We've established that heights of 10-year-old boys vary
according to a Normal distribution with mu = 138cm and sigma = 7cm
What proportion is between 152 and 124 cm ?

The parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4 are:

x = (208 + 70t) / 52

y = (13 + 19t) / 13

z = t

To find the parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4, we can solve these two equations simultaneously.

Step 1: Set up a system of equations:

4x - 3y + z = 1

3x + y - 4z = 4

Step 2: Solve the system of equations to find the values of x, y, and z. One way to solve the system is by using the method of elimination:

Multiply the first equation by 3 and the second equation by 4 to eliminate the y term:

12x - 9y + 3z = 3

12x + 4y - 16z = 16

Subtract the first equation from the second equation:

12x + 4y - 16z - (12x - 9y + 3z) = 16 - 3

12x + 4y - 16z - 12x + 9y - 3z = 13y - 19z = 13

Step 3: Express y and z in terms of a parameter, let's call it t:

13y - 19z = 13

y = (13 + 19z) / 13

We can take z as the parameter t:

z = t

Substituting the value of z in terms of t into the equation for y:

y = (13 + 19t) / 13

Step 4: Express x in terms of t:

From the first equation of the original system:

4x - 3y + z = 1

4x - 3((13 + 19t) / 13) + t = 1

4x - (39 + 57t) / 13 + t = 1

4x - (39 + 57t + 13t) / 13 = 1

4x - (39 + 70t) / 13 = 1

4x = (39 + 70t) / 13 + 1

x = ((39 + 70t) / 13 + 13) / 4

x = (39 + 70t + 169) / 52

x = (208 + 70t) / 52

Therefore, the parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4 are:

x = (208 + 70t) / 52

y = (13 + 19t) / 13

z = t

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We can conclude that the null hypothesis is rejected. There is sufficient evidence to support the claim that the mean cost in Utah hospitals is below the national mean of $1231.

H₀: μ ≥ 1231 (The mean cost in Utah hospitals is greater than or equal to the national mean)

Hₐ: μ < 1231 (The mean cost in Utah hospitals is below the national mean)

Given

The test statistic for a one-sample t-test is given by

t = (x - μ) / (σ / √n)

Substituting we have

t = (1103 - 1231) / (252 / √25)

≈ -6.103

To determine the critical value, we need to find the critical t-value at the 5% significance level with degrees of freedom

(df) equal to (n - 1)

= (25 - 1)

= 24.

Using a t-distribution table or calculator, the critical value is approximately -1.711.

Since the calculated test statistic (-6.103) is smaller than the critical value (-1.711) and falls into the critical region, we reject the null hypothesis.

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(i) For X ~ Laplace(0,1):

E(X) = 0, Var(X) = 2.

(ii) If X ~ Laplace(0,1) and Y = bX + μ:

Y ~ Laplace(μ, b).

(iii) For W ~ Laplace(2,8):

E(W) can be approximated numerically.

Var(W) = 128.

(i) If X ~ Laplace(0,1), we need to find the expected value (E(X)) and variance (Var(X)).

The Laplace(0,1) distribution has μ = 0 and b = 1. Substituting these values into the PDF, we have:

fX(x) = (1/2) * exp(-|x|)

To find E(X), we integrate x * fX(x) over the entire range of X:

E(X) = ∫x * fX(x) dx = ∫x * [(1/2) * exp(-|x|)] dx

Since the Laplace distribution is symmetric about the mean (μ = 0), the integral of an odd function over a symmetric range is zero. Therefore, E(X) = 0 for X ~ Laplace(0,1).

To find Var(X), we use the formula:

Var(X) = E(X^2) - [E(X)]^2

First, let's find E(X^2):

E(X^2) = ∫x^2 * fX(x) dx = ∫x^2 * [(1/2) * exp(-|x|)] dx

Using the symmetry of the Laplace distribution, we can simplify the integral:

E(X^2) = 2 * ∫x^2 * [(1/2) * exp(-x)] dx (integral from 0 to ∞)

Solving this integral, we get:

E(X^2) = 2

Now, substitute the values into the variance formula:

Var(X) = E(X^2) - [E(X)]^2 = 2 - 0 = 2

Therefore, for X ~ Laplace(0,1), E(X) = 0 and Var(X) = 2.

(ii) To show that Y = bX + μ follows a Laplace(μ, b) distribution, we need to find the probability density function (PDF) of Y.

Using the transformation method, let's express X in terms of Y:

X = (Y - μ)/b

Now, calculate the derivative of X with respect to Y:

dX/dY = 1/b

The absolute value of the derivative is |dX/dY| = 1/b.

To find the PDF of Y, substitute the expression for X and the derivative into the Laplace(0,1) PDF:

fY(y) = fX((y-μ)/b) * |dX/dY| = (1/2) * exp(-|(y-μ)/b|) * (1/b)

Simplifying this expression, we get:

fY(y) = 1/(2b) * exp(-|y-μ|/b)

This is the PDF of a Laplace(μ, b) distribution, thus showing that Y ~ Laplace(μ, b).

(iii) For W ~ Laplace(2,8), we need to find E(W) and Var(W).

The PDF of W is given by:

fW(w) = (1/16) * exp(-|w-2|/8)

To find E(W), we integrate w * fW(w) over the entire range of W:

E(W) = ∫w * fW(w) dw = ∫w * [(1/16) * exp(-|w-2|/8)] dw

This integral can be challenging to solve analytically. However, we can approximate the expected value using numerical methods or software.

To find Var(W), we can use the property that the variance of the Laplace distribution is given by 2b^2, where b is the scale parameter.

Var(W) = 2 * b^2

= 2 * (8^2)

= 2 * 64

= 128

Therefore, Var(W) = 128 for W ~ Laplace(2,8).

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Answer:

1st, 2nd, 3rd

Step-by-step 1explanation:

30+40+5=75

30+40=70

70+5=75

20+1+50+4

=20+50=70

1+4=5

70+5=75

50+30-5  

50+30=80

80-5=75

I hope this helps :)

Answer:

one solution.            

Answer:

x = -7.5

Step-by-step explanation:

12(x+5) = 4x

12x+ 60 = 4x

60 = -8x

-7.5 = x

The probability that the patient has breast cancer given a positive result is 62.2%.

The probability of testing positive given the patient has breast cancer is:

P(P|C) = 0.869

The specificity of the test is 88.9% or 0.889, meaning that the test will correctly identify 88.9% of patients who do not have breast cancer as not having the disease.

So, the probability of testing negative given the patient does not have breast cancer is:

P(N|N) = 0.889

Now, using Bayes' theorem:

P(C|P) = P(P|C) * P(C) / P(P)

where,P(P) = P(P|C) * P(C) + P(P|N) * P(N)

Here, P(P|N) is the probability of testing positive given that the patient does not have breast cancer. This is equal to 1 - specificity = 1 - 0.889 = 0.111.

So, P(P) = P(P|C) * P(C) + P(P|N) * P(N) = 0.869 * 0.13 + 0.111 * (1 - 0.13) = 0.1823

So,P(C|P) = 0.869 * 0.13 / 0.1823 = 0.622 or 62.2%

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Answer: You will pay $510 for a item with original price of $600 when discounted 15%.

A survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table is as follows:

a) The probability that a randomly selected​ person's gender is​ female is 270/570 or 0.474, which is approximately 47.4%.Formula used: P (Female) = Number of Females/Total Number of Individuals

b) The probability that a randomly selected person feels tense or stressed out at work and is​ female is 145/570 or 0.254, which is approximately 25.4%. Formula used: P (Female and Tense) = Number of Females who are Tense/Total Number of Individuals

c) The probability that a randomly selected person feels tense or stressed out at work or is​ female is: P (Female or Tense) = P(Female) + P(Tense) - P(Female and Tense)P(Tense) = (245/570) or 0.43, which is approximately 43%P(Female or Tense) = 0.47 + 0.43 - 0.254 = 0.646, which is approximately 64.6%.

d) The distinction between the outcomes in​ (b) and​ (c) is that the former shows the likelihood of being female and tense at work, whereas the latter shows the likelihood of being female or tense at work.

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The principal part at the isolated singular point -1 is not provided, so we cannot determine its nature or residue. And b) The principal part at the isolated singular point 1 is (x - 1)^2022 exp(-1). It is a pole of order 2022, and its residue is 0.

a) The principal part at the isolated singular point -1 is not provided, so we cannot determine its nature (removable singular point, essential singular point, or pole) or calculate its residue without additional information.

b) The given function is (x - 1)^2022 exp(-1). At the isolated singular point x = 1, the principal part of the function is (x - 1)^2022 exp(-1). Here, (x - 1)^2022 represents the pole part of the function, and exp(-1) represents the non-pole part.

Since the term (x - 1)^2022 dominates near x = 1, we can conclude that x = 1 is a pole. The order of the pole is determined by the exponent of (x - 1), which is 2022 in this case.

To calculate the residue, we need more information about the function, specifically the coefficients of the Laurent series expansion near the singular point. Without that information, we cannot determine the residue at x = 1.

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Step-by-step explanation:

1st expression

= px^4 + px

= px ( x³ + 1 )

= px ( x + 1) (x² - x + 1)

2nd expression

= qx³ - qx

= qx ( x² - 1 )

= qx ( x + 1) ( x - 1)

HCF = x ( x + 1)

Hope it will help :)❤

Answer:

The radius is 3.5 inches I think.

Step-by-step explanation:

Hope this helped Mark BRAINLIEST!!!

Answer:

3.5

Step-by-step explanation:

You would simply divide 7 inches by 2 because the radius is one-half the measure of the diameter.

Set up a ratio of height over shadow for each :

12/8 = 4/x

Cross multiply:

12x = 32

Divide both sides by 12:

X = 2 2/3 feet

The shadow is 2 2/3 feet.

Answer:

2.2y=1x or just x

Step-by-step explanation:

Answer: y=2.2x

Step-by-step explanation:

Answer:

Variant b

Step-by-step explanation:

If you multiply it should be a+b but if you divided

A+b-c

We have: u''(x) = ƒ(x)cot(x) - 2u'(x)cot(x).Thus, we can find a particular solution of this differential equation by using variation of parameters.

Let G(x) = ƒ(s)sin(x - s) ds.

Then, by the product rule, we have: G' = ƒ(s)cos(x - s) ds - ƒ(s)sin(x - s) ds, and G'' = -ƒ(s)sin(x - s) ds - ƒ(s)cos(x - s) ds. Hence, we have:G'' + G = ƒ(s)sin(x - s) ds - ƒ(s)cos(x - s) ds + ƒ(s)sin(x - s) ds = ƒ(s)sin(x - s) ds = G.

So, G is indeed a solution of y'' + y = ƒ(x).Next, we will use variation of parameters to find a second solution of the same differential equation.

Let us suppose that we have another solution of the form y = u(x) sin(x).

Then, y' = u(x)cos(x) + u'(x)sin(x), and y'' = - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x).

Substituting these into the differential equation, we get:- u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x) + u(x)sin(x) = ƒ(x)2u'(x)cos(x) + u''(x)sin(x) = ƒ(x)

Dividing by sin(x), we get:2u'(x)cot(x) + u''(x) = ƒ(x)cot(x).

Now, let us assume that the second solution is of the form y = u(x)sin(x), where u is a function to be determined.

Then, we have: y' = u(x)cos(x) + u'(x)sin(x) and y'' = - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x).

Substituting these into the differential equation, we get: - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x) + u(x)sin(x) = ƒ(x)2u'(x)cos(x) + u''(x)sin(x) = ƒ(x)

Dividing by sin(x), we get:2u'(x)cot(x) + u''(x) = ƒ(x)cot(x).

Hence, we have: u''(x) = ƒ(x)cot(x) - 2u'(x)cot(x).Thus, we can find a particular solution of this differential equation by using variation of parameters.

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Answer:

For square matrices A and B of equal size, the determinant of a matrix product equals the product of their determinants: det (A.B) = det (A) det (B) 1. A is invertible if and only if its determinant is nonzero 1.

Step-by-step explanation:

The required solutions are:

a) Yes, the relation R is an equivalence relation.

b)The equivalence class of -17 is {-17, -23, -29, -35, ...}.

a) In order to determine whether R is an equivalence relation or not, we need to check if it satisfies the following three conditions:

We can see that the relation 'R' satisfies all the conditions ( reflexibility, symmetry, and transitivity), so R is an equivalence relation.

b) In order to find the equivalence class of -17, we need to find all integers that are related to -17 under the relation R.

We can rewrite the relation as mRn if and only if m' - n' = 6k for some integer k.

In this case, -17Rn if and only if (-17)' - n' = -17 - n = 6k for some integer k.

To find all integers n that satisfy this equation, we can rearrange it as n = -17 - 6k.

By putting in different values of k, we can find all the integers n that are in the equivalence class of -17.

For example, when k = 0, n = -17 - 6(0) = -17. So, -17 is in the equivalence class of -17.

We can also see that when k = 1, n = -17 - 6(1) = -23. So, -23 is also in the equivalence class of -17.

The equivalence class of -17 consists of all integers that can be obtained by subtracting multiples of 6 from -17. So, the equivalence class of -17 is {-17, -23, -29, -35, ...}.

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Answer:

B. A = 1/2(7)h

Step-by-step explanation:

Formula for area of triangle = 1/2 x base x height

H is the height of the triangle.

7cm is identified as the base of the triangle.

1/2(7)h is also the same thing as 1/2 x 7 x h basically.

Answer:

B

Step-by-step explanation:

The area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )

Here b = 7 and h = h , then

A = [tex]\frac{1}{2}[/tex] (7) h → B

Answer:

A

Step-by-step explanation:

the mean is what occurs most often

Therefore, the variance of the time it takes for someone to finish a bowl of ramen is 4.6875.

Given, The moment generating function of the time it takes for someone to finish a bowl of ramen is

M(t)= 1/ (1−0.05t​)1​,t<0.05 We have to find the variance of the time it takes for someone to finish a bowl of ramen.

The variance of the random variable can be calculated by the formula Variance = M''(0) - [M'(0)]^2 where M(t) is the moment generating function of the random variable M'(t) is the first derivative of M(t)M''(t) is the second derivative of M(t)

We need to find M''(t) and M'(t)M(t) = 1/(1 - 0.05t)M'(t) = [0.05/(1 - 0.05t)^2]M''(t) = [0.1/(1 - 0.05t)^3] Now, at t = 0, M(0) = 1, M'(0) = 1.25, M''(0) = 6.25 Variance = M''(0) - [M'(0)]^2 Variance = 6.25 - (1.25)^2 Variance = 6.25 - 1.5625 Variance = 4.6875

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Given: The time it takes for someone to finish a bowl of ramen can be modeled by a random variable with the following moment generating function: M(t)= 1/ (1−0.05t​)1​,t<0.05. The variance of the time it takes for someone to finish a bowl of ramen is 400.

The moment generating function of a random variable is defined as [tex]$M(t) = \mathbb{E}(e^{tX})$[/tex] for all t in an open interval around 0 which X is a random variable.

We are given that the moment generating function of the random variable T is given by:

[tex]$$M(t)= \frac{1}{1-0.05t} ,\ t < 0.05$$[/tex]

The [tex]$n^{th}$[/tex] derivative of M(t) at 0 is given by:

[tex]$$\frac{d^n}{dt^n} M(t) \biggr|_{t=0} = \mathbb{E}(X^n)$$[/tex]

We differentiate $[tex]M(t)$[/tex] with respect to $t$ to get [tex]$$M'(t) = \frac{0.05}{(1 - 0.05t)^2}$$[/tex].

Differentiating [tex]$M'(t)$[/tex] with respect to [tex]$t$[/tex] we get [tex]$$M''(t) = \frac{2(0.05)^2}{(1-0.05t)^3}$$[/tex].

Differentiating [tex]$M''(t)$[/tex] with respect to [tex]$t$[/tex] we get [tex]$$M'''(t) = \frac{6(0.05)^3}{(1-0.05t)^4}$$[/tex].

Substituting t = 0, we get [tex]$$M'(0) = \frac{1}{0.05} = 20$$[/tex]

[tex]$$M''(0) = \frac{2}{(0.05)^3} = 800$$[/tex]

[tex]$$M'''(0) = \frac{6}{(0.05)^4} = 4800$$[/tex]

Using the following formula to calculate the variance of X: [tex]$$Var(X) = \mathbb{E}(X^2) - [\mathbb{E}(X)]^2$$[/tex], where [tex]$$\mathbb{E}(X^2) = M''(0) = 800$$[/tex].

[tex]$$[\mathbb{E}(X)]^2 = [M'(0)]^2 = 400$$[/tex]

Hence, we get:$$Var(X) = 800 - 400 = \boxed{400}$$.

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Q1.1.1 The probability of randomly selecting a male respondent from the sample is 0.4. Q.1.1.2 The probability of randomly selecting a respondent who is female and prefers HP is 0.275. Q.1.1.3 The probability of selecting a male respondent, given that the preferred brand is Lenovo is 0.4545. Q.1.1.4 The probability of selecting a respondent who is male or who prefers HP is 0.575. Q.1.1.5 The probability of selecting a respondent who does not prefer Lenovo is 0.725.

Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?

The probability of randomly selecting a male respondent is given by the number of male respondents divided by the total number of respondents:

Probability = No. of Males / Total = 160 / 400 = 0.4

Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?

The probability of randomly selecting a respondent who is female and prefers HP is given by the number of females who prefer HP divided by the total number of respondents:

Probability = No. of Females who prefer HP / Total = 110 / 400 = 0.275

Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?

The probability of selecting a male respondent, given that the preferred brand is Lenovo, is given by the number of males who prefer Lenovo divided by the total number of respondents who prefer Lenovo:

Probability = No. of Males who prefer Lenovo / Total No. of respondents who prefer Lenovo = 50 / 110 = 0.4545

Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?

The probability of selecting a respondent who is male or who prefers HP is given by the sum of the probabilities of selecting a male respondent and selecting a respondent who prefers HP, minus the probability of selecting both (to avoid double counting):

Probability = (No. of Males / Total) + (No. of Females who prefer HP / Total) - (No. of Males who prefer HP / Total)

Probability = (160 / 400) + (110 / 400) - (40 / 400) = 0.4 + 0.275 - 0.1 = 0.575

Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?

The probability of selecting a respondent who does not prefer Lenovo is given by the number of respondents who do not prefer Lenovo divided by the total number of respondents:

Probability = (Total - No. of respondents who prefer Lenovo) / Total

Probability = (400 - 110) / 400 = 290 / 400 = 0.725

The complete question is:

The following contingency table gives the results of a sample survey of South African male and female respondents with regard to their preferred brand of notebook:

                              HP  Lenovo  Dell  Total

No. of Females        110    60   70    240

No. of Males             40    50   70    160

Total                        150    110  140    400

Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?

Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?

Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?

Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?

Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?

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i have now attached the picture but it can be wrong!

Answer:

angle C = 36°

Step-by-step explanation:

Fun fact that I found out:

all interior angles of a triangle added together = 180°

5x + 3x + 2x = 180°

combine like terms:

10x = 180°

divide both sides of the equation by 10:

x = 18°

angle C = 2(18°) = 36°

Answer:

175 degrees Fahrenheit

Step-by-step explanation:

We are to find the difference between the two temperatures

125 - (-50)

two minuses gives a plus

125 = 50 = 175

Answer:

F

Step-by-step explanation:

1(10) + 12= 22

2(10) + 12= 32

etc.....