Problem 3 (20 points) In this problem we aim at utilizing the kerenl trick in Ridge regression and propose its kernalized version. Recall the Ridge regression training objective function:
f(w)= ||Xw - y|| 2 ^ 2 + lambda||w|| 2 ^ 2
for lambda > 0
a) Show that for w to be a minimizer of f(w) we must have X^ top Xw + lambda*Iw =X^ top y where X in mathbb R ^ (nd) is the data matrix with n samples each with d features, and I is iden- tity matrix (please check lectures for more details). Show that the minimizer of f(w) is w=(X^ top X + lambda*I )^ -1 X^ top y. Justify that the matrix X^ top X + lambda*I is invertible, for lambda > 0 (Hint: use SVD decomposition of data matrix X= U*Sigma V^ top and show all the eigenvalues of X^ top X + lambda*I are larger than zero).
b) Rewrite X^ top Xw + lambda*Iw =X^ top u as w= 1/lambda (X^ top y-X^ top Xw) . Based on this, show that we can write w =X^ top alpha for some alpha in mathbb R ^ n , and give an expression for a.
c) Based on the fact that w =X^ top alpha. explain why we say w is "in the span of the data."
d) Show that alpha=( lambda*I +XX^ top )^ -1 y. Note that X X^ top is the nn Gram (kernel) matrix for the standard vector dot product. (Hint: Replace w by X ^ top alpha in the expression for a, and then solve for a.)
e) Give a kernelized expression for the Xw, the predicted values on the training points. (Hint: Replace w by X ^ top alpha and a by its expression in terms of the kernel matrix X X^ overline top )
f) Give an expression for the prediction w * ^ top x for a test sample æ, not in the training set, where w * is the optimal solution. The expression should only involve a via inner products training data samples x_{i}, i = 1 ,...,n.
g) Based on (f), propose a kernalized version of the Ridge regression.

The solution of the Cauchy problem is y = (sina - 1/(2u - 1))e^(ux)cos(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

To solve the Cauchy problem 2ux + y = cos x, we first need to find the general solution of the corresponding homogeneous equation 2ux + y = 0.

The characteristic equation is r^2 - 2ur + 1 = 0, which has roots r = u ± sqrt(u^2 - 1).

Case 1: u^2 < 1

In this case, the roots are complex conjugates, so the general solution of the homogeneous equation is

y = c₁e^(ux)cos(sqrt(1 - u^2)x) + c₂e^(ux)sin(sqrt(1 - u^2)x).

Case 2: u^2 > 1

In this case, the roots are real and distinct, so the general solution of the homogeneous equation is

y = c₁e^(r1x) + c₂e^(r2x),

where r1 = u + sqrt(u^2 - 1) and r2 = u - sqrt(u^2 - 1).

Case 3: u^2 = 1

In this case, the root is r = u, so the general solution of the homogeneous equation is

y = c₁e^(ux) + c₂xe^(ux).

Now, we can find the particular solution of the non-homogeneous equation using the method of undetermined coefficients.

Assuming a particular solution of the form y = Asin(x) + Bcos(x), we have

2uB - Asin(x) - Bcos(x) = cos(x).

Matching coefficients, we get A = 0 and 2uB - B = 1, so B = 1/(2u - 1).

Therefore, the particular solution is y = (1/(2u - 1))cos(x).

The general solution to the Cauchy problem is then

y = c₁e^(ux)cos(sqrt(1 - u^2)x) + c₂e^(ux)sin(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

To determine the constants c₁ and c₂, we use the initial condition y(2,0) = sina.

Substituting x = 0, we get

c₁ + (1/(2u - 1)) = sina.

Substituting x = pi/2sqrt(1 - u^2), we get

c₂sqrt(1 - u^2) = 0.

Since sqrt(1 - u^2) ≠ 0, we have c₂ = 0.

Therefore, c₁ = sina - 1/(2u - 1), and the solution of the Cauchy problem is

y = (sina - 1/(2u - 1))e^(ux)cos(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

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The 95% confidence interval for the population mean is (54.70, 70.96).

To find the 95% confidence interval for the population mean, we need to first find the sample mean and sample standard deviation.

Sample Mean (x) = (60.3 + 65.2 + 60 + 70.6 + 62 + 55.9) / 6 = 62.33

Sample Standard Deviation (s) = sqrt([(60.3 - 62.33)^2 + (65.2 - 62.33)^2 + (60 - 62.33)^2 + (70.6 - 62.33)^2 + (62 - 62.33)^2 + (55.9 - 62.33)^2] / (6 - 1)) = 5.28

Next, we can use the formula for the confidence interval:

CI = x ± t*(s/sqrt(n))

where x is the sample mean, s is the sample standard deviation, n is the sample size, and t is the t-value for the 95% confidence level with (n-1) degrees of freedom.

From a t-distribution table with 5 degrees of freedom (n-1 = 6-1 = 5) and a 95% confidence level, we find that the t-value is 2.571.

Plugging in the values, we get:

CI = 62.33 ± 2.571*(5.28/sqrt(6))

CI = (54.70, 70.96)

Therefore, the 95% confidence interval for the population mean is (54.70, 70.96).

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The radius of the circle is 24 units.

We know that for an arc defined by an angle θ on a circle of radius R, the area is:

A = (θ/360°)*π*R²

Here we can see that the area of the sector is 432π and the angle is 270°, then we can replace that in the formula above so we get:

432π = (270°/360°)*π*R²

432 = (3/4)*R²

(4/3)*432 = R²

√576 = R

24 = R

The radius is 24 units.

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The argument is not valid.

The argument assumes that a flood occurs only if the global temperature has increased, but this is not necessarily true. A flood can occur due to other reasons such as heavy rain, land use changes, or other natural disasters. Therefore, the conclusion "a flood occurs" does not necessarily follow from the premises given.

This means that if either the ice melts or a flood occurs, then the global temperature must have increased. However, the converse of this statement is not necessarily true. That is, the global temperature can increase for reasons other than the ice melting or a flood occurring. Therefore, it is not valid to conclude that a flood must occur just because the global temperature has not increased.

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The slope of the line is 3 which represents that the time is rising as the distance increases.

The slope of the line that passes through the pair of points (3, 6) and (1, 3) is calculated as follows:

slope = (6 - 3) / (3 - 2)

slope = (3) / (1)

slope = 3

In this case, the slope of 3 means that the time is rising as the distance increases.

Therefore, the slope of the line that passes through the pair of points  (3, 6) and (1, 3) is 3.

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To decrease the width of a confidence interval while maintaining the same level of confidence, you can increase your sample size i.e., Option C is the correct answer.

A confidence interval is a range of values that is likely to contain the true population parameter with a certain level of confidence. The width of the confidence interval is influenced by several factors, including the sample size, level of confidence, and population standard deviation.

When the sample size is increased, the standard error of the sample mean decreases, which in turn decreases the width of the confidence interval. This means that a larger sample size provides more precise estimates of the population parameter and reduces the variability of the sample means. As a result, a larger sample size allows for a narrower confidence interval while maintaining the same level of confidence.

On the other hand, decreasing the sample size can widen the confidence interval, as there are less data available to estimate the population parameter. Increasing the significance level or decreasing the population standard deviation may also widen the confidence interval, as this increases the range of plausible values for the population parameter.

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The correct interpretation is option (a). It states that we are 90% certain that the difference in percentages of men and women experiencing food cravings falls within the specified interval.

Your question is asking for the correct interpretation of a 90% confidence interval for the difference in percentages of men and women experiencing food cravings. Here are the interpretations of each statement:

a. We are 90% certain that the difference in percentages of men and women experiencing food cravings falls within the specified interval.
b. We are 90% certain that the percentages of men and women experiencing food cravings both fall within the specified interval.
c. 90% of men and women feel that the difference report experiencing food cravings a percentage of the time, specified within the given interval.
d. The specified interval represents 90% of the data for men and women having food cravings.

This interpretation accurately reflects the confidence interval for the difference in percentages between the two groups.

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The measure of ∠WOV is 60°. You would use complementary angles that are adjacent (∠WOV, and ∠XOW)

The measure of ∠YOZ is 60°. You would use the vertical angles that are non-adjacent (∠WOV, and ∠YOZ). These two angles are congruent so they would have the same measure. These angles combined also create supplementary angles

Another way to find the measure of ∠YOZ would be to make/write an equation and solve for x. For example, (3x+30)°=60°. x would equal 10 because 10x3=30+30=60°

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Since we do not have information on a previous survey, we cannot calculate this percentage

i. To find the number of employees who prefer LEXUS only, we need to subtract the number of employees who preferred both LEXUS and MERCEDES, and the number of employees who preferred all three brands, from the total number of employees who prefer LEXUS:

LEXUS only = Total number of LEXUS preference - (Number of employees who prefer both LEXUS and MERCEDES) - (Number of employees who prefer all three brands)

LEXUS only = (8 + 4) - 8 - 4

LEXUS only = 8

Therefore, 8 employees prefer LEXUS only.

ii. To find the number of employees who prefer AMAROK only, we need to subtract the number of employees who preferred both MERCEDES and AMAROK, and the number of employees who preferred all three brands, from the total number of employees who prefer AMAROK:

AMAROK only = Total number of AMAROK preference - (Number of employees who prefer both MERCEDES and AMAROK) - (Number of employees who prefer all three brands)

AMAROK only = (8 + 4) - 4 - 4

AMAROK only = 4

Therefore, 4 employees prefer AMAROK only.

iii. We are given that 5 employees dislike all three brands. Since this information is only from the Private Sector, we cannot determine the number of employees who dislike all three brands in the total of 175 employees from both Public and Private Sectors.

iv. The percentage increase in the number of employees who preferred all three brands is the difference between the number of employees who preferred all three brands in the current survey and the number of employees who preferred all three brands in a previous survey, divided by the number of employees who preferred all three brands in the previous survey, multiplied by 100%. Since we do not have information on a previous survey, we cannot calculate this percentage.

v. The proportion of employees who preferred all three brands in the current survey is 4/23 or approximately 0.174. To compute the percentage decrease in the proportion of employees who preferred all three brands, we need to compare this proportion with the proportion of employees who preferred all three brands in a previous survey. Since we do not have information on a previous survey, we cannot calculate this percentage.

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The Set A defined as A = { 0,3,8} is not a subset of which set is set rational numbers. So, option(a) is right one.

Set is defined as a well defined collection of objects. It is denoted by capital letters of alphabets. The set represented by clearly brackets. We have a set A = { 0,3,8}

Subset is a also a set. A set A is called subset of another set B iff all elements of the set A are elements of the set B. Now,

The counting set is defined as a set with finite number of elements. It has range set {-∞, ∞}. As we see set A is consists three elements that is counted set. Also, these elements belong to whole and integers set. Hence, set A is subset of whole numbers, natural numbers and counting set.

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Complete question :

Set A = 0,3,8 Set A is Not a subset of which set of numbers?

a) Rational numbers

b) Integers

c) Whole numbers

d) Counting numbers

The volume of the figure is given as follows:

V = 132 ft³.

The volume of a triangular prism is given as half the multiplication of the dimensions of the triangle, as follows:

V = 0.5 x l x w x h.

The dimensions of the triangle in this problem are given as follows:

3 ft, 8 ft and 11 ft.

Hence the volume of the prism is given as follows:

V = 0.5 x 3 x 8 x 11

V = 132 ft³.

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Answer:

.40x < 15

x < 37.5 copies

The maximum number of copies is 37.

37 × $.40 = $14.80

The value of x is 88.22 units.

Given is right triangle, we need to find the value of x,

tan 60° = 85 / a

a = 85 / √3

a = 49

Now,

tan 30° = 85 / a + x

a+x = 85 ÷ 1/√3

a+x = 147.22

x = 147.2-49

x = 88.22

Hence, the value of x is 88.22 units.

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In a perfectly competitive market, when the price is greater than the minimum average total cost for all firms, it indicates that the firms are experiencing economic profits. Here's a step-by-step explanation:

1. In a perfectly competitive market, there are numerous firms producing a homogeneous product with free entry and exit for new firms.

2. The demand curve for each firm is perfectly elastic (horizontal), which means that firms can sell any quantity of the product at the market price without affecting the price.

3. The average total cost (ATC) includes both the average variable cost (AVC) and the average fixed cost (AFC). The ATC curve is U-shaped, with the minimum ATC representing the lowest average cost per unit.

4. When the market price is greater than the minimum ATC, it means that the firms are covering all their variable and fixed costs, with additional revenue left over as profit.

5. In this scenario, the firms are earning economic profits, which can be calculated by subtracting the ATC from the market price and multiplying the result by the quantity produced.

6. Economic profits attract new firms to enter the market, leading to increased competition. This, in turn, drives down the market price, reducing economic profits.

7. Over time, as more firms enter and exit the market, the price will eventually reach a point where it is equal to the minimum ATC. This is the long-run equilibrium, where firms earn zero economic profits, covering all their costs but not earning any additional profit.

To summarize, in a perfectly competitive market, when the price is greater than the minimum average total cost for all firms, it indicates that the firms are experiencing economic profits. These profits will attract new firms to enter the market, which will eventually lead to the price falling to the minimum ATC in the long-run equilibrium.

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To test your hypothesis that the average time to deliver pizza, once the order is placed, is greater than 25 minutes in the population, you would perform a one-sample T-test.

1. Formulate the null hypothesis (H0) and the alternative hypothesis (H1).

In this case, H0: the average delivery time is equal to 25 minutes, and H1: the average delivery time is greater than 25 minutes.

2. Collect a sample of delivery times and calculate the sample mean and sample standard deviation.

3. Determine the appropriate T-distribution based on your sample size (degrees of freedom = sample size - 1).

4. Calculate the T-statistic using the sample mean, sample standard deviation, and sample size.

5. Determine the critical T-value based on your chosen level of significance (e.g., 0.05) and the one-tailed T-distribution.

6. Compare the calculated T-statistic to the critical T-value. If the T-statistic is greater than the critical T-value, you can reject the null hypothesis in favor of the alternative hypothesis.

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y=x(a+b)

Divide both sides by (a+b)

x=y/(a+b)

The formula for standard error that applies to this problem is SE = sqrt[p(1-p)/n], where SE represents the standard error, p represents the probability of success, and n represents the sample size. In this case, N represents the population size, but it is not necessary for calculating the standard error.

Substituting the values given in the problem, we have:

SE = sqrt[0.45(1-0.45)/44] = 0.0717 (rounded to four decimal places)

Therefore, the standard error for this problem is 0.0717. This value represents the degree of variability or uncertainty in the sample proportion, or the degree to which the sample proportion is likely to deviate from the true population proportion. A larger sample size or a more extreme probability of success (closer to 0 or 1) would result in a smaller standard error, indicating greater precision in the estimate of the population proportion.

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For theoretical probability of each letter is 0.2, the experimental probability of drawing a card with the letter for which the experimental probability is closest to the theoretical probability is equals the 0.5.

Theoretical Probability: The theoretical probability of an event is the probability of a particular event based on mathematical calculations, assuming ideal conditions. The theoretical probabilities do not take into account the flaws of the system.

Experimental probability: The experimental probability of an event is the actual probability of an event occurring based on the results of facts rather than mathematical calculations

The theoretical probability of each letter = 0.2 = 2/10

We have to determine the experimental probability of drawing a card with the letter for which the experimental probability is near to the theoretical probability. Now, total possible outcomes = 10 and each letter comes two times

so, the experimental probability = 10/20

= 0.5

Hence, required value is 0.5.

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The total number of fourth-grade students in the school is 144

From the question, we have the following parameters that can be used in our computation:

This means that

Fourth-grade students = 6 * Ms. Smyth's fourth-grade class.

Substitute the known values in the above equation, so, we have the following representation

Fourth-grade students = 6 * 24

Evaluate

Fourth-grade students = 144

Hence, the total number of students is 144

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Option E is incorrect because we have enough evidence to reject the null hypothesis.

The correct answer is (C) There is evidence that μ ≠ 70.

The null hypothesis is that the true mean tire pressure for the population of tires on city buses in this city is equal to 70psi, i.e., H0: μ = 70. The alternative hypothesis is that it is not equal to 70psi, i.e., Ha: μ ≠ 70.

The p-value of 0.043 is less than the significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is evidence that the true mean tire pressure is not equal to 70psi.

Option A is incorrect because we cannot conclude that the true mean tire pressure is less than 70psi. Option B is incorrect because we cannot conclude that the true mean tire pressure is equal to 70psi. Option D is incorrect because the p-value is not the probability that μ = 70. Option E is incorrect because we have enough evidence to reject the null hypothesis.

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The probability of getting heads or tails when flipping a coin is 0.5 or 50%. The final probability will give you the chance of getting between 4900 and 5100 heads when flipping a fair coin 10,000 times.

However, the probability of getting a certain number of heads when flipping a coin, a certain number of times can be calculated using probability theory. In this case, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times can be calculated using the binomial distribution formula.

The binomial distribution formula is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where P(X=k) is the probability of getting k heads, n is the number of coin flips, p is the probability of getting a head (0.5 in this case), and (n choose k) is the binomial coefficient.

Using this formula, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times is approximately 0.023 or 2.3%. This means that out of 1000 trials, you can expect to get between 4900 and 5100 heads around 23 times.

In summary, the probability of getting between 4900 and 5100 heads when flipping a fair coin 10000 times is around 2.3%. This probability can be calculated using the binomial distribution formula, which takes into account the number of coin flips and the probability of getting a head.

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Confidence interval for the true mean weight is (964, 1043.8)

P-value= 0.5836≥0.5 then conclude that null hypothesis is rejected.

What is null hypothesis?

The assertion that there is no link between the two sets of data or variables under investigation is known as the null hypothesis. The word "null" refers to the idea that any empirically observed difference is entirely attributable to chance and that no underlying causal link exists.

Given,

Data= 956, 1012, 954, 988, 973, 1048, 1075, 1064, 856, 1026, 1031, 1064

n=12

Mean = sum(x)/n=sum(x)/12= 12047/12= 1003.9167

We get the standard deviation from the given data by using formula

Std. deviation= 62.7976

C=95%= 0.95

α= 1-0.95=0.05

Critical value= 2.201

The 95% confidence interval for the true mean value is=

=(1003.9167±2.201*62.7976/√12)

= (1003.9167±39.9)

= (964, 1043.8)

2)The null and alternative hypothesis,

H0:μ=1000

Hα: μ<1000

The statistic is (t)= (1003.9167-1000)/ 62.7976/√12

=0.216

The P-value from online calculator

P-value= 0.5836≥0.5

Then conclude that null hypothesis is rejected.

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Answer:

11.4583 = 114583 / 10000

Step-by-step explanation:

To perform a hypothesis test on the proportion of people who subscribe to a cable television service versus others, we need to check the following conditions:

1. Random Sampling: Ensure that the sample is randomly selected, which means each person has an equal chance of being selected. This is important to avoid biases and ensure the sample is representative of the population.

2. Independence: If sampling without replacement, the sample size (n) should be less than 10% of the population size (N) to assume independence. This is called the 10% Rule. Independence is important because the outcome of one person's choice should not affect another person's choice.

3. Normality: We need to ensure that the sampling distribution of the sample proportion (p-hat) is approximately normal. This can be checked using the success-failure condition, which states that both np and n(1-p) should be greater than or equal to 10. This ensures that the distribution is sufficiently normal to apply the z-test.

To help the student, remind them to:

1. Verify that the sample is random.
2. Check the independence condition by confirming the sample size is less than 10% of the population size.
3. Check the normality condition by calculating np and n(1-p) and making sure both values are greater than or equal to 10.

It is crucial to check these conditions because if they are not met, the hypothesis test's results may not be valid or reliable. Meeting these conditions ensures that the statistical methods applied are appropriate and provide accurate results for making conclusions.

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The cost of the carpeting Gabriela will use in her living room is  $3,305.47 and Area of living room is 346.41 sq ft

Area of rectangular room = length x width = 25 ft x 16 ft = 400 sq ft

Area of quarter-circle = (1/4) x pi x r^2 = (1/4) x pi x 8^2 = 16 pi sq ft

So the area of the living room is:

Area of living room = Area of rectangular room - Area of quarter-circle

= 400 sq ft - 16 pi sq ft

= 346.41 sq ft

The cost of carpeting this area is:

Cost of carpeting = Area of living room x Cost per square foot

= 346.41 sq ft x $9.55/sq ft

≈ $3,305.47

Therefore, the cost of the carpeting Gabriela will use in her living room is $3,305.47.

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Answer:

[tex]3x - 12 = 4x - 24[/tex]

[tex]x = 12[/tex]

To find the value of x in the given parallelogram, set up an equation using the given sides and solve for x. The value of x is 12.

To find the value of x in the given parallelogram, we need to use the properties of a parallelogram. In a parallelogram, opposite sides are equal in length. So, we can set up an equation using the given sides:

3x - 12 = 4x - 24

Now, we can solve the equation to find the value of x:

3x - 4x = -24 + 12

-x = -12

x = 12

Therefore, the value of x is 12.

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Answer:

The given expression can be simplified using the rule for multiplying powers with the same base by adding their exponents.

(9^-3)(9^12) can be written as (1/9^3)(9^12)

Now, using the rule of adding exponents, (1/9^3)(9^12) can be simplified as 9^(12-3) = 9^9.

Therefore, the expression (9^-3)(9^12) simplified as 9^9.

Step-by-step explanation:

Answer:

11

Step-by-step explanation:

If quilt has 8 rows of "quilt-blocks" and has an area of 7,744 square inches, then the "side-length" of one quilt blocks is 11 inches.

Let us assume that each quilt block has a side length of "x" inches.

Since the quilt has 8 rows of quilt blocks, the total number of quilt blocks in the quilt is 8 times the number of quilt blocks in each row, which is 8x.

The shape of a quilt is a square and has a "total-area' of 7744 square inches, we equate the area equation;

We get,

⇒ (x × 8)² = 7744,

Simplifying the equation,

We get,

⇒ 64x² = 7744,

Dividing both sides by 64,

We get,

⇒ x² = 121,

⇒ x = 11,

Therefore, each quilt-block has a side length of 11 inches.

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The correct equation representing the total number, s, of stickers is option C. s/3 = 4.

The formula or relation that can accurately indicate the total number of stickers, number of girls and number of stickers received by girl is as follows -

Total number of stickers/number of girls = number of stickers received by each girl

(We will perform division as number of stickers must be more than number of girls)

Keep the representative value of each

s/3 = 4

Solving it to know the number of stickers

s = 4 × 3

s = 12

Hence, there were 12 stickers divided four each among three girls. Thus, the correct answer is C. s/3 = 4.

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The complete question is -

Elizabeth has some stickers. She divide her stickers equally amiung herself and two friends. Each person get 4 stickers. Which equation repersents the total number,s,of stickers?

A. s + 3 = 4

B. s 3 = 4

C. s/3 = 4

D. 3s = 4

It has been 16 hours since the time of death.

Stage 1: First 12 hours

12 hours * 1.4F per hour = 16.8F

Stage 2: After 12 hours

Since the body has already cooled 16.8F in the first 12 hours, we only need to find the remaining temperature drop:

19.6F - 16.8F = 2.8F

Now, we can calculate the time taken for the remaining temperature drop using the second cooling rate:

2.8F / 0.7F per hour = 4 hours

Adding the time taken for both stages, we get:

12 hours (Stage 1) + 4 hours (Stage 2) = 16 hours

So, it has been 16 hours since the time of death.

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