, we get:

y' = 0 - x^2 + x^5/2 - x^8/(3·2) + ...

y" = 0 - 0 + 5x^4/2 - 8x^7/(3·2) + ...

Multiplying x and y(x) term by term, we get:

xy = x - x^4/3 + x^7/(3·5) - x^10/(3·5·7) + ...

Taking the derivative of xy with respect to x, we get:

(xy)' = 1 - 4x^3/3 + 7x^6/(3·5) - 10x^9/(3·5·7) + ...

Differentiating xy' with respect to x again, we get:

(xy)" = - 4x^2 + 21x^5/(3·5) - 30x^8/(3·5·7) + ...

Adding xy to -x(xy)' and simplifying, we get:

y" + xy = 5x^4/2 - 8x^7/(3·2) - 4x^3/3 + 7x^6/(3·5) - 10x^9/(3·5·7) - 4x^3 + 7x^6/(3·5) - 10x^9/(3·5·7) + ...

Collecting like terms, we get:

y" + xy = 5x^4/2 - 8x^7/(3·2) - 8x^3/3 + 14x^6/(3·5) - 20x^9/(3·5·7

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The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population.

For both a sample and a population, the formula value = mean + (#ofSTDEVs)(standard deviation) may be used. The calculation of the standard deviation for a sample vs a population does, however, differ slightly. The standard deviation is written as s for a sample and is determined using the following formula:

[tex]s = sqrt(sum((xi - x)^2) / (n - 1))[/tex]

When n is the sample size, x is the sample mean, and xi represents each unique data point. When calculating the population standard deviation from a sample, the degrees of freedom are taken into account using the denominator (n - 1).

Thus,

value = mean + (#ofSTDEVs)(s)

[tex]σ = √sum((xi - μ)^2) / N)[/tex]

value = mean + (#ofSTDEVs)(σ)

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a) If V is a finite-dimensional vector space, then V cannot contain an infinite linearly independent subset. This statement is false.

(b) If Vi and V2 are vector spaces and dim(V1) < dim (V2), then V1 ⊂ V2. This statement is false.

(a) If xo is a solution to Ax=b, then xo is in row(A). This statement is false.

(b) If A is a 4x13 matrix, then the nullity of A could be equal to 5. This statement is false.

(a) False. A vector space can have an infinite linearly independent subset, but only if it is infinite-dimensional. For example, the vector space of polynomials has an infinite linearly independent subset {1, x, [tex]x^2[/tex], [tex]x^3[/tex], ...}.
(b) False. Just because the dimension of V1 is less than V2 does not necessarily mean V1 is a subset of V2. For example, V1 could be the x-axis in [tex]R^2[/tex] and V2 could be the entire plane.
(a) False. Just because xo is a solution to Ax=b does not mean xo is in row(A). xo could be any vector that satisfies the equation Ax=b.
(b) False. The nullity of matrix A is equal to the dimension of its null space. The null space is the set of all solutions to the homogeneous equation Ax=0. Since A has 13 columns, the maximum rank it can have is 13. Therefore, the nullity of A can be at most 13-4=9. It cannot be equal to 5.

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The multiplication of 165×3 is 321

To start, we can break down 165 into 100+60+5. This is because multiplication is distributive over addition. That means we can multiply each part separately and then add the results. So,

165×3 = (100+60+5)×3

= 100×3 + 60×3 + 5×3

Now, we can use the fact that 65×3=195 to find the answer to each of these products.

100×3 = 100×(65/65)×3 = (100×65)/65×3 = 65×3

60×3 = 60×(65/65)×3 = (60×65)/65×3 = 39×3

5×3 = 5×(65/65)×3 = (5×65)/65×3 = 3×3

Putting it all together,

165×3 = 65×3 + 39×3 + 3×3

= 195 + 117 + 9

= 321

Therefore, 165×3 = 321.

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