32. Jarak kota Yogyakarta dan Surabaya adalah 325 km, Andi berangkat dari kota
Yagyakarta menuju Surabaya dengan kecepatan 48 km/ jam, sedangkan Dody
berangkat dari kota Surabaya menuju Yogyakarta dengan kecepatan 52 km/jam.
Jika rute jalan yang dilalui sama dan keduanya berangkat bersamaan pukul 08.15
WIB. Pukul berapa mereka berpapasan ?

The x-component of the normal vector is -42, the y-component is 0, and the z-component is -14.

A vector is a quantity that describes not only the magnitude of an object but also its movement or position with respect to another point or object. It is sometimes referred to as a Euclidean vector, a geometric vector, or a spatial vector.

To find the normal vector to the tangent plane of [tex]z = 7e^{(x^2-6y)[/tex] at the point (12, 24, 7), we first need to find the partial derivatives of the function with respect to x and y evaluated at this point.

Taking the partial derivative with respect to x, we get:

[tex]∂z/∂x = 14xe^{(x^2-6y)[/tex]

Evaluating this at the point (12, 24), we get:

[tex]∂z/∂x = 14(12)e^{(12^2-6(24))} = 0[/tex]

Taking the partial derivative with respect to y, we get:

[tex]∂z/∂y = -42e^{(x^2-6y)[/tex]

Evaluating this at the point (12, 24), we get:

[tex]∂z/∂y = -42e^{(12^2-6(24))} = -42[/tex]

Therefore, the normal vector to the tangent plane at the point (12, 24, 7) is given by:

(0, 0, -1) x (-14, 0, 42) = (-42, 0, -14)

So, the x-component of the normal vector is -42, the y-component is 0, and the z-component is -14.

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The recommended model for trend and seasonality is the Seasonal-Trend Decomposition using Loess (STL) regression model.

The variables in the model are time (t), trend (Tt), seasonality (St), and residual (Rt).

The number of independent variables depends on the frequency of data and degree of seasonality, and can be determined by the formula 2q x m.

To develop a model for trend and seasonality, we can use a regression model known as the Seasonal-Trend Decomposition using Loess (STL).

The variables in the model are:

The number of independent variables in the regression depends on the frequency of the data and the degree of seasonality. If the data has a daily frequency and exhibits daily seasonality, the regression model will have 365 independent variables (one for each day of the year). If the data has a monthly frequency and exhibits monthly seasonality, the regression model will have 12 independent variables (one for each month of the year).

The number of independent variables can be determined by the formula 2q × m, where q is the number of harmonics (usually set to 1 or 2) and m is the number of observations per season (e.g., 12 for monthly data).

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The correct answer is: D.

The system has only one solution, and it is viable because it results in positive side lengths.

Let x be the shorter leg of the triangle, and y be the area. The longer leg is 4 + 6x, and the area of the triangle is y = (1/2) * x * (4 + 6x).

For the rectangle, the width is 5 + x, the length is 3, and its area is also y = (5 + x) * 3.

The system of equations is:

y = (1/2) * x * (4 + 6x)

y = (5 + x) * 3

Substitute equation (2) into equation (1) and solve for x:

(5 + x) * 3 = (1/2) * x * (4 + 6x)

30 + 6x = 4x + 6x^2

6x^2 - 2x - 30 = 0

Using the quadratic formula, we find two solutions for x:

x1 ≈ 2.62

x2 ≈ -1.95

Since x represents the length of the shorter leg, we discard the negative solution. Thus, there is only one viable solution for x: x ≈ 2.62. Now find y using equation (2): y ≈ 22.86.

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A carpenter is creating two new templates for his designs. One template will be in the shape of a right triangle, where the longer leg is 4 inches more than six times the shorter leg.

The second template will be in the shape of a rectangle, where the width is 5 inches more than the triangle’s shorter leg, and the length is 3 inches.

The carpenter needs the areas of the two templates to be the same. Write a system of equations to represent this situation, where y is the area, and x is the length of the shorter leg of the triangle. Which statement describes the number and viability of the system’s solutions?

A.

The system has two solutions, but only one is viable because the other results in negative side lengths.

B.

The system has two solutions, and both are viable because they result in positive side lengths.

C.

The system has only one solution, but it is not viable because it results in negative side lengths.

D.

The system has only one solution, and it is viable because it results in positive side lengths.

Answer:

g(x) = (x/2)^2

Step-by-step explanation:

if we look at the Y values for each point they remain the same, however the X values change, which tells us that the X variable is being 'tampered' with.

to find the scale factor, we divide 4/2, and we get 2.

so g(x) = (x/2)^2

remember that when applying a scale factor to the X value the effect will be inversed, so we multiply the X value with 1/2. you can use desmos to double check.

Answer:

a).141.5 b).125.8 c).875.4

a).34 b).154.5 c). 49

Step-by-step explanation:

to find the height you would simply divide the volume by pie times the radius squared.      

The gradient of a function points in the direction in which the function increases the fastest because it represents the direction of greatest increase of the function.

The gradient of a function is a vector that points in the direction of the steepest increase of the function at a particular point. This means that if we move in the direction of the gradient, the value of the function increases the fastest.

To understand why this is true, let's consider the definition of the gradient. The gradient of a function f(x) is defined as a vector of partial derivatives:

∇f(x) = (∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn)

Each component of the gradient vector represents the rate of change of the function with respect to the corresponding variable. In other words, the gradient tells us how much the function changes as we move a small distance in each direction.

When we take the norm (or magnitude) of the gradient vector, we get the rate of change of the function in the direction of the gradient. This means that if we move in the direction of the gradient, the value of the function changes the fastest, because this is the direction in which the function is most sensitive to changes in the input variables.

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Answer:

2n+3 for ≥0

Step-by-step explanation:

the formula that generate the sequence is

 2n+3

the difference between the arithmetic sequence is 2

2(0)+3

0+3=3

2(1)+3

2+3=5

2(2)+3

4+3=7

2(3)+3

6+3=9

n≥0 means n can start from 0 till infinity

if n is substitute into the formula, it will give the arithmetic sequence which means formula generated is correct



Hope this helps!

i. cos 0 =  [tex]\sqrt{3}[/tex]

ii. tan 0 = 1/ [tex]\sqrt{3}[/tex]

Trigonometric functions are a set of given functions which are required in determining the value(s) of the sides or internal angle(s) of a given right angled triangle; when the value of one none right angle is given.

In the given question, we have;

sin 0 = 1/2

This implies that;

sin 0 = opposite/ hypotenuse = 1/2

So that;

opposite = 1

hypotenuse = 2

Apply the Pythagorean's theorem so as to determine its adjacent, we have;

adjacent = [tex]\sqrt{3}[/tex]

Then,

cos 0 = adjacent/ hypotenuse

         =  [tex]\sqrt{3}[/tex] / 1

cos 0 =  [tex]\sqrt{3}[/tex]

ii. tan 0 = opposite/ adjacent

            = 1/  [tex]\sqrt{3}[/tex]

tan 0 = 1/ [tex]\sqrt{3}[/tex]

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Answer:

Step-by-step explanation:

To convert the equation r sin theta = 4 to Cartesian coordinates, we use the identities r^2 = x^2 + y^2 and y/x = tan theta. Substituting r sin theta = 4 into these identities, we get:

x^2 + y^2 = (r sin theta)^2 = 16

y/x = sin theta/ cos theta = tan theta

Squaring both sides of the second equation and substituting y^2/x^2 = 1 + tan^2 theta, we get:

y^2/x^2 = 1 + (y/x)^2

x^2 + y^2 = 16(1 + (y/x)^2)

Simplifying this equation, we get:

x^2 + y^2 = 16 + 4y^2/x^2

Multiplying both sides by x^2, we get:

x^2 y^2 + y^2 = 16x^2 + 4y^2

Bringing all the terms to one side, we get:

x^2 y^2 - 16x^2 = 3y^2

This is the Cartesian equation of the curve. To describe the curve, we can rewrite this equation as:

y^2/x^2 - 16/x^2 = 3

This is the equation of a hyperbola with center at the origin, vertical axis, and asymptotes given by y/x = ±4/sqrt(3).

The limit doesn't exist for lim(x,y)→(0,0) [(x² + 3y²)/(2x² + 23y²)]² because the limit along the x-axis and the limit along the y-axis give different values.

To find the limit of lim(x,y)→(0,0) [(x² + 3y²)/(2x² + 23y²)]², we can first simplify the expression inside the parentheses by dividing both the numerator and denominator by y²:

[(x²/y² + 3)/(2(x/y)² + 23)]²

As (x,y) approaches (0,0), both x and y approach 0, so x²/y² approaches 0/0, which is an indeterminate form. To resolve this, we can use L'Hôpital's rule, taking the partial derivative with respect to x and y:

lim(x,y)→(0,0) [(x²/y² + 3)/(2(x/y)² + 23)]²

= [lim(x,y)→(0,0) 2(x/y)² / 4(x²/y²) ]² (using L'Hôpital's rule)

= [lim(x,y)→(0,0) x² / 2y² ]²

= [lim(x,y)→(0,0) (x/y)² / 2 ]²

Since (x,y) approaches (0,0), we have (x/y)² approaching 0/0, another indeterminate form. Using L'Hôpital's rule again, we get:

lim(x,y)→(0,0) (x/y)² / 2

= lim(x,y)→(0,0) 2x / (2y)

= lim(x,y)→(0,0) x / y

Now, we have two paths to consider: approaching along the x-axis (y = 0) and approaching along the y-axis (x = 0). Along the x-axis, the limit is:

lim(x,0)→(0,0) x / 0

which does not exist, since the expression approaches infinity as x approaches 0 from either direction. Similarly, along the y-axis, the limit is lim(0,y)→(0,0) 0 / y which is 0.

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Step-by-step explanation:

v = h(πr^2)

= 13.8 cm (3.14(8.9)^2)

= 3432.32 ~3432.3cm ^3

so the volum of the cub will be 3432.3cm ^3

a) At least 5,675 adults.

b) if we use the results from the 2014 survey, we still need to survey at least 5,675 adults.

c) It does not have much of an effect on the sample size.

Sample size refers to the number of observations or participants included in a study or survey. In statistical analysis, the size of the sample is an important consideration as it can affect the accuracy and reliability of the results. A larger sample size generally leads to more precise estimates and increased statistical power, while a smaller sample size may be more susceptible to sampling errors and variability.

(a) To find the minimum sample size needed, we can use the formula:

n = (z² × p × (1-p)) / E²

where z is the z-score corresponding to the desired confidence level (99%), p is the estimated proportion of e-cigarette users (3.65% or 0.0365), and E is the desired margin of error (3 percentage points or 0.03).

Plugging in these values, we get:

n = (2.576² × 0.0365 × 0.9635) / 0.03²

n = 5,674.85

Rounding up to the nearest integer, we get:

n = 5,675

Therefore, we need to survey at least 5,675 adults to obtain today's e-cigarette usage rate with a 99% confidence level and a margin of error of 3 percentage points.

(b) If we use the results from the 2014 survey, we can estimate the population proportion of e-cigarette users as 0.0365. Using the same formula as above, we get:

n = (2.576² × 0.0365 × 0.9635) / 0.03²

n = 5,674.85

Rounding up to the nearest integer, we get:

n = 5,675

Therefore, even if we use the results from the 2014 survey, we still need to survey at least 5,675 adults to obtain today's e-cigarette usage rate with a 99% confidence level and a margin of error of 3 percentage points.

(c) The use of the results from the 2014 survey does not have much of an effect on the sample size. This is because the desired confidence level and margin of error are fixed, and the estimated proportion from the 2014 survey is relatively close to the true proportion (since e-cigarette use is still a relatively new phenomenon).

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The Mean Value Theorem applies on [0 , 16].

The x-value with 0 < c < 16 such that f'(c) = average rate of change of f(x) on [0 , 16].

If f(x) is a function that satisfies below conditions;

i) f(x) is Continuous in [a,b]

ii) f(x) is Differentiable in (a,b)

Then, there exists a number c, such that  a < c < b and

f(b) – f(a) = f ‘(c) (b – a)

The given function f(x) has the power and it is a power function

This has even denominator in the exponent.

=> f(x) is continuous on [0 , ∞) and differentiable on (0 , ∞)

Thus, the Mean Value Theorem applies on [0 , 16].

f(0) = 0 ;

f(16) = 4th root of 16³ = 8

Average rate of change of f(x) on [0 , 16]

= (8 - 0) / (16 - 0)

= 1/2

Now, differentiate the given function

[tex]f'(x) = 3/4x^{(-1/4)} = 1/2[/tex]

=>[tex]x^{(-1/4)} = 2/3[/tex]

=> [tex]x^{(1/4)} = 3/2[/tex]

Thus, x = 81/16 = c

The x-value with 0 < c < 16 such that f'(c) = average rate of change of f(x) on [0 , 16].

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The value of x that make the area of the compound shape as 24 mm² is 1.5 mm

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional state.

The area of the compound shape is 24 mm².

For the first rectangle:

Area = x * (2x + 6) = 2x² + 6x

For the second rectangle:

Area = x * (7) = 7x

The area of compound shape = 2x² + 6x + 7x = 2x² + 13x

Since the area is 24 mm², hence:

2x² + 13x = 24

2x² + 13x - 24 = 0

x = 1.5 mm

The value of x is 1.5 mm

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Answer:

x = 10 , arc GF = 65°

Step-by-step explanation:

the measure of the chord- chord angle GEF is half the sum of the arcs intercepted by the angle and its vertical angle, that is

∠ GEF = [tex]\frac{1}{2}[/tex] (GF + HL) , so

60 = [tex]\frac{1}{2}[/tex] ( 5x + 15 + 55) ← multiply both sides by 2 to clear the fraction

120 = 5x + 70 ( subtract 70 from both sides )

50 = 5x ( divide both sides by 5 )

10 = x

Then

arc GF = 5x + 15 = 5(10) + 15 = 50 + 15 = 65°

Answer: 5/12 or 41.67% chance

Step-by-step explanation:

number > 1 is 5/6 chance

heads on a coin is 1/2 chance

5/6 * 1/2 = 5/12

A CR less than or equal to 0.1 is considered acceptable, indicating a consistent set of judgments in comparing the criteria. If the CR is greater than 0.1, it is advised to revise the pairwise comparisons to improve consistency.

The Consistency Ratio (CR) in the context of the GEAR Matrix (which is related to bikes, not fruit) measures the level of consistency in judgments made when comparing criteria in a decision-making process, such as the Analytic Hierarchy Process (AHP). To calculate the CR for the Criteria in the GEAR Matrix, follow these steps:

1. Determine the pairwise comparison matrix by comparing the importance of each criterion against the others.
2. Calculate the weights of each criterion by normalizing the columns and finding the average for each row.
3. Multiply the pairwise comparison matrix by the weight vector to obtain a new vector.
4. Divide each element of the new vector by its corresponding weight to obtain the Consistency Vector.
5. Calculate the average of the Consistency Vector to get the Consistency Index (CI).
6. Divide the CI by the Random Index (RI) for the specific matrix size (this value can be found in AHP literature) to obtain the Consistency Ratio (CR).

A CR less than or equal to 0.1 is considered acceptable, indicating a consistent set of judgments in comparing the criteria. If the CR is greater than 0.1, it is advised to revise the pairwise comparisons to improve consistency.

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The population whose properties are analyzed by the data can be described as households in City A.

Given numerical information is,

58% of households in City A were online.

We have to describe what describes this numerical information.

Here, it is described that a certain percent of households in a city A are online.

So the description is about the households of the city A.

So this is the population.

Hence the best description is households in City A.

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The support of the association rule is 0.0895. The closest answer choice is (B) 0.0895.

What is Association Rule?

An association rule is a relationship between two or more variables or items that are frequently found together in a dataset. In data mining and machine learning, association rules are used to discover interesting relationships and patterns among large sets of data. Association rules are often expressed in the form "if X, then Y" where X and Y are sets of items or variables, and the rule indicates that there is a strong correlation between X and Y. The strength of an association rule is typically measured in terms of its support and confidence, which are statistical measures that indicate how often the rule is true in the dataset.

According to the given information:

The support of an association rule is defined as the proportion of transactions in the dataset that contain both the antecedent and the consequent of the rule.

In this case, the antecedent is "coffee, ice cream, and chips", and the consequent is not specified. Therefore, we can assume that we are interested in the support of the rule "if a transaction contains coffee, ice cream, and chips, then it also contains [some item]".

The support of this rule is equal to the number of transactions that contain coffee, ice cream, and chips, divided by the total number of transactions in the dataset. Therefore, the support of the rule is:

support = 895 / 10,000

support = 0.0895

Rounding to four decimal places, the support of the association rule is 0.0895. The closest answer choice is (B) 0.0895.

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The support of the association rule is given by option b. 0.0895.

In association rule mining, the two most crucial metrics are support and confidence. Support is a metric for how frequently a specific itemset or association rule appears in the dataset. It is the percentage of transactions that either include the itemset or adhere to the association rule. Support therefore gauges how well-liked an itemset or rule is throughout the dataset.

The support of association is given by the formula:

support = 895 / 10,000

support = 0.0895

Hence, the support of the association rule is given by option b. 0.0895.

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The circumference of a circle with diameter of 21 cm is given as follows:

C = 65.97 cm.

The circumference of a circle of radius r is given by the equation presented as follows:

[tex]C = 2\pi r[/tex]

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, while the diameter of the circle is the distance between two points on the circumference of the circle that pass through the center. Hence, the diameter’s length is twice the radius length.

The radius is half the diameter of 21 cm, hence it is given as follows:

r = 10.5 cm.

Then the circumference of the circle is given as follows:

C = 2π x 10.5

C = 65.97 cm.

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The dy/dx of the equation  x⁴ * xy - y⁴ = x * y² is (y² - 4x³ * xy) / (x⁴ - 4y³ + 2xy).

To find dy/dx of the given equation x⁴ * xy - y⁴ = x * y², we'll first differentiate both sides of the equation with respect to x.

Using the product rule for differentiation (uv)' = u'v + uv', we have:

d/dx (x⁴ * xy) - d/dx (y⁴) = d/dx (x * y²)

Differentiating each term, we get:

(x⁴)'(xy) + (x⁴)(xy)' - (y⁴)' = (x)'(y²) + (x)(y²)'

Now, we'll find the derivatives:

4x^3 * xy + x⁴ * (y + x(dy/dx)) - 4y³(dy/dx) = y² + x * (2y * (dy/dx))

Now, we'll solve for dy/dx. First, let's collect the terms containing dy/dx on one side:

x⁴(dy/dx) - 4y³dy/dx) + 2xy(dy/dx) = y² - 4x³ * xy

Next, we factor out dy/dx:

dy/dx (x⁴ - 4y³ + 2xy) = y² - 4x³ * xy

Finally, we'll divide both sides by the expression in parentheses to isolate dy/dx:

dy/dx = (y² - 4x³ * xy) / (x⁴ - 4y³ + 2xy)

This is the expression for dy/dx.

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The value of the side x is 8cm

Note that the different trigonometric identities are;

From the information shown in the diagram, we have that;

The angle, theta = 60 degrees

the Hypotenuse side = xcm

the adjacent side = 4cm

Using the cosine identity, written as;

cos θ  = adjacent/hypotenuse

cos 60 = 4/x

cross multiply

x = 8cm

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The critical values for this test are -2.201 and 2.201. If your test statistic falls outside this range, you would reject the nullSo, the critical values for this test are -2.201 and 2.201. If your test statistic falls outside this range, you would reject the null hypothesis in favor of the alternative hypothesis (H1: μ ≠ μ0). in favor of the alternative hypothesis (H1: μ ≠ μ0).

To determine the critical values for a two-tailed test with a sample size of n = 12 and a significance level of α = 0.05, we need to consult a t-distribution table.

First, we need to find the degrees of freedom, which is equal to n - 1 = 12 - 1 = 11.

Next, we look up the t-value for a two-tailed test with a 0.025 level of significance (0.05/2) and 11 degrees of freedom. From the t-distribution table, we find that the t-value is 2.201.

Therefore, the critical values for a two-tailed test of a population mean at the α = 0.05 level of significance based on a sample size of n = 12 are -2.201 and +2.201.

This means that if our calculated t-value falls outside of this range, we can reject the null hypothesis (H0: μ = μ0) in favor of the alternative hypothesis (H1: μ ≠ μ0) at the 0.05 level of significance.

To determine the critical values for a two-tailed test of a population mean at the α = 0.05 level of significance with a sample size of n = 12, you'll need to use a t-distribution table.

Since it's a two-tailed test, you'll need to find the t-score that corresponds to α/2, which is 0.025 in each tail. With a sample size of 12, you have 11 degrees of freedom (df = n - 1).

Looking up the t-distribution table with df = 11 and α/2 = 0.025, you'll find the critical t-value to be approximately ±2.201.

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partA - if we raise $500 out of a goal of $750, we would raise 66.67% of the goal.

partB - our contributions would fall into the Silver Award category, since we raised between 60% and 79% of the goal. This is because 66.67% falls within the range of 60% to 79%.

what is range ?

In mathematics, range refers to the set of all output values that a function can take . It is the set of all possible values of the dependent variable in a function, given all possible values of the independent variable.

In the given question,

Part A:

Let's say we want to raise $500 for our favorite charity. To find the percentage of the goal we would raise, we can use the following formula:

percentage = (amount raised / goal) x 100%

Substituting the given values, we get:

percentage = (500 / 750) x 100% = 66.67%

Therefore, if we raise $500 out of a goal of $750, we would raise 66.67% of the goal.

Part B:

Based on the percentage found in Part A, our contributions would fall into the Silver Award category, since we raised between 60% and 79% of the goal. This is because 66.67% falls within the range of 60% to 79%.

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The null hypothesis for the type of test from Exercise 11.53 is that there is no difference between the observed distribution and the expected distribution. In other words, the observed distribution is the same as the expected distribution.

The test used to decide whether the observed distribution is the same as the expected distribution is the chi-square test. This test is used to determine if there is a significant difference between the observed distribution and the expected distribution based on the assumption that the two distributions are independent of each other. The chi-square test compares the observed values with the expected values, and if the difference between them is statistically significant, the null hypothesis is rejected.

On the other hand, if the difference between the observed and expected values is not statistically significant, the null hypothesis cannot be rejected. In conclusion, the null hypothesis for the chi-square test used in Exercise 11.53 is that there is no significant difference between the observed and expected distributions.

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The null hypothesis for the type of test from Exercise 11.53 is that there is no difference between the observed distribution and the expected distribution. In other words, the observed distribution is the same as the expected distribution.

The test used to decide whether the observed distribution is the same as the expected distribution is the chi-square test. This test is used to determine if there is a significant difference between the observed distribution and the expected distribution based on the assumption that the two distributions are independent of each other. The chi-square test compares the observed values with the expected values, and if the difference between them is statistically significant, the null hypothesis is rejected.

On the other hand, if the difference between the observed and expected values is not statistically significant, the null hypothesis cannot be rejected. In conclusion, the null hypothesis for the chi-square test used in Exercise 11.53 is that there is no significant difference between the observed and expected distributions.

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Answer:

254.3

Step-by-step explanation:

A= 3.14*9^2=3.14*81=254.34

254.34 rounded is 254.3

We cannot say with absolute certainty that the true relationship between the LHS and RHS variables is not influenced by other factors.

If we say that an estimate is statistically significant, it means that the difference or relationship observed between the left-hand side (LHS) variable and the right-hand side (RHS) variable is unlikely to have occurred by chance alone.

However, it does not necessarily mean that we are 100% certain of the true relationship between the LHS and RHS variables. There may still be other factors or variables that were not accounted for in the analysis that could affect the relationship between the two variables.

Therefore, we cannot say with absolute certainty that the true relationship between the LHS and RHS variables is not influenced by other factors.

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Mean of x for sample size is [28.34, 32.66]

The closest option is (c) 30.5.

The mean of the sample means will be the same as the population mean, which is 30.5.

The standard error of the mean, which is the standard deviation of the sampling distribution of the mean, can be calculated as:

SE = σ / sqrt(n)

where σ is the population standard deviation and n is the sample size.

SE = 3.5 / sqrt(10) = 1.108

The mean of x for sample size of 10 can be calculated as:

x = μ ± z*(SE)

where μ is the population mean, z is the z-score corresponding to the desired level of confidence (we'll assume 95% here), and SE is the standard error of the mean.

Using a z-score of 1.96 for a 95% confidence interval, we have:

x = 30.5 ± 1.96*(1.108) = [28.34, 32.66]

Therefore, the closest option is (c) 30.5.

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A particular solution of the given differential equation y'' + 3y' + 4y = 8x + 2 can be found using the method of undetermined coefficients. The correct answer is: a. y_p = 2x + 1

The correct answer is b. y_p = 8x + 2. To find a particular solution of the differential equation, we can use the method of undetermined coefficients. Since the right-hand side of the equation is a polynomial of degree 1 (8x + 2), we assume that the particular solution has the same form, i.e. y_p = Ax + B. We then substitute this into the differential equation and solve for the constants A and B. Plugging in y_p = Ax + B, we get:

y" + 3y' +4y = 8x + 2
2A + 3(Ax + B) + 4(Ax + B) = 8x + 2
(2A + 3B) + (7A + 4B)x = 8x + 2

Since the left-hand side and right-hand side must be equal for all values of x, we can equate the coefficients of x and the constant terms separately:

7A + 4B = 8  (coefficient of x)
2A + 3B = 2  (constant term)

Solving these equations simultaneously, we get A = 8 and B = 2/3. Therefore, the particular solution is y_p = 8x + 2.

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