G and H are mutually exclusive events.

• P(G) = 0.5
• P(H) = 0.3

1.) Explain why the following statement MUST be false: P(H | G) = 0.4. (Select a letter for your answer)

(A.)The events are mutually exclusive, which makes P(H AND G) = 0; therefore, P(H | G) = 0.

(B.)The statement is false because P(H | G) = P(H)/P(G)= 0.6

(C.) The events are mutually exclusive, which means they can be added together, and the sum is not 0.4.

(D.)To find conditional probability, divide P(G AND H) by P(H), which gives 0.5.

2.) Find P(H OR G).

3.) Are G and H independent or dependent events? Explain. (Select a letter for your answer)

(A.) G and H are dependent events because P(G OR H) ≠ 1.

(B.) G and H are independent events because they are mutually exclusive.

(C.) G and H are dependent events because they are mutually exclusive.T

(D.) There is not enough information to determine if G and H are independent or dependent events

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:

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°

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.

https://brainly.com/question/32015929

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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) 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:

a) Empty set

b)  [tex]\{x : x \in N \text{ and } x < 13\}[/tex]                                        

Step-by-step explanation:

Roster form is a comma separated list form of set.

a) The set of natural numbers x that satisfy x + 4 = 1.  

[tex]x + 4 = 1\\x = -3 \notin N[/tex]

Thus, x is an empty set.

b) set-builder notation for the set  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

We use x to represent this set. Now x belongs to natural number and is less than equal to 12.

Thus, it can be written as:

[tex]\{x : x \in N \text{ and } x < 13\}[/tex]

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:

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:

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:

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 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:

a) The person traveled 2.83 hours.

b) The person travels 220.17 kilometers.

Step-by-step explanation:

We have that the speed is the distance divided by the time. Mathematically, that is

[tex]s = \frac{d}{t}[/tex]

(a) how much time is spent on the trip and

The peron's average speed is 77.8 km/h, which means that [tex]s = 77.8[/tex]

The person distance traveled is:

22 min is 22/60 = 0.37h.

So for  the time t1, the person traveled at a speed of 89.5 km/h. Which has a distance of 89.5*t1.

For 0.37h, the person was at a stop, so she did not travel. This means that the total distance is

[tex]d = 89.5t1 + 0 = 89.5t1[/tex]

The total time is the time traveling t and the stoppage time 0.37. So

[tex]t = t1 + 0.37[/tex]

We want to find t1, which is the time that the person was driving.

So

[tex]s = \frac{d}{t}[/tex]

[tex]77.8 = \frac{89.5t1}{t1 + 0.37}[/tex]

[tex]77.8t1 + 77.8*0.37 = 89.5t1[/tex]

[tex]11.7t1 = 28.786[/tex]

[tex]t1 = \frac{28.786}{11.7}[/tex]

[tex]t1 = 2.46[/tex]

The total time is

[tex]t = t1 + 0.37 = 2.46 + 0.37 = 2.83[/tex]

The person traveled for 2.83 hours.

(b) how far does the person travel?

The person traveled 2.46 hours at an average speed of 77.8 km/h. So

[tex]s = \frac{d}{t}[/tex]

[tex]77.8 = \frac{d}{2.83}[/tex]

[tex]d = 77.8*2.83 = 220.17[/tex]

The person travels 220.17 kilometers.

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:

c₁ = 1/2

c₂ = - e²/2

y = (1/2)*(eˣ - e²⁻ˣ)

Step-by-step explanation:

Given

y = c₁eˣ + c₂e⁻ˣ

y(1) = 0

y'(1) = e

We get y' :

y' = (c₁eˣ + c₂e⁻ˣ)'  ⇒  y' = c₁eˣ - c₂e⁻ˣ

then we find y(1) :

y(1) = c₁e¹ + c₂e⁻¹ = 0

⇒  c₁ = - c₂/e² (I)

then we obtain y'(1):

y'(1) = c₁e¹ - c₂e⁻¹ = e    (II)

⇒  (- c₂/e²)*e - c₂e⁻¹ = e

⇒  - c₂e⁻¹ - c₂e⁻¹ = - 2c₂e⁻¹ = e

⇒  c₂ = - e²/2

and

c₁ = - c₂/e² = - (- e²/2) / e²

⇒  c₁ = 1/2

Finally, the equation will be

y = (1/2)*eˣ - (e²/2)*e⁻ˣ = (1/2)*(eˣ - e²⁻ˣ)

Applying the initial conditions, it is found that the solution is:

[tex]y = \frac{1}{2}e^{x} - \frac{e^2}{2}e^{-x}[/tex]

------------------------

The solution for the PVI is given by:

[tex]y = c_1e^{x} + c_2e^{-x}[/tex]

------------------------

The condition [tex]y(1) = 0[/tex] means that when [tex]x = 0, y = 1[/tex], and thus, we get:

[tex]c_1e + c_2e^{-1} = 0[/tex]

[tex]c_1e+ \frac{c_2}{e} = 0[/tex]

[tex]c_1e^{2} + c_2 = 0[/tex]

[tex]c_2 = -c_1e^{2}[/tex]

------------------------

The derivative is:

[tex]y^{\prime}(x) = c_1e^{x} - c_2e^{-x}[/tex]

Applying the condition [tex]y^{\prime}(1) = e[/tex], we get:

[tex]c_1e - \frac{c_2}{e} = e[/tex]

Considering [tex]c_2 = -c_1e^{2}[/tex]:

[tex]c_1e + c_1\frac{e^2}{e} = e[/tex]

[tex]c_1e + c_1e = e[/tex]

[tex]2c_1e = e[/tex]

[tex]2c_1 = 1[/tex]

[tex]c_1 = \frac{1}{2}[/tex]

------------------------

The second constant is:

[tex]c_2 = -c_1e^{2} = -\frac{e^2}{2}[/tex]

And the solution is:

[tex]y = \frac{1}{2}e^{x} - \frac{e^2}{2}e^{-x}[/tex]

A similar problem is given at https://brainly.com/question/13244107

Answer:

The unit rate is 6 1/4 cups of soda per cup of pineapple juice

Step-by-step explanation:

we know that

To find out the unit rate of soda to pineapple juice in the punch, divide the cups of soda by the cups of pineapple juice

so

[tex]2\frac{1}{2} :\frac{2}{5}[/tex]

Convert mixed number to an improper fraction

[tex]2\frac{1}{2}=2+\frac{1}{2}=\frac{2*2+1}{2}=\frac{5}{2}[/tex]

substitute

[tex]\frac{5}{2} :\frac{2}{5}[/tex]

Multiply in cross

[tex]\frac{25}{4}= 6.25[/tex]

Convert to mixed number

[tex]6.25=6+0.25=6+\frac{1}{4}= 6\frac{1}{4}[/tex]

That means

The unit rate is 6 1/4 cups of soda per cup of pineapple juice

Answer:

6 1/4

Step-by-step explanation:

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:

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:

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.

Answer:

A = -10

Step-by-step explanation:

A/2 = -5

Multiply both sides by the denominator of the fraction

We have A/2 x2 = -5 x 2

A = -10

Answer:

-10

Step-by-step explanation:

It is the easiest equation.

A/2= -5

At first, we have to multiply both the sides by 2. Therefore, we can get,

[tex]\frac{A*2}{2}[/tex] = (-5 × 2)

or, A = -10

Therefore, the value of A is -10. It remains negative because we cannot multiply both the sides by -1. If we do that, we cannot determine the constant.

Answer: A = -10

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:

[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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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:

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

Answer:

Correct option is (B) descriptive statistical method

Step-by-step explanation:

Descriptive statistics branch in statistics deals with the representation of the data using distinct brief coefficients. These coefficients are used as either the representative of the sample or the population.

The descriptive statistics branch is divided into two sub branches:

The three measures of central tendency are:

The measures of dispersion are:

The education researcher computes the average number of hours student study at various local colleges.

The average of a data is the mean value which is the measure of central tendency.

Thus, the researcher is using descriptive statistical method to summarize the data.

Final answer:

The education researcher is using descriptive statistical methods by calculating an average of study hours, which is a measure of central tendency, a fundamental aspect of descriptive statistics.

Explanation:

An education researcher who collects data on how many hours students study at various local colleges and then calculates an average to summarize this data is using descriptive statistical methods. Descriptive statistics involve organizing and summarizing data to provide a clear overview of its characteristics. Examples of descriptive statistics include measures of central tendency (mean, median, mode), which indicate the typical value within a data set, and measures of variability (range, variance, standard deviation), which show how spread out the data points are. The calculation of an average, or mean, falls under the measure of central tendency, making it a key component of descriptive statistics.

The probability that at least one member of a married couple will vote is 0.34 or 34%. The probability of a wife voting given that her husband will vote is approximately 0.7143 or 71.43%. The probability of a husband voting given his wife will not vote is 0.06 or 6%.

The subject of this question is probability within the realm of mathematics. To find the probability of at least one member of a married couple voting, we can use the formula P(A or B) = P(A) + P(B) - P(A and B).

Therefore, the probability is 0.21 (husband voting) + 0.28 (wife voting) - 0.15 (both voting), which equals 0.34.

For (b), the probability that the wife will vote, given that her husband will vote, is P(Wife|Husband) = P(Wife and Husband)/P(Husband).

So, this probability is 0.15/0.21, which equals approximately 0.7143.

For (c), the probability that the husband will vote, given that his wife will not vote, is P(Husband|Wife not voting) = P(Husband) - P(Husband and Wife).

So, this probability is 0.21 - 0.15, which yields 0.06 or 6%.

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.

To learn more about conditional probability;

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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:

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:

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 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