"I am struggling with calculating the p-value. I am using z as
the test statistic and have found that z=-4.22. Please help with
finding the p-value. Thank you.
The output voltage for an electric circuit is specified to be 130. A sample of 40 independent readings on the voltage for this circuit gave a sample mean 128.6 and standard deviation 2.1. (a) Test the hypothesis that the average output voltage is 130 against the alternative that it
is less than 130. Use a test with level.05. Report the p-value as well.

7) Given the extreme values in a set of data as 5 and 19, the truth about the data set is C. The range is 14.

8) The mean of the data set 5, 9, 15, 18, 22 is B. 14.

9) The mode for the following set of data, 4, 5, 5, 6, 7, 7, 8, 12, is A. there is none.

The range is the difference between the extreme values of a data set.

This difference is computed by subtracting the minimum value from the maximum value.

The mean represents the average value of a data set, computed by dividing the total value by the number of items.

The mode is one value in the data set that occurs most.  There cannot be more than one mode in a data set.

Range between 5 and 19 = 14 (19 - 5)

Mean of 5, 9, 15, 18, 22 = 13.8 (69 ÷ 5) = 14

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Santiago used 185 cell phone minutes over the course of a month.

To determine the total number of cell phone minutes that Santiago used over the course of a month, you would use the operation of addition.

You would add up the number of minutes that Santiago spoke with his mom, dad, and friends:

Total cell phone minutes = 45 minutes + 30 minutes + 110 minutes

Total cell phone minutes = 185 minutes

Therefore, Santiago used 185 cell phone minutes over the course of a month.

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The number of tables that have 2 chairs each, if there are 50 tables at the restaurant and 40% have 2 chairs each, based on the percentage, therefore is 20 tables

A percentage is a representation of a part of a quantity, expressed as a fraction of 100.

The number of tables in the restaurant = 50 tables

The percentage of the table that have 2 chairs = 40%

The percentage of the table that have 4 chairs = 60%

The percentage of the tables that have 2 chairs each indicates;

The number of tables that have 2 chairs = (40/100) × 50 = 20

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The quantity that maximizes total revenue is 40. The correct option is d. 40.

To maximize total revenue, we need to find the quantity where marginal revenue (MR) equals zero.

Given the MR equation:

MR = 40 - Q

Set MR to zero and solve for Q:

0 = 40 - Q

Q = 40

So, the quantity that maximizes total revenue is 40. The correct option is d. 40.

Maximizing total revenue requires optimizing prices, bundling products, cross-selling/upselling, improving customer retention, expanding the customer base, and reducing costs. Make sure to regularly check on revenue performance and modify strategies accordingly.

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Suppose a monopolist has the following equations: P=40-0.5Q MR=40-Q What is the quantity that maximizes total revenue? a. 25 10 30 d.40 e. 20 T1e MC=10

The number of those games won in that season are: 22 games

Percentage is defined a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. It is given by:

Percentage = (value / total value) * 100%

We are given:

Total number of games played through the season = 40 games

Percentage of games won = 40%

Thus:

Number of games won = 40% * 55

= 22

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Since 504 > 100, Town A had a greater average rate of change.

The given function is P(x)=10x²+8x+600 represents the population of Town A.

Here, x represent the number of years.

In year 0, Town B had a population of 400 people. Town B's population increased by 100 people each year.

P(x)=400+100x

We can calculate the average rate of change for each town by finding the difference between the population in year 8 and the population in year 4 and dividing by the number of years (4).

For Town A, we have:

P(8) - P(4) = 10(8² + 8(8) + 600 - (10(4²) + 8(4) + 600) = 2016

Average rate of change = 2016/4 = 504

For Town B, we have:

400 + (100×4) = 800

Average rate of change = 400/4 = 100

Since 504 > 100, Town A had a greater average rate of change.

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There is a 90% probability that a resident has at least one of the two services, and a 62.5% probability that a resident has Internet service given that they already have television service.

To answer your question, we will use the formula for the probability of the union of two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B), where A represents having Internet service and B represents having television service.

The probability of having at least one of the two services is:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
= 0.60 (Internet) + 0.80 (television) - 0.50 (both)
= 1.40 - 0.50
= 0.90 or 90%

Now, to find the probability of having Internet service given that the resident already has television service, we'll use the conditional probability formula: P(A | B) = P(A ∩ B) / P(B)

P(Internet | Television) = P(Internet ∩ Television) / P(Television)
= 0.50 (both) / 0.80 (television)
= 0.625 or 62.5%

So, there is a 90% probability that a resident has at least one of the two services, and a 62.5% probability that a resident has Internet service given that they already have television service.

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The estimated number of bacteria after 12 hours would be 4083.

The growth of bacteria in this experiment follows exponential growth, where the number of bacteria doubles over a certain time period. The formula for exponential growth will be given by;

N(t) = N0 × [tex]2^{(t/h)}[/tex]

where[tex]N_{(t)}[/tex] is the final number of bacteria after time period t, N0 is the initial number of bacteria, t is the time period, and h is the doubling time (time it takes for the population to double).

Given; N0 = 3100 (initial number of bacteria)

t = 12 hours (time period)

h = 26 hours (doubling time)

Plugging these values into the formula;

[tex]N_{(12)}[/tex] = 3100 × [tex]2^{(12/26)}[/tex]

Calculating; [tex]N_{(12)}[/tex] = 3100 × [tex]2^{(0.4615)}[/tex]

[tex]N_{(12)}[/tex] ≈ 3100 × 1.317

[tex]N_{(12)}[/tex] ≈ 4082.7

Rounding to the nearest whole number, the estimated number of bacteria after 12 hours would be 4083.

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(a) The is the solution to the integral equation is:

f(t) = (t-2)/2 + (1/2) e^(7-t) f(t-7) - (1/2) S e^(7-t) f'(t-7) dt

(b) The is the solution to the integral equation is:

f(t) = L^-1[F(s)] = (1/2) sin(t) + (1/2) cos(t-T) u(t-T)
where u(t-T) is the unit step function.

To solve integral equations, we need to use techniques such as integration by substitution or integration by parts. Let's start with the given equations:

(a) t - 2f(2)= S e---)f(t – 7)dt

To solve this integral equation, we need to integrate the function on the right-hand side with respect to t. Let u = t - 7, then du = dt. The integral becomes:

S e---)f(t – 7)dt = S e---)f(u)du

We can then apply integration by parts, using u = f(u) and dv = e^-u du, which gives us:

S e^-u f(u) du = -e^-u f(u) + S e^-u f'(u) du

Substituting back in for u, we get:

S e---)f(t – 7)dt = -e^(7-t) f(t-7) + S e^(7-t) f'(t-7) dt

Now we can substitute this into the original equation:

t - 2f(2) = -e^(7-t) f(t-7) + S e^(7-t) f'(t-7) dt

To solve for f(t), we need to isolate it on one side of the equation. Rearranging, we get:

f(t) = (t-2)/2 + (1/2) e^(7-t) f(t-7) - (1/2) S e^(7-t) f'(t-7) dt

This is the solution to the integral equation (a).

(b) f(t) = cost + Stef(t – T)dt = е

To solve this integral equation, we can take the derivative of both sides with respect to t. Using the chain rule, we get:

f'(t) = -sinf(t) + s e^(-T) f(t-T)

Now we can substitute this back into the original equation:

f(t) = cost + S e^(-T) f(t-T)dt

To solve for f(t), we need to isolate it on one side of the equation. Rearranging, we get:

f(t) - S e^(-T) f(t-T) = cost

Now we can take the Laplace transform of both sides of the equation:

L[f(t) - S e^(-T) f(t-T)] = L[cos(t)]

Using the properties of the Laplace transform, we get:

F(s) - e^(-Ts) F(s) e^(-Ts) = s/(s^2 + 1)

Simplifying, we get:

F(s) = s/(s^2 + 1) / (1 - e^(-Ts))

Now we can take the inverse Laplace transform to get the solution to the integral equation:

f(t) = L^-1[F(s)] = (1/2) sin(t) + (1/2) cos(t-T) u(t-T)

where u(t-T) is the unit step function. This is the solution to the integral equation (b).

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Answer:

Assuming that those are fraction the simplest form would be --> 7/4

Hope this helps

The number of people that are expected to go to the youth wing on Sunday would be = 543.

For Saturday;

The number of people that went to the youth wing = 497

The number of people that went to social issues = 369

The number of people that went to Fiction and Literature = 416

The total number of people that went to the central library = 497+369+416 = 1,282

For Sunday, the number of people that would visit the youth wing ;

= 497/1282× 1400/1

= 695800/1282

= 543 (to the nearest whole number)

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The choice that matches the graph of the function as is defined to us is:  Graph A.

We are given a function f(x) as:

    f(x)=   2x    if x < 3

 and        4     if  x ≥ 3

This means that in the region (-∞,3) the graph of a function is a straight line that passes through the origin and has a open circle at x=3.

Also, in the region [3,∞) the graph is a straight horizontal line i.e. y=4.

Hence, the graph of this function is Graph A.

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On a piece of paper, graph f(x)={2x if x <3

{4 if x >3. Then determine which answer choice matches the graph you drew

There are three types of arcs in a circle: minor arcs, major arcs, and semicircles. The method for finding the measure of each arc depends on its type.

1. Minor arcs are arcs that measure less than 180 degrees. To find the measure of a minor arc, simply measure the angle that it subtends at the center of the circle. This angle is equal to the arc's measure.

2. Major arcs are arcs that measure greater than 180 degrees but less than 360 degrees. To find the measure of a major arc, subtract the measure of its corresponding minor arc from 360 degrees. For example, if the minor arc measures 60 degrees, the major arc measures 360 - 60 = 300 degrees.

3. Semicircles are arcs that measure exactly 180 degrees. To find the measure of a semicircle, simply divide the measure of the full circle (360 degrees) by 2. Therefore, a semicircle always measures 180 degrees.

Remember, when finding the measure of an arc, it is important to identify the type of arc and use the appropriate method.

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Therefore, the five one-foot rulers laid end-to-end would be equal to 60 inches.

One foot or 12 inches is equivalent to one ruler. Three feet make up a yard. Three rulers make up a yardstick. To measure shorter distances, use rulers.  A foot is made up of 12 inches. Typically, a ruler is 12 inches long. Yardsticks are longer rulers with a length of 3 feet (or 36 inches, which is equivalent to one yard).

Larger things like this teacher's desk are measured in length using a ruler, which is commonly used to represent one foot. The length of the teacher's desk is equal to the edge of five rulers, or around five feet.

There are 12 inches in one foot, so five one-foot rulers laid end-to-end would be equal to:

5 feet × 12 inches/foot = 60 inches

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Answer:

The expected profit from the game can be calculated as the revenue from ticket sales minus the investment cost:

Expected profit = $89,399 - $23,197 = $66,202

The expected loss if it rains can be calculated as the investment cost:

Expected loss = $23,197

To find the net result, we need to use the probability of the game happening (1 - 0.28 = 0.72) and the probability of it raining (0.28):

Net result = (0.72) * (Expected profit) + (0.28) * (Expected loss)

Net result = (0.72) * ($66,202) + (0.28) * ($23,197)

Net result = $47,683.44

Since the net result is positive, the expected outcome is a profit of $47,683.44.

There is an expected profit of $57,963.32.

To calculate the expected profit or loss, we need to consider two possible scenarios:

Scenario 1: It doesn't rain on the day of the game, and the club is able to sell tickets worth $89,399.

Scenario 2: It rains on the day of the game, and the club loses the entire investment of $23,197.

To calculate the expected profit, we need to multiply the revenue from scenario 1 by the probability of it happening, which is (1 - 0.28) = 0.72 (since there's a 28% chance of rain). So, the expected profit is:

Expected profit = 0.72 x $89,399 = $64,451.28

To calculate the expected loss, we need to multiply the investment from scenario 2 by the probability of it happening, which is 0.28 (since there's a 28% chance of rain). So, the expected loss is:

Expected loss = 0.28 x $23,197 = $6,487.96

To find the net result, we subtract the expected loss from the expected profit:

Net result = Expected profit - Expected loss = $64,451.28 - $6,487.96 = $57,963.32

Therefore, there is an expected profit of $57,963.32.

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Answer:   4.

Step-by-step explanation: y = kx

36 = k(9)

k = 4

The percent of container A that is empty after the pumping is complete is approximately 35.4%.

We have,

The volume of water in container A is given by:

V(A) = πr²h

= π(6 ft)²(7 ft)

= 882π cubic feet

The volume of water in container B is given by:

V(B) = πr²h

= π(5 ft)²(10 ft)

= 250π cubic feet

When container A is emptied into container B, the total volume of water becomes:

= V(A) + V(B)

= 882π + 250π

= 1132π cubic feet

The volume of container B is 250π cubic feet, so the remaining volume of water in container A is:

= 1132π - 250π

= 882π cubic feet

The percent of container A that is empty after the pumping is complete is:

= (882π / (πr²h)) x 100%

= (882 / (6² x 7)) x 100%

= 35.4%

Therefore,

The percent of container A that is empty after the pumping is complete is approximately 35.4%.

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Algebra is used to solve the mathematical problems, the total amount spent by Andy on lunches in this week is equals to $195.

Algebra is the branch of mathematics that use in the representation of problems or situations in the form of mathematical expressions. Mathematical ( arithmetic) operations say multiplication (×), division (÷), addition (+), and subtraction (−) are used to form a mathematical expression.

We have, a data of amount spent by Andy on lunches in a week. Let the total amount spent by him in this week be "x dollars". Using algebra of mathematics, we can written as x = sum of amounts spent by him in whole week so, x = $50 + $20 + $10 + $25 + $25 + $15 + $50 = $195

Hence, required total amount value is $195.

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Complete question:

Andy spent the following amounts on lunches this week,

day. amount

Sunday $50

Monday. $20

Tuesday $10

Wednesday $25

Thursday $25

Friday $15

Saturday $50

Calculate total amount he spent in this week.

We are given two sets A and B, and we are asked to describe some sets that can be formed using these sets. We will use set operations such as union and intersection to form new sets and provide descriptions of these sets in words.

a. A ∩ B: This set includes all current students who are attending both ABC and Harvard at the same time.

b. A ∪ B: This set includes all current students who are attending either ABC, Harvard, or both.

c. A ∩ B: (same as 'a') This set includes all current students who are attending both ABC and Harvard at the same time.

d. (A ∩ B): (same as 'a') This set includes all current students who are attending both ABC and Harvard at the same time.

e. (A ∪ B): (same as 'b') This set includes all current students who are attending either ABC, Harvard, or both.

3. Let A = {2 ∈ Z | x = 6a for some integer a} and B = {y ∈ Z | y = 36 for some integer b}. To prove that A ⊂ B, we need to show that every element of A is also an element of B.

Let x be an arbitrary element of A.

Since x = 6a for some integer a, we can write x as 6a = 2 * 3a.

Because 3a is also an integer (since a is an integer), we can say x = 2 * (3 * a), which implies x = 36 * a for some integer a. Thus, x ∈ B.

Since every element of A is also an element of B, we have proven that A ⊂ B.

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The sample space for choosing one card is, S={I, L, L, M, P, T, U, Y}

Since, We know that;

A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.

Given that,

A set of eight cards were labeled with M, U, L, T, I, P, L, Y.

Here, the sample space is;

{ S, U, B, T, R, A, C, T}

Now, Elements in order is,

⇒ S = {I, L, L, M, P, T, U, Y}

Therefore, the sample space for the given cards is,

S = {I, L, L, M, P, T, U, Y}

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The value of the iterated integral is [tex]7/3[/tex] √2 in the given case

To convert to polar coordinates, we need to express the integrand and the limits of integration in terms of polar coordinates. Let's start by finding the limits of integration:

0 ≤ y ≤ √2 - y[tex]^2[/tex]

0 ≤ x ≤ 1

The first inequality can be rewritten as [tex]y^2 + x^2[/tex] ≤ 2, which is the equation of a circle centered at the origin with a radius √of 2. Therefore, the limits of integration in polar coordinates are:

0 ≤ r ≤ √2

0 ≤ θ ≤ π/2

Now, let's express the integrand in polar coordinates:

7xy = 7r cos(θ) sin(θ)

And the differential area element in polar coordinates is:

dA = r dr dθ

Therefore, the integral becomes:

= [tex]7/3[/tex] √2

Therefore, the value of the iterated integral is [tex]7/3[/tex] √2.

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This strategy ensures that every multiple of gcd(a, b) up to max(a, b) is eventually on the board, and since the player who cannot make a move loses, you will always win.

In mathematics, a linear combination is a sum of scalar multiples of one or more variables.

(a) To show that every number on the board at the end of the game is a multiple of gcd(a, b), we will use mathematical induction.

First, note that any number that is a multiple of gcd(a, b) can be written as a linear combination of a and b. That is, for any positive integer k, there exist integers x and y such that k*gcd(a,b) = xa + yb.

Now, suppose that after some number of turns, the numbers on the board are c and d, where c is a multiple of gcd(a, b) and d is some other number. Then, we can write c = xa + yb and d = wa + zb for some integers x, y, w, and z.

On the next turn, a player must choose a number that is the difference of two numbers already on the board. Thus, the only possible choice is |c - d| = |xa + yb - wa - zb|.

We can rewrite this as |(x-w)a + (y-z)b|. Note that (x-w) and (y-z) are integers, so this number is a linear combination of a and b, and therefore a multiple of gcd(a, b). Thus, the new number on the board is a multiple of gcd(a, b).

By induction, every number on the board at the end of the game is a multiple of gcd(a, b).

(b) To show that every positive multiple of gcd(a, b) up to max(a, b) is on the board at the end of the game, we will again use induction.

First, note that gcd(a, b) itself must be on the board, since it is a multiple of gcd(a, b) and can be written as a linear combination of a and b.

Now, suppose that after some number of turns, all multiples of gcd(a, b) up to k are on the board, where k is a positive integer less than or equal to max(a, b).

Consider the next turn. The player must choose a number that is the difference of two numbers already on the board. Let c and d be the two numbers chosen. Then, we know that c - d is a multiple of gcd(a, b) by part (a).

Thus, every multiple of gcd(a, b) up to k + (c - d) is on the board. If k + (c - d) is greater than max(a, b), then we are done, since all multiples of gcd(a, b) up to max(a, b) are on the board.

Otherwise, we can continue the game and use induction to show that all multiples of gcd(a, b) up to max(a, b) will eventually be on the board.

(c) To win the game every time, always start by choosing gcd(a, b). This is a legal move, since it can be written as a linear combination of a and b.

From then on, always choose a number that is the difference of the two numbers on the board, except when that number is already on the board. In that case, choose any other number that is a multiple of gcd(a, b) that is not already on the board.

This strategy ensures that every multiple of gcd(a, b) up to max(a, b) is eventually on the board, and since the player who cannot make a move loses, you will always win.

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The 95% confidence interval for the mean amount (in milligrams) of nicotine in the sampled brand of cigarettes is C.39.2 to 40.8

We can use a t-table or a calculator to calculate the t-score. The t-score for a 95% confidence interval with 22 degrees of freedom (n-1) is around 2.074.

When we plug in the values, we get:

CI = 40 ± 2.074 * 1.8/√23 = 40 ± 0.763 = (39.237, 40.763)

As a result, the 95% confidence interval for the mean nicotine content of the studied cigarette brand is (39.237, 40.763) mg.

Because 39.2 to 40.8 is the closest response choice, the answer is 39.2 to 40.8.

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There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

The probability that exactly two kittens will be chocolate brown is 1/12.

The probability that at least two kittens will be chocolate brown is 7/12.

Priya can increase the accuracy of the simulation by increasing the number of trials.

We have,

To determine if a trial was a success, we need to count the number of chocolate brown kittens in each trial.

If a trial has at least two chocolate brown kittens, it is considered a success.

Now,

Using the table provided, we can count the number of chocolate brown kittens in each trial:

trial number outcome count of chocolate brown kittens

1 ggggg 0

2 gggwg 1

3 wgwgw 0

4 gwggg 1

5 gggwg 1

6 wwggg 0

7 gwggg 1

8 ggwgw 1

9 wwwgg 0

10 ggggw 2

11 wggwg 1

12 gggwg 1

So,

There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

To estimate the probability that exactly two kittens will be chocolate brown, we need to count the number of trials where exactly two chocolate brown kittens were born and divide it by the total number of trials.

From the table, we can see that there is only one trial where exactly two chocolate brown kittens were born (trial 10).

The estimated probability.

=  1/12

= 0.0833.

To estimate the probability that at least two kittens will be chocolate brown, we need to count the number of trials where at least two chocolate brown kittens were born and divide it by the total number of trials.

From the table, we can see that there are 7 successful trials.

The estimated probability.

= 7/12

= 0.5833.

Another question Priya could answer using this simulation is:

Question:

What is the probability that all five kittens will be white?

Answer:

We need to count the number of trials where all five cubes drawn were white (trial 6 and trial 9) and divide it by the total number of trials.

The estimated probability.

= 2/12

= 0.1667.

To increase the accuracy of the simulation, Priya could increase the number of trials conducted.

The more trials conducted, the more accurate the estimated probabilities will be.

Thus,

There are 8 successful trials (trials 2, 4, 5, 7, 8, 10, 11, and 12).

The probability that exactly two kittens will be chocolate brown is 1/12.

The probability that at least two kittens will be chocolate brown is 7/12.

Priya can increase the accuracy of the simulation by increasing the number of trials.

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Finally, using the commutative and associative properties of Boolean addition, we can group the terms to get AB + AC + B.

Q1:

Using De Morgan's theorem, we have:

F = XYZ + XYZ = XYZ(1 + 1) = XYZ

Taking the complement of F, we get:

F' = (XYZ)'

= (X'+Y'+Z')

= X'Y'Z'

Now, let's find the complement of F';

F' = X(YZ + Y'Z')

Taking the complement of F', we get:

F'' = (X(YZ + Y'Z'))'

= (X(YZ)')(Y(Y')Z')'

= (X'(Y'+Z))(YZ)

= X'YZ + XYZ'

Therefore, the complement of F is X'Y'Z', and the complement of F'; is X'YZ + XYZ'.

Q2:

ABC + ABC + ABC + ABC + ABC = ABC + ABC + ABC = ABC

Explanation: Using the associative property of Boolean addition, we can group the terms to get ABC + ABC + ABC = ABC.

AB + A(B+C) + B(B+C) = AB + AB + AC + BB + BC

= AB + AC + B

Explanation: Using the distributive property of Boolean multiplication over addition, we can expand the second and third terms to get AB + AC + BB + BC. Using the identity law, BB can be simplified to B. Finally, using the commutative and associative properties of Boolean addition, we can group the terms to get AB + AC + B.

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The number of miles is 7 while the number of minutes for this trip is 21.

Let x represent the number of minutes and y represent the number of miles. According to the given information, we have two equations:

1) 0.20x + 1.40y = $14
2) x = 3y

First, we will solve equation (2) for x:

x = 3y

Next, substitute this value of x into equation (1):

0.20(3y) + 1.40y = $14

Now, simplify and solve for y:

0.60y + 1.40y = $14
2.00y = $14
y = 7

Now that we have the value for y (number of miles), we can find the value for x (number of minutes) using equation (2):

x = 3y
x = 3(7)
x = 21

So, the number of minutes for this trip is 21, and the number of miles is 7.

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(a) The joint probability mass function of X and Y x1=0 x2=2 x3=4

y1=0 1/8 0 PY(y1)=1/2

y2=1 1/8 1/4 PY(y2)=1/4

y3=2 0 0 PY(y3)=1/4

(b) X and Y are uncorrelated. (c) The random variables X and Y are not independent with each other.

(a) We know that P(X=x2,Y=y1) = 0, since there are no entries in the table where X=x2 and Y=y1. Therefore,

x1=0 x2=2 x3=4

y1=0 1/8 0 PY(y1)=1/2

y2=1 1/8 1/4 PY(y2)=1/4

y3=2 0 0 PY(y3)=1/4

(b) The covariance of X and Y is :

Cov(X,Y) = E[XY] - E[X]E[Y]

where E[XY] is the expected value of the product XY,

E[X] = x1P(X=x1) + x2P(X=x2) + x3P(X=x3) = 0(1/4) + 2(1/2) + 4(1/4) = 2

E[Y] = y1P(Y=y1) + y2P(Y=y2) + y3P(Y=y3) = 0(1/2) + 1(1/4) + 2(1/4) = 1

Now,

E[XY] = x1y1P(X=x1,Y=y1) + x2y1P(X=x2,Y=y1) + x2y2P(X=x2,Y=y2) + x3y2P(X=x3,Y=y2) = 0(1/8) + 2(0) + 2(1/8) + 4(1/4) = 1.5

Therefore,

Cov(X,Y) = E[XY] - E[X]E[Y] = 1.5 - 2(1) = -0.5

Since the covariance is negative, hence X and Y are negatively correlated.

(c) To determine whether X and Y are independent, check whether:

P(X=x,Y=y) = P(X=x)P(Y=y)

for all possible values of x and y.

Using the joint probability mass function we determined in part (a), we can check this condition:

P(X=0,Y=0) = 1/8 ≠ (1/4)(1/2) = P(X=0)P(Y=0)

P(X=2,Y=1) = 1/8 ≠ (1/2)(1/4) = P(X=2)P(Y=1)

P(X=4,Y=2) = 1/4 ≠ (1/4)(1/4) = P(X=4)P(Y=2)

Therefore, X and Y are not independent.

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The total change to his score as an integer is -115

The total change in Juan score is calculated as follows;

Let Juan's initial score = x

when Juan finished the next level of his video game, he lost 10 points for each of the two targets he missed.

total points deducted = 20 points.

New score = x - 20

He was also penalized 95 points for taking too long.

His final score;

(x - 20) - 95

= x - 115.

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Part(a),

The average rate of change in annual sales between 2001 and 2002 was $14$ million per year.

Part(b),

The average rate of change in annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

a) Compute the difference in sales between 2001 and 2002 and divide it by the total number of years in order to determine the rate of change between those two years:

Rate of change = (Sales in 2002 - Sales in 2001) / (2002 - 2001)

Rate of change = (255 - 241) / 1 = 14 million/year

Therefore, the average rate of change in annual sales between 2001 and 2002 was $14$ million per year.

b) Calculate the difference in sales between 2001 and 2004 and divide it by the total number of years to determine the rate of change between those two years:

Rate of change = (Sales in 2004 - Sales in 2001) / (2004 - 2001)

Rate of change = (231 - 241) / 3 = -3.33 million/year

Therefore, the average rate of change of annual sales between 2001 and 2004 was a decrease of $3.33$ million per year.

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