A circle has the center of (1,-5) and a radius of 5 determine the location of the point (4,-1)

Answer:

a. P(X = 0)= 0.001

b. P(X = 1)= 0.001

c. P(X=2)= 0.044

d. P(X=3)= 0.117

e. P(X=4)= 0.205

f. P(X=5)= 0.246

g. P(X=6)= 0.205

h. P(X=7)= 0.117

i. P(X=8)= 0.044

j. P(X=9)= 0.001

k. P(X=10)= 0.001

Step-by-step explanation:

Hello!

You have the variable X with binomial distribution, the probability of success is 0.5 and the sample size is n= 10 (I suppose)

If the probability of success p=0.5 then the probability of failure is q= 1 - p= 1 - 0.5 ⇒ q= 0.5

You are asked to calculate the probabilities for each observed value of the variable. In this case is a discrete variable with definition between 0 and 10.

You have two ways of solving this excersice

1) Using the formula

[tex]P(X)= \frac{n!}{(n-X)!X!} * (p)^X * (q)^{n-X}[/tex]

2) Using a table of cummulative probabilities of the binomial distribution.

a. P(X = 0)

Formula:

[tex]P(X=0)= \frac{10!}{(10-0)!0!} * (0.5)^0 * (0.5)^{10-0}[/tex]

P(X = 0) = 0.00097 ≅ 0.001

Using the table:

P(X = 0) = P(X ≤ 0) = 0.0010

b. P(X = 1)

Formula

[tex]P(X=1)= \frac{10!}{(10-1)!1!} * (0.5)^1 * (0.5)^{10-1}[/tex]

P(X = 1) = 0.0097 ≅ 0.001

Using table:

P(X = 1) = P(X ≤ 1) - P(X ≤ 0) = 0.0107-0.0010= 0.0097 ≅ 0.001

c. P(X=2)

Formula

[tex]P(X=2)= \frac{10!}{(10-2)!2!} * (0.5)^2 * (0.5)^{10-2}[/tex]

P(X = 2) = 0.0439 ≅ 0.044

Using table:

P(X = 2) = P(X ≤ 2) - P(X ≤ 1) = 0.0547 - 0.0107= 0.044

d. P(X = 3)

Formula

[tex]P(X = 3)= \frac{10!}{(10-3)!3!} * (0.5)^3 * (0.5)^{10-3}[/tex]

P(X = 3)= 0.11718 ≅ 0.1172

Using table:

P(X = 3) = P(X ≤ 3) - P(X ≤ 2) = 0.1719 - 0.0547= 0.1172

e. P(X = 4)

Formula

[tex]P(X = 4)= \frac{10!}{(10-4)!4!} * (0.5)^4 * (0.5)^{10-4}[/tex]

P(X = 4)= 0.2051

Using table:

P(X = 4) = P(X ≤ 4) - P(X ≤ 3) = 0.3770 - 0.1719= 0.2051

f. P(X = 5)

Formula

[tex]P(X = 5)= \frac{10!}{(10-5)!5!} * (0.5)^5 * (0.5)^{10-5}[/tex]

P(X = 5)= 0.2461 ≅ 0.246

Using table:

P(X = 5) = P(X ≤ 5) - P(X ≤ 4) = 0.6230 - 0.3770= 0.246

g. P(X = 6)

Formula

[tex]P(X = 6)= \frac{10!}{(10-6)!6!} * (0.5)^6 * (0.5)^{10-6}[/tex]

P(X = 6)= 0.2051

Using table:

P(X = 6) = P(X ≤ 6) - P(X ≤ 5) = 0.8281 - 0.6230 = 0.2051

h. P(X = 7)

Formula

[tex]P(X = 7)= \frac{10!}{(10-7)!7!} * (0.5)^7 * (0.5)^{10-7}[/tex]

P(X = 7)= 0.11718 ≅ 0.1172

Using table:

P(X = 7) = P(X ≤ 7) - P(X ≤ 6) = 0.9453 - 0.8281= 0.1172

i. P(X = 8)

Formula

[tex]P(X = 8)= \frac{10!}{(10-8)!8!} * (0.5)^8 * (0.5)^{10-8}[/tex]

P(X = 8)= 0.0437 ≅ 0.044

Using table:

P(X = 8) = P(X ≤ 8) - P(X ≤ 7) = 0.9893 - 0.9453= 0.044

j. P(X = 9)

Formula

[tex]P(X = 9)= \frac{10!}{(10-9)!9!} * (0.5)^9 * (0.5)^{10-9}[/tex]

P(X = 9)=0.0097 ≅ 0.001

Using table:

P(X = 9) = P(X ≤ 9) - P(X ≤ 8) = 0.999 - 0.9893= 0.001

k. P(X = 10)

Formula

[tex]P(X = 10)= \frac{10!}{(10-10)!10!} * (0.5)^{10} * (0.5)^{10-10}[/tex]

P(X = 10)= 0.00097 ≅ 0.001

Using table:

P(X = 10) = P(X ≤ 10) - P(X ≤ 9) = 1 - 0.9990= 0.001

Note: since 10 is the max number this variable can take, the cummulated probability until it is 1.

I hope it helps!

Answer:

Step-by-step explanation:

The formula for finding the sum of the measure of the interior angles in a regular polygon is expressed as (n - 2) × 180. Therefore,

(n - 2) × 180 = 9000

180n - 360 = 9000

180n = 9000 + 360 = 9360

n = 9360/180

n = 52

The regular polygon has 52 sides

7) The sum of the angles in the quadrilateral is 360°. Let x represent the missing angle. Therefore,

64 + 116 + 120 + x = 360

300 + x = 360

x = 360 - 300

x = 60°

8a) let x represent the missing side. Therefore,

24/15 = x/10

Cross multiplying,

15x = 240

x = 240/15 = 16

8b) let x represent the missing side. Therefore,

6/12 = 5/x

6x = 60

x = 60/6 = 10

Answer:

a) A^² is a Hermitian operator

b) A^B^ is not a Hermitian operator

c)  A^B^+ B^A^  is a Hermitian operator

d) It is not possible to be complex it must be a real number

Step-by-step explanation:

In order to understand this solution we need to define the concept Hermitian

HERMITIAN

 This can be defined as a matrix whose elements are real and symmetrical i.e. it a square matrix that is equal to its own conjugate, or we can simply put that its a matrix in which those pairs of element that are symmetrically placed with respect to the principal diagonal are complex conjugates.i.e the diagonal elements( Hermitian operators) are real numbers while others are complex numbers.

The solution to the question above are on the first and second uploaded image.

Answer:

Part a: The two events are termed as disjoint because the event B clearly rules out the obese person

Part b: In the plain language, the event "A or B" means that the person is  overweight or obese. Its probability is 0.74.

Part c: If C is the event that the person chosen has normal weight or less, its probability is 0.26.

Step-by-step explanation:

As per the question obtained from the google search, the question has 3 parts as follows:

Part a

Explain why events A and B are disjoint.

Solution

The two events are termed as disjoint because the event B clearly rules out the obese person so the events are disjoint. so the correct option as given in the complete question is A.

Part b

Say in plain language what the event "A or B" is.

What is P(A or B)? (Enter your answer to two decimal places.)

Solution

In the plain language, the event "A or B" means that the person is  overweight or obese. The correction option as given in the complete question is a.

P(A or B) is given as

P(A or B)=P(AUB)=P(A)+P(B)-P(A∩B)

Here from the data of

P(A)=0.41

P(B)=0.33

P(A∩B)=0 (As the events are disjoint)

P(A or B)=P(AUB)=0.41+0.33-0

P(AUB)=0.74

So the probability of A or B is 0.74.

Part c

If C is the event that the person chosen has normal weight or less, what is

P(C)? (Enter your answer to two decimal places.)

Solution

P(C) is given as

P(C)=1-P(AUB)

P(C)=1-0.74

P(C)=0.26

So the probability of event C is 0.26.

To find the angle of intersection (θ) between the curves r1(t) and r2(t) at the origin, we calculate the dot product of their tangent vectors and use the arccosine formula. θ ≈ 79 degrees.

To find the angle of intersection (θ) between the curves r1(t) = (3t, [tex]t^2[/tex], [tex]t^4[/tex]) and r2(t) = (sin(t), sin(2t), 5t) at the origin, we can use the dot product formula for angles between vectors.

First, we need to calculate the tangent vectors at the origin for both curves. The tangent vector for r1(t) is (3, 2t, [tex]4t^3[/tex]), and for r2(t), it is (cos(t), 2cos(2t), 5).

Next, evaluate these vectors at t = 0 (the origin) to get the tangent vectors at the point of intersection: r1'(0) = (3, 0, 0) and r2'(0) = (1, 2, 5).

Now, calculate the dot product of these vectors:

r1'(0) · r2'(0) = (3 × 1) + (0 × 2) + (0 × 5) = 3.

The magnitude of r1'(0) is [tex]\sqrt{ (3^2 + 0^2 + 0^2)[/tex] = 3, and the magnitude of r2'(0) is [tex]\sqrt{(1^2 + 2^2 + 5^2[/tex]) = [tex]\sqrt{(1 + 4 + 25)[/tex] = √30.

Now, use the dot product formula for angles:

cos(θ) = (r1'(0) · r2'(0)) / (|r1'(0)| ×|r2'(0)|)

cos(θ) = 3 / (3 × [tex]\sqrt{30}[/tex]) = 1 / [tex]\sqrt30}[/tex]

Now, find θ:

θ = arc cos(1 / [tex]\sqrt{30[/tex])

Using a calculator, θ ≈ 79 degrees (rounded to the nearest degree).

So, the angle of intersection θ is approximately 79 degrees.

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Answer:

Therefore the measurement of EF,

[tex]EF=1.98\ units[/tex]

Step-by-step explanation:

Given:

In Right Angle Triangle DEF,

m∠E=90°

m∠D=26°

∴sin 26 ≈ 0.44

DF = Hypotenuse = 4.5    

To Find:

EF = ? (Opposite Side to angle D)

Solution:

In Right Angle Triangle DEF, Sine Identity,

[tex]\sin D= \dfrac{\textrm{side opposite to angle D}}{Hypotenuse}\\[/tex]

Substituting the values we get

[tex]\sin 26= \dfrac{EF}{DF}=\dfrac{EF}{4.5}[/tex]

Also,  sin 26 ≈ 0.44 .....Given

[tex]EF = 4.5\times 0.44=1.98\ units[/tex]

Therefore the measurement of EF,

[tex]EF=1.98\ units[/tex]

Answer:

please see answers are as in the explanation.

Step-by-step explanation:

As from the data of complete question,

[tex]0\leq x\leq 1\\0\leq y\leq 1\\0\leq z\leq 1\\u= \alpha x\\v=\beta y\\w=0[/tex]

The question also has 3 parts given as

Part a: Sketch the deformed shape for α=0.03, β=-0.01 .

Solution

As w is 0 so the deflection is only in the x and y plane and thus can be sketched in xy plane.

the new points are calculated as follows

Point A(x=0,y=0)

Point A'(x+αx,y+βy)

Point A'(0+(0.03)(0),0+(-0.01)(0))

Point A'(0,0)

Point B(x=1,y=0)

Point B'(x+αx,y+βy)

Point B'(1+(0.03)(1),0+(-0.01)(0))

Point B'(1.03,0)

Point C(x=1,y=1)

Point C'(x+αx,y+βy)

Point C'(1+(0.03)(1),1+(-0.01)(1))

Point C'(1.03,0.99)

Point D(x=0,y=1)

Point D'(x+αx,y+βy)

Point D'(0+(0.03)(0),1+(-0.01)(1))

Point D'(0,0.99)

So the new points are A'(0,0), B'(1.03,0), C'(1.03,0.99) and D'(0,0.99)

The plot is attached with the solution.

Part b: Calculate the six strain components.

Solution

Normal Strain Components

                             [tex]\epsilon_{xx}=\frac{\partial u}{\partial x}=\frac{\partial (\alpha x)}{\partial x}=\alpha =0.03\\\epsilon_{yy}=\frac{\partial v}{\partial y}=\frac{\partial ( \beta y)}{\partial y}=\beta =-0.01\\\epsilon_{zz}=\frac{\partial w}{\partial z}=\frac{\partial (0)}{\partial z}=0\\[/tex]

Shear Strain Components

                             [tex]\gamma_{xy}=\gamma_{yx}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}=0\\\gamma_{xz}=\gamma_{zx}=\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}=0\\\gamma_{yz}=\gamma_{zy}=\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}=0[/tex]

Part c: Find the volume change

[tex]\Delta V=(1.03 \times 0.99 \times 1)-(1 \times 1 \times 1)\\\Delta V=(1.0197)-(1)\\\Delta V=0.0197\\[/tex]

Also the change in volume is 0.0197

For the unit cube, the change in terms of strains is given as

             [tex]\Delta V={V_0}[(1+\epsilon_{xx})]\times[(1+\epsilon_{yy})]\times [(1+\epsilon_{zz})]-[1 \times 1 \times 1]\\\Delta V={V_0}[1+\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}+\epsilon_{xx}\epsilon_{zz}+\epsilon_{yy}\epsilon_{zz}+\epsilon_{xx}\epsilon_{yy}\epsilon_{zz}-1]\\\Delta V={V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]

As the strain values are small second and higher order values are ignored so

                                      [tex]\Delta V\approx {V_0}[\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\ \Delta V\approx [\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}]\\[/tex]

As the initial volume of cube is unitary so this result can be proved.

Answer:

0.9688

Step-by-step explanation:

For each time the coin is tossed, there are only two possible outcomes. Either it is heads, or it is tails. The probabilities for each coin toss are independent from each other. So 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 combinations 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:

For each time the coin is tossed, heads or tails are equally as likely, since the coin is fair. So [tex]p = \frac{1}{2} = 0.5[/tex]

You toss a fair coin 5 times. What is the probability of at least one head?

Either there are no heads, or there is at least one head. The sum of the probabilities of these events is decimal 1. So

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

We want to find [tex]P(X \geq 1)[/tex], when [tex]n = 5[/tex].

So

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

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

[tex]P(X = 0) = C_{5,0}.(0.5)^{0}.(0.5)^{5} = 0.03125[/tex]

[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.03125 = 0.96875[/tex]

Rounding to the nearest ten-thousandth(four decimal places), this probability is 0.9688.

The probability of getting at least one head when tossing a fair coin 5 times is 31/32 or 0.96875 when rounded to the nearest ten-thousandth.

The subject of this question is probability, a field within Mathematics. To answer your question: the probability of getting at least one head when tossing a fair coin 5 times can be found by calculating the probability of not getting a head (which is tossing tails 5 times in a row) and then subtracting that from 1 (representing certainty). Each toss of the fair coin has two outcomes, heads or tails, with equal probability of 1/2. If you toss the coin 5 times, the total number outs comes is 2^5, or 32. The chance of getting all tails is (1/2)^5, which is 1/32.

So, the probability of not getting a head (only tails) in 5 tosses is 1/32. Subtract this from 1 to find the probability of getting at least one head: This equals 1 - 1/32 = 31/32 = 0.96875, when rounded to the nearest ten-thousandth.

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Answer:

The original​ values are : 16, 25, 25, 49, 49, 64, 121.

Step-by-step explanation:

We know that  a researcher took square roots of the counts. Some of the resulting​ values, which are reasonably​ symmetric, were 4​, 5​, 5​, 7​, 7​, 8​, and 11.  We calculate the original​ values:

[tex]4^2=16\\5^2=25\\5^2=25\\7^2=49\\7^2=49\\8^2=64\\11^2=121\\[/tex]

The original​ values are : 16, 25, 25, 49, 49, 64, 121.

We conclude that the  original data is not simmetric.

The original values obtained by reversing the square root transformation are 16, 25, 25, 49, 49, 64, and 121. These values show variability in the number of employees across different small companies in the town. The transformed dataset was made more symmetrical for statistical analysis.

When a researcher applies a square root transformation to a dataset, the purpose is often to make the data more symmetrical and easier to analyze using certain statistical methods.

Given the transformed values 4, 5, 5, 7, 7, 8, and 11, we can reverse the transformation to find the original values.

The square of each transformed value yields the original data points:

Thus, the original values are 16, 25, 25, 49, 49, 64, and 121. These values are distributed with some repeated data points and a range from 16 to 121.

This distribution indicates variability in the number of employees across the small companies studied.

Answer:

The answer to your question is below

Step-by-step explanation:

43.-

x = [tex]\sqrt{27^{2} + 22^{2}- 2(27)(22)cos 73}[/tex]

x = [tex]\sqrt{865.66}[/tex]

x = 29.42

44.-

x = [tex]\sqrt{10^{2} + 14^{2} -2(10)(14)cos 66}[/tex]

x = [tex]\sqrt{182.11}[/tex]

x = 13.49

45.-

cos x = [tex]\frac{11^{2} - 17^{2} - 10^{2}}{-2(11)(17)}[/tex]

cos x = 0.7166

     x = 44.22°

46.-

x = [tex]\sqrt{16^{2} + 12^{2} -2(16)(12)cos75}[/tex]

x = [tex]\sqrt{300.61}[/tex]

x = 17.34

47.-

cos P = [tex]\frac{6^{2} - 13^{2} -11^{2}}{-2(13)(11)}[/tex]

cos P = 0.888

     P = 27.36°

sinR/13 = sinP/6

sin R = 13sin27.36/6

sinR = 0.996

   R = 84.71°

Q = 180 - 84.71 - 27.36

Q = 67.93°

48.-

D = 180 - 25 - 113

D = 42°

CD = 9Sin113/sin42

CD = 12.38

ED = 9sin25/sin42

ED = 5.68

Answer:

Option a

Step-by-step explanation:

Given that all of the seniors and none of the freshmen voted for it.

Some of the juniors and some of the sophomores voted for the proposal and some voted against it.

If the proposal was to be adopted then a simple of majority of votes should have been cast.

Since all seniors voted for it, and no freshmen number of freshmen has to be less than seniors then only majority would have voted.

Regarding juniors and sophomores we assume some voted and some against thus approximately nullifying their votes.

So the correct option would be

a. There are more seniors than freshmen on the council.

Option b wrong since only some voted for it.

C is wrong because actual seniors were not given.

D is wrong because while seniors voted for sophomores voted against.

e is wrong since sophomores and freshmen nullified each other

Answer:

Parameter                                            

Step-by-step explanation:

We are given the following in the question:

The average age of men who had walked on the moon was 39 years, 11 months, 15 days.

Population and sample:

Parameter and statistic:

Population of interest:

men who had walked on the moon

Value:

average age of men who had walked on the moon

Thus, the give value describes a population and hence, it is a parameter.

The student is undertaking an English language assignment designed to strengthen their understanding of vocabulary, word formation, and spelling through a variety of exercises including word scrambles, pattern recognition, spelling reviews, and word categorization.

The student appears to be working on a language arts activity related to vocabulary and word structure. The question likely requires them to engage with various linguistic exercises such as word scrambles, identifying patterns in spelling or pronunciation, reviewing correct spellings, and categorizing words. Such tasks are designed to help students learn about word formation, synonyms, antonyms, and the nuances of English spelling.

Word Scrambles and Patterns

For word scrambles, students are expected to rearrange the letters to form meaningful words. In doing this, they might uncover a hidden word that pertains to the lesson's focus. Identifying patterns in words might involve recognizing prefixes, suffixes, or roots that appear consistently across different words.

Reviewing Spelling

The student is also asked to choose the word with the correct spelling. This likely involves comparing similar words and identifying the correctly spelled one, perhaps with the aid of a dictionary for verification.

Categorizing Words

In the task of sorting words into groups, students might have to classify words based on different criteria such as part of speech, phonetic features, or spelling patterns. This reinforces their understanding of language structure and proper spelling conventions.

Answer:

a.) 0.4565

b.) 0.8656

c.) 0.4615

Step-by-step explanation:

We solve this using the probability distribution formula of combination.

nCr * p^r * q^n-r

Where

n = number of trials

r = successful trials

probability of success = p = 77% =0.77

Probability of failure= q = 1-0.77 = 0.23

a.) When exactly 3 opponents are defeated, When n = 3 and r = 3, probability becomes:

= 3C3 * 0.77³ * 0.23^0

= 1 * 0.456533 * 1

= 0.456533 = 0.4565 (4.d.p)

b.) When at least 2 opponents are defeated, that is when r = 2 and when r = 3,

When r = 2, probability becomes:

= 3C2 * 0.77² * 0.23¹

= 3 * 0.5929 * 0.23

= 0.409101

When 3 opponents are defeated, we calculated it earlier to be 0.456533

Hence, probability that at least 2 opponents are defeated

= 0.409101 + 0.456533

= 0.865634 = 0.8656(2.d.p)

c.) If 2 games are played, probability he defeat all 3 at least once in the game will be the sum (probability of defeating all 3 opponents in the first game and not defeating all 3 in the second game) + (probability of defeating all three opponents in both games)

Probability of defeating all three opponents in the first game = 0.456533

Probability of not defeating all three opponents in the second game = 1 - 0.456533 = 0.543467

Hence ,

probability of defeating all 3 opponents in the first game and not defeating all 3 in the second game = 0.465633 * 0.543467 = 0.253056

probability of defeating all three opponents in both games

= 0.456533 * 0.456533

=0.208422

Probability he defeats all three opponents at least once in 2games

= 0.253056 + 0.208422

=0.461478 = 0.4615(4.d.p)

Answer:

The answer to your question is 13 u²

Step-by-step explanation:

We know that the small triangle is surrounded by right triangles so we can use the Pythagorean theorem to find the lengths  of the small triangle

                 AD² = 3² + 2²

Simplify

                 AD² = 9 + 4

                 AD² = 13

                 AD = [tex]\sqrt{13}[/tex]

Find the area of the square

Area = side x side

Area = AD x AD

Area = [tex]\sqrt{13} x \sqrt{13}[/tex]

Area = 13 u²                  

Answer:

18 in^2

Step-by-step explanation:

1  )The three sides of the gutter add up to 12

            2x+ y = 12  

2)  Subtract 2x from both sides.

           y = 12 — 2x  

3 )Find the area of the rectangle in terms of x and simplify.

    Area = xy = x(12 — 2x) = -2x^2+12x = f(x)  

4 )  x=-b/2a

     x co-ordinate of the vertex= -12/2(-2)=3

5 )Plug in 3 for x into they equation.

    y co-ordinate of the vertex= 12 — 2(3) = 6  

6  )  Plug in 3 for x and 6 for y.  

      Area= xy = 3(6) = 18  

RESULT  

            18 in^2

Answer:

[tex]t=\frac{11-8}{\frac{6}{\sqrt{9}}}=1.5[/tex]    

[tex]p_v =P(t_{(8)}>1.5)=0.086[/tex]  

If we compare the p value and the significance level assumed [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis, we can conclude that the mean is higher than 8 at 5% of signficance.  

Step-by-step explanation:

Data given and notation  

[tex]\bar X=11[/tex] represent the mean height for the sample  

[tex]s=6[/tex] represent the sample standard deviation for the sample  

[tex]n=9[/tex] sample size  

[tex]\mu_o =8[/tex] represent the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.   (assumed)

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 higher than 8, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 8[/tex]  

Alternative hypothesis:[tex]\mu > 8[/tex]  

If we analyze the size for the sample is < 30 and 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{11-8}{\frac{6}{\sqrt{9}}}=1.5[/tex]    

P-value

The first step is calculate the degrees of freedom, on this case:  

[tex]df=n-1=9-1=8[/tex]  

Since is a one side upper test the p value would be:  

[tex]p_v =P(t_{(8)}>1.5)=0.086[/tex]  

Conclusion  

If we compare the p value and the significance level assumed [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis, we can conclude that the mean is higher than 8 at 5% of signficance.  

To determine if the population standard deviation is known, we can use the formula for the standard error of the mean (SEM) and use the t-distribution for a sample size of 9.

To determine if the population standard deviation LaTeX: \sigma\sigma is known, we can use the formula for the standard error of the mean (SEM):

SE = \frac{s}{\sqrt{n}}

If the sample size is less than or equal to 30, we can use the t-distribution to find the critical value for a given level of significance. If the sample size is greater than 30, we can use the z-distribution. In this case, since the sample size is 9, the t-distribution should be used.

Thus, with a sample size of 9 and the population standard deviation not known, \sigma\sigma is not known.

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The histogram is shown below.

You'll have 10 bars. Under each bar is a label from 0 to 9. The height of each bar represents the frequency of each units digit.

The histogram shows two bars that are relatively large compared to the rest. These bars have frequency of 16 and 14 (for units digits of 0 and 5 respectively). The distribution is nearly bimodal. If the two frequencies mentioned were the same, say both 16, then it would be exactly bimodal. As you can probably guess, bimodal means "two modes".

The rest of the bar heights are nearly the same, so the remaining portion of the distribution is nearly uniform. A uniform distribution has every bar the same height. Collectively, all the bars of a uniform distribution forms a larger rectangle.

Answer:

The chi-square null hypothesis for the study that "socioeconomic status is related to smoking behavior" is False

Step-by-step explanation:

The chi-square null hypothesis is false because the chi-square null hypothesis states that no relationship exists on the categorical variables in a population, they are all independent of each other.

Answer:

The variance of this distribution is 0.0036.

Step-by-step explanation:

The variance of n binomial distribution trials with p proportion is given by the following formula:

[tex]Var(X) = \frac{p(1-p)}{n}[/tex]

In this problem, we have that:

About 90% of defendants are found guilty in criminal trials. This means that [tex]p = 0.9[/tex]

Suppose we take a random sample of 25 trials. This means that [tex]n = 25[/tex]

Based on a proportion of .90, what is the variance of this distribution?

[tex]Var(X) = \frac{p(1-p)}{n}[/tex]

[tex]Var(X) = \frac{0.9*0.1}{25} = 0.0036[/tex]

The variance of this distribution is 0.0036.

a. The probability is 0.30. The advertisement seems to have a positive effect on customer purchases.

b. The market share is 0.20. The advertisement could potentially increase the company's market share

c. The conditional purchase probability of 0.333. The second advertisement seems to have had a slightly bigger effect on customer purchases.

a. We are asked to find the probability of an individual purchasing the product given that the individual recalls seeing the advertisement, i.e., P(B|S).

Using the formula for conditional probability:

P(B|S) = P(B∩S) / P(S)

Given:

P(B∩S) = 0.12

P(S) = 0.40

So, P(B|S)

= 0.12 / 0.40

= 0.30

Seeing the advertisement increases the probability that an individual will purchase the product from 0.20 (P(B)) to 0.30 (P(B|S)). Therefore, the advertisement seems to have a positive effect on customer purchases.

As a decision maker, if the cost of the advertisement is reasonable, it would be recommended to continue the advertisement since it increases the likelihood of product purchases.

b. The company's market share can be estimated by considering the probability of individuals purchasing the company's soap product and the probability of individuals purchasing from competitors.

The probability of an individual purchasing from competitors

= 1 - P(B)

= 1 - 0.20

= 0.80

Therefore, the company's market share is:

Market Share = P(B)

                      = 0.20

Continuing the advertisement could potentially increase the company's market share since the advertisement has a positive effect on customer purchases, as shown in part (a).

c. For the other advertisement:

Given:

P(S) = 0.30

P(B∩S) = 0.10

Using the formula for conditional probability:

P(B|S) = P(B∩S) / P(S)

So, P(B|S)

= 0.10 / 0.30

= 0.333

Comparing the two advertisements, the first advertisement had a conditional purchase probability of 0.30, while the second advertisement had a conditional purchase probability of 0.333. Therefore, the second advertisement seems to have had a slightly bigger effect on customer purchases.

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The advertisements increase the chance for an individual to purchase the product. The second advertisement yields a higher likelihood of purchase, implying it's more effective.

The problem is related to conditional probability. To solve the question:

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Answer with Step-by-step explanation:

We are given that

[tex]yx+9y=0[/tex]

[tex]-4x+5y=3[/tex]

We have to find the angle between the lines.

[tex]y(x+9)=0[/tex]

[tex]y=0,x+9=0\implies x=-9[/tex]

[tex]y=0[/tex]..(1)

[tex]x=-9[/tex]..(2)

[tex]-4x+5y=3[/tex]..(3)

The angle between two lines

[tex]a_1x+b_1y+c_1=0[/tex]

[tex]a_2x+b_2y+c_2=0[/tex]

[tex]tan\theta=\mid \frac{a_1b_2-b_1a_2}{a_1a_2+b_1b_2}\mid[/tex]

By using the formula the angle between equation (1) and equation (2) is given by

[tex]tan\theta_1=\mid\frac{0\times 0-1\times 1}{0+0}\mid=\infty=90^{\circ}[/tex]degree

[tex]tan90^{\circ}=\infty[/tex]

It is not possible because we are given that the acute angle between intersecting lines that do not cross at right angles is same as the angle determined by vectors that either normal to the lines or parallel to lines.

By using the formula the angle between equation (2) and equation(3)

[tex]tan\theta_2=\mid\frac{1(5)-0(4)}{-4(1)+5(0)}\mid=\frac{5}{4}[/tex]

[tex]\theta_2=tan^{-1}(1.25)[/tex] degree

By using the formula the angle between equation (3) and equation(1)

[tex]tan\theta_3=\mid\frac{-4(1)-5(0)}{-4(0)+5(1)}\mid=\frac{4}{5}[/tex]

[tex]\theta_3=tan^{-1}(\frac{4}{5})[/tex]degree

Answer:

A. 0.52

Step-by-step explanation:

Let D be the event that person used Debit card and C b the event that person used coupon.

We have to find the probability of customer does not use coupons given that a customer does not pay with a debit card,

P(C'/D')=P(C')P(D'/C')/[P(C')P(D'/C')+P(D')P(D'/C')]

We are given that P(D)=0.35, P(C)=0.30 and P(D/C)=0.5.

P(D')=1-0.35=0.65

P(C')=1-0.3=0.7

P(D'/C')=0.5.

P(C'/D')=0.7(0.5)/[0.7(0.5)+0.65(0.5)]

P(C'/D')=0.35/[0.35+0.325]

P(C'/D')=0.35/[0.35+0.325]

P(C'/D')=0.35/0.675

P(C'/D')=0.5185=0.52

Thus, the probability of customer does not use coupons given that a customer does not pay with a debit card is 0.52.

Answer:

a. 59049 ways

b. [tex]1.69\times10^{-5}[/tex]

Step-by-step explanation:

If the multiple choice test has 10 questions with 3 choices each, for the each question there are 3 ways to answer the test. Then for n question there are [tex]3^n[/tex] ways to answer the test

The total number of ways to answer 10 questions at 3 choices each is

[tex]3^{10} = 59049[/tex]ways

b. There are 59049 ways to answer the test but if one test must match one other test then the probability for that to happen is

[tex]\frac{1}{59049} = 1.69\times10^{-5}[/tex]

There are 59,049 ways to answer a 10-question test with 3 choices per question. The probability of two tests having the same answers, with all answer patterns being equally likely, is 1/59,049.

This problem belongs to the domain of combinatorics. (a) As there are 3 choices for each question and 10 questions, there are 3^10 or 59,049 ways to answer the test. This assumes you are answering every question. (b) This is essentially the question of the probability that two randomly selected tests are identically answered. Assuming all ways to answer a test are equally likely, there are 59,049 different ways to answer, so the probability that two randomly answered tests are the same is 1/59,049 or approximately 0.000017.

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Answer:

6 fl. oz

Step-by-step explanation:

The most accurate measurement for the volume a teacup can hold is 6 fluid ounces.

To determine the most accurate measurement for the amount that a teacup can hold, we need to consider the typical volume of a teacup and understand the relationship between different units of capacity. A standard teacup holds about 6 fluid ounces. Given the unit conversion factors, we know that:

1 cup = 8 fluid ounces

1 pint = 2 cups = 16 fluid ounces

1 quart = 2 pints = 32 fluid ounces

1.5 quarts = 48 fluid ounces

Considering these units:

6 fluid ounces is less than 1 cup (8 fl oz).

2 cups equal 16 fluid ounces, which is more than a typical teacup can hold.

1 pint (16 fluid ounces) is also more than what a teacup can hold.

1.5 quarts (48 fluid ounces) is significantly more than a teacup's capacity.

Therefore, the most accurate measurement to describe the amount a teacup can hold is 6 fluid ounces.

Answer:

A number to the third power would be a number cubed, so the answer is "Cubed".

Answer:

6.56% probability that a real emergency situation exists.

Step-by-step explanation:

We have these following probabilities:

A 0.4% probability that a real emergency situation exists.

A 99.6% probability that a real emergency situation does not exist.

If an emergency situation exists, a 95% probability that the alarm sounds.

If an emergency situation does not exist, a 2% probability that the alarm sounds.

The problem can be formulated as the following question:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

In this problem:

What is the probability of a real emergency situation existing, given that the alarm has sounded.

P(B) is the probability of there being a real emergency situation. So [tex]P(B) = 0.004[/tex].

P(A/B) is the probability of the alarm sounding when there is a real emergency situation. So P(A/B) = 0.95.

P(A) is the probability of the alarm sounding. This is 95% of 0.4%(real emergency situation) and 2% of 99.6%(no real emergency situation). So

P(A) = 0.95*0.04 + 0.02*0.996 = 0.05792

Given that the alarm has just sounded, what is the probability that a real emergency situation exists?

[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.004*0.95}{0.05792} = 0.0656[/tex]

6.56% probability that a real emergency situation exists.

Answer:

a)  D_u .  ∀ f ( 2 , 1 ) = 3 , Q = 72 degrees

b) D_u .  ∀ f ( 2 , 1 ) = 4*sqrt(2) , Q = 80 degrees

c) D_u .  ∀ f ( 2 , 1 ) = - sqrt(2) , Q = -54.74 degrees

d) u = 1 / sqrt(34) *(-3i +5j) ,  D_u .  ∀ f ( 2 , 1 )  = +/- sqrt (34) , Q = 80.27 degrees

Step-by-step explanation:

Given:

- The terrain is modeled as a surface:

                                   f(x , y) = x*y + y^3 - x^2

At point P( 2 , 1 , -1 ) is the current position:

Find:

(a) Determine the slope you would encounter if you headed due West from your position. What angle of inclination does this correspond to?

(b) Determine the slope you would encounter if you headed due North-West from your position. What angle of inclination does this correspond to?

(c) Determine the slope you would encounter if you headed due South-West from your position. What angle of inclination does this correspond to?

(d) Determine the steepest slope you could encounter from your position and the direction of that slope (as a unit vector).

Solution:

- We will compute the gradient of the vector ∀ f @ point P( 2 , 1 , -1 ) as follows:

                                ∀ f ( x , y ) = (y - 2x ) i + ( x + 3y^2) j

- Evaluate at point P ( 2 , 1 ):

                               ∀ f ( 2 , 1 ) = (1 - 2*2 ) i + ( 2 + 3*1^2) j    

                               ∀ f ( 2 , 1 ) = -3 i + 5 j        

- We will use the result of ∀ f @ point P( 2 , 1 , -1 ) for the all the parts.

a)

- The direction due west can be written as a unit vector u_w = - i .

- Now compute the directional derivative in the direction of u_w = - i

                                D_u .  ∀ f ( 2 , 1 ) =-3 i + 5 j . - i

                                D_u .  ∀ f ( 2 , 1 ) = 3

- Now compute the angle of inclination Q for the following direction:

                                Q = arctan(3) = 71.57 = 72 degrees

 Hence, along the direction due west from position we ascend with an inclination of approximately 72 degrees.

b)

- The direction due north-west can be written as a unit vector u_w = - i + j .

- Now compute the directional derivative in the direction of u_w = - i + j

                                D_u .  ∀ f ( 2 , 1 ) =-3 i + 5 j . (- i + j) / sqrt(2)

                                D_u .  ∀ f ( 2 , 1 ) = 8 / sqrt(2) = 4*sqrt(2)

- Now compute the angle of inclination Q for the following direction:

                                Q = arctan(4*sqrt(2)) = 79.98 = 80 degrees

 Hence, along the direction due north-west from position we ascend with an inclination of approximately 80 degrees.    

c)

- The direction due south-west can be written as a unit vector u_w = - i - j .

- Now compute the directional derivative in the direction of u_w = - i - j

                                D_u .  ∀ f ( 2 , 1 ) =-3 i + 5 j . (- i - j) / sqrt(2)

                                D_u .  ∀ f ( 2 , 1 ) = -2 / sqrt(2) = - sqrt(2)

- Now compute the angle of inclination Q for the following direction:

                                Q = arctan(-sqrt(2)) = -54.74 degrees

 Hence, along the direction due south-west from position we descend with an inclination of approximately 55 degrees.    

d)

- We know that ∀ f ( 2 , 1 ) gradient points in the direction greatest increase, hence, So from P the direction of greatest increase is  ∇f(P) = -3i +5j. The unit vector pointing in this direction is:

                                  u = 1 / sqrt(34) *(-3i +5j)

- so we have:

                     D_u .  ∀ f ( 2 , 1 ) =  u . ∀ f ( 2 , 1 ) = (-3i +5j) . (-3i +5j) / sqrt(34)

                     = +/- sqrt (34)

- Hence, the steepest ascent or decent is of :

                     Q = arctan ( sqrt(34)) = 80.27 degrees

Answer:

a(n)=14+6(n-1)

Step-by-step explanation:

The first term is 14. This would be an arithmetic sequence, so you will add 6 to every term: the common difference is 6.

14+6= 20

20+6=26

You have the formula a(n)= a(1)+d(n-1)

a(n)= 14+6(n-1)

14 for the first term, 6 for the common difference.

Answer:

The measure is 6°.

Step-by-step explanation:

If an angle of 96 degrees is rotated 90 degrees clockwise then the measure of the new angle will be given by

= 96° - 90° = 6°