A poll asked the question, "What do you think is the most important problem facing this country today?" Twenty percent of the respondents answered "crime and violence." The margin of sampling error was plus or minus 4 percentage points. Following the convention that the margin of error is based on a 95% confidence interval, find a 95% confidence interval for the percentage of the population that would respond "crime and violence" to the question asked by the pollsters.

Answer:

68%

Step-by-step explanation:

It is given that the diameters of bearing have a normal distribution.

Mean = u = 1.2 cm

Standard deviation = [tex]\sigma[/tex] = 0.03 cm

We have to find the proportion of values which falls in between 1.17 to 1.23

In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:

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

For 1.17:

[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]

For 1.23:

[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]

So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question.

According to the empirical rule:

68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1.

Therefore, the answer to this question is 68%

Answer: (a) 19448 ways

(b) 5292 ways

(c) 0.2721

Step-by-step explanation:

(a) 10 county council members be randomly elected out of the 17 candidates in the following ways:

= [tex]^{n}C_{r}[/tex]

= [tex]^{17}C_{10}[/tex]

= [tex]\frac{17!}{10!7!}[/tex]

= 19448 ways

(b) 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men in the following ways:

we know that there are 7 women and 10 men in total, so

= [tex]^{7}C_{5}[/tex] × [tex]^{10}C_{5}[/tex]

= [tex]\frac{7!}{5!2!}[/tex] × [tex]\frac{10!}{5!5!}[/tex]

= 21 × 252

= 5292 ways

(c) Now, the probability that 5 are women and 5 are men are selected:

= [tex]\frac{ ^{7}C_{5} * ^{10}C_{5}}{^{17}C_{10}}[/tex]

= [tex]\frac{5292}{19448}[/tex]

= 0.2721

Answer:

For any value of C1 and C2, [tex]y = C1 + C2*e^{2x}[/tex] is a solution.

Step-by-step explanation:

Let's verify the solution, but first, let's find the first and second derivatives of the given solution:

[tex]y = C1 + C2*e^{2x}[/tex]

For the first derivative we have:

[tex]y' = 0 + C2*(2x)'*e^{2x}[/tex]

[tex]y' = C2*(2)*e^{2x}[/tex]

For the second derivative we have:

[tex]y'' = C2*(2)*(2x)'*e^{2x}[/tex]

[tex]y'' = C2*(2)*(2)*e^{2x}[/tex]

[tex]y'' = C2*(4)*e^{2x}[/tex]

Let's solve the ODE by the above equations:

[tex]y'' - 2y' = 0[/tex]

[tex]C2*(4)*e^{2x} - 2*C2*(2)*e^{2x} = 0[/tex]

[tex]C2*(4)*e^{2x} - C2*(4)*e^{2x} = 0[/tex]

From the above equation we can observe that for any value of C2 the equation is solved, and because the ODE only involves first (y') and second (y'') derivatives, C1 can be any value as well, because it does not change the final result.  

Answer:  No, the given transformation T is NOT a linear transformation.

Step-by-step explanation:  We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :

[tex]T(x,y)=(x,y^2).[/tex]

We know that

a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R

[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]

So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have

[tex]T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)[/tex]

and

[tex]aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).[/tex]

Therefore, we get

[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]

Thus, the given transformation T is NOT a linear transformation.

Answer:

540

Step-by-step explanation:

we have given E=0.3

σ = 27 hours

100(1-α)%=99%

from here α=0.01

using standard table [tex]Z_\frac{\alpha }{2}=Z_\frac{0.01}{2}=2.58[/tex]

[tex]n=\left ( Z_\frac{\alpha }{2}\times \frac{\sigma }{E} \right )^{2}[/tex] =

[tex]\left ( 2.58\times \frac{27}{3} \right )^{2}[/tex]

n = [tex]23.22^{2}[/tex]

n=539.16

n can not be in fraction so n=540

To obtain a 99% confidence interval with a margin of error of 3 hours, at least 602 components must be sampled.

In order to determine the number of components that must be sampled so that a 99% confidence interval will have a margin of error of 3 hours, we can use the formula:

n = (z * s / E)^2

Where:

n = sample size

z = z-value corresponding to the desired confidence level (in this case, 99% confidence level)

s = population standard deviation

E = margin of error

Plugging in the given values, we have:

n = (2.576 * 27 / 3)^2

n = 601.3696

Rounding up to the nearest whole number, we need to sample at least 602 components.

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Answer:

B'(3, -8)

Step-by-step explanation:

The image is the mirror image of the trapezoid below the x-axis.

Each x-coordinate remains the same. Each y-coordinate becomes the opposite.

B'(3, -8)

Answer:

(3 , -8)

Step-by-step explanation:

The current coordinates of B. are (3,8). The x-axis is the horizontal line that runs across. This means that if the trapezoid were to be reflected, it would end up upside down. When this happens, only the y value changes its sign.

In short, your Y value would become negative, making 8 change to -8

Answer: Option 'B' is correct.

Step-by-step explanation:

Let the number of 10 cents pcs be 'x'.

Let the number of 25 cents pcs be 'y'.

Since we have given that

Total number of coins = 32

Sum of money = $5.15

As we know that

$1 = 100 cents

$5.15 = 5.15×100 = 515 cents

According to question, we get that

[tex]x+y=32-----------(1)\\\\10x+25y=\$515------------(2)[/tex]

Using the graphing method, we get that

x = 19

y = 13

So, there are 13 pcs of 25 cents.

Hence, Option 'B' is correct.

By creating a system of equations based on the total number of coins (32) and their total value (P5.15), we calculate that there are 13 pieces of 25-cent coins.

The student's question involves figuring out the number of 25-cent coins among a total of 32 coins which altogether amount to P5.15. This problem can be solved by setting up a system of equations to account for the total number of coins and the total value in pesos.

Let's denote the number of 10-cent coins as t and the number of 25-cent coins as q. We know from the problem that there are 32 coins in total, so:

(1) t + q = 32

We also know that the total value of the coins is P5.15, or 515 cents. Therefore:

(2) 10t + 25q = 515

By solving this system of equations, we can find the value of q, the number of 25-cent coins. First, we can multiply equation (1) by 10 to eliminate t when we subtract the equations:

10t + 10q = 320

Subtracting this from equation (2) gives us:

15q = 195

Dividing both sides by 15, we find that:

q = 13

So, there are 13 pieces of 25-cent coins, which corresponds to option B.

Answer:

$15.30

30% of 60 is 18

Step-by-step explanation:

To find the maximum amount he can spend on a meal, you have to find how  much he is going to tip.

So to find the tip you multiply 15% by 18 and you get 2.7

Then you subtract 18 by 2.7 to find out how much he can spend on the meal.

18 - 2.7 = 15.30

So he can spend $15.30 on his meal and tip $2.70

To find what percent of 60 is 18, you have to use this equation:

is over of equals percent over 100

So is/of = x/100 We have the x as the percent because that's what you're trying to figure out.

You would put 18 as is because it has the word is before it and put 60 as of because it has of before it.

So 18/60 = x/100

Now you would do Cross Product Property

18*100 = 1800

60*x = 60x

60x = 1800

Now divide 60 by itself and by 1800

1800/60 = 30

x = 30%

Answer: 0.064

Step-by-step explanation:

Binomial probability formula :-

[tex]P(X)=^nC_x \ p^x\ (1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.

Given : The proportion of the homes constructed in the Quail Creek area include a security system :  [tex]p=0.40[/tex]

Now, if three homes are selected at random, then the probability all three of the selected homes have a security system is given by :-

[tex]P(3)=^3C_3 \ (0.40)^3\ (1-0.40)^{3-3}\\\\=(0.40)^3=0.064[/tex]

Hence, the probability all three of the selected homes have a security system = 0.064

The probability that all three homes selected at random in the Quail Creek area have a security system is 6.4%.

The probability that all three of the selected homes in the Quail Creek area have a security system, given that 40% of the homes have a security system, can be calculated by using the rule for independent events in probability.

Since each house is selected at random, we can multiply the probability of each house having a security system together:

P(all three homes have a security system) = P(home 1 has security system) × P(home 2 has security system) × P(home 3 has security system)

As every home has a 40% (or 0.40) chance of having a security system:

P(all three) = 0.40 × 0.40 × 0.40 = 0.064

Therefore, there is a 6.4% chance that all three homes selected will have a security system.

Answer:

Standard deviation.

Step-by-step explanation:

The standard deviation measures how accurate the point estimate is likely to be in estimating a parameter.

The confidence interval measures how accurate the point estimate is likely to be in estimating a parameter.

A confidence interval communicates how accurate our estimate is likely to  be.

The confidence interval is a range of of all plausible values of the random variable under test at a given confidence level which is expressed in percentage such as 98%, 95% and 90% of confidence level.  

The standard deviation is the parameter to signify the  dispersion of data around  the mean value of the data.

Researchers prefer it because    on the basis of the percentage of certainty in the test result of  null hypothesis are accepted or rejected as it includes some chance for errors too. (example  95% sure means 5% not sure) also this gives a range of values and hence good chance to normalize errors.    

An interval estimate is typically preferred over a point estimate because

i) it gives us a sense of accuracy of the point estimate

ii) we know the probability that it contains the parameter (e.g., 95%)

iii) it provides us with more possible parameter values

I only

II only  

both I and II  

all of these

III only

All three statements above are true hence all of these is  the answer.

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Answer:

Mustapha can arrange 15 bouquets per day.

Step-by-step explanation:

Neneh can arrange 20 bouquets per day.

Together Neneh and Mustapha can arrange 35 bouquets per day.

So,  Mustapha can arrange [tex]35-20=15[/tex] bouquets per day.

Therefore, Mustapha’s marginal product is 15 bouquets.

Answer:

Jayanta needs 2 labor union groups and 8 church groups.

Step-by-step explanation:

Let c denotes churches .

Let l denotes labor unions.

We know that Jayanta can only spend 20 hours letter writing and 14 hour of follow-up.

So, equations becomes:

[tex]2c+2l=20[/tex]

[tex]c+3l=14[/tex]

And total money raised can be shown by = [tex]150c+175l[/tex]

We have to maximize [tex]150c+175l[/tex] keeping in mind that [tex]2c+2l \leq 20[/tex] and [tex]c+3l \leq 14[/tex]

We will solve the two equations:  [tex]2c+2l=20[/tex] and [tex]c+3l=14[/tex]

We get l = 2 and c = 8

And total money raised is [tex]150\times8 + 175\times2[/tex] = [tex]1200+350=1550[/tex] dollars.

Hence, Jayanta needs 2 labor union groups and 8 church groups.

  f : X → Y  and  g: Y → Z

Now we have to show:

If gof is one-to-one then f must be one-to-one.

Given:

gof is one-to-one

To prove:

f is one-to-one.

Proof:

Let us assume that f(x) is not one-to-one .

This means that there exist x and y such that x≠y but f(x)=f(y)

On applying both side of the function by the function g we get:

g(f(x))=g(f(y))

i.e. gof(x)=gof(y)

This shows that gof is not one-to-one which is a contradiction to the given statement.

Hence, f(x) must be one-to-one.

Let A={1,2,3,4}  B={1,2,3,4,5} C={1,2,3,4,5,6}

Let f: A → B

be defined by f(x)=x

and g: B → C be defined by:

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

is not a one-to-one function.

since 4≠5 but g(4)=g(5)

Also, gof : A → C

is a one-to-one function.

Answer:[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]

Step-by-step explanation:

Given two upward facing parabolas  with equations

[tex]y=6x^2 & y=x^2+2[/tex]

The two intersect at

[tex]6x^2=x^2+2[/tex]

[tex]5x^2=2[/tex]

[tex]x^2[/tex]=[tex]\frac{2}{5}[/tex]

x=[tex]\pm \sqrt{\frac{2}{5}}[/tex]

area  enclosed by them is given by

A=[tex]\int_{-\sqrt{\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left [ \left ( x^2+2\right )-\left ( 6x^2\right ) \right ]dx[/tex]

A=[tex]\int_{\sqrt{-\frac{2}{5}}}^{\sqrt{\frac{2}{5}}}\left ( 2-5x^2\right )dx[/tex]

A=[tex]4\left [ \sqrt{\frac{2}{5}} \right ]-\frac{5}{3}\left [ \left ( \frac{2}{5}\right )^\frac{3}{2}-\left ( -\frac{2}{5}\right )^\frac{3}{2} \right ][/tex]

A=[tex]\frac{8}{3}\times \sqrt{\frac{2}{5}}[/tex]

The area of the enclosed region is -√2/15 square units.

To find the area of the enclosed region, we need to find the points of intersection between the two equations y=6x^2 and y=x^2+2. Setting them equal to each other:

6x^2 = x^2 + 2

5x^2 = 2

x^2 = 2/5

x = ±√(2/5)

Substituting these values of x back into one of the equations, we can find the corresponding y values:

For x = √(2/5), y = 6(√(2/5))^2 = 6(2/5) = 12/5

For x = -√(2/5), y = 6(-√(2/5))^2 = 6(2/5) = 12/5

Now we can find the area of the enclosed region by calculating the definite integral of y=6x^2 - (x^2+2) from x = -√(2/5) to x = √(2/5). This can be done using the fundamental theorem of calculus:

∫[√(2/5), -√(2/5)] [6x^2 - (x^2+2)] dx = ∫[√(2/5), -√(2/5)] (5x^2 - 2) dx = [5/3x^3 - 2x] [√(2/5), -√(2/5)] = 2(√(2/5))^3 - 5/3(√(2/5))^3 = 4√2/15 - 5√2/15 = -√2/15

Therefore, the area of the enclosed region is -√2/15 square units.

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Answer:

a) reject null hypothesis since p < 0.01

Step-by-step explanation:

Given that the  average assembly time for a Ford Taurus is

[tex]μ = 38 hrs[/tex]

Sample size [tex]n=36[/tex]

[tex]x bar = 37.5\\s=1.2\\SE = 1.2/6 = 0.2[/tex]

Test statistic t = mean diff/se = 0.5/0.2 = 2.5

(Here population std dev not known hence t test is used)

df = 35

p value = 0.008703

a) reject null hypothesis since p < 0.01

Answer:

[tex]a) \quad A=\left[\begin{array}{cc}8&5\\5&-1\end{array}\right] \\\\\\b) \quad \{-1+5x, 2+x\}[/tex]

Step-by-step explanation:

To compute the representation matrix A of f with respect the basis {1,x} we first compute

[tex]f(1)=f(1+0x)=(8\cdot 1 + 2 \cdot 0) + (5 \cdot 1 - 0)x=8+5x \\\\f(x)=f(0 + 1\cdot x)=(8 \cdot 0 + 2\cdot 1)+(5 \cdor 0 - 1)x = 2-1[/tex]

The coefficients of the polynomial f(1) gives us the entries of the first column of the matrix A, where the first entry is the coefficient that accompanies the basis element 1 and the second entry is the coefficient that accompanies the basis element x. In a similar way, the coefficients of the polynomial f(x) gives us the the entries of the second column of A. It holds that,

[tex]A=\left[\begin{array}{ccc}8&5\\2&-1\end{array}\right][/tex]

(b) First, note that we are using a one to one correspondence between the basis {1,x} and the basis {(1,0),(0,1)} of [tex]\mathbb{R}^2[/tex].

To compute a basis P1 with respect to which f has a diagonal matrix, we first have to compute the eigenvalues of A. The eigenvalues are the roots of the characteristic polynomial of A, we compute

[tex]0=\det\left[\begin{array}{ccc}8-\lambda & 2\\ 5 & -1-\lambda \end{array}\right]=(8 - \lambda)(-1-\lambda)-18=(\lambda - 9)(\lambda +2)[/tex]

and so the eigenvalues of the matrix A are [tex]\lambda_1=-2 \quad \text{and} \quad \lambda_2=9[/tex].

After we computed the eigenvalues we use the systems of equations

[tex]\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right] = \left[\begin{array}{c}-2x_1\\-2x_2\end{array}\right] \\\\\text{and} \\\\\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1 \\ x_2\end{array}\right]=\left[\begin{array}{cc}9x_1\\2x_1\end{array}\right][/tex]  

to find the basis of the eigenvalues. We find that [tex]v_1=(-1,5 )[/tex] is an eigenvector for the eigenvalue -2 and that [tex]v_2=(2,1)[/tex] is an eigenvector for the eigenvalue 9. Finally, we use the one to one correspondence between the [tex]\mathbb{R}^2[/tex] and the space of liear polynomials to get the basis [tex]P1=\{-1+5x, 2+x\}[/tex] with respect to which f is represented by the diagonal matrix [tex]\left[\begin{array}{ccc}-2&0\\0&9\end{array}\right][/tex]

Interpreting the situation, we can conclude that the statement means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

Thus, N(8) is the number of students who got a grade of 8, while N(5) is the number of students who got a grade of 5.

[tex]N(8) > 2N(5)[/tex]

It means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

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The statement [tex]\( N(8) > 2 \cdot N(5) \)[/tex] means that the number of students who scored a grade of 8 on the exam is greater than twice the number of students who scored a grade of 5 on the exam.

To understand this, let's break down the notation:

-  N(g)  represents the number of students who scored (g) on the exam.

-  N(8)  is the number of students who scored an 8.

-  N(5)  is the number of students who scored a 5.

The inequality [tex]\( N(8) > 2 \cdot N(5) \)[/tex] compares these two quantities. It states that the count of students with a grade of 8 exceeds two times the count of students with a grade of 5. This indicates that a higher number of students performed better (scoring an 8) than those who scored a 5, with the difference being more than the number of students who scored a 5. In other words, if we were to take the number of students who scored a 5 and double it, there would still be more students who scored an 8. This could be an indicator of the overall performance of the class, suggesting that more students achieved a higher grade than those who scored in the middle range of the grading scale.

Answer:

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

Step-by-step explanation:

It is given that a data set lists earthquake depths. The summary statistics are

[tex]n=300[/tex]

[tex]\overline{x}=5.89km[/tex]

[tex]s=4.44km[/tex]

Level of significance = 0.01

We need to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 5.00.

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The formula for z-value is

[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]z=\frac{5.89-5.00}{\frac{4.44}{\sqrt{300}}}[/tex]

[tex]z=\frac{0.89}{0.25634351952}[/tex]

[tex]z=3.4719[/tex]

The p-value for z=3.4719 is 0.000517.

Since the p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

The null and alternative hypotheses are H0: µ = 5.00 km and HA: µ ≠ 5.00 km. A t-test is used to calculate the test statistic, and the p-value is compared to the significance level of 0.01 to either reject or not reject the null hypothesis. The final conclusion is made in consideration of the original claim.

In statistics, hypothesis testing is a tool for inferring whether a particular claim about a population is true. For this question about earthquake depths, we would start by setting our null hypothesis (H0) and our alternative hypothesis (HA).

The null hypothesis would be H0: µ = 5.00 km, and the alternative hypothesis would be HA: µ ≠ 5.00 km.

The test statistic can be calculated using a t test, since we are dealing with a sample mean and we know the sample standard deviation (sequals4.44 km).

The p-value associated with this test statistic would then be calculated, and compared to the significance level of 0.01. If the p-value is less than 0.01, we reject the null hypothesis. If, however, the p-value is greater than 0.01, we cannot reject the null hypothesis.

The final conclusion must be stated in terms of the original claim. If we reject the null hypothesis, we conclude that the evidence supports the claim that the mean earthquake depth is not equal to 5.00 km (supporting the alternative hypothesis). If we do not reject the null hypothesis, we conclude that the evidence does not support the claim that the mean is not 5.00 km. The data does not provide sufficient evidence to support a conclusion that the mean earthquake depth is different than 5.00 km.

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Answer:

  13

Step-by-step explanation:

The product of two negative numbers is positive. The absolute value of a number is its magnitude written with a positive sign.

  6 -2(-1) +|-5|

  = 6 + 2 + 5

  = 13

If [tex]W[/tex] is a random variable representing your winnings from playing the game, then it has support

[tex]W=\begin{cases}43&\text{if you draw something with value at most 3}\\-11&\text{otherwise}\end{cases}[/tex]

There are 52 cards in the deck. Only the 1s, 2s, and 3s fulfill the first condition, so there are 12 ways in which you can win $43. So [tex]W[/tex] has PMF

[tex]P(W=w)=\begin{cases}\frac{12}{52}=\frac3{13}&\text{for }w=43\\1-\frac{12}{52}=\frac{10}{13}&\text{for }w=-11\\0&\text{otherwise}\end{cases}[/tex]

You can expect to win

[tex]E[W]=\displaystyle\sum_ww\,P(W=w)=\frac{43\cdot3}{13}-\frac{11\cdot10}{13}=\boxed{\frac{19}{13}}[/tex]

or about $1.46 per game.

The expected value of the proposition is $7.31.

To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and then sum them up.

The probability of drawing a card with a value of three or less is 12/52 since there are 12 cards with values of three or less in a standard deck of 52 cards. The probability of not drawing a card with a value of three or less is 40/52.

Using these probabilities and the given payoffs, we can calculate the expected value as follows:
Expected Value = (Probability of Winning * Payoff if Win) + (Probability of Losing * Payoff if Lose)
Expected Value = (12/52 * 43) + (40/52 * -11)

Calculating this expression gives us an expected value of $7.31 (rounded to two decimal places).

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Answer: (2.2, 5.8)

Step-by-step explanation:

The confidence interval for standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),\alpha/2}}} , \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),1-\alpha/2}}}\right )[/tex]

Given :  Sample size : 16

Mean height : [tex]\mu=67.5[/tex] inches

Standard deviation : [tex]s=3.2[/tex] inches

Significance level : [tex]1-0.99=0.01[/tex]

Using Chi-square distribution table ,

[tex]\chi^2_{(15,0.005)}=32.80[/tex]

[tex]\chi^2_{(15,0.995)}=4.60[/tex]

Then , the 99% confidence interval for the population standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(15)(3.2)^2}{32.80}} , \sqrt{\dfrac{(15)(3.2)^2}{4.6}}\right )\\\\=\left ( 2.1640071232,5.77852094812\right )\approx\left ( 2.2,5.8 \right )[/tex]

Answer:

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

Step-by-step explanation:

We can write the equation of a circle in the form

(x-h)^2 + (y-k)^2 = r^2

Where (h,k) is the center and r is the radius

(x--7)^2 + (y--4)^2 = 6^2

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

Answer:

Parent function f(x) is inverted, stretched vertically by 1 : 3, shifted 5 units right and 4 units upwards to form new function g(x).

Step-by-step explanation:

The parent function graphed is f(x) = x²

This graph when inverted (parabola opening down)function becomes

g(x) = -x²

Further stretched vertically by a scale factor of 1:3 then new function becomes as g(x) = -3x²

Then we shift this function by 5 units to the right function will be

g(x) = -3(x - 5)²

At last we shift it 4 units vertically up then function becomes as

g(x) = -3(x - 5)² + 4

Answer:

  2

Step-by-step explanation:

The z-score is the number of standard deviations the value is above the mean. If that number is 2, then Z=2. If that number is -2, then Z=-2.

___

Both Z=2 and Z=-2 are values that are two standard deviations from the mean.

Answer:

see below

Step-by-step explanation:

The horizontal line will have the same y and the y value will be constant

y = -7

The vertical line will have the same x and the x value will be constant

x = -1

Answer:

Solution is [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]

Step-by-step explanation:

the given equation y''-4y[tex]=e^{2x}[/tex] can be written as

[tex]D^{2}y-4y=e^{2x}\\\\(D^{2}-4)y=e^{2x}\\\\[/tex]

The Complementary function thus becomes

y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}

where [tex]m_{1} , m_{2}[/tex] are the roots of the [tex]D^{2}-4[/tex]

The roots of [tex]D^{2}-4[/tex] are +2,-2 Thus the comlementary function becomes

[tex]y=c_{1}e^{2x}+c_{2}e^{-2x}[/tex]

here [tex]c_{1},c_{2}[/tex] are arbitary constants

Now the Particular Integral becomes using standard formula

[tex]y=\frac{e^{ax}}{f(D)}\\\\y=\frac{e^{ax}}{f(a)} (f(a)\neq 0)\\\\y=x\frac{e^{ax}}{f'(a)}(f(a)=0)[/tex]

[tex]y=\frac{e^{2x}}{D^{2}-4}\\\\y=\frac{e^{2x}}{(D+2)(D-2)}\\\\y=\frac{1}{D-2}\times \frac{e^{2x}}{2+2}\\\\y=\frac{1}{4}\times \frac{e^{2x}}{D-2}\\\\y=\frac{1}{4}xe^{2x}[/tex]

Hence the solution is = Complementary function + Particular Integral

Thus Solution becomes [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]

The final general solution is [tex]y(x) = C1e^2x + C2e^-2x + 1/2xe^2x[/tex].

To find the general solution of the given differential equation: y'' - 4y = e2x, we will follow these steps:

1. Solve the Homogeneous Equation

First, solve the homogeneous part: y'' - 4y = 0

The characteristic equation is: r2 - 4 = 0

Solving for r, we get: r = ±2

Thus, the general solution to the homogeneous equation is: yh(x) = C1e2x + C2e-2x

2. Find a Particular Solution

Next, find a particular solution, yp(x), to the non homogeneous equation through the method of undetermined coefficients. Assume a particular solution of the form: yp(x) = Axe2x

Differentiating, we get: yp' = Ae2x + 2Axe2x and yp'' = 4Axe2x + 2Ae2x

Substitute these into the original equation:

4Axe2x + 2Ae2x - 4(Axe2x) = e2x

which simplifies to: 2Ae2x = e2x

Thus, A = 1/2

So, the particular solution is: yp(x) = (1/2)xe2x

3. Form the General Solution

The general solution to the nonhomogeneous differential equation is the sum of the homogeneous solution and the particular solution:

y(x) = yh(x) + yp(x)

Therefore, the general solution is: [tex]y(x) = C1e2x + C2e-2x + (1/2)xe2x[/tex].

Answer: This is a combination.

There are 1947792 ways to choose his students.

Step-by-step explanation:

Since we have given that

Number of students in a class = 36

Number of students selected for his English class = 6

We would use "Combination" .

As permutation is used when there is an arrangement.

whereas Combination is used when we have select r from group of n.

So, Number of ways that Prof. Jones can choose his students is given by

[tex]^{36}C_6=1947792[/tex]

Hence, there are 1947792 ways to choose his students.

Answer:

[tex]2x+y=3[/tex]

Step-by-step explanation:

Here we aer given a point (2,-1) and a line [tex]2y=x-4[/tex]. We are supposed to find the equation of the line passing through this point and perpendicular to this line.

Let us find the slope of the line perpendicular to [tex]2y=x-4[/tex]

Dividing above equation by 2 we get

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

Hence we have this equation in slope intercept form and comparing it with

[tex]y=mx+c[/tex] , we get Slope [tex]m = \frac{1}{2}[/tex]

We know that product of slopes of two perpendicular lines in -1

Hence if slope of line perpendicular to [tex]y=\frac{1}{2}x-2[/tex] is m' then

[tex]m\times m' =-1[/tex]

[tex]\frac{1}{2} \times m' =-1[/tex]

[tex]m'=-2[/tex]

Hence the slope of the line we have to find is -2

now we have slope and a point

Hence the equation of the line will be

[tex]\frac{y-(-1)}{x-2}=-2[/tex]

[tex]y+1=-2(x-2)[/tex]

[tex]y+1=-2x+4[/tex]

adding 2x and subtracting  on both sides we get

[tex]2x+y=3[/tex]

Which is our equation asked

The answer does not exist.

Note - The statement has typing mistakes. Correct form is presented below:

Let [tex]f(x) = (x-3)^{-2}[/tex]. Find all values of [tex]c[/tex] in (2, 5) such that [tex]f(5) - f(2) = f'(c) \cdot (5-2)[/tex]. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

In this question we should use the Mean Value Theorem, which states that given a secant line between points A and B, there is at least a point C that belongs to the curve whose derivative exists.

We begin by calculating [tex]f(2)[/tex] and [tex]f(5)[/tex]:

[tex]f(2) = (2-3)^{-2}[/tex]

[tex]f(2) = 1[/tex]

[tex]f(5) = (5-3)^{-2}[/tex]

[tex]f(5) = 1[/tex]

And the slope of the derivative is:

[tex]f'(c) = \frac{f(5) - f(2)}{5-2}[/tex]

[tex]f'(c) = 0[/tex]

Now we find the derivative of the function:

[tex]f'(x) = -2\cdot (x-3)^{-3}[/tex]

[tex]-2\cdot (x-3)^{-3} = 0[/tex]

[tex]-2 = 0[/tex] (ABSURD)

Hence, we conclude that the answer does not exist.

We kindly invite to see this question on Mean Value Theorem: https://brainly.com/question/3957181