Verify that y1 = x and y2 = x ln x are solutions to x 2y ′′ − xy′ + y = 0. b) Use the Wronskian to show that y1 and y2 are linearly independent. c) Find the particular solution to the differential equation with initial conditions y(1) = 7, y′ (1) = 2

Answer:  0.0013

Step-by-step explanation:

Given : The test scores are normally distributed with

Mean : [tex]\mu=\ 60,000[/tex]

Standard deviation :[tex]\sigma= 4,000[/tex]

Sample size : [tex]n=4[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

For x = 66,000

[tex]z=\dfrac{66000-60000}{\dfrac{4000}{\sqrt{4}}}=3[/tex]

The p-value = [tex]P(z>3)\=1-P(z<3)=1- 0.9986501\approx0.0013[/tex]

Hence, the likelihood the mean tire life of these four tires is more than 66,000 miles = 0.0013

Final answer:

If 3 letters are followed by 2 numbers and repetitions are allowed, there are a total of 1757600 possible different account numbers. Each letter position has 26 choices and each number position has 10 choices.

Explanation:

The student has asked to determine the number of different customer account "numbers" a company can have if the accounts consist of 3 letters followed by 2 numbers, with repetitions allowed for both letters and numbers.

To calculate the total number of possible account numbers, we can use the multiplication principle of counting. The number of options for each position of the account number is multiplied together to get the total number of combinations.

For the 3 letters, each position can contain any letter from A-Z, which gives us 26 choices per position. Since repeats are allowed, each of the 3 positions has 26 possible choices.

For the 2 numbers, each position can contain any digit from 0-9, which gives us 10 choices per position.

Therefore, to find the total number of possible account numbers, we calculate:

26 × 26 × 26 × 10 × 10 = 1757600 possible account numbers.

Step-by-step explanation:

There are 20 chips in total.

P(red) = 8/20 = 2/5

P(not green) = 16/20 = 4/5

P(yellow or green) = 9/20

The probability that a truck drives between 100 and 157 miles in a day within a normal distribution can be calculated using z-scores. The z-scores for 100 and 157 miles are computed relative to the mean and standard deviation, and the corresponding probabilities are obtained from the standard normal distribution table. The final probability is the difference of these two probabilities.

Given that the distribution of trucks' daily mileage is normally distributed, we can approach this problem by using the principles of normal distribution and z-scores. The z-score is a measure of how many standard deviations an element is from the mean.

First, we calculate the z-scores for both 100 miles and 157 miles:

Next, we look up these z-scores in the standard normal distribution table (or use a calculator with a normal distribution function), which will give us the probabilities P(Z To arrive at four decimal places precision, this process typically involves using a statistical calculator or software.

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Answer:

Amount theta she is putting in Checking account is 2272.80

Step-by-step explanation:

Given:

Amount on check = 2941

Amount that he want in cash = 100

Amount she put in saving account = 20% of remaining after getting cash

Remaining Amount she put in checking account.

To find: Amount in her Checking Account.

Amount left after taking cash = 2941 - 100 = 2841

Amount that she put in saving account = 20% of 2841 = [tex]\frac{20}{100}\times2841[/tex] =  568.20

Amount in her checking account = 2941 - 100 - 568.20  = 2272.8

Therefore, Amount theta she is putting in Checking account is 2272.80

[tex]_7C_3\cdot {_{13}C_2}=\dfrac{7!}{3!4!}\cdot\dfrac{13!}{2!11!}=\dfrac{5\cdot6\cdot7}{2\cdot3}\cdot\dfrac{12\cdot13}{2}=2730[/tex]

The number of ways the student can select 5 books such that at least 3 are English novels can be calculated as the sum of combinations of 3 English and 2 American, 4 English and 1 American, and all 5 being English.

The subject matter of this question is based in the mathematics field, specifically combinatorics. To tackle this problem, we will utilize the concept of combination, which is a way of selecting items from a larger set where order does not matter.

The student has to select 5 books out of 16 (9 American and 7 English novels). But at least 3 should be English novels. It means the student can pick 3, 4 or all 5 novels as English novels. Let's calculate each possibility:

So, the total number of ways = [C(7,3)*C(9,2)] + [C(7,4)*C(9,1)] + C(7,5). Here C(n,r) denotes combination and is equal to n! / [(n-r)!*r!], where '!' denotes factorial.

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Answer: 0.116

Step-by-step explanation:

The Poisson distribution probability formula is given by :-

[tex]P(X=x)=\dfrac{e^{-\lambda}\lambda^x}{x!}[/tex], where \lambda is the mean of the distribution and x is the number of success

Given : The number of inclusions in one cubic millimeter = 3.2

Then , the number of inclusions in two cubic millimeters=[tex]\lambda=2\times3.2=6.4[/tex]

Now, the probability of exactly four inclusions in 2.0 cubic millimetres is given by :-

[tex]P(X=4)=\dfrac{e^{-6.4}(6.4)^4}{4!}\\\\=0.11615127195\approx0.116[/tex]

Hence, the probability of exactly four inclusions in 2.0 cubic millimetres = 0.116

In matrix form, the system is

[tex]\dfrac{\mathrm d}{\mathrm dt}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}1&1\\4&1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}[/tex]

First find the eigenvalues of the coefficient matrix (call it [tex]\mathbf A[/tex]).

[tex]\det(\mathbf A-\lambda\mathbf I)=\begin{vmatrix}1-\lambda&1\\4&1-\lambda\end{vmatrix}=(1-\lambda)^2-4=0\implies\lambda^2-2\lambda-3=0[/tex]

[tex]\implies\lambda_1=-1,\lambda_=3[/tex]

Find the corresponding eigenvector for each eigenvalue:

[tex]\lambda_1=-1\implies(\mathbf A+\mathbf I)\vec\eta_1=\vec0\implies\begin{bmatrix}2&1\\4&2\end{bmatrix}\begin{bmatrix}\eta_{1,1}\\\eta_{1,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]

[tex]\lambda_2=3\implies(\mathbf A-3\mathbf I)\vec\eta_2=\vec0\implies\begin{bmatrix}-2&1\\4&-2\end{bmatrix}\begin{bmatrix}\eta_{2,1}\\\eta_{2,2}\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}[/tex]

[tex]\implies\vec\eta_1=\begin{bmatrix}1\\-2\end{bmatrix},\vec\eta_2=\begin{bmatrix}1\\2\end{bmatrix}[/tex]

Then the system has general solution

[tex]\begin{bmatrix}x\\y\end{bmatrix}=C_1\vec\eta_1e^{\lambda_1t}+C_2\vec\eta_2e^{\lambda_2t}[/tex]

or

[tex]\begin{cases}x(t)=C_1e^{-t}+C_2e^{3t}\\y(t)=-2C_1e^{-t}+2C_2e^{3t}\end{cases}[/tex]

Given that [tex]x(0)=1[/tex] and [tex]y(0)=2[/tex], we have

[tex]\begin{cases}1=C_1+C_2\\2=-2C_1+2C_2\end{cases}\implies C_1=0,C_2=2[/tex]

so that the system has particular solution

[tex]\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}e^{3t}\\2e^{3t}\end{bmatrix}[/tex]

Final answer:

The linear system of differential equations can be written in matrix form as [dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]. By solving the system with the given initial conditions x(0) = 1 and y(0) = 2, the values of x and y at different time points can be determined.

Explanation:

To write the linear system of differential equations in matrix form, we can express the given equations as:

[dx/dt, dy/dt] = [1, 1; 4, 1] * [x, y]

Using the initial conditions x(0) = 1 and y(0) = 2, we can solve the system of equations to find the values of x and y at different time points.

Answer:

x = 250 units

Step-by-step explanation:

We can easily solve this problem by using a graphing calculator or any plotting tool.

We must find the minimum point in the graph. This corresponds to the number of machines that produce the minimum cost.

The equation is

C (x) = 1.2x^2 -600x + 89,966

Please see attached image below

By producing x = 250 units, we obtain the minimum cost

[tex]\dfrac{x^4-2x^3+x^2+3x-2}{x^2-2x+1}[/tex]

The degree of the numerator exceeds the degree of the denominator, so first you have to divide:

[tex]x^2+\dfrac{3x-2}{x^2-2x+1}[/tex]

Now, [tex]x^2-2x+1=(x-1)^2[/tex], so the remainder term can be expanded to get

[tex]\boxed{x^2+\dfrac a{x-1}+\dfrac b{(x-1)^2}}[/tex]

Answer:

x^2+y^2 = 3^2

Step-by-step explanation:

We need to eliminate the parameter t

Given:

x = 3 cos t

y = 3 sin t

Squaring the above both equations

(x)^2=(3 cos t)^2

(y)^2 =(3 sin t)^2

x^2 = 3^2 cos^2t

y^2=3^2 sin^2t

Now adding both equations

x^2+y^2=3^2 cos^2t+3^2 sin^2t

Taking 3^2 common

x^2+y^2=3^2 (cos^2t+sin^2t)

We know that cos^2t+sin^2t = 1

so, putting the value

x^2+y^2=3^2(1)

x^2+y^2 = 3^2

Hence the parameter t is eliminated.

To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we can substitute cos(t) and sin(t) in terms of x and y to eliminate the parameter. The resulting equations represent the line y = x.

To eliminate the parameter in the given equations x = 3 cos t and y = 3 sin t, we need to express x and y in terms of each other without the parameter 't'. Using the identity cos^2(t) + sin^2(t) = 1, we can solve for cos(t) and sin(t), and substitute them into the equations to eliminate the parameter.

Using the fact that cos(t) = x/3 and sin(t) = y/3, we can rewrite the equations as x = 3 cos(t) = 3(x/3) = x and y = 3 sin(t) = 3(y/3) = y. Therefore, eliminating the parameter results in x = x and y = y, which simply means that the equations represent the line y = x.

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ANSWER

[tex]36k^{8} -13{k}^{6} -64k^{4} + 30 {k}^{2} [/tex]

EXPLANATION

Recall the distributive property:

[tex](a + b + c)(d + e) = a(d + e) + b(d + e) + c(d + e)[/tex]

We apply this property multiple times to simplify

[tex](9k^{6}+8k^{4}-6k^{2})(4k^{2}-5)[/tex]

This implies that:

[tex]9k^{6}(4k^{2}-5)+8k^{4}(4k^{2}-5)-6k^{2}(4k^{2}-5)[/tex]

We apply the distributive property again:

This time: a(b+c)=ac+ab

[tex] \implies \: 9k^{6} \times 4k^{2}-5 \times 9 {k}^{6} +8k^{4} \times 4k^{2}-5 \times 8 {k}^{4} -6k^{2} \times 4k^{2} + 5 \times 6 {k}^{2} [/tex]

[tex]\implies \: 36k^{8} -45{k}^{6} +32k^{6} -40 {k}^{4} -24k^{4} + 30 {k}^{2} [/tex]

[tex]\implies 36k^{8} -13{k}^{6} -64k^{4} + 30 {k}^{2} [/tex]

NB: [tex]k^{n}\times{k}^{m}=k^{m+n} [/tex]

Answer: 19.6 million

Step-by-step explanation:

The exponential growth function is given by :-

[tex]A=A_0(1+r)^x[/tex], where A is the initial amount , r is rate of interest and x is time period.

Given : The automobile sales in a country this year : A= 20.6 million

The rate of increase : r = 4.9 %=0.049

For last year , we take x = 1 , then the required exponential equation will be :-

[tex]20.6=A_0(1+0.049)^1\\\\\Rightarrow\ A_0=\dfrac{20.6}{1.049}=19.63775\approx19.6[/tex]

Hence, the  number of auto sales in the country last year = 19.6 million.

Final answer:

To find last year's auto sales, the formula original amount = final amount / (1 + rate of increase) is used. The sales last year, before a 4.9% increase to 20.6 million, were approximately 19.6 million when rounded to the nearest tenth.

Explanation:

To find the number of automobile sales last year before the increase, we can use the formula: original amount = final amount / (1 + rate of increase).

Given that the sales this year were 20.6 million and the rate of increase was 4.9%, the calculation for last year's sales would be as follows:

Original sales = 20.6 million / (1 + 0.049) = 20.6 million / 1.049

After performing the division, we get:

Original sales = 19.638 million

Rounding to the nearest tenth, the number of auto sales in the country last year was 19.6 million.

Answer:

Given relations defined on the set {a, b, c, d},

S= { (a, b), (a, c), (c, d), (c, a)}

R={ (b, c), (c, b), (a, d), (d, b)},

Since, SoR(x) = S(R(x)),

So, SoR(a) = S(R(a)) = S(d) = ∅,

SoR(b) = S(R(b)) = S(c) = d and a,

SoR(c) = S(R(c)) = S(b) = ∅,

SoR(d) = S(R(d)) = S(b) = ∅,

Thus, SoR = { (b,d), (b,a) }

RoS(a) = R(S(a)) = R(b) = c and RoS(a) = R(S(a)) = R(c) = b,

RoS(b) = R(S(b)) = R(∅) = ∅,

RoS(c) = R(S(c)) = R(d) = b and RoS(c) = R(S(c)) = R(a) = d

RoS(d) = R(S(d)) = R(∅) = ∅,

Thus, RoS = { (a, c), (a, b), (c,d), (c, b) },

SoS(a) = S(S(a)) = S(b) = ∅ and SoS(a) = S(S(a)) = S(c) = d and a

SoS(b) = S(S(b)) = S(∅) = ∅,

SoS(c) = S(S(c)) = S(d) = ∅ and SoS(c) = S(S(c)) = S(a) = b and c

SoS(d) = S(S(d)) = S(∅) = ∅,

SoS = { (a, d), (a, a), (c, b), (c, c) }

The composition of relations S and R mentioned in the question are SoR: { (a, c), (c, b)}, RoS: { (b, d), (a, b)} and SoS: { (a, d), (c, b)}.

The question is asking for the composition of relations. So, composition of relations S and R, denoted as 'SoR' or 'S ◦ R', is the set of ordered pairs where the first element is related to the second element through the combination of relations S and R. In this case the relations S and R on the set {a, b, c, d} are: S= { (a, b), (a, c), (c, d), (c, a)} and R={ (b, c), (c, b), (a, d), (d, b)}.

By the rule of composition SoR will be: { (a, c), (c, b)}.

Similarly, for RoS will be: { (b, d), (a, b)}.

And for SoS it will be: { (a, d), (c, b)}.

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Answer:

6 people

Step-by-step explanation:

Suppose A represents the event of going on Saturday,

B represents the event of going on Sunday,

According to the question,

n(A)=11

n(B)=14

n(A∪B)=20

We know that,

n(A∪B) = n(A) + n(B) - n(A∩B)

By substituting values,

20 = 11 + 14 - n(A∩B)

⇒ n(A∩B) = 25 - 20 = 5,

Hence, the number of people who can only go to the game on Saturday = n(A) - n(A∩B) = 11 - 5 = 6.

Answer:

its E none of the above

Step-by-step explanation:

We know that speed is defined as the ratio of distance to time.

i.e.

[tex]Speed=\dfrac{Distance}{Time}[/tex]

Let the distance traveled to work be: x m.

Now, while going to work it takes a person 20 minutes.

This means that the speed of the person while going to work is:

[tex]S_1=\dfrac{x}{20}[/tex]

Also, the time taken to come back home is: 30 minutes.

This means that the speed of person while riding to home is:

[tex]S_2=\dfrac{x}{30}[/tex]

Also, it is given that  the rate back is 8 mph slower than the trip to work.

This means that:

[tex]S_1-S_2=8[/tex]

i.e.

[tex]\dfrac{x}{20}-\dfrac{x}{30}=8\\\\i.e.\\\\\dfrac{30x-20x}{600}=8\\\\i.e.\\\\\dfrac{10x}{600}=8\\\\i.e.\\\\\dfrac{x}{60}=8\\\\i.e.\\\\x=480\ \text{m}[/tex]

Hence, the distance to work is:  480 m.

Also, the rate while going to work is:

[tex]=\dfrac{480}{20}\\\\=24\ \text{mph}[/tex]

and the trip back to home is covered with the speed:

[tex]=\dfrac{480}{30}\\\\=16\ \text{mph}[/tex]

Final answer:

There are approximately 132489 prime numbers between 2³¹and 2³², with a ratio of primes to all numbers being approximately 0.1156.

Explanation:

To find the number of primes between 2³¹and 2³², we can use the Sieve of Eratosthenes algorithm. With this algorithm, we can mark all the multiples of each prime number, and the remaining unmarked numbers will be prime.

Using this method, we can calculate that there are approximately 132489 primes between 2³¹ and 2³². The ratio of primes to all the numbers between 2³¹and 2³²is approximately 0.1156.

Answer:

6

Step-by-step explanation:

96x5

Answer:

32x(2)         (squared)

Step-by-step explanation:

GCF of 96 and 64:

  64 = (2)(2)(2)(2)(2)(2)

  96 = (2)(2)(2)(2)(2)(3)

  GCF = (2)(2)(2)(2)(2) = 32

GCF of x5 and x2:

x5 = (x)(x)(x)(x)(x)

x2 = (x)(x)

GCF = (x)(x) = x2

Step-by-step explanation:

Consider the first sequence:

15, 20, 25, 30, 35,...

Note that each term is increased by 5 from its previous term.

Therefore,

[tex]a_n=a_{n-1}+5[/tex]

If the pattern continue, the next three term of the sequence will be:

[tex]a_6=a_{6-1}+5[/tex]

[tex]a_6=a_{5}+5[/tex]

[tex]a_6=35+5[/tex]

[tex]a_6=40[/tex]

Similarly,

[tex]a_7=a_{7-1}+5[/tex]

[tex]a_7=a_{6}+5[/tex]

[tex]a_7=40+5[/tex]

[tex]a_7=45[/tex]

Similarly,

[tex]a_8=a_{8-1}+5[/tex]

[tex]a_8=a_{7}+5[/tex]

[tex]a_8=45+5[/tex]

[tex]a_8=50[/tex]

Thus, the next three term of the sequence 15, 20, 25, 30, 35,... is 40, 45, and 50.

Now, consider the second sequence:

5, 9, 13, 17, 21,...

Note that each term is increased by 4 from its previous term.

Therefore,

[tex]a_n=a_{n-1}+4[/tex]

If the pattern continue, the next three term of the sequence will be:

[tex]a_6=a_{6-1}+4[/tex]

[tex]a_6=a_{5}+4[/tex]

[tex]a_6=21+4[/tex]

[tex]a_6=25[/tex]

Similarly,

[tex]a_7=a_{7-1}+4[/tex]

[tex]a_7=a_{6}+4[/tex]

[tex]a_7=25+4[/tex]

[tex]a_7=29[/tex]

Similarly,

[tex]a_8=a_{8-1}+4[/tex]

[tex]a_8=a_{7}+4[/tex]

[tex]a_8=29+4[/tex]

[tex]a_8=33[/tex]

Thus, the next three term of the sequence 5, 9, 13, 17, 21,... is 25, 29, and 33.

[tex]2^{2013}=2^{4\cdot503+1}\\\\2^4=16\equiv 1\pmod{15}\\2^{4\cdot 503}\equiv 1\pmod{15}\\2^{4\cdot 503+1}\equiv 2\pmod{15}\\\\2^{2013}\equiv 2\pmod{15}[/tex]

Answer with Step-by-step explanation:

Consider,

[tex](AB)^{-1}(AB)=I[/tex] (Identity rule)

Multiplying by B⁻¹ on the both the sides, we get that

[tex](AB)^{-1}(AB)B^{-1}=IB^{-1}\\\\(AB)^{-1}A(BB^{-1})=B^{-1}[/tex]

And we know that BB⁻¹ = I

So, it becomes,

[tex](AB)^{-1}A=B^{-1}[/tex]

Now, multiplying by A⁻¹ on both the sides, we get that

[tex](AB)^{-1}AA^{-1}=B^{-1}A^{-1}\\\\(AB)^{-1}=B^{-1}A^{-1}[/tex] (AA⁻¹=I)

Hence, proved.

Answer:

  The junction law is based on the conservation of charge.

Step-by-step explanation:

Kirchhoff's current law, or junction law, (1st Law) states that current flowing into a node (or a junction) must be equal to current flowing out of it. This is a consequence of charge conservation—charge is not created or destroyed in a closed system.

Answer:

The percent of Increase is of 11.08% (0.1108)

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

Since we have two different values for two different years, we can use the following Algebraic Expression to calculate the percent difference of sales between both years. The Expression would be the following,

[tex]370 * (x+1) = 411[/tex]

Where x is the percent difference. Now we solve for x,

[tex]370 * (x+1) = 411[/tex]

[tex]x+1 = 1.1108[/tex]

[tex]x = 0.1108[/tex]

so now we see that the percent of Increase is of 11.08% (0.1108)

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The number of units sold increased by about 11.08%.

To find the percent increase or decrease in the number of units sold, we need to calculate the difference between the number of units sold in 2012 and 2011, and then divide that difference by the number of units sold in 2011.

The amount of increase or decrease is calculated as: (Number of units sold in 2012 - Number of units sold in 2011)/Number of units sold in 2011 x 100

In this case, the calculation is: (411 - 370)/370 x 100 = (41/370) x 100 = 11.08%

Therefore, the number of units sold increased by about 11.08%.

To calculate the value of 715 × 211, you can use the standard multiplication method by multiplying each digit of the two numbers and summing up the results.

To find the value of 715 × 211, you can use the standard multiplication method. Start by multiplying the ones digit of 715 (5) by each digit of 211 (1, 1, and 2), and write down the results. Then, multiply the tens digit of 715 (1) by each digit of 211, and write down the results one place to the left of the previous results. Finally, multiply the hundreds digit of 715 (7) by each digit of 211 and write down the results two places to the left. Sum up the columns and you will get the final product.

Here's how it looks:

   715
× 211
--------
 715
 1430  
+1425
--------
 150665

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Answer:  216

Step-by-step explanation:

Given : We like to make a salad that consists of​ lettuce, tomato,​ cucumber, and onions.

The number of varieties of lettuce = 9

The number of varieties of tomatoes = 4

The number of varieties of cucumbers = 2

The number of varieties of onions = 3

Now, the number of different salads we can make is given by :-

[tex]9\times4\times2\=216[/tex]

Hence, we can make 216 different types of salads.

To find power series solutions of the differential equation y'' − y' = 0, we can assume a power series solution and find the recurrence relation. Two power series solutions are found by choosing different initial conditions. The power series solutions are equivalent to the exponential solutions obtained using another method.

To find power series solutions of the differential equation y'' − y' = 0, we can assume a power series solution of the form y(x) = ∑(n=0)∞ a_nx^n. Substituting this into the differential equation and simplifying, we find that the power series satisfies the recurrence relation a_{n+2} = a_{n+1} in terms of a_0 and a_1.

By letting a_0 = 0 and a_1 = 1, we obtain the power series solution y_1(x) = x. Alternatively, by letting a_0 = 1 and a_1 = 0, we obtain the power series solution y_2(x) = 1.

Comparing these power series solutions with the solutions obtained using the method of Section 4.3, we see that the power series solutions are polynomials. In this case, the power series solutions are equivalent to the solutions obtained using the method of Section 4.3, which are exponential functions.

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Answer:

The flow rate is 267ml/hour

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Equations.

To solve this we first need to find out how many ml the Rottweiler needs over 12 hours. We do this by using the Rule of Three property.

[tex]\frac{40ml}{1kg} = \frac{x}{80kg}[/tex]

[tex]\frac{40ml*80kg}{1kg} =x[/tex]

[tex]3200ml = x[/tex]

So the Rottweiler needs 3200 ml over a 12 hour period. We now need to find the flow rate per hour. We can solve this by simply dividing 3200 ml by 12 hours.

[tex]3200ml / 12hours = 266.67ml/hour[/tex]

So the flow rate is 267 ml/hour (rounded to the nearest whole number)

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

Answer: The z score that corresponds to the value x=22 is 0.4 .

Step-by-step explanation:

Given : A normal distribution has a mean 20 and standard deviation 5.

i.e. [tex]\mu=20[/tex]

[tex]\sigma=5[/tex]

Let x be the random selected variable.

We know that to find the z-score corresponds to the value x is given by :-

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

For x = 22, we have

[tex]z=\dfrac{22-20}{5}=\dfrac{2}{5}\\\\\Rightarrow\ z=0.4[/tex]

Hence, the z score that corresponds to the value x=22 is 0.4

A z-score in a normal distribution measures the number of standard deviations a value is from the mean. To calculate it, use the formula z = (x - μ) / σ for the specific values provided, such as half a standard deviation below the mean, 5 points above the mean, three standard deviations above the mean, and 22 points below the mean.

The calculation of a z-score within a normal distribution is a common task in statistics, allowing one to determine how many standard deviations a particular value, x, is from the mean, μ, of the distribution. The z-score is calculated using the formula:

z = (x - μ) / σ

where x is the value in question, μ is the mean, and σ is the standard deviation. Now, we will calculate the z-scores for the given situations:

Remember, when you use these calculations for specific numerical values, you need to insert the actual values of mean and standard deviation into the formula.