Here are the number of hours that 9 students spend on the computer on a typical day: 1 6 7 6 8 11 6 12 15 The data from the above 9 students form what type of distribution? Unimodal Ultimodal Bimodal

Answer: Sample size n is 9.

The sample mean M is 84.

Step-by-step explanation:

Given : A professor gives a statistics exam.

The exam has 100 possible points.

The scores for the students in the second classroom are as follows:

88 84 84 52 100 84 92 84 88

Since , the number of scores of different students= 9

∴ Sample size is n= 9.

Now , Sample mean = [tex]\dfrac{\text{Sum of all data values}}{\text{Sample size}}[/tex]

[tex]=\dfrac{88+ 84 +84 +52 +100+ 84+ 92+ 84+ 88 }{9}\\\\=\dfrac{756}{9} =84[/tex]

Hence, the sample mean M is 84.

We have that the sample size, n, and the sample mean, M. is mathematically given as

From the question we are told

A professor gives a statistics exam.

Generally the equation for the  Mean is mathematically given as

[tex]M=\sum/n\\\\\Therefore\\\\M=\frac{88 +84 +84+ 52+ 100+ 84+ 92+ 84+ 88 }{9}\\\\M=84[/tex]

Therefore

New mean

M_n=84+4

M_n=88

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Answer:

[tex] cos \theta = -\frac{8}{17}[/tex]

Step-by-step explanation:

For this case we know that:

[tex] sin \theta = \frac{15}{17}[/tex]

And we want to find the value for [tex] cos \theta[/tex], so then we can use the following basic identity:

[tex] cos^2 \theta + sin^2 \theta =1 [/tex]

And if we solve for [tex] cos \theta [/tex] we got:

[tex] cos^2 \theta = 1- sin^2 \theta[/tex]

[tex] cos \theta =\pm \sqrt{1-sin^2 \theta}[/tex]

And if we replace the value given we got:

[tex] cos \theta =\pm \sqrt{1- (\frac{15}{17})^2}=\sqrt{\frac{64}{289}}=\frac{\sqrt{64}}{\sqrt{289}}=\frac{8}{17}[/tex]

For our case we know that the angle is on the II quadrant, and on this quadrant we know that the sine is positive but the cosine is negative so then the correct answer for this case would be:

[tex] cos \theta = -\frac{8}{17}[/tex]

Answer:

It is D

Step-by-step explanation:

EDGE 2021

The expressions can be used to calculate volume if they incorporate height, length, and width. They could calculate surface area if they involve length and width or the sum of all faces of a 3D object, without height. If these measurements are not present, they may not calculate surface area or volume.

The original question appears to be a task related to a mathematical exercise instead of a traditional query. It's asking whether certain expressions can be used to calculate volume or surface area, or if they're not meant to calculate either.

If the expression used includes measurements for length, width, and height and involves their multiplication, it's likely being used to calculate the volume of a 3D object. For a cube, volume is calculated as length x width x height, for example.

If the expression incorporates the multiplication of length and width, without incorporating height, it's being used to calculate surface area. The expression might also calculate the sum of all the faces of a 3D object to find total surface area.

If there aren't any apparent calculations for length, width, or height, it could be that these expressions aren't being used for calculating surface area or volume.

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Answer:

Option B is right

Step-by-step explanation:

P(A or B)

In set theory if A and B are two sets we have either A occurs or B occurs or both as

P(A or B) = P(AUB)

This implies that A occurs or B occurs

Probability is calculated as P(A)+P(B)-P(AB)

P(AB) here represents both occurring and this is subtracted as this was added two times

Option A is wrong because this is AB

Option C is wrong because it is not necessary A occurs first

D is wrong because this is A'UB' where A' represents the complement of A

So correct answer out of four options is

B. the probability that in a single​ trial, event A​ occurs, event B​ occurs, or they both occur.

The notation P(A or B) in probability specifies the probability that in a single trial, either event A happens, event B happens, or they both happen.

The notation P(A or B) in probability theory denotes the probability that event A occurs, event B occurs, or both events occur in a single trial. If we look at the four options provided, option B fits this definition. Hence, P(A or B) indicates the probability that in a single trial, event A occurs, event B occurs, or they both occur. It does not imply that both events A and B have to occur concurrently in the same trial (option A), nor that event A has to occur in one trial followed by event B in another trial (option C), and certainly not the probability that event A or event B does not occur in a single trial (option D).

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Answer:

There is a 99.99998% probability that at least one valve opens.

Step-by-step explanation:

For each valve there are only two possible outcomes. Either it opens on demand, or it does not. This means that we use the binomial probability distribution to solve this problem.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

In this problem we have that:

[tex]n = 5, p = 0.93[/tex]

Calculate P(at least one valve opens).

This is [tex]P(X \geq 1)[/tex]

Either no valves open, or at least one does. The sum of the probabilities of these events is decimal 1. So:

[tex]P(X = 0) + P(X \geq 1) = 1[/tex]

[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]

So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 0) = C_{5,0}.(0.93)^{0}.(0.07)^{5} = 0.0000016807[/tex]

Finally

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0000016807 = 0.9999983193[/tex]

There is a 99.99998% probability that at least one valve opens.

Answer:

0.930000

Step-by-step explanation:

Answer: Yes, you are correct it's $8.95 because like they stated in the word problem p= the price of each bowl and like they said in the word problem they charge $8.95 a bowl. So, $8.95 is the answer.

Step-by-step explanation:

The time needed to run the second half of the race is 8.44 s

Step-by-step explanation:

The total distance covered by the athlete during the race is

d = 200 m

And the total time taken to cover this distance is

T = 19.19 s

We also know that the time the athlete needs to cover the first half of the race is

[tex]t_1 = 10.75 s[/tex]

Also, we know that

[tex]T=t_1 + t_2[/tex]

where [tex]t_2[/tex] is the time the athlete takes to cover the second half of the race.

Re-arranging this equation and susbtituting the values, we find the value of t2:

[tex]t_2 = T-t_1 = 19.19-10.75=8.44 s[/tex]

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Answer:

[tex]\frac{1}{7}[/tex]

Step-by-step explanation:

The table containing information about the sexes of twins and and the type of twins are given in the table below with corresponding weightage.

Now, since we know that she is expecting twins, the probability of the event

                [tex]A - '\text{the woman will have identical twins }'[/tex]

is calculated as follows

[tex]Pr(A) = \dfrac{\text{total number of boy/boy, boy/girl, girl/boy and girl/girl identical twins}}{\text{total number of both identical and frathernal twins}}[/tex]

By using the informations from the given table, we obtain

                                        [tex]Pr(A) = \frac{2 + 0 + 0 +2}{28} = \frac{4}{28} = \frac{1}{7}[/tex]

Answer:

The probability that they get a sampling error of greater than 1.0 mg/L is 0.04.

To calculate this value, I assume that the samples are randomly collected

Step-by-step explanation:

Sampling error can be calculated using the formula

[tex]\frac{t*s}{\sqrt{n}}[/tex]  where

For the sampling error of 1.0mg/L we have

[tex]1=\frac{t*1.60}{\sqrt{10}}[/tex]    

Solving for t we have t≈1.976

Then the probability that they get a sampling error of greater than 1.0 mg/L is

P(t>1.976) ≈ 0.04.

In other words, we are 96% confident that the sampling error is within 1.0 mg/L.

To calculate this value, I assume that the samples are randomly collected.

Answer:

Roughly the same as what?

Step-by-step explanation:

Answer:

A. 400 GRAMS

B. 120 GRAMS

C. 356 GRAMS

D. 340 GRAMS

PENN FOSTER

ANSWER IS D 340 GRAMS

Step-by-step explanation:1 ounce = 28.3495 12 ounces = 28.3495 * 12 = 340.194 grams

Answer:

Red Marble is the color in the bag

Step-by-step explanation:

Total number of marbles ; 201

number of red marbles ; 101

number of blue marbles ; 101

first possibility ; From what is said as the condition, it can be inferred that the number of blue marbles is reduced by a factor of 1 , while the number of red marbles remains constant. this shows that the two marbles of the same color that was removed is most likely BLUE.

Second possibility ; Two red marbles are added (of the same color) and add a blue marbles, implies the the red marbles reduce by a factor of 2 while the blue marbles increase by a factor of 1. this is also possible.

Third possibility ; Two marbles of different color implies 1R and 1B , and then a red marble is added. this shows that the number of red marble still remains constant ( 1R removed, 1R added), while the number of blue reduce by 1 ( remove 1B, and not replaced back).

From the various possibilities, it can be inferred that the number of red marbles is either increasing by a factor of 2 or remains constant. this further shows that in all three , 1R marble will always remain in the bag no matter the possibility as such in all three scenerios, it further shows that if there is only a marble remaining, it would have been the RED MARBLE WHICH IS AUTOMATICALLY THE LAST MARBLE from the analysis done above.

Answer:

For First Solution: [tex]y_1(t)=e^t[/tex]

[tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.

For 2nd Solution:[tex]y_2(t)=cosht[/tex]

[tex]y_2(t)=cosht[/tex]  is the solution of equation y''-y=0.

Step-by-step explanation:

For First Solution: [tex]y_1(t)=e^t[/tex]

In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

[tex]y_1(t)=e^t[/tex]

First order derivative:

[tex]y'_1(t)=e^t[/tex]

2nd order Derivative:

[tex]y''_1(t)=e^t[/tex]

Put Them in equation y''-y=0

e^t-e^t=0

0=0

Hence [tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.

For 2nd Solution:

[tex]y_2(t)=cosht[/tex]

In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.

[tex]y_2(t)=cosht[/tex]

First order derivative:

[tex]y'_2(t)=sinht[/tex]

2nd order Derivative:

[tex]y''_2(t)=cosht[/tex]

Put Them in equation y''-y=0

cosht-cosht=0

0=0

Hence [tex]y_2(t)=cosht[/tex]  is the solution of equation y''-y=0.

The functions [tex]y(t)=e^t[/tex] and [tex]y(t)=cosht[/tex] are solutions of given differential equation.

We have to prove that given function is a solution of the differential equation.

Given function,   [tex]y(t)=e^t[/tex]

Given differential equation are, [tex]y''-y=0[/tex]

  So that,  [tex]y'(t)=e^t , y''(t)=e^t[/tex]

Substitute values in given differential equation.

                    [tex]e^t - e^t=0[/tex]

Thus, function [tex]y(t)=e^t[/tex] is solution of given differential equation.

Another function is,  [tex]y(t)=cosht[/tex]

So that,  [tex]y'(t)=sinht , y''(t)=cosht[/tex]

Substitute values in given differential equation.

                    [tex]cosht-cosht=0[/tex]

Thus, function [tex]y(t)=cosht[/tex] is solution of given differential equation.

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Answer:

a)  m = -9 or m = 1

b) m =  1 + √6 or m = 1 -√6

Step-by-step explanation:

for

Φ(x) = x^m

then

dΦ/dx (x) = m*x^(m-1)

d²Φ/dx² (x) = m*(m-1)*x^(m-2)

then

for a

3x^2 (d^2y/dx^2) + 11x(dy/dx) - 3y = 0

3x^2*m*(m-1)*x^(m-2) + 11*x* m*x^(m-1) - 3*x^m = 0

3*m*(m-1)*x^m + 11*m*x^m- 3*x^m = 0

dividing by x^m

3*m*(m-1) + 11*m - 3 =0

3*m² + 8 m - 3 =0

m= [-8 ± √(64 + 4*3*3)]/2 = (-8±10)/2  

m₁ = -9 , m₂= 1

then Φ(x) = x^m is a solution for the equation a , when m = -9 or m = 1

for b)

x^2 (d^2y/dx^2) - x(dy/dx) - 5y = 0

x^2*m*(m-1)*x^(m-2) -  x* m*x^(m-1) - 5*x^m = 0

m*(m-1)*x^m -m *x^m- 5*x^m = 0

dividing by x^m

m*(m-1) -m - 5 =0

m² - 2 m - 5 =0

m= [2 ± √(4 + 4*1*5)]/2 = (2±√24)/2 = 1 ±√6

m₁ =  1 + √6 , m₂ =  1 - √6

then Φ(x) = x^m is a solution for the equation b , when m =  1 + √6 or m = 1 - √6

Answer

a) m = -3 or 1/3

b) m = 1 + root 6 or 1 - root 6

Step-by-step explanation:

The step by step calculation is as shown in the attachment.

Answer:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

Step-by-step explanation:

First we can define some notation useful:

C ="represent the event of incurring in operating charges"

R= represent the event of emergency rooms charges"

For this case we are interested on P(C) since they want "the probability that a claim submitted to the insurance company includes operating room charges."

We have some probabilities given:

[tex] P(R') = 0.25 , P(C \cup R) =0.85[/tex]

Solution to the problem

By the complement rule we have this:

[tex] P(R') = 0.25 =1-P(R)[/tex]

[tex] P(R) = 1-0.25 = 0.75[/tex]

Since the two events C and R are considered independent we have this:

[tex]P(C \cap R) = P(C) *P(R)[/tex]

Now we can use the total probability rule like this:

[tex] P(C \cup R) = P(C) + P(R) - P(R)*P(C)[/tex]

And if we replace we got:

[tex] 0.85 = P(C) + 0.75 -0.75 P(C)[/tex]

[tex]0.1 = 0.25 P(C)[/tex]

[tex] P(C) = 0.4[/tex]

Answer:

We can assume that the sample size is large enough to use the z distribution as an approximation of the t distribution

Now we need to find on the z distribution a value that accumulates 0.02 of the area on the left and this value is [tex]z_{crit}=-2.05[/tex]

We can use the following excel code to verify it: "=NORM.INV(0.02,0,1)"Reject H0 if test statistic < -2.05

And for our case our calculated value <-2.05 so we have enough evidence to reject the null hypothesis at 2% of significance.

Step-by-step explanation:

Data given and notation  

[tex]\bar X=13.8[/tex] represent the sample mea   n

[tex]s=0.3[/tex] represent the sample standard deviation

[tex]n=45[/tex] sample size  

[tex]\mu_o =68[/tex] represent the value that we want to test

[tex]\alpha=0.02[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is at least 14 oz, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \geq 14[/tex]  

Alternative hypothesis:[tex]\mu < 14[/tex]  

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

[tex]t=\frac{13.8-14}{\frac{0.3}{\sqrt{45}}}=-4.47[/tex]    

Critical region

We can assume that the sample size is large enough to use the z distribution as an approximation of the t distribution

Now we need to find on the z distribution a value that accumulates 0.02 of the area on the left and this value is [tex]z_{crit}=-2.05[/tex]

We can use the following excel code to verify it: "=NORM.INV(0.02,0,1)"

So then the correct answer for this case would be:

Reject H0 if test statistic < -2.05

And for our case our calculated value <-2.05 so we have enough evidence to reject the null hypothesis at 2% of significance.

The decision criterion to reject the null hypothesis that the mean weight is at least 14 oz at the 0.02 level of significance is to reject H0 if the test statistic is less than -2.33, as derived from the standard normal z-table.

To determine the criterion for rejecting the null hypothesis in a hypothesis test like the one presented - where a cereal company claims that the mean weight of the cereal in its packets is at least 14 oz - we use the level of significance (alpha). Since the level of significance in this problem is 0.02, we refer to the standard normal z-table to find the critical z-value that corresponds to this significance level for a one-tailed test, which in this case is a left-tailed test because we are checking if the mean weight is less than the claim (14 oz).

Using the z-table, we find that the critical z-value for an alpha of 0.02 in a one-tailed test is approximately -2.33. If our calculated test statistic is less than this critical value, we would reject the null hypothesis in favor of the alternative. Therefore, the decision criterion is: Reject H0 if test statistic < -2.33.

Since the given options are 'Reject H0 if test statistic < 2.05', 'Reject H0 if test statistic > -2.05', 'Reject H0 if test statistic < -2.05', and 'Reject H0 if test statistic < -2.33', the correct criterion for this test, considering the level of significance is 0.02, is 'Reject H0 if test statistic < -2.33'.

Answer:

635599482kg

Step-by-step explanation:

A gourmet chocolate candy has 7.00 gg of dietary fat in each 22.7-gg piece

for the 22.7 piece , the gg is 7 x 22.7  = 158.9gg in total

1gg = 2204622.6218488 lbs

158.9gg in total  = 158.9 x 2204622.6218488 = 350314534.612

for the box of 4.00 lblb of candy

it is 350314534.612 x 4 = 1401258138.45lb

since our answer is needed in kilograms we convert 1401258138.45lb to kilograms

1lb = 0.453592kg

1401258138.45lb =  1401258138.45lb x 0.453592kg = 635599481.513 ≈ 635599482kg

Answer:

$34 billion

Step-by-step explanation:

Data provided in the question:

Total number of inhabitants of a country = 2 million

1 mean gross income = $20,000 per year

Tax paid by mean individual = $3000 per year

Now,

Mean disposable income

= Mean gross income - Mean tax

= $20,000 - $3,000

= $17,000 per year

Therefore,

Total disposable income per year

= Mean disposable income × Total number of inhabitants of a country

= $17,000 × 2 million

= $34 billion

Answer:

a) The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people

b) For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.

Step-by-step explanation:

Part a

The target population of interest on this case represent the "customers who bought a car during the previous year". They want to analyze all the people that satisfy this condition in order to see the satisfaction rate of these people

Part b

For this case the population is not hypothetical since is well defined and they have all the customers who bought a car during the last year, since that info is on the records of the manufacturer from people who bought a car. So then the population is available in order to analyze the question desired.

a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.

b.   The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.

the process of ensuring that research data is stored.

a. The customer who bought the car the previous year can get feedback from them so that we can evaluate the status of customer satisfaction.

b.   The customer who bought the car the previous year, so info is on the record of the manufacturer from the people who bought a car.

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Answer:

Step-by-step explanation:

Given that a die is thrown twice.  

X1, X2 are the outcomes in 2 throws

X1 = minimum of two

X1 can be either 1 or 2...6

Total outcomes are 36.

For x1 =1, fav ourable outcomes are (1,1) (1,2)...(1,6) (6,1)...(2/,1)= 11

P(X1=1) = [tex]\frac{11}{36}[/tex]\

P(X1=2) =Prob for one die showing two and other die showing 2 to 6

= [tex]\frac{9}{36}[/tex]

P(x1=3) = Prob for one die showing three and other die showing 3 to 6

=[tex]\frac{7}{36}[/tex]

thus we find that probability is reducing by 2 in the numerator

P(x1=4) = 5/36 followed by 3/36 for 5 and 1/36 for 6

Final answer:

The random variable X represents the minimum outcome when rolling a die twice. The values of X range from 1 to 6 and its distribution is given.

Explanation:

The random variable X represents the minimum outcome when rolling a die twice. In other words, X is the smaller of the two outcomes.

X can take on values from 1 to 6 since the outcomes of rolling a die are integers between 1 and 6. The minimum value of X is 1, which occurs when both dice show a 1.

The distribution of X is as follows:

Final answer:

The question translates to the inequality (1/5)n - 8 ≤ 96, which is solved by adding 8 to both sides and then multiplying by 5, resulting in n ≤ 520.

Explanation:

The question involves translating a word problem into a mathematical inequality. The phrase 'Eight less than the product of a number n and 1/5' can be written as (1/5)n - 8. When it states that this is 'no more than 96', it implies that the expression should be less than or equal to 96. Therefore, the inequality we need to solve is (1/5)n - 8 ≤ 96.

Now, let's solve this inequality step-by-step:

Add 8 to both sides of the inequality: (1/5)n ≤ 104.

Multiply both sides by 5 to solve for n: n ≤ 520.

This gives us the solution to the inequality, indicating that the number n can be any value less than or equal to 520 to satisfy the initial condition.

Answer:

Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.

Step-by-step explanation:

Let the number of months be 'x'.

Given:

Money Geno has = 'y' dollars.

Total Fitness charges:

Monthly fee = $30

Initial fee = $100

Gymania charges:

Monthly fee = $50

Initial fee = $25

Total charges is equal to the sum of initial fee and monthly fee multiplied by number of months.

So, for 'x' months, monthly fee charged by Total Fitness = [tex]30x[/tex]

For 'x' months, monthly fee charged by Gymania = [tex]50x[/tex]

Now, total charge by Total Fitness = Initial fee + Fee for 'x' months

Total charge by Total Fitness = [tex]100+30x[/tex]

Now, total charge by Gymania = Initial fee + Fee for 'x' months

Total charge by Gymania = [tex]25+50x[/tex]

Now, Geno has only 'y' dollars to spend. So, 'y' must be less than or equal to the total charge.

Therefore, the total charge for each membership is:

[tex]y=30x+100\\\\y= 50x+25[/tex]

Now, we graph both the equations. The graph is shown below.

From the graph, it is clear that, the total cost for Gymania (blue line) is less than that of Total Fitness (red line) till number of months equals 3.75 or 3 months. After 3.75 months, the graph of Gymania is above Total Fitness. So, if the membership is 4 months or above, then Total Fitness is more efficient.

Therefore, Gymania is a better deal if the membership is for 3 months and below.

Total Fitness is a better if the membership is for 4 months and above.

To determine which company offers the better deal for a gym membership, set up a system of equations for the two companies' costs and compare. Total Fitness charges $30 per month, while Gymania charges $50 per month. The equation is solved to find the break-even point where their costs are equal.

To determine which company offers the better deal, we can set up a system of equations based on the given information:

Let x be the total number of months for the gym membership.

Total Fitness charges an initial fee of $100 plus $30 per month, so the total cost can be represented by the equation: y = 30x + 100.

Gymania charges an initial fee of $25 plus $50 per month, so the total cost can be represented by the equation: y = 50x + 25.

To compare the two deals, we need to find the values of x where the total cost is the same for both companies. We can set up the following equation:

30x + 100 = 50x + 25.

Simplifying, we get:

20x = 75.

Dividing both sides by 20, we find that x = 3.75.

Since x represents the number of months, it cannot be a decimal, so we round up to the nearest whole number. Therefore, Geno should join Total Fitness if he plans to have the membership for 4 or more months, and Gymania if he plans to have the membership for 3 or fewer months.

Answer:

[tex] (\forall x) S(x)\rightarrow A(x) [/tex]

[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]

Step-by-step explanation:

Everyone who studied for the test received an A on the test.

That means- if you studied for the test you will recived an A, and it is hold for everyone, so we will use quantifire [tex]\forall[/tex].

[tex] (\forall x) S(x)\rightarrow A(x) [/tex]

It means: for every student holds- If it is correct that student x studied then  the student got an A.

Someone who did not study for the test received an A on the test.

It means that, there is at least one student hwo didn't studie but student got an A. So we have conjuction of two sentences (negation of the S(x) and A(x) for some student- for that we use existential quantifie).

[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]

Final answer:

The logical expression involving predicates S(x) and A(x) can be represented by ∀x(S(x) → A(x)), stating that all students who studied for the test received an A.

Explanation:

The question involving the predicates S(x): x studied for the test and A(x): x received an A on the test revolves around predicate logic, where we aim to understand and analyze the logical relations of sentences with subjects and predicates within a specified domain of discourse.

To define the logical expression that describes a relationship between studying for a test and receiving an A would depend on the specific relationship we want to express. For instance, a possible logical expression could be ∀x(S(x) → A(x)), which translates to 'For all students x in the class, if x studied for the test, then x received an A on the test.'

Answer:

Option a) Rejected              

Step-by-step explanation:

We define critical region as:

Thus,

If a test statistic falls in the critical region the null hypothesis is rejected.

Answer:

8 points below the mean.

Step-by-step explanation:

Let 'X' be the score of the third person, and let 'M' be the mean score of the sample.

If the other two people scored 5 and 3 above the mean, in order to maintain consistency, the following expression must be true:

[tex]M+5+(M+3)+X =3M\\X=3M-2M-8\\X=M-8[/tex]

Therefore, the third person has a score 8 points below the mean.

Given that the other two people's scores are collectively 8 points above the mean, the third person's score must be 8 points below the mean to balance out the total deviation.

This question is about determining the score of the third person in a sample relative to the mean of the sample. We know that the total deviation from the mean must equal zero because the mean is the average of all the scores. Given the first person has a score 5 points above the mean, and the second person has a score 3 points above the mean, the total deviation above the mean is (5+3)=8 points. For the score deviations to balance out to zero, the third person's score must be 8 points below the mean.

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Answer:

a) 5 units/s

b) yes

c) counter-clockwise

d) yes

Step-by-step explanation:

part a

r(t) = cos (5t) i + sin (5t)j

v(t) = dr(t) / dt = -5sin(5t) i + 5cos(5t)j

[tex]mag( v(t)) = \sqrt{(-5sin(5t))^2 + (5cos(5t))^2} \\mag( v(t)) = \sqrt{25sin^2(5t) + 25cos^2(5t)} \\ \\mag( v(t)) = \sqrt{25*(sin(5t)^2 + cos(5t)^2)} \\\\mag( v(t)) = \sqrt{25} \\\\mag( v(t)) = 5 units/s[/tex]

Hence, the particle has a constant speed of 5 units/s

part b

a(t) = dv(t) / dt = -25cos(5t) i - 25sin(5t)j

To check orthogonality of two vectors their dot product must be zero

a(t) . v(t) = (-25cos(5t) i - 25sin(5t)j) . (-5sin(5t) i + 5cos(5t)j)

= 125cos(5t)*sin(5t) -125cos(5t)*sin(5t)

= 0

Yes, the particles velocity vector is always orthogonal to acceleration vector.

part c

Use any two values of t and compute results of r(t)

t = 0 , r(0) = 1 i

t = pi/2, r(0) = j

Hence we can see that the particle moves counter-clockwise

part d

Find the value r(t) at t=0

r(0) = cos (0) i + sin (0) j

r(0) = 1 i + 0 j

Yes, the particle starts at point ( 1, 0)

Answer:

a) [tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]

b) [tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]

c) [tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]

Step-by-step explanation:

For this case we know that the average scores for the class are given by the following model:

[tex] f(t) = 76-18 log_{10} (t+1) , 0 \leq t \leq 12[/tex]

Where t is in months. The graph attached illustrate the function for this case

And for this case we can answer the questions like this:

Part a

We just need to replace t =0 into the model and we got:

[tex] f(t) = 76-18 log_{10} (0+1)= 76 - 18 log_{10} 1= 76[/tex]

Part b

We just need to replace t =2 into the model and we got:

[tex] f(t) = 76-18 log_{10} (2+1)= 76 - 18 log_{10} 3= 76-8.588=67.41[/tex]

Part c

We just need to replace t =11 into the model and we got:

[tex] f(t) = 76-18 log_{10} (11+1)= 76 - 18 log_{10} 12= 76-19.425=56.574[/tex]

Final answer:

To evaluate the manufacturer's claim, a one-sample t-test can be conducted. The test statistic can be calculated using the given information.

Explanation:

To conduct a hypothesis test to evaluate the manufacturer's claim, we can use a one-sample t-test. The null hypothesis (H0) is that the mean MPG is equal to 39.0, and the alternative hypothesis (Ha) is that the mean MPG is not equal to 39.0. We can calculate the test statistic using the formula:

test statistic = (sample mean - population mean) / (standard deviation / sqrt(sample size))

Using the given values:

sample mean = 38.8

population mean = 39.0

standard deviation = 2.2

sample size = 120

Substituting these values, the test statistic is:

test statistic = (38.8 - 39.0) / (2.2 / sqrt(120))

Answer:

Ace Truck

[tex]C(m) = 30 + 0.5*m[/tex]

Acme Truck

[tex]C(m) = 25 + 0.55*m[/tex]

Step-by-step explanation:

The cost function to lease a box truck from a company has the following format:

[tex]C(m) = F + a*m[/tex]

In which F is the fixed cost and a is the cost per mile m.

(a) Find the daily cost of leasing from each company as a function of the number of miles driven.

Ace Truck

$30/day and $0.50/mi. This means that [tex]F = 30, a = 0.50[/tex]. So

[tex]C(m) = 30 + 0.5*m[/tex]

Acme Truck

$25/day and $0.55/mi. This means that [tex]F = 25 a = 0.55[/tex]. So

[tex]C(m) = 25 + 0.55*m[/tex]

Answer:

4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.

Step-by-step explanation:

In bag of Vigoro Ultra Turf fertilizer:

29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash

In bag of Parkera's Premium Starter:

18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash

Let the mass of Vigoro Ultra Turf fertilizer = x

Amount of nitrogen in x amount = 29 x

Amount of phosphoric acid  in  x amount = 3x

Let the mass of Parkera's Premium Starter= y

Amount of nitrogen in y amount = 18 y

Amount of phosphoric acid  in y  amount = 25 y

Amount of nitrogen in desired mixture formed by combing x and y fertilizers bag : 134 pounds

29 x + 18 y = 134 ..[1]

Amount of phosphoric acid  in desired mixture formed by combing x and y fertilizers bag : 37 pounds

3x + 25 y = 37 ..[2]

Solving [1] anf [2] by substitution method :

[tex]x=\frac{134-18y}{29}[/tex]

[tex]3\times \frac{134-18y}{29}+25y=37[/tex]

y = 1

[tex]x=\frac{134-18y}{29}=4[/tex]

4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.

To create a mixture containing 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash, 400 pounds of Vigoro Ultra Turf and 100 pounds of Parker's Premium Starter are required, based on solving a system of linear equations.

We are tasked with finding the amount of two different fertilizers needed to obtain a mixture with 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash. The two fertilizers are Vigoro Ultra Turf and Parkerâ's Premium Starter. We'll use a system of linear equations to solve this problem.

Let V represent the amount (in pounds) of Vigoro Ultra Turf and P represent the amount (in pounds) of Parkerâ's Premium Starter. The equations based on the given information are:

Now we solve these equations simultaneously to determine V and P. After solving, we find that V = 400 and P = 100. Thus, we need 400 pounds of Vigoro Ultra Turf and 100 pounds of Parkerâ's Premium Starter to obtain the desired mixture.

Answer:

a) (-∞,-9)∪(-9,9)∪(9,+∞)

b) (-∞,+∞)

Step-by-step explanation:

The domain of a real function is the largest subset og the real line in which it is defined.

a) The function [tex]f(x)=\frac{81-e^{x^2}}{1-e^{81-x^2}}[/tex] is defined for all values of x in which the denominator is not zero. The denominator is zero if [tex]0=1-e^{81-x^2}[/tex], that is,  [tex]e^{81-x^2}=1[/tex]. The exponential function is equal to one only if the exponent is equal to zero, then the values of x that nullify the denominator satisfy [tex]81-x^2=0[/tex], thus x=9 or x=-9. Then the domain of f is the set of all real numbers such that x≠9 and x≠-9, which is the set (-∞,-9)∪(-9,9)∪(9,+∞).

b) In this case the denominator is [tex]e^{\cos x}[/tex] which is always positive. Thus the denominator is not zero for all real x, then the fomain is the real line, which is the interval (-∞,+∞).

Final answer:

The domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2) is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞). The domain of the function f(x) = 7 + x/e^cos(x) is (-∞, ∞).

Explanation:

(a) To find the domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator is not zero. So, we need to solve the equation 1 - e^(81-x^2) = 0 to find where the denominator equals zero. Simplifying this equation, we get e^(81-x^2) = 1. Taking the natural logarithm of both sides, we have 81-x^2 = 0. Therefore, x^2 = 81, and x = ±9. The domain of the function is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞).

(b) To find the domain of the function f(x) = 7 + x/e^cos(x), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator e^cos(x) is not zero. Since the exponential function is always positive, we know that e^cos(x) will never be zero. Therefore, the domain of the function is the set of all real numbers, (-∞, ∞).