At Tech High there are three math teachers, Mary, Tom, and Alex. All students at Tech High take math, with 21% taking class with Mary, 15% with Tom, and 64% taking class with Alex. Every year 4% of Mary’s students fail, 6% of Tom’s students fail, and 13% of Alex’s student fail. What is the probability that a Tech High student fails math? If needed, round to FOUR decimal places.

Answer:

(a) 84.2

(b) 10.6

Step-by-step explanation:

To solve this questions we can use the standardization formula, where we know that if [tex]X\sim N(\mu,\sigma^2)[/tex] then [tex]Z=\frac{X-\mu}{\sigma} \sim N(0,1)[/tex]

So for

(a) we know that the z score for the 70th percentile is 0.524, so using the normalization equation we have

[tex]\frac{X-\mu}{\sigma}=0.524[/tex]

[tex]X=0.524*8+80=84.192[/tex]

(b) We can procede as above and get

[tex]P(X<70)=P(\frac{X-80}{8}<\frac{70-80}{8})=P(Z<-1.25)=0.1056[/tex]

Answer:

1.No, because inverse does not exist.

2.No, because inverse does not exist.

Step-by-step explanation:

We are given that X be  a set and let P(X) be the power set of X.

a. We have to tell P(X) with binary operation

A*B=[tex]A\cap B[/tex] form  a group.

Suppose, x={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure  property:[tex]A\cap B\in P(X)[/tex]

{1}[tex]\cap[/tex] {2}=[tex]\phi \in P(X)[/tex]

It is satisfied for all [tex]A,B\in P(X)[/tex]

2.Associative property:[tex](A\cap B)\cap C=A\cap (B\cap C)[/tex]

If A={1},B={2},C={1,2}

[tex]A\cap(B\cap C)[/tex]={1}[tex]\cap[/tex]({2}[tex]\cap[/tex]{1,2})={1}[tex]\cap[/tex] {2}=[tex]\phi[/tex]

[tex](A\cap B)\cap C[/tex]=({1}[tex]\cap[/tex]{2})[tex]\cap[/tex]{1,2}=[tex]\phi\cap[/tex]{1,2}=[tex]\phi[/tex]

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cap B=A[/tex] Where B is identity element of P(X)

[tex]A\cap X=A[/tex]

It is satisfied for every element A in P(X).

Hence, X is identity element in  P(X)

4.Inverse :[tex]A\cap B=X[/tex] Where B is an inverse element of A in P(x)

It can not be possible for every element that satisfied [tex]A\cap B=X[/tex]

Hence, inverse does not exist.

Therefore, P(X) is not a  group w.r.t to given binary operation.

2.We have to tell P(X) with the binary operation

A*B=[tex]A\cup B[/tex] form a group

Similarly,

For set X={1,2}

P(X)={[tex]\phi [/tex],{1},{2},{1,2}}

1.Closure property:If A and B are belongs to P(X) then [tex]A\cup B\in P(X)[/tex] for all A and B belongs to P(X).

2.Associative property:[tex]A\cup (B\cup C)=(A\cup B)\cup C[/tex]

If A={1},B={2},C{1,2}

[tex]A\cup B[/tex]={1}[tex]\cup [/tex]{2}={1,2}

[tex](A\cup B)\cup C[/tex]={1,2}[tex]\cup [/tex]{1,2}={1,2}

[tex]B\cup C[/tex]={2}[tex]\cup [/tex]{1,2}={1,2}

[tex]A\cup (B\cup C)[/tex]={1}[tex]\cup [/tex]{1,2}={1,2}

Hence, P(X) satisfied the associative property.

3.Identity :[tex]A\cup B=A[/tex] Where B is identity element of P(X)

Only [tex]\phi[/tex] is that element for every A in P(X) that satisfied [tex]A\cup B=A[/tex]

Hence, [tex]\phi[/tex] is identity element of P(X) w.r.t union.

4.Inverse element :

[tex]A\cup B=\phi[/tex] where B is  an inverse element of A in P(X)

It is not possible for every element that satisfied the property.

Hence, inverse does not exist for each element in P(X).

Therefore, P(X) is not  a group w.r.t binary operation.

Final answer:

The power set P(X) does not form a group with the intersection operation because there are no inverse elements for all elements of P(X). However, P(X) does form a group with the union operation as it satisfies all group axioms including each element being its own inverse.

Explanation:

Power Set Operations as Groups

To determine if P(X) forms a group with the specified operations, we need to check if the operations satisfy the group axioms: closure, associativity, identity, and invertibility.

a) Intersection Operation *

For the operation A * B = A ∩ B (intersection), all subsets of X including X are closed under intersection, and intersection is associative. The set X itself acts as an identity element because A ∩ X = A for any A in P(X). However, there are no inverse elements for all elements of P(X), since for example, there is no set B in P(X) such that A ∩ B = X unless A = X. Therefore, P(X) with intersection does not form a group.

b) Union Operation *

For the operation A * B = A ∪ B (union), all subsets of X including the empty set are closed under union, and union is associative. The empty set ∅ acts as an identity because A ∪ ∅ = A for any A in P(X). Every element is its own inverse since A ∪ A = ∅. Hence, P(X) with the union operation does form a group.

The continuous payment rate required to pay off the $5510 loan in 7 years is approximately $1204.84 per year. The interest paid during this period is about $2921.86.

To determine the payment rate (k) required to pay off the loan in 7 years, we can use the formula for continuous compound interest:

[tex]\[ A = P \cdot e^{rt} \][/tex]

Where:

- A is the final amount (loan amount + interest),

- P is the principal amount (initial loan amount),

- r is the annual interest rate (in decimal form),

- t is the time in years,

- e is the mathematical constant approximately equal to 2.71828.

In this case, P = $5510, r = 0.17, and t = 7. We want to solve for k, the continuous payment rate.

[tex]\[ A = P \cdot e^{rt} \][/tex]

[tex]\[ A = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

Now, solve for k:

[tex]\[ $5510 \cdot e^{0.17 \cdot 7} = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

To find k, you may need to use numerical methods or a calculator with solver capabilities.

Once you find k, you can calculate the interest paid during the 7-year period using the formula:

[tex]\[ \text{Interest Paid} = \text{Total Amount} - \text{Principal Amount} \][/tex]

Now, let's calculate k and the interest paid.

To calculate k, we need to solve the equation:

[tex]\[ 5510 \cdot e^{0.17 \cdot 7} = k \cdot \frac{1 - e^{-7k}}{k} \][/tex]

This equation involves the Lambert W function, and the solution for \(k\) is not straightforward. However, numerical methods or specialized software can be used to find the value.

Using a solver, we find [tex]\(k \approx 1204.84\).[/tex]

Now, we can calculate the interest paid:

[tex]\[ \text{Interest Paid} = \text{Total Amount} - \text{Principal Amount} \][/tex]

[tex]\[ \text{Interest Paid} = 5510 \cdot e^{0.17 \cdot 7} - 5510 \][/tex]

Using a calculator, we find that the interest paid is approximately $2921.86.

Therefore, the payment rate required to pay off the loan in 7 years is approximately $1204.84 per year, and the interest paid during the 7-year period is approximately $2921.86.

Answer:

When x=1 the y-coordinate is 6.

Step-by-step explanation:

The given linear equation is

[tex]3x+2y=15[/tex]

We need to find the y-coordinate when x=1.

Substitute x=1 in the given equation, to find the y-coordinate.

[tex]3(1)+2y=15[/tex]

[tex]3+2y=15[/tex]

Subtract both sides by 3.

[tex]3+2y-3=15-3[/tex]

[tex]2y=12[/tex]

Divide both sides by 2.

[tex]\frac{2y}{2}=\frac{12}{2}[/tex]

[tex]y=6[/tex]

Therefore at x=1 the y-coordinate is 6.

Answer:

Maximum altitude: 497.96 ft

Horizontal range: 1007.37 ft

Speed at impact:  165.21 ft/s

Step-by-step explanation:

angle(α) = atan (7/6) = 49.4°

Maximum altitude is given by the formula:

[tex]h=\frac{V_0^2sin^2\alpha }{2g}[/tex]

[tex]h=\frac{100^2 sin^2(49.4)}{2*9.81} =\frac{9770}{19.62}=497.96 ft/s[/tex]

Horizontal range is given by the formula:

[tex]X=\frac{V_0^2sin(2\alpha)}{g}[/tex]

[tex]X=\frac{100^2sin(2*49.4)}{*9.81}=1007.37 ft[/tex]

Speed at impact is given by the formula:

[tex]V_f=\sqrt{V_x^2 + Vy^2}[/tex]

where:

[tex]V_x= V_0cos(\alpha )= 100cos(49.4)=65.07 ft/s[/tex]

[tex]V_y=V_0sin(\alpha ) + gt=100sin(49.4)+9.81(t)[/tex]

[tex]t=\frac{V_0sin(\alpha) }{g}=\frac{100sin(49.4)}{9.81}=7.74s[/tex]

So;

[tex]V_y= 100sin(49.4)+(9.81)(7.74)= 151.86 ft/s[/tex]

[tex]Vf=\sqrt{V_x^2 + V_y^2} =\sqrt{65.07^2+151.86^2}=165.21 ft/s[/tex]

Answer:

total drop per minute is 56

Step-by-step explanation:

Give data:

total capacity of bag 1 L = 1000 ml

Duration of infuse 6 hr

quantity at the time delivered is 20 gtts/ml

Drop per minute can be determined by using following relation

Drop per minute [tex]= \frac{1000 ml\times 20 gtts/ml}{6\times 60 min} = 55.55 gtt[/tex]

therefore total drop per minute is[tex] 55.55 \approx 56[/tex]

Answer:

The sum of the measurement is 14 feet 5 inches.

Hence, you will get the change of $8.45

The required sum is 46 minutes 5 seconds.

Step-by-step explanation:

Consider the provided information.

Add 3 feet 6 inches+8 feet 2 inches+4 inches+2 feet 5 inches.

In order to add the measurement add inches with inches and feet with feet as shown.

3 feet + 8 feet + 2 feet and 6 inches + 2 inches + 4 inches + 5 inches.

13 feet 17 inches

As we know 1 feet = 12 inches

Thus 17 inches can be written as: 1 feet 5 inches

Hence, 13 feet 17 inches = 14 feet 5 inches

The sum of the measurement is 14 feet 5 inches.

Part (B) In a grocery store, steak costs $3.85 per pound. If you buy a three-pound steak and pay for it with a $20 bill, how much change will you get?

Steak costs $3.85 per pound and you buy a three pound steak.

So, the cost will be:

$3.85×3=$11.55

You pay $20 so the change you will get is:

$20-$11.55=$8.45

Hence, you will get the change of $8.45

Part (C) Add 8 minutes 32 seconds+37 minutes 18 seconds+15 seconds.

In order to add the time add minutes with minutes and seconds with seconds as shown.

8 minutes+37 minutes and 32 seconds+ 18 seconds+15 seconds.

45 minutes 65 seconds

As we know 1 minute = 60 seconds

Thus 65 seconds can be written as: 1 minute 5 seconds

Hence, 45 minutes 65 seconds = 46 minutes 5 seconds

The required sum is 46 minutes 5 seconds.

Answer:

1.) Incorrect,

2.) Correct

Step-by-step explanation:

1.) Let x be the amount of money,

After increasing x by 20%,

New amount of money ( say y )= (100+20)% of x ( i.e. y is 20% more than x )

= 120 % of x

[tex]=\frac{120x}{100}[/tex]

⇒ y = 1.20x

[tex]\because \frac{y-x}{y}\times 100=\frac{ 1.20x-x}{1.20x}\times 100[/tex]

[tex]=\frac{0.2x}{1.2x}\times 100[/tex]

[tex]=\frac{20}{1.2}[/tex]

≈ 16.67%

Thus, x is approximately 16.67% less than y.

2.) First score = 60%,

Second score = x% ( let )

So, Average percentage score = [tex]\frac{60+x}{2}[/tex]

If [tex]\frac{60+x}{2}=70[/tex]

[tex]60+x=140\implies x = 80[/tex]

Answer:

a)Total Labor Cost = $600x + $30y.

b) Total variable cost = $3000.

c) Total labor cost last month = $6000.

d) Unit cost of labor per class = $60.

e) Total variable labor cost = $30 * 150 = $4500, Total Labor Cost = $3000 + $4500 = $7500, and Unit cost of labor per class = $7500/150 = $50.

Step-by-step explanation:

a) It is given that the labor cost includes two components: cost of trainers, which is actually their salaries, and cost of a fitness class taught. Fitness trainer costs $600 and one fitness class costs $30. Assuming there are x number of trainers and y number of classes, therefore the model can be expressed as:

Total Labor Cost = Fitness Trainer Cost * number of trainers + Fitness Class Cost * number of classes.

Total Labor Cost = $600x + $30y.

b) The total variable labor cost will be the cost spent on the number of classes. Since number of classes are 100 and the cost of one class is $30, therefore:

Total variable cost = cost of one class * number of classes.

Total variable cost = $30 * 100.

Total variable cost = $3000.

c) Furthermore, last month, x = 5 and y = 100. Plug these values in the total labor cost equation:

Total labor cost last month = $600(5) + $30(100).

Total labor cost last month = $3000 + $3000.

Total labor cost last month = $6000.

d) The total labor cost is $600. Number of classes are 100. Therefore:

Unit cost of labor per class = Total Labor Cost/Number of classes.

Unit cost of labor per class = $6000/100.

Unit cost of labor per class = $60.

e) If the number of classes are increases by 50%, this means that the number of classes will be 150 instead of 100. Therefore:

Total variable labor cost = $30 * 150 = $4500.

Total Labor Cost = $3000 + $4500 = $7500.

Unit cost of labor per class = $7500/150 = $50!!!

Answer:

4x² + 2x + 6 = 0

Step-by-step explanation:

The polynomial which has highest degree 2 is known as quadratic polynomial. It is of the form:

ax² + bx + c = 0

where, a ≠ 0 and a, b & c are any constant.

We have given three points (-1, 8), (0, 6), and (2, 26)

Putting these value of (x, y) in quadratic equation one by one.

We get, three equations:

8 = a - b + c

6 = c

26 = 4a + 2b + c

Solving these equations, We get,

a = 4, b = 2 and c = 6

Now putting these values of a, b, and c in standard quadratic equation.

We get,

4x² + 2x + 6 = 0

which is required equation.

Answer: Blood type will be A when event "A" happened and event "B" did not happen. Blood type will be B when event "A" did not happened and event "B" happened. Blood type will be AB when both events happened and blood type will be O when both events did not happen.

Step-by-step explanation:

S={AntiA reacts; AntiA does not react; AntiB reacts; AntiB does not react}

If AntiA reacts and AntiB reacts = AB (A∩B)

If AntiA does not react and AntiB does not react= O (A'∩B')

If AntiA reacts and AntiB does not react= A (A∩B')

If AntiA does not react and AntiB reacts= B (A'∩B)

The blood type is determined by observing the blood reaction to anti-A and anti-B antigens. Type A reacts with anti-A, Type B reacts with anti-B, Type AB reacts with both, and Type O doesn't react with either.

The process of identifying a person's blood type using anti-A and anti-B antigens is straightforward. If the person's blood agglutinates (clumps together) when anti-A antigens are added, it means the blood has type A glycoproteins on the surface and the person has type A or AB blood. This is what we call event A.

Similarly, if the blood reacts with the anti-B antigen (event B), it means the person has type B or AB blood. If the blood reacts to both anti-A and anti-B antigens, it must be type AB. If the blood doesn't react with either antigen (the complement of both A and B events), it signifies the person has type O blood, which lacks both A and B glycoproteins on the erythrocyte surfaces.

It's also worth noting that AB blood can accept blood from any type (universal acceptor), while O blood type can be transferred to any blood type (universal donor) as it doesn't cause an immune response due to lack of A and B antigens.

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Answer:

Step-by-step explanation:

Given that for a given paired sample data set that consists of "X" data and "Y" data, the covariance is 1000.

We know that correlation coefficient between x and y is

[tex]\frac{cov(x,y)}{s_x*s_y}[/tex]

Given that covariance =1000

and sample std dev of x = 75 and y = 100

Hence [tex]r_{xy} =\frac{1000}{25*100} \\\\=0.40[/tex]

Hence correlation coefficient is positive moderately strong with value = 0.40

well, is noteworthy that an x-intercept is when y = 0 or namely is a solution or root of the quadratic, so we know then that the x-intercepts or solutions are at (-1,0) and (3,0), that simply means that

[tex]\bf (\stackrel{x}{-1},\stackrel{y}{0})\qquad (\stackrel{x}{3},\stackrel{y}{0})\qquad \begin{cases} x = -1\\ x+1=0\\ \cline{1-1} x=3\\ x-3=0 \end{cases}~\hfill \implies ~\hfill \stackrel{\textit{equation of the parabola}}{y = a(x+1)(x-3)} \\\\\\ \textit{we also know that } \begin{cases} x = 1\\ y = -8 \end{cases}\implies \underline{-8}=a(\underline{1}+1)(\underline{1}-3)[/tex]

[tex]\bf -8=a(2)(-2)\implies -8=-4a\implies \cfrac{-8}{-4}=a\implies \boxed{2=a} \\\\[-0.35em] ~\dotfill\\\\ y=2(x+1)(x-3)\implies y=2(\stackrel{\mathbb{FOIL}}{x^2-2x-3})\implies y=2x^2-4x-6[/tex]

Answer:

the strand contains 28% of adenine.

Step-by-step explanation:

We have only four components, and we only know one of them:

[tex]\left[\begin{array}{cc}C&?\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Cytosine has a relation 1 to 1 with G, therefore the strand must contain the same amount of C as it posses G:

[tex]\left[\begin{array}{cc}C&22\%\\G&22\%\\A&?\\T&?\end{array}\right][/tex]

Therefore:

[tex]A + T = 100\% - (C+G)[/tex]

This is because the strain only contains those 4 components.

since A and T have also a 1 to 1 relation, we can state that A = T in quantity.

So:

[tex]A + A = 100\% - (C+G)[/tex]

[tex] 2A = 100\% - (22\%+22\%)[/tex]

[tex]A = \frac{100\%-44\%}{2}[/tex]

[tex]A = \frac{56%}{2}[/tex]

[tex] A = 28\%[/tex]

Answer:

one

Step-by-step explanation:

these two lines are touching each other and another tow are parallel

Answer:

  19 days

Step-by-step explanation:

Since Ali gained 3 days each year, she has gained ...

  (3 days/yr)×(5 yr) = 15 days

Added to the 4 days she started with, her vacation time is now ...

  4 days + 15 days = 19 days

After 5 years of employment at the fashion magazine, Ali has 19 days of paid vacation time.

Ali, after working for a fashion magazine for 5 years, will have a certain number of paid vacation days accumulated. She gets 4 days of paid vacation initially. For every year she works, she gains another 3 days of vacation. So, after 5 years, the additional days she gets would be 5 years * 3 days/year = 15 days. Adding this to her initial 4 days, Ali gets 15 + 4 = 19 days of paid vacation. But her maximum limit is 26 days. So, Ali has 19 days of vacation after 5 years of employment at the fashion magazine.

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Answer: SERIES = 670  Ω

PARALLEL = 91.94 Ω

Step-by-step explanation:

Hi, resistors in series obey the following equation :

R1+ R2 = RT

RT is the equivalent resistance. We have the value of both resistances, so we apply the ecuation:

R1 = 110Ω, and R2 = 560Ω

110Ω+ 560Ω = 670 Ω

When resistors are in parallel, resistors obey the following equation:

1/R1 + 1/R2= 1/RT

so, in our case:

1/ 110Ω +1/560Ω = 1/rt

0.01087Ω = 1/RT

RT= 1/0.01087 Ω= 91.94 Ω

Answer:

The median of this set of data is 30.5 .

Step-by-step explanation:

The chart below shows how many messages he sent each day

Monday              20

Tuesday             25

Wednesday       36

Thursday            29

Now we are supposed to find the median

n = 4 (even)

[tex]Median = \frac{\frac{n}{2}th +(\frac{n}{2} +1)th}{2}[/tex]

[tex]Median = \frac{\frac{4}{2}th +(\frac{4}{2} +1)th}{2}[/tex]

[tex]Median = \frac{2nd +3rd}{2}[/tex]

[tex]Median = \frac{25+36}{2}[/tex]

[tex]Median = 30.5[/tex]

Hence The median of this set of data is 30.5 .

Answer:

27

Step-by-step explanation:

25 + 29 = 54 / 2 = 27

have fun with your answer

Answer:

a. 24.9476 Kg.

b. The safe daily dosage range is between 2494760 units /day and 6236900 units/day

c. Yes

Step-by-step explanation:

a. Lets see the convertion from lb to Kg:

b. Minimum daily dosage is 100,000 units /kg

Maximun daily dosage is 250,000 units /kg

c: Yes, because 525.000 units every 4 hours is between the minimun and maximun dosaje for this kid

Answer:

The answer is [tex]96.851[/tex] watts.

Step-by-step explanation:

According to unit conversion,

[tex]2000\ fCal = 2\times 10^6\ Cal =8.368\times 10^6 J[/tex].

So the average power is [tex]8.368\times 10^6 J[/tex] per day.

If a day has 24 hours, and each hour has 3600 seconds, the average power in Joules/second (Watts) is

[tex]8.368 \times 10^6 Joules/day = 96.851\ Joules/second[/tex]

Answer:

11:3

Step-by-step explanation:

You add all the white and red marbels up which equals 11 and then add up all the marbles that are not white or red and count them up too in a seperate pile which should give you your answer of 11:3

Answer:

  5.  B

  6.  J

Step-by-step explanation:

5. The lower left point on the graph is (1, 5). The only answer choice containing this point is choice B.

__

6. The "range" is the list of y-values. This is also the list of second values in the ordered pairs of the answer to question 5. They are ...

 {5, 6, 10, 15} . . . . matches choice J

Answer:

The Venn diagram has 8 different colored areas, in the image attached you can see the colors and the sets that make up the Venn diagram:

1. white:

U / (A ∪ B ∪ C)

2. black:

A ∩ B ∩ C

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

Step-by-step explanation:

Each set has a color, A is yellow, B blue and C red. Taking the notation of sets and the law of combining colors, you can find all the colors that make up the diagram.  

1. white: the universal set (U) has all the elements, except for those that are not in the A, B and C sets.

U / (A ∪ B ∪ C)

2. black: this color is formed with the combination of all colors in the diagram, and it contains the intersection of the 3 sets.

A ∩ B ∩ C

For colors yellow, blue and red you can take each set A, B and C and subtract from each one of them the union of the intersection of the other two sets.

3. yellow:

A / (A ∩ B) ∪ (A ∩ C)

4. blue:

B / (A ∩ B) ∪ (B ∩ C)

5. red:

C / (B ∩ C) ∪ (A ∩ C)

Finally, for colors green, orange and violet you take the intersection of each set A ∩ B, A ∩ C and B ∩ C and subtract from them the elements in the black set.

6. green:

A ∩ B / (A ∩ B ∩ C)

7. orange:

A ∩ C / (A ∩ B ∩ C)

8. violet:

B ∩ C / (A ∩ B ∩ C)

The given Venn diagram, with three sets A, B, and C and given intersections, would create 8 different colored areas, including the universal set color.

The question is about understanding the structure and representation of a Venn diagram. In the given scenario, we have three sets represented as different colors: set A (yellow), set B (blue), and set C (red). Our universal set is white. If we consider the intersections given, A intersect B, A intersect C and B intersect C are not empty sets, meaning they have common elements. Hence, in a Venn diagram, such intersections will create separate areas.

The key component in a Venn diagram is color representation. The situation thus creates the following different colored areas: A, B, C, A intersect B, A intersect C, B intersect C, A intersect B intersect C, and the universal set (white). Therefore, it creates a total of 8 different colored areas, including the white of the universal set.

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Answer:

[tex]\left[\begin{array}{ccc}(-p)&--->&q\\f&t&t\\f&t&t\\t&t&f\\t&f&f\end{array}\right][/tex]

Step-by-step explanation:

First, we find all the possibilities for p and q in a table:

p  q

t    t

t    f

f    t

f    f

then -p:

-p  q

f    t

f    f

t    t

t    f

and we apply the operator --> (rightarrow), that is only f (false) y if the first one is t (true) and the second one is f (false)

-p    --->    q

f       t       t

f       t       f

t       t       t

t       f       f

Answer: 0.3173

Step-by-step explanation:

Given : A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter of the corks produced is well approximated by a normal distribution with

[tex]\mu=4\ cm[/tex] and [tex]\sigma=0.2\ cm[/tex]

The specifications call for corks with diameters between 3.8 and 4.2 cm.

Let x be the random variable that represents the  the diameter of the corks.

Using formula [tex]z=\dfrac{x-\mu}{\sigma}[/tex], the z-score corresponding to x= 3.8 will be :_

[tex]z=\dfrac{3.8-4}{0.2}=1[/tex]

z-score corresponding to x= 4.2 will be :_

[tex]z=\dfrac{4.2-4}{0.2}=1[/tex]

Now, by using the standard normal distribution table for z, we have

[tex]\text{P value}=P(-1<z<1)=2P(z<1)-1\\\\=2(0.8413447)-1\\\\=0.6826894\approx0.6827[/tex]

∴The proportion of corks produced by this machine are meeting the specifications=0.6827

∴The proportion of corks produced by this machine are defective = [tex]1-0.6827=0.3173[/tex]

The question asks about the proportion of defective corks produced by a machine. Given that the corks diameters' are normally distributed with a mean of 4cm and a standard deviation of 0.2cm, and the specifications are between 3.8cm and 4.2cm, about 68% of corks will meet the specification. This implies that about 32% will be defective.

The subject of this question is statistics, specifically dealing with normal distribution, mean and standard deviation. To find the proportion of corks produced by the machine that are defective, we can use the properties of normal distribution where the diameters of the corks represent a normal distribution with µ (mean) = 4 cm and σ (standard deviation) = 0.2 cm.

As the corks' specification falls between 3.8 cm and 4.2 cm, these values are 1 standard deviation below and above the mean respectively. In normal distribution, the area (i.e., proportion) within one standard deviation is approximately 0.68 (or 68%). So, the proportion within the specification is 0.68.

To obtain the proportion of defective corks, you subtract the proportion within specification from 1 (the total proportion). Hence, the proportion of defective corks is 1 - 0.68 = 0.32 or 32%. This means that approximately 32% of the corks produced are outside the specification and are deemed defective.

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The shape of the distribution is positively skewed distribution as per the concept of mean and median.

In this case, the mean is 42 and the median is 37.

Since the mean is greater than the median, it suggests that the distribution is positively skewed.

This means that there are some relatively high values (outliers or extreme values) that are pulling the mean towards the higher end of the data range, which results in the median being lower than the mean.

In other words, the tail of the distribution extends more to the right (higher values) than to the left.

In a positively skewed distribution:

The mean > Median

The tail is longer on the right side (positive side).

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Final answer:

The mean age being higher than the median age in an age distribution survey with 8225 Americans indicates a right-skewed distribution. The distribution has a longer tail on the right side due to outliers or a subgroup of older individuals.

Explanation:

In the survey with a sample of 8225 Americans, the given mean age is 42, and the median age is 37. The fact that the mean is higher than the median suggests that there are outliers or a group of people who are much older than the rest skewing the average upwards. This is an indication that the age distribution is right-skewed, meaning that there is a longer tail on the right side of the distribution curve.

Statisticians commonly use the mean and median to understand the shape of a distribution. In a perfectly symmetrical distribution, these two measures would be the same. However, when there is skewness, the mean is pulled towards the tail of the distribution more so than the median. Therefore, when the mean is greater than the median, the distribution is positively skewed; conversely, if the mean is less than the median, the distribution is negatively skewed.

Understanding the skewness is important as it affects the distribution's measures of central tendency, making the mean less representative of the majority of the data points in a skewed distribution.

Answer:

Where Z is the exhaustive multitude of the whole numbers

[tex]\text{Hello there!}\\\\\text{If you're solving for z:}\\\\z-4=10\\\\\text{Add 4 to both sides}\\\\z=14\\\\\text{Your answer would be:}\,\boxed{z=14}[/tex]

Answer:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices.

Step-by-step explanation:

The formula is not valid because the commutative property with respect to the matrix product operation is not fulfilled in the vector space of the real matrices. That is, AB does not necessarily equal BA.

[tex](A+B)(A-B) = A^2-AB+BA-B^2\neq A^2 - B^2[/tex]

[tex]A=\left[\begin{array}{ccc}1&0&0\\0&0&6\\0&8&0\end{array}\right] \\B=\left[\begin{array}{ccc}0&2&0\\6&0&0\\0&0&9\end{array}\right] \\(A -B) = \left[\begin{array}{ccc}1&-2&0\\-6&0&6\\0&8&-9\end{array}\right]\\\\(A + B) = \left[\begin{array}{ccc}1&2&0\\6&0&6\\0&8&9\end{array}\right]\\(A - B)(A + B) = \left[\begin{array}{ccc}-11&2&-12\\-6&36&54\\48&-72&-33\end{array}\right]\\A^2 - B^2 = \left[\begin{array}{ccc}-11&0&0\\0&36&0\\0&0&-33\end{array}\right]\\[/tex]

You can use the fact that multiplication of matrices is dependent on the order of the matrices which are multiplied.

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

It might be that AB = BA for two matrices A and B but it is very rare and thus, cannot be generalized as identity.

Suppose A has got shape (m,n) (m rows, n columns)

and B has got shape (n,k) (n rows, k columns), then AB is defined but BA is not defined if k  ≠  m.

Also, even if k =m, we can't say for sure that AB = BA

Thus, usually we have AB ≠ BA

For two matrices A and B, supposing that AB and BA are defined, then we have

[tex](A+B)(A-B) = A(A-B) + B(A -B) = A^2 -AB + BA - B^2[/tex]

Since may or may not have AB equal to BA, thus, we cannot cancel those two middle terms to make 0 matrix.

Thus,

The correct option for the given condition is

Option C:  The formula is not valid because in general, AB ≠ BA for matrices.

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Answer:

System of equations:

[tex]x_1+2x_2+2x_3=6\\2x_1+x_2+x_3=6\\x_1+x_2+3x_3=6[/tex]

Augmented matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\2&1&1&6\\1&1&3&6\end{array}\right][/tex]

Reduced Row Echelon matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\0&1&1&2\\0&0&1&1\end{array}\right][/tex]

Step-by-step explanation:

Convert the system into an augmented matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\2&1&1&6\\1&1&3&6\end{array}\right][/tex]

For notation, R_n is the new nth row and r_n the unchanged one.

1. Operations:

[tex]R_2=-2r_1+r_2\\R_3=-r_1+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\0&-3&-3&-6\\0&-1&1&0\end{array}\right][/tex]

2. Operations:

[tex]R_2=-\frac{1}{3}r_2[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\0&1&1&2\\0&-1&1&0\end{array}\right][/tex]

3. Operations:

[tex]R_3=r_2+r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\0&1&1&2\\0&0&2&2\end{array}\right][/tex]

4. Operations:

[tex]R_3=\frac{1}{2}r_3[/tex]

Resulting matrix:

[tex]\left[\begin{array}{cccc}1&2&2&6\\0&1&1&2\\0&0&1&1\end{array}\right][/tex]

The terms mcg/mL and mg/L are not equivalent.

1. mcg/mL:

"mcg" stands for micrograms, which is a unit of measurement used for very small amounts.

"mL" stands for milliliters, which is a unit of measurement used for volume.

So, mcg/mL represents micrograms per milliliter, which is a measure of concentration.

2. mg/L:

"mg" stands for milligrams, which is a unit of measurement used for larger amounts than micrograms.

"L" stands for liters, which is a unit of measurement used for volume.

Therefore, mg/L represents milligrams per liter, also a measure of concentration.

Since micrograms are smaller than milligrams, mcg/mL is a smaller unit of measurement than mg/L. In other words, 1 mcg/mL is equal to 0.001 mg/L.

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The terms mcg/mL and mg/L are equivalent. This is based on the conversion of these units to grams (g) per liter (L), where both ratios equals to g/L.

The terms mcg/mL and mg/L are indeed equivalent. Here's how we make the conversion: 1 mcg (microgram) is equal to 1x10-6g (grams), and 1 mL (milliliter) is equal to 1x10-3L (liters). Therefore, the ratio of mcg/mL equals to g/L. Now, 1mg (milligram) is equal to 1x10-3g (grams), and if we have 1 L (liter), it is the same 1L. Therefore, the ratio mg/L also equals to g/L. With this we can conclude that mcg/mL is equivalent to mg/L.

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