One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 93% detection rate for carriers and a 2% false positive rate. Suppose that an individual is tested. What is the probability that an individual who tests negative does not carry the disease? What is the specificity of the test?

Answer:

The proof is shown in the explanation below.

Step-by-step explanation:

Analysis:

The proof by induction focuses on n. In this case, let n = 1, and [tex]L^{1}[/tex] will be a linear operator since [tex]L^{1} = L[/tex]

The exercise will show that [tex]L^{n}[/tex] is a linear operator on V and that [tex]L^{n+1}[/tex] is also a linear operator on V. This, follows that:

[tex]L^{n+1} (av) = L(L^{m}(v_{1}+v_{2})\\ = L(L^{m} (v_{1} + L^{m}v_{2})\\ = L(L^{m}v_{1} + L(L^{m}v_{2})\\ = L^{m+1}(v_{1}) + L^{m+1}(v_{2})[/tex]

Answer/Step-by-step explanation:

For the mathematical induction,

We show that the equation

L (α1v1 + α2v2 +· · ·+αnvn)= α1L (v1) + α2L (v2)+· · ·+αnL (vn) is true for

L = 1,

Assume it is true for L = n and show that it is true for L = n + 1.

If L = 1, the equation become

(α1v1 + α2v2 +· · ·+αnvn)= α1(v1) + α2 (v2)+· · ·+αn (vn). Therefore, the Right Hand side(RHS) = Left Hand side(LHS)

When L = n, we assume the following is true

(α1nv1 + α2nv2 +· · ·+αnvn)= α1n(v1) + α2n (v2)+· · ·+αn (vn)

Then, when L = n + 1,

n +1 (α1v1 + α2v2 +· · ·+αvn)= α1(n +1) (v1) + α2(n +1) (v2)+· · ·+αn(n + 1)(vn).

Open the bracket,

n(α1v1 + α2v2 +· · ·+αvn) + α1v1 + α2v2 +· · ·+αnvn = α1n (v1) + α2v2 +· · ·+αvn ) + α1(v1) + α2v2+· · ·+αn(vn)

Since we assume the the equation is true for L = n, and eliminating some terms, then

L (α1v1 + α2v2 +· · ·+αnvn)= α1L (v1) + α2L (v2)+· · ·+αnL (vn)

Answer:

Therefore the required probability is 0.6472.

Step-by-step explanation:

Poisson distribution: A Poisson distribution is discrete distribution.

Let X be a discrete variable the number of event. Let λ be the mean of X.

Then the probability of k event is

[tex]P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}[/tex]

Here mean of each page is =0.03

Mean of 100 pages = (0.03×100)= 3

λ = 3

The required probability

P(X≤3)  

= P(X=0)+P(X=1)+P(X=2)+P(X=3)

[tex]=\frac{3^0e^{-3}}{0!}[/tex] [tex]+\frac{3^1e^{-3}}{1!}[/tex][tex]+\frac{3^2e^{-3}}{2!}[/tex][tex]+\frac{3^3e^{-3}}{3!}[/tex]

[tex]=e^{-3}(1+3+\frac{9}{2}+\frac{27}{6})[/tex]

[tex]=e^{-3}(13)[/tex]

≈ 0.6472

Therefore the required probability is 0.6472

Answer:

Step-by-step explanation:

Hello!

The objective is to study the semicircular canals (SC) of the inner ear of the three-toed sloth (Bradypus variegatus) to see if it is weak compared to faster animals.

The study variable is X: length to width of the anterior semicircular canal of a three-toed sloth.

A sample of 7 sloths was taken and the semicircular canal was measured:

1.52, 1.06, 0.93, 1.38, 1.47, 1.20, 1.16

∑X= 8.72

∑X²= 11.15

a.

[tex]S^2= \frac{1}{n-1} * [sumX^2-\frac{(sumX)^2}{n} ]= \frac{1}{6} *[11.15-\frac{(8.72)^2}{7} ][/tex]

S²= 0.047≅0.05

S=0.218≅ 0.22

Comparing the estimation of the variance of the length to width of the anterior semicircular canal of three-toed sloths with the known number of length to width of the anterior semicircular canal of faster animals (S=0.09), it appears that the variability os length to width of the anterior semicircular canal of sloths is greater than the length to width of the anterior semicircular canal of faster animals.

b. and c.

To estimate the most plausible range of values of the population standard deviation of the anterior semicircular canal of the sloths, you have to do an estimation per confidence interval.

To be able to make this estimation we have to assume that the variable of interest has a normal distribution. With this assumption, it is valid to use a Chi-Square statistic to estimate the population standard deviation.

[tex]X^2= \frac{(n-1)S^2}{Sigma^2} ~X^2_{n-1}[/tex]

I'll choose a confidence level of 95%

The formula for the interval is:

[tex][\frac{(n-1)S^2}{X^2_{n-1;1-\alpha /2}} ;\frac{(n-1)S^2}{X^2_{n-1;\alpha /2}} ][/tex]

[tex]X^2_{n-1;1-\alpha /2}= X^2_{6;0.975}= 14.449[/tex]

[tex]X^2_{n-1;\alpha /2}}= X^2_{6;0.025}= 1.2373[/tex]

[tex][\frac{6*0.05}{14.449} +\frac{6*0.05}{1.2373} ][/tex]

[0.2076;2.4246] ⇒This confidence interval is for the population variance, calculating the square root of each bond gives us the CI for the population standard deviation:

√[0.2076;2.4246]= [0.1441;1.5571]

The 95% CI [0.1441;1.5571] is expected to contain the true value of the population standard deviation of the length to width of the anterior semicircular canal of the three-toed sloths.

As you can see this interval does not contain the known value of the population standard deviation for faster animals, which leads to thinking there is a difference between the standard deviation of the anterior semicircular canal in both species.

I hope it helps!

Answer: x = - 16

y = - 11

Step-by-step explanation:

The given system of linear equations is expressed as

2x - 3y = 1- - - - - - - - - - - - -1

x - 2y = 6 - - - - - - - - - - - -2

We would eliminate x by multiplying equation 1 by 1 and equation 2 by 2. It becomes

2x - 3y = 1- - - - - - - - - - -3

2x - 4y = 12- - - - - - - - - - - -4

Subtracting equation 4 from equation 3, it becomes

y = - 11

Substituting y = - 11 into equation 2, it becomes

x - 2 × - 11 = 6

x + 22 = 6

Subtracting 22 from the left hand side and the right hand side of the equation, it becomes

x + 22 - 22 = 6 - 22

x = - 16

Answer:

Part (a) : 0.2297

Part (b) :  0.1112

Part (c) : 0.1469

Part (d) : 77,171

Step-by-step explanation:

Given info on Health Sciences:

Mean = $51,541

Standard Deviation = $11,000

Given info on Business:

Mean = $53,901

Standard Deviation = $15,000

Part (a)

Let X represents the new college graduate in business,

P (X ≥ 65,000) = 1 - P (X < 65,000)

= 1 - P ( z < [tex]\frac{65,000 - 53,901}{15,000}[/tex] )

= 1 - P ( z < 0.74)

= 1 - 0.77035

= 0.2297

Part (b)

Let Y represents the new college graduate in Health Sciences,

P (Y ≥ 65,000) = 1 - P (Y < 65,000)

= 1 - P ( z < [tex]\frac{65,000 - 51,541}{11,000}[/tex] )

= 1 - P ( z < 1.22)

= 1 - 0.88877

= 0.1112

Part (c)

Let Y represents the new college graduate in Health Sciences,

P (Y  < 40,000) = P (Y < [tex]\frac{40,000-51,541}{11,000}[/tex])

= P ( z < -1.05 )

= 0.1469

Part (d)

To have a starting salary higher than 99%, the z-score = 2.33. Let A represents the salary of a new college graduate in health sciences higher than 99% of all starting salaries.

[tex]2.33 = \frac{A - 51,541}{11,000}[/tex]

[tex]A = 77,171[/tex]

77,171 new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in health sciences.

Final answer:

To obtain the desired mixture, you will need 24 lbs of the first type of chocolate and 6 lbs of the second type of chocolate.

Explanation:

To find the amount of each chocolate needed to obtain the desired mixture, we can set up a system of equations. Let's say x represents the pounds of the first type of chocolate and y represents the pounds of the second type of chocolate. We know that the total weight of the mixture is 30 lbs, so we have the equation:

x + y = 30

We also know that the cost per pound of the mixture is $4.20, so the total cost is:

2.40x + 9.90y = 4.20 * 30

Simplifying the second equation, we get:

2.40x + 9.90y = 126

We can solve this system of equations using substitution, elimination, or a calculator. The solution is x = 24 lbs and y = 6 lbs. Therefore, you will need 24 lbs of the first type of chocolate and 6 lbs of the second type of chocolate to obtain the desired mixture.

Answer:

mL

Step-by-step explanation:

Ml

Answer:

No

Step-by-step explanation:

Because the model was developed using only pre-statistics course grades between 70% and 95% so it is risky to assume that linear trend will continue far beyond that span of values.

Step-by-step explanation:

x² + y² − 8x + 10y − 8 = 0

Rearrange:

x² − 8x + y² + 10y = 8

To complete the square, take half of the x and y coefficients, square it, then add the result to both sides.

(-8/2)² = 16

(10/2)² = 25

x² − 8x + 16 + y² + 10y + 25 = 8 + 16 + 25

x² − 8x + 16 + y² + 10y + 25 = 49

Factor the squares:

(x − 4)² + (y + 5)² = 49

The center is (4, -5) and the radius is 7.

Answer:

16.38%

Step-by-step explanation:

Given

Frequency of homozygous recessive individuals for the character extra-long eyelashes is 90 per 1000

Let p = Probability of homozygous recessive individuals having character extra-long eyelashes = 0.09

p + q = 1 where q = Probability of homozygous recessive individuals not having character extra-long eyelashes

0.09 + p = 1

p = 1 - 0.09

p = 0.91

The frequency of the carrier individual is calculated by npq

Where n = mating partners = 2

Frequency = 2 * 0.09 * 0.91

Frequency = 0.1638 or 16.38%

Answer:

The percentage of the population carries this trait but displays the dominant phenotype, short eyelashes is the frequency of hetrozygous individuals which is

0.42 or 42 %

Step-by-step explanation:

To solve the question, we note that

p²+2pq+q²=1  and p + q =1

where

p = frequency of the dominant allele

q= frequency of the recessive allele

p² = frequency of homozygous dominant allele

q² = frequency of homozygous recessive allele

2·p·q = frequency of hetrozygous individuals

The frequency of the homozygous recessive individuals q² is 0.09

Therefore the frequency of q = √(0.09) = 0.3, therefore p = 1 - 0.3 =0.7

The  percentage of the population carries this trait but displays the dominant phenotype, short eyelashes is the frequency of hetrozygous individuals or 2×p×q = Ae = 2*0.3*0.7 = 0.42

→ 0.42× 100 = 42 %

Answer:

I. $291.67 per passenger

II. $583.33 per passenger

Step-by-step explanation:

The cost of flying the passenger plane from A to B is $70,000.

If the Capacity of the plane is 240 people.

(I)At 7A.M and 4P.M., the average cost per passenger is given as:

Average Cost=Total Cost/Number of Passengers

=70000/240=$291.67 per passenger

(II)At 10AM and 1PM, the flight is only half full, it has 120 passengers.

Average Cost = 70000/120=$583.33 per passenger

Answer: The cost will be $291.70 and $583.40 for the first and the last and for the second and third flights, respectively.

Step-by-step explanation: The cost is $70,000, the full capacity of seats is 240 and half capacity is 120.

In the first and last flights, which have the full capacity, the average cost per passenger is:

C = [tex]\frac{70,000}{240}[/tex] = 291.70

In the second and third ones, the average cost is:

C = [tex]\frac{70,000}{120}[/tex] = 583.4

The average cost per passenger for the first and the last flight are $291.70, while for the second and third are $583.4

Answer:

ht j5yh bgmn gmjdt ky

Step-by-step explanation:

w245358697809908

The cross sections of the given mathematical functions provide a range of shapes, from parabolas to circles or lines, depending on whether x, y or z are fixed.

Interpreting and sketching the three-dimensional functions in question gives us an understanding of the cross-sections. For the first function i) z = x2 + y2 + 6, when x is fixed, the cross section gives us a parabola facing upwards in yz-plane. Moving ahead, when y is fixed, it gives us a similar parabola in the xz-plane. When z is fixed, we end up with a circle in xy-plane.

Going to the second function ii) z = 3x2, cross sections with x fixed results in a vertical line in yz-plane as z is not a function of y here. For y being fixed, it will again give us a vertical line but in the xz-plane because z and y are again unrelated. If z is fixed, we obtain a parabola opening upwards (or downwards depending on the sign) parallel to xy-plane.

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Answer:

The probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is [tex]\frac{14}{29}[/tex].

Step-by-step explanation:

The conditional probability of an events A given that another events B has already occurred is given by:

[tex]P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A^{c})P(A^{c})}[/tex]

The estimation of the past presidential election is provided.

Denote the events as follows:

R = a Republican candidate would be elected

D = a Democratic candidate would be elected

C = a conservative judge would be appointed to the Supreme Court

M = a moderate judge would be appointed to the Supreme Court

L = a liberal judge would be appointed to the Supreme Court

The information provided is:

[tex]P(R)=\frac{5}{7},\ P(D)=\frac{2}{7}[/tex]

[tex]P(C|R)=\frac{1}{7},\ P(M|R)=\frac{1}{7},\ P(L|R)=\frac{5}{7}[/tex]

[tex]P(C|B)=\frac{1}{3},\ P(M|D)=\frac{1}{6},\ P(L|D)=\frac{1}{2}[/tex]

It is provided that a conservative judge was appointed to the Supreme Court during the presidential term.

Compute the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court as follows:

[tex]P(D|C)=\frac{P(C|D)P(D)}{P(C|D)P(D)+P(C|R)P(R)}[/tex]

             [tex]=\frac{(\frac{1}{3}\times \frac{2}{7})}{(\frac{1}{3}\times \frac{2}{7})+(\frac{1}{7}\times \frac{5}{7})}[/tex]

             [tex]=\frac{\frac{2}{21}}{\frac{2}{21}+\frac{5}{49}}[/tex]

             [tex]=\frac{2}{21}\div \frac{29}{147}[/tex]

             [tex]=\frac{14}{29}[/tex]

Thus, the probability that the Democratic candidate was elected given that a conservative judge was appointed to the Supreme Court is [tex]\frac{14}{29}[/tex].

Answer:

The argument is valid.

Step-by-step explanation:

We are told "Loretta's hobby is stamp collecting."

"If her husband likes to fish, then Loretta's hobby is not stamp collecting."

Her hobby is stamp collecting, so her husband does not like to fish.

"If her husband does not like to fish, then Nathan likes to read."

Her husband does not like to fish; therefore, Nathan likes to read.

The argument is valid.

The argument is invalid as it fails the informal test of validity; the premises about Loretta and her husband's hobbies do not logically lead to the conclusion about Nathan's reading habits.

The task involves assessing the validity of an argument by applying the informal test of validity. Let's deconstruct the argument provided. It states that Loretta's hobby is stamp collecting, and presents two conditional statements involving her husband's hobbies and Nathan's interest. Finally, it concludes that Nathan likes to read.

To apply the informal test of validity, we must determine if it's possible for all premises to be true while the conclusion could still be false. In this argument, even if all the stated conditions about Loretta and her husband's hobbies are true, they don't logically necessitate the conclusion about Nathan's reading habits. Thus, the argument is invalid because it's possible to imagine a scenario where all premises are true, but Nathan does not like to read. The premises about Loretta and her husband have no logical connection to Nathan's interests, making it invalid.

A valid argument ensures that if the premises are true, the conclusion must also be true. However, this argument fails to meet that criterion, indicating its invalidity based on the informal test.

Answer:

(D) 9a

Step-by-step explanation:

Given

4a^2 + 9

Add 9a to the expression

4a^2 + 9 + 9a

This can be written as

(2a)^2 + (3)^2 + (3)^2 a

Answer:

E. 12a

Step-by-step explanation:

TBH no idea why thats correct.

Answer:

41 degrees

Step-by-step explanation:

step 1

Find the measure of arc DGF

we know that

The inscribed angle is half that of the arc comprising

so

[tex]m\angle E=\frac{1}{2}[arc\ DGF][/tex]

we have

[tex]m\angle E=123^o[/tex]

substitute

[tex]123^o=\frac{1}{2}[arc\ DGF][/tex]

[tex]arc\ DGF=246^o[/tex]

step 2

Find the measure of arc EF

we know that

[tex]arc\ DGF+arc\ DE+arc\ EF=360^o[/tex] ----> by complete circle

substitute the given values

[tex]246^o+73^o+arc\ EF=360^o[/tex]

[tex]319^o+arc\ EF=360^o[/tex]

[tex]arc\ EF=360^o-319^o=41^o[/tex]

Answer: A)41 Degrees

Step-by-step explanation:

It was right on edu 2023

Answer:

Step-by-step explanation:

Check attachment for solution

Answer:

Step-by-step explanation:

Given that of all airline flight requests received by a certain discount ticket broker, 90% are for domestic travel (D) and 10% are for international flights (I).

Let x be the number of requests among the next three requests received that are for domestic flights.

X can take values as 0,1, 2 or 3.

Since independence of successive requests is assumed we find that X having two outcomes and independence

X is binomial with n =3 and p= 0.90 (constant for each trial)

PDF of X would be

[tex]P(X=x) = 3Cx (0.9)^r (0.1)^{3-r} , r=0,1,2,3[/tex]

Thus pdf of X is

X           0           1             2            3

p         0.001    0.027   0.243   0.729

Final answer:

The probability distribution of x, where x is the number of domestic flight requests among the next three requests, follows a binomial distribution with n=3 and p=0.9. It can be calculated using the formula for binomial probabilities, considering the value of x can be 0, 1, 2, or 3. We get P = 0.081.

Explanation:

The question involves calculating the probability distribution of the random variable x, which represents the number of domestic flight requests among the next three requests received by a discount ticket broker. Since each request is independent and only two outcomes are possible (domestic or international), a binomial distribution can be used to determine the probabilities for x. For instance, the probability for one particular sequence of requests such as DID (domestic, international, domestic) is calculated as (0.9)(0.1)(0.9), which is the product of the probabilities for each request in that sequence. To find the probability distribution of x, we consider all possible sequences of three requests and their associated probabilities, which are determined by multiplying the individual probabilities for each request in the sequence following the rules of binomial distribution.

The random variable x can take on the following values: 0, 1, 2, or 3, representing the number of domestic flight requests out of three. We calculate probabilities for each of these values using the binomial formula:

[tex]P(x=0) = (0.1)^3\\P(x=1) = 3 * (0.9)(0.1)^2\\P(x=2) = 3 * (0.9)^2(0.1)\\P(x=3) = (0.9)^3[/tex]

P = 0.081.

Answer:

b. 10.4

c.  26.9

Step-by-step explanation:

Let the universal set U = 100% which is the total no of people in the American community

Let A = 14.6% which is the total no of people living below poverty line

Let B = 20.7% which is the total no of people speaking foreign Language

C = 4.2% no of people who both speak foreign language and live below poverty line

X = no of people who neither live below poverty line nor speak foreign language

P (A) = 14.6%

P (B) = 20.7%

P (C) = P (A ∩ B) = 4.2%

P (A – C) = P (A ∩ U) = 14.6 – 4.2 = 10.4%

P (B – C) = P (B ∩ U) = 20.7 – 4.2 = 16.5%

P (X) = P (A ᴜ B) c =100 – (10.4 + 4.2 + 16.5) = 68.9%

a. The venn diagram is as shown above

b. Percent of Americans who live below poverty line and Speak English at home(minus foreign lang speakers living below poverty line) that is A only

= A – C

= 14.6 – 4.2

= 10.4%

c. Percentage of Americans Living below poverty line or Speaking foreign language

= A only + B only

A only = A – C ( People living below poverty line only)

= 14.6 -4.2

= 10.4%

B only = B – C ( people speaking foreign languages only)

= 20.7 – 4.2

= 16.5%

Hence

A only + B only = 10.4 + 16.5 = 26.9%

Answer with Step-by-step explanation:

S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),..,(2,6),(3,1),(3,2),...,(3,6),(4,1),..,(4,6),(5,1),(5,2),(5,3),...,(5,6),(6,1),(6,2),..,(6,6)}

Total number of cases=36

M=Maximum of the two tosses

M=1,

(1,1)

Therefore, [tex]P(1)=\frac{1}{36}[/tex]

Using the formula, P(E)=[tex]\frac{favorable\;cases}{Total\;number\;of\;cases}[/tex]

M=2

(1,2),(2,1),(2,2)

Favorable cases=3

[tex]P(2)=\frac{3}{36}=\frac{1}{12}[/tex]

M=3

(1,3),(3,1),(3,2),(2,3),(3,3)

[tex]P(3)=\frac{5}{36}[/tex]

M=4

(1,4),(4,1),(2,4),(3,4),(4,2)(4,3),(4,4)

[tex]P(4)=\frac{7}{36}[/tex]

M=5

(1,5),(2,5),(3,5),(4,5),(5,1),(5,2),(5,3),(5,4),(5,5)

[tex]P(5)=\frac{9}{36}=\frac{1}{4}[/tex]

M=6

(1,6),(2,6),(3,6),(4,6),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

[tex]P(6)=\frac{11}{36}[/tex]

Answer:

(C) Systematic Sampling

Step-by-step explanation:

The reporter decides to interview every fifth student of Ohio State University that passes by. This is referred to as Systematic Sampling.

In Systematic Sampling, the members of the population are place in an ordered list and the members of the sample are taken at a stated interval. This interval is called the Sampling Interval.

Although the members of the sample are random, they are chosen at stated intervals.

Answer:

Face validity.

Step-by-step explanation:

Face validity is considered the weakest form of validity as it does a superficial, subjective assessment which does not involve objective approach, revealing the deeper intent of the test being used. It involved a process similar to skimming the surface of an item or a book to make opinions. Though it is a weak form of validity, it is the easiest to apply to research. Face validity is an informal approach to identifying how suitable the content of a test is to the research.

It generally asks the question:

Is the content of the test suitable to its aims?

Answer:

(a) 35 heads correspond to -3 on the standard scale.

(b) z = 2.4 corresponds to 62 heads on the number of heads scale.

Step-by-step explanation:

(a) If we flip a fair coin once, probability of getting head = 0.5

If we flip a fair coin 100 times, mean number of heads = 100(0.5) = 50

If there are N draws with a P probability of success, the standard deviation (SD) is given as:

[tex]SD = \sqrt{(N)(P)(1 - P)}[/tex]

Here, the probability of getting a head (P) is 0.5 while the number of draws (N) is 100. So,

[tex]SD = \sqrt{(100)(0.5)(1-0.5)}[/tex]

SD = 5

The standard scale value is: (35 - 50) / 5 = -3

Hence, 35 heads correspond to -3 on the standard scale.

(b) The standard scale value is 2.4 and we need to find the number of heads.

(X - 50) / 5 = 2.4

X - 50 = 12

X = 62

Hence, z = 2.4 on the standard scale corresponds to 62 on the number of heads scale.

The 35 heads corresponds to -3 on the standard scale, and the number of heads is 62

The head on the standard scale

The given parameter is:

n = 100

In a flip of a coin, the probability of a head is:

p = 1/2

So, the mean of the distribution is:

[tex]\bar x = np[/tex]

[tex]\bar x = 100 * 1/2[/tex]

[tex]\bar x = 50[/tex]

The standard deviation is:

[tex]\sigma = \sqrt{np(1 - p)}[/tex]

So, we have:

[tex]\sigma = \sqrt{100 * 1/2 * (1 - 1/2)}[/tex]

[tex]\sigma = 5[/tex]

The corresponding value on the standard scale is then calculated as:

[tex]z= \frac{x - \mu}{\sigma}[/tex]

For 35 heads, we have:

[tex]z= \frac{35 - 50}{5}[/tex]

[tex]z = -3[/tex]

Hence, the 35 heads corresponds to -3 on the standard scale

(b) The number of heads

We have:

[tex]z= \frac{x - \mu}{\sigma}[/tex]

When z = 2.4, the equation becomes

[tex]2.4= \frac{x - 50}{5}[/tex]

Multiply both sides by 5

[tex]12= x - 50[/tex]

Add 50 to both sides

[tex]x = 62[/tex]

Hence, the number of heads is 62

Read more about z-scores at:

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Answer:

Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is [tex]-\frac{5200\sqrt6}{3}e^{-43}[/tex]  °C/m.

Step-by-step explanation:

Given that, the temperature at a point (x,y,z) is

[tex]T(x,y,z)=200e^{-x^2-3y^2-9z^2}[/tex].

Rate change of temperature at the point P(2,-1,2) in the direction toward the point Q (3,-3,3) is [tex]D_uT(2,-1,2)[/tex]

[tex]T_x= 200(-2x)e^{-x^2-3y^2-9z^2}[/tex][tex]=-400x200.2xe^{-x^2-3y^2-9z^2}[/tex]

[tex]T_y = 200.(-6y)e^{-x^2-3y^2-9z^2}[/tex][tex]=-1200ye^{-x^2-3y^2-9z^2}[/tex]

[tex]T_z=200(-18z)e^{-x^2-3y^2-9z^2}[/tex][tex]=-3600ze^{-x^2-3y^2-9z^2}[/tex]

The gradient of the temperature

[tex]\bigtriangledown T(x,y,z)= (-400x200.2xe^{-x^2-3y^2-9z^2},-1200ye^{-x^2-3y^2-9z^2},-3600ze^{-x^2-3y^2-9z^2})[/tex]                   [tex]=-400e^{-x^2-3y^2-9z^2}(x,3y,9z)[/tex]

[tex]\bigtriangledown T(2,-1,2) = -400e^{-2^2-3(-1)^2-9.2^2}(2,3.(-1),9.2)[/tex]

                   [tex]=-400 e^{-43}(2,-3,18)[/tex]

V=[tex]\overrightarrow {PQ}= \vec{Q}-\vec{P}[/tex]=(3,-3,3)-(2,-1,2)=(1,-2,1)

The unit vector of V is [tex]\frac{(1,-2,1)}{\sqrt{1^2+(-2)^2+1^2}}[/tex]

                                   [tex]=\frac{1}{\sqrt6}(1,-2,1)[/tex]

Therefore,

[tex]D_uT(2,-1,2) = \bigtriangledown T(2,-1,2).\frac{1}{\sqrt6}(1,-2,1)[/tex]

                       [tex]=-400 e^{-43}(2,-3,18).\frac{1}{\sqrt6}(1,-2,1)[/tex]

                       [tex]=-\frac{400}{\sqrt6}e^{-43}(2.1+(-3).(-2)+18.1)[/tex]

                       [tex]=-\frac{10400}{\sqrt6}e^{-43}[/tex]  

                        [tex]=-\frac{5200\sqrt6}{3}e^{-43}[/tex] °C/m

Therefore the rate change of temperature at the point P(2,-1,2) in the direction toward the point (3,-3,3) is [tex]-\frac{5200\sqrt6}{3}e^{-43}[/tex]  °C/m.

Answer:

a.[tex]\mu=15[/tex]

b.[tex]\mu=7.8586 \ and \ \mu=22.1414[/tex]

c. Choice A- Based on limits above, it is unlikely that we would see x = 45, so it might be possible that the trials are not independent.

Step-by-step explanation:

a.Binomial distribution is defined by the expression

[tex]P(X=k)=C_k^n.p^k.(1-p)^{n-k}[/tex]

Let n be the number of trials,[tex]n=100[/tex]

and p be the probability of success,[tex]p=15\%[/tex]

The mean of a binomial distribution is the probability x sample size.

[tex]\mu=np=100\times0.15=15[/tex]

b.Limits within which p is approximately 95%

sd of a binomial distribution is given as:[tex]\sigma=\sqrt npq\\q=1-p[/tex]

Therefore, [tex]\sigma=\sqrt(100\times0.015\times0.85)=3.5707[/tex]

Use the empirical rule to find the limits. From the rule, approximately 95% of the observations are within to standard deviations from mean.

[tex]sd_1=>\mu-2\sigma=15-2\times3.3507=7.8586\\sd_2=>\mu-2\sigma=15+2\times3.3507=22.1414[/tex]

Hence, approximately 95% of the observations are within 7.8586 and 22.1414 (areas of infestation).

c.  [tex]x=45[/tex] is not within the limits in b above (7.8586,22.1414). X=45 appears to be a large area of infestation. A.Based on limits above, it is unlikely that we would see x = 45, so it might be possible that the trials are not independent.

Final answer:

Using the empirical rule, we can determine the limits within which we would expect to find the number of infested fields with a 95% probability. If we found a number of infested fields that is higher than expected, it suggests that one of the characteristics of a binomial experiment may not be satisfied.  Correct answer is B.

Explanation:

In this problem, we are given that 15% of the fields in a given agricultural area are infested with the sweet potato whitefly. We are asked to determine the limits within which we would expect to find the number of infested fields with a probability of approximately 95%. This situation can be modeled using the binomial distribution, as each field can be considered as a separate trial with two possible outcomes, infested or not infested.

According to the empirical rule, for a binomial distribution, we can expect about 95% of the outcomes to fall within two standard deviations of the mean. The mean number of infested fields can be calculated as 15% of the total number of fields, which is 15. The standard deviation of the number of infested fields can be determined using the formula σ = [tex]\sqrt{(npq)}[/tex], where n is the number of trials, p is the probability of success, and q is the probability of failure. In this case, n = 100, p = 0.15, and q = 0.85.

Using these values, we can calculate the standard deviation as σ = [tex]\sqrt{(100 * 0.15 * 0.85)}[/tex] ≈ 3.150. Therefore, we would expect to find about 95% of the number of infested fields within two standard deviations of the mean, which is between 15 - (2 * 3.150) = 8.700 and 15 + (2 * 3.150) = 21.300.

If we found that x = 45 fields were infested, we can conclude that this number is higher than what we would expect based on the binomial distribution. It is unlikely to observe such a high number of infested fields if the trials were independent and had only two possible outcomes. Therefore, we might suspect that one of the characteristics of a binomial experiment is not satisfied in this experiment.

Based on the limits above, it is unlikely that we would see x = 45, so it might be possible that the trials have more than two possible outcomes. Therefore, the correct answer is B.

The slope of the line is [tex]m=-4[/tex]

Explanation:

Given that the pair of points are [tex](1,-19)[/tex] and [tex](-2,-7)[/tex]

We need to determine the slope of the line.

The slope of the line can be determined using the formula,

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Substituting the points [tex](1,-19)[/tex] and [tex](-2,-7)[/tex] for the coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] in the above formula, we have,

[tex]m=\frac{-7-(-19)}{-2-1}[/tex]

Simplifying the terms, we get,

[tex]m=\frac{-7+19}{-2-1}[/tex]

Adding the terms, we have,

[tex]m=\frac{12}{-3}[/tex]

Dividing, we get,

[tex]m=-4[/tex]

Thus, the slope of the line between the pair of points is [tex]m=-4[/tex]

Answer:

[tex]\frac{dV(t)}{dt} =[/tex] - 1675.38

Step-by-step explanation:

In 2017, the vakue of the kitchen equipment was $14550

V(0)=$14550

Its value after then was modelled by [tex]V(t)=14550e^{-0.158t[/tex]

We are required to find the rate of change in value on January 1, 2019

[tex]V(t)=14550e^{-0.158t[/tex]

[tex]\frac{dV(t)}{dt} =\frac{d}{dt}14550e^{-0.158t[/tex]

[tex]\frac{dV(t)}{dt} =14550 \frac{d}{dt}e^{-0.158t[/tex]

[tex]\\Let u= -0.158t,\frac{du}{dt}=-0.158[/tex]

[tex]\frac{dV(t)}{dt} =14550 \frac{d}{du}e^u\frac{du}{dt}[/tex]

[tex]\frac{dV(t)}{dt} =14550 X -0.158 e^{-0.158t}=-2298.9e^{-0.158t}[/tex]

In 2019, i.e. 2 years after, t=2

The rate of change of the value

[tex]\frac{dV(t)}{dt} =-2298.9e^{-0.158X2}[/tex]

=[tex]\frac{dV(t)}{dt} =-2298.9e^{-0.316}[/tex]= - 1675.38

The rate of change in the value of the equipment on January 1, 2019, is -2,302.9e^(-0.158t).

The rate of change in the value of the equipment on January 1, 2019, can be found by taking the derivative of the given model equation V(t) = 14,550e^(-0.158t).

The derivative of V(t) with respect to t is dV/dt = -0.158(14,550)e^(-0.158t), which simplifies to dV/dt = -2,302.9e^(-0.158t).

Therefore, the rate of change in the value of the equipment on January 1, 2019, is -2,302.9e^(-0.158t).

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Answer: the height of the building is 40ft

Step-by-step explanation:

Looking at the right angle triangle formed,

With angle P as the reference angle, the length shadow of the building on ground represents the adjacent side of the right angle triangle.

The height of the building represents the opposite side of the right angle triangle.

a) to determine the height of the building, x, we would apply the tangent trigonometric ratio which is expressed as

Tan θ, = opposite side/adjacent side. Therefore, the equation becomes

Tan P = x/50

b) 0.8 = x/50

x = 50 × 0.8

x = 40 ft

Answer:

Step-by-step explanation:

Given that an ANOVA procedure is used for data obtained from four populations. Four samples, each comprised of 30 observations, were taken from the four populations.

Hence total observations are 30*4 =120

No of groups = 3

Hence numerator df = 3-1 =2

Now total degrees of freedom = 120-1 =119

So denominator degrees of freedom = 119-2 = 117

Thus F statistic will have numerator as 2 degrees of freedom and denominator as 117 degrees of freedom.

Answer:

[tex]z=-0.674<\frac{a-12}{1.25}[/tex]

And if we solve for a we got

[tex]a=12 -0.674*1.25=11.157[/tex]

So the value of height that separates the bottom 25% of data from the top 75% is 11.157.  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the mean life of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(12,1.25)[/tex]  

Where [tex]\mu=12[/tex] and [tex]\sigma=1.25[/tex]

For this part we want to find a value a, such that we satisfy this condition:

[tex]P(X>a)=0.75[/tex]   (a)

[tex]P(X<a)=0.25[/tex]   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75

If we use condition (b) from previous we have this:

[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.25[/tex]  

[tex]P(z<\frac{a-\mu}{\sigma})=0.25[/tex]

But we know which value of z satisfy the previous equation so then we can do this:

[tex]z=-0.674<\frac{a-12}{1.25}[/tex]

And if we solve for a we got

[tex]a=12 -0.674*1.25=11.157[/tex]

So the value of height that separates the bottom 25% of data from the top 75% is 11.157.  

The lifespan of the dishwashers in the bottom 25% is approximately 11.16 years, found by calculating the corresponding value to the 25th percentile z-score from the normal distribution of the appliance's lifespan.

The problem is asking for the life of the dishwasher that falls in the bottom 25% of the normally distributed lifespan range. In other words, we need to find the dishwasher lifespan that is the cutoff for the bottom quartile. For a normal distribution, the 25th percentile (the cutoff for the bottom quartile) is commonly found by using a z-score, which is a measure of how many standard deviations a value is from the mean.

The z-score for the 25th percentile is typically -0.675. We can find the lifespan corresponding to this z-score using the formula: lifespan = mean + z*sd. Here, 'mean' is the average lifespan, 'z' is the z-score, and 'sd' is the standard deviation.

Therefore, substituting the known values into the formula, we get: lifespan = 12 + (-0.675)*1.25 = 11.15625 years.

Therefore, the number of years the bottom 25% of dishwashers would last is approximately 11.16 years.

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