If sin2 x + cos2 y = 2 sec2 z, then general solution of triplets (x, y, z) is

the corresponding angles should be equal

so, [tex] 5x+15=90 \implies 5x=75\implies x=15^{\circ}[/tex]

Answer:

a) In the interval  (  55,9  ;   456,7 ) we will find 99,7 % of all values

b) In the interval  (  122,7  ;  389,9 ) we find 95,4 % of all values

Step-by-step explanation:

For a Normal distribution N (μ ; σ ) the Empirical rule establishes that the intervals:

( μ  ±  σ  )          contains 68,3 % of all values

( μ  ±  2σ  )        contains 95,4 % of all values

( μ  ±  3σ  )        contains 99,7 % of all values

If   N ( 256,3 ; 66,8 )

σ  =  66,8        ⇒   3*σ  = 3 * 66,8  = 200,4

Then:     256,3 - 200,4  =  55,9

And        256,3 + 200,4 = 456,7

a) In the interval  (  55,9  ;   456,7 ) we will find 99,7 % of all values

b) 2*σ  = 2 * 66,8  = 133,6

Then  256,3 - 133,6  = 122,7

And    256,3 + 133,6 = 389,90

Then in the interval  (  122,7  ;  389,9 ) we find 95,4 % of all values

Answer:

{June, Janet, Jill, Justin, Jeffrey, Jelly,Irina, Irena, Arena, Arina, }

Step-by-step explanation:

A ∪ C

This means union so we join the sets together

A = {June, Janet, Jill, Justin, Jeffrey, Jelly} + C = {Irina, Irena, Arena, Arina, Jelly}

A U C =  {June, Janet, Jill, Justin, Jeffrey, Jelly,Irina, Irena, Arena, Arina, Jelly}

We get rid of repeats

A U C =  {June, Janet, Jill, Justin, Jeffrey, Jelly,Irina, Irena, Arena, Arina, }

Answer:

the value of the 3 is 30

Step-by-step explanation:

the second digit to the left of a decimal is always tens column

Answer:

14 miles

Step-by-step explanation:

Since we know that the distance of the paths from Cedarburg to Allenville is 22 and 13/16 miles, and we know the distance from Cedarburg to Lakeside is 8 and 13/16 miles.

We know that the total distance is made up of the distance from C to L and L to A.

So 22 and 13/16 = 8 and 13/16 + L to A

We can subtract 22 and 13/16 by 8 and 13/16 to get 14 miles.

Hope this helps.

Answer:

Please see the Step-by-step explanation for the answers

Step-by-step explanation:

1)

∑[tex]\left \ {{5} \atop {j=1}} \right.[/tex] 2j

The sum of series from j=1 to j=5 is:

∑ = 2(1) + 2(2) + 2(3) + 2(4) + 2(5)

  =  2 + 4 + 6 + 8 + 10

∑ = 30

2)

This question is not given clearly so i assume the following series that will give you an idea how to solve this:

∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] 2k²

The sum of series from k=1 to j=4 is:

∑ = 2(1)² + 2(2)² + 2(3)² + 2(4)²

  = 2(1) + 2(4) + 2(9) + 2(16)

  =  2 + 8 + 18 + 32

∑ = 60

∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] (2k)²

∑ = (2*1)² + (2*2)² + (2*3)² + (2*4)²

  = (2)² + (4)² + (6)² + (8)²

  = 4 + 16 + 36 + 64

∑ = 120

∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] (2k)²- 4

∑ = (2*1)²-4 + (2*2)²-4 + (2*3)²-4 + (2*4)²-4

  = (2)²-4 + (4)²-4 + (6)²-4 + (8)²-4

  = (4-4) + (16-4) + (36-4) + (64-4)

  = 0 + 12 + 32 + 60

∑ = 104

∑[tex]\left \ {{4} \atop {k=1}} \right.[/tex] 2k²- 4

∑ = 2(1)²-4 + 2(2)²-4 + 2(3)²-4 + 2(4)²-4

  = 2(1)-4 + 2(4)-4 + 2(9)-4 + 2(16)-4

  = (2-4) + (8-4) + (18-4) + (32-4)

  = -2 + 4 + 14 + 28

∑ = 44

3)

∑[tex]\left \ {{6} \atop {k=3}} \right.[/tex] (2k-10)

∑ = (2×3−10) + (2×4−10) + (2×5−10) + (2×6−10)  

  = (6-10) + (8-10) + (10-10) + (12-10)

  = -4 + -2 + 0 + 2  

∑ = -4

4)

1+1/2+1/4+1/8+1/16+1/32+1/64

This is a geometric sequence where first term is 1 and the common ratio is 1/2 So

a = 1

This can be derived as

1/2/1 = 1/2 * 1 = 1/2

1/4/1/2 = 1/4 * 2/1 = 1/2

1/8/1/4 = 1/8 * 4/1  = 1/2

1/16/1/8 = 1/16 * 8/1  = 1/2

1/32/1/16 = 1/32 * 16/1  = 1/2

1/64/1/32 = 1/64 * 32/1  = 1/2

Hence the common ratio is r = 1/2

So n-th term is:

[tex]ar^{n-1}[/tex] = [tex]1(\frac{1}{2})^{n-1}[/tex]

So the answer that represents the series in sigma notation is:

∑[tex]\left \ {{7} \atop {j=1}} \right.[/tex] [tex](\frac{1}{2})^{j-1}[/tex]

5)

−3+(−1)+1+3+5

This is an arithmetic sequence where the first term is -3 and the common difference is 2. So  

a = 1

This can be derived as

-1 - (-3) = -1 + 3 = 2

1 - (-1) = 1 + 1 = 2

3 - 1 = 2

5 - 3 = 2

Hence the common difference d = 2

The nth term is:

a + (n - 1) d

= -3 + (n−1)2

= -3 + 2(n−1)

= -3 + 2n - 2

= 2n - 5

So the answer that represents the series in sigma notation is:

∑[tex]\left \ {{5} \atop {j=1}} \right.[/tex] (2j−5)

For (1) the sum is 30, for (2) the sum is 90, for (3) the sum is -4, for(4) the sigma notation is  [tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex]  where j = 1 to j = 7, and for (5) the sigma notation is  [tex]\rm\sum j = (2j-5)[/tex]  where j = 1 to j = 5.

We have different series in the question.

It is required to find the sum of all series.

In mathematics, a series can be defined as a group of data that followed certain rules of arithmetic.

1) We have:

[tex]\rm \sum j=2j[/tex]   where j = 1 to j = 5

After expanding the series, we get:

= 2(1)+2(2)+2(3)+2(4)+2(5)

=2(1+2+3+4+5)

= 2(15)

=30

2) We have:

[tex]\rm \sum k=(2k^2-4)[/tex]  where k = 1 to k = 4

After expanding the series, we get:

[tex]\rm = (2(1)^2-4)+(2(2)^2-4)+(2(3)^2-4)+(2(4)^2-4)+(2(5)^2-4)\\[/tex]

[tex]\rm = 2[1^2+2^2+3^2+4^2+5^2]-4\times5\\\\\rm=2[55]-20\\\\\rm = 90[/tex]

3) We have:

[tex]\rm \sum k= (2k-10)[/tex]  where k = 3 to k = 6

After expanding the series, we get:

[tex]= (2(3)-10)+(2(4)-10)+(2(5)-10)+(2(6)-10)\\\\=2[3+4+5+6] - 10\times4\\\\=2[18] - 40\\\\= -4[/tex]

4) The series given below:

[tex]1, \frac{1}{2} ,\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\frac{1}{64}[/tex]

It is a geometric progression:

[tex]\rm n^t^h[/tex] for the geometric progression is given by:

[tex]\rm a_n = ar^{n-1}[/tex]

[tex]\rm a_n = 1(\frac{1}{2})^{n-1}\\\\\rm a_n = (\frac{1}{2})^{n-1}\\[/tex]

In sigma notation we can write:

[tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex]  where j = 1 to j = 7

5) The given series:

−3+(−1)+1+3+5, it is arithmetic series.

[tex]\rm n^t^h[/tex] for the arithmetic progression is given by:

[tex]\rm a_n = a+(n-1)d[/tex]

[tex]\rm a_n = -3+(n-1)(2)\\\\\rm a_n = 2n-5[/tex]

In sigma notation we can write:

[tex]\rm\sum j = (2j-5)[/tex]  where j = 1 to j = 5

Thus, for (1) the sum is 30, for (2) the sum is 90, for (3) the sum is -4, for(4) the sigma notation is  [tex]\rm \sum j = 1(\frac{1}{2})^{j-1}\\[/tex]  where j = 1 to j = 7, and for (5) the sigma notation is  [tex]\rm\sum j = (2j-5)[/tex]  where j = 1 to j = 5.

Learn more about the series here:

https://brainly.com/question/10813422

Answer:

[tex]r=\frac{\sqrt{3} }{4}[/tex]    and    [tex]r=-\frac{\sqrt{3} }{4}[/tex]

Step-by-step explanation:

when you solve for r in the given equation, you need to apply the square root property, which gives positive and negative answers (both should therefore be considered):

[tex]r^2=\frac{3}{16} \\r=+/-\sqrt{\frac{3}{16}} \\r=+/-\frac{\sqrt{3} }{4}[/tex]

then you need to include these two possible solutions:

[tex]r=\frac{\sqrt{3} }{4}[/tex]    and    [tex]r=-\frac{\sqrt{3} }{4}[/tex]

Answer:

3/10

Step-by-step explanation:

6+3+1=10

since there are 3 blue marbles, we put the 3 into the place of the numerator

and since there is 10 marbles in total it goes into the denominator

The probability that a randomly selected marble is not blue will be 0.70.

Its basic premise is that something will almost certainly happen. The percentage of favorable events to the total number of occurrences.

A bag contains 6 red marbles, 3 blue marbles and 1 green marble.

The total number of the event will be

Total event = 6 + 3 + 1

Total event = 10

Then the probability that a randomly selected marble is not blue will be

Favorable event = 7 {red, green}

Then the probability will be

P = 7 / 10

P = 0.70

More about the probability link is given below.

https://brainly.com/question/795909

#SPJ2

Hey there! I'm happy to help!

We want to isolate y on one side of the equation to see what it equals. To do this, we use inverse operations to cancel out numbers on the y side and find the correct value.

1/3y+4=16

We subtract 4 from both sides, canceling out the +4 on the right but keeping the same y-value by doing the same to the other side.

1/3y=12

We divide both sides by 1/3 (which is multiplying both sides by 3) which will cancel out the 1/3 and tell us what y is equal to.

y=36

Now you know how to solve basic equations! Have a wonderful day! :D

Step-by-step explanation:

Hello, there!!!

Let ABC be a Right angled triangle,

where, AB = 3

BC= 10

and AC= x

now,

As the triangle is a Right angled triangle, taking angle C asrefrence angle. we get,

h= AC = x

p= AB = 3

b= BC= 10

now, by Pythagoras relation we get,

[tex]h = \sqrt{ {p}^{2} + {b}^{2} } [/tex]

[tex]or ,\: h = \sqrt{ {3}^{2} + {10}^{2} } [/tex]

by simplifying it we get,

h = 10.44030

Therefore, the answer is x= 10.

Hope it helps...

Answer:

$198

Step-by-step explanation:

198x.07=13.86

198+13.86=211.86

Answer:

Hello the options to your question is missing below are the options

 A) if sample means were obtained for a long series of samples, approximately 95 percent of all sample means would be between 10 and 16 miles

B.the unknown population mean is definitely between 10 and 16 miles

C.if these intervals were constructed for a long series of samples, approximately 95 percent would include the unknown mean commute distance for all Californians

D.the unknown population mean is between 10 and 16 miles with probability .95

Answer : if these intervals were constructed for a long series of samples, approximately 95 percent would include the unknown mean commute distance for all Californians  ( c )

Step-by-step explanation:

95%  confidence

interval = 10 to 16 miles

To have 95% confidence signifies that if these intervals were constructed for a long series of samples, approximately 95 percent would include the unknown mean commute distance for all Californians

confidence interval covers a range of samples/values in the interval and the higher the % of the confidence interval the more precise the interval is,

Answer:

  71°

Step-by-step explanation:

Consider triangle BDH. x is the external angle that is remote to internal angles B and D, so is equal to their sum:

  x° = 41° +30°

  x° = 71°

Answer:

point A is the rays endpoint

Step-by-step explanation:

Answer:

The "endpoint" of a ray is the origin point of the ray, or the point at which the ray starts.

Step-by-step explanation:

A ray starts at a given point, the endpoint, and then goes in a certain direction forever ad infinitum.  The origin point of a ray is called "the endpoint".

Cheers.

Answer:

[tex]y = -x - 3[/tex]

Step-by-step explanation:

We are trying to get the equation [tex]3x + 3y = -9[/tex] into the form [tex]y = mx+b[/tex], aka slope-intercept form.

To do this we are trying to isolate y.

[tex]3x + 3y = -9[/tex]

Subtract 3x from both sides:

[tex]3y = -9 - 3x[/tex]

Rearrange the terms:

[tex]3y = -3x - 9[/tex]

Divide both sides by 3:

[tex]y = -x - 3[/tex]

Hope this helped!

Answer:

C

Step-by-step explanation:

The first sentence basically sets up the equation which is given, so we can read it for knowledge but it is not crucial to solve the problem.

We start here:

we are given: $120 - 0.2($120)

= 120 - (0.2)(120)   (factoring out 120)

= 120 (1 - 0.2)

= 120 (0.8)

= 0.8 (120)     (answer c)

Answer:

five and forty-four hundredths

Step-by-step explanation:

Answer:

five point four four

Step-by-step explanation:

Step-by-step explanation:

I used an app called DESMOS It Is usually super helpful!!!

Answer:

20 cm

Step-by-step explanation:

20 cm

8/2 = 4

6/2 = 3

3 and 4 are the sides of the triangle (four triangles in rhombus)

a²+b²=c²

4³+3²=c²

c = 5

5 x 4 = 20

Hope this helped

Answer:

perimeter = 20 cm

Step-by-step explanation:

consider breaking the rhombus into four equal parts.

and that gives you a triangle.

(refer to image attached for more clarification)

let a = 3, b = 4

to get the side c, use Pythagorean theorem = c² = a² + b²

c = sqrt (3² + 4²)

side c = 5

therefore,

perimeter = 4 x sides (c)

perimeter = 4 x 5

perimeter = 20 cm

Answer:

Stratified Random sampling.

Step-by-step explanation:

As per the scenario, It is stratified random sampling as it divides students into strata which represent Sophomores, Juniors, and Seniors.

Simple random samples of the given sizes of the proportional to the size of the stratum which is to be taken from every stratum that is to be about 10 percent of students from every class that is selected here.

Hence, according to the given situation, the correct answer is a random stratified sampling.

Answer:

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

Step-by-step explanation:

[tex]\frac{3x-2}{2}[/tex] = [tex]\frac{x-1}{2}[/tex]

Cross-multiply:

2(3x-2) = 2(x-1)

Simplify:

6x - 4 = 2x - 2

Subtract 2x from both sides:

4x - 4 = -2

Add 4 to both sides:

4x = 2

Divide both sides by 4:

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

Answer:

75

Step-by-step explanation:

4:3

4x25=100

3x25=75

Answer:

50ft by 120ft

Step-by-step explanation:

Area of a rectangle = L × W

6000ft² = L × W

L = 6000/W

When a diagonal line divides a rectangle into 2 right angled triangles, the diagonal line = Hypotenuse of either of the triangle and it is the longest side.

The formula for a right angle triangle =

a² + b² = c²( c = hypotenuse)

We are told in the question that:

A diagonal between opposite corners is measured to be 10 ft longer than one side of the parcel

Let us assume the side that the hypotenuse is longer than = Width

Hence, the Diagonal = (W + 10)²

Therefore

L² + W² = (W + 10)²

Since L = 6000/W

W² + (6000/W)² = (W + 10)²

W² + (6000/W)² = (W + 10) (W + 10)

W² + (6000/W)² = W² + 10W + 10W + 100

W² + (6000/W)² = W² + 20W + 100

W² - W² + (6000/W)² = 20W+ 100

6000²/W² = 20W + 100

6000² = W²( 20W + 100)

6000² = 20W³ + 100W²

20W³ + 100W² - 6000² = 0

20W³ + 100W² - 36000000 = 0

20(W³ + 5W² - 1800000) = 0

Factorising the quadratic equation,

20(W − 120)(W² + 125W + 15000) = 0

W - 120 = 0

W = 120

Therefore,

W(Width) = 120feet

Since the Width = 120 feet

We can find the length

6000ft² = L × W

L = 6000/W

L = 6000/120

L = 50 feet

The dimensions of the land, correct to the nearest foot is 50ft by 120ft

Answer:

Step-by-step explanation:

Given that:

[tex]\mathsf{f(x) = x^2 -5 } \\ \\ \mathsf{x_1 = 2}[/tex]

The derivative of the first function of (x) is:

[tex]\mathsf{f'(x) =2x }[/tex]

According to Newton's Raphson method for function formula:

[tex]{\mathrm{x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}}[/tex]

where;

[tex]\mathbf{x_1 =2}[/tex]

The first iteration is as follows:

[tex]\mathtt{f(x_1) = (2)^2 - 5} \\ \\ \mathbf{f(x_1) = -1}[/tex]

[tex]\mathtt{f'(x_1) = 2(2)} \\ \\ \mathbf{ = 4}[/tex]

[tex]\mathtt{\dfrac{f(x_1)}{f'(x_1)}} = \dfrac{-1}{4}}[/tex]

[tex]\mathbf{\dfrac{f(x_1)}{f'(x_1)} =-0.25}[/tex]

[tex]\mathtt{x_1 - \dfrac{f(x_1)}{f'(x_1)}} = \mathtt{2 - (-0.25)}}[/tex]

[tex]\mathbf{x_1 - \dfrac{f(x_1)}{f'(x_1)} = 2.25}[/tex]

Therefore;

[tex]\mathbf{x_2 = 2.25}[/tex]

For the second iteration;

[tex]\mathtt f(x_2) = (2.25)^2 -5}[/tex]

[tex]\mathtt f(x_2) = 5.0625-5}[/tex]

[tex]\mathbf{ f(x_2) =0.0625}[/tex]

[tex]\mathtt{f'(x_2)= 2(2.25)}[/tex]

[tex]\mathbf{f'(x_2)= 4.5}[/tex]

[tex]\mathtt{ \dfrac{f(x_2)}{f'(x_2)}} = \dfrac{0.0625}{4.5}}[/tex]

[tex]\mathbf{ \dfrac{f(x_2)}{f'(x_2)} = 0.01389}[/tex]

[tex]\mathtt{x_2 - \dfrac{f(x_2)}{f'(x_2)}} = \mathtt{2.25 -0.01389}}[/tex]

[tex]\mathbf{x_2 - \dfrac{f(x_2)}{f'(x_2)} = 2.2361}}[/tex]

Therefore, [tex]\mathbf{x_3 = 2.2361}[/tex]

Answer:

3.25 + 5.50S ≤ 30

Step-by-step explanation:

Given:

Chocolate milk cost $3.25.

Maximum bill that Benjamin wants = $30

Cost of a stack pancake(44) = $5.50

Let number of stacks of pancakes bought = S

Benjamin will spend all the money available on 1 chocolate milk and S number of stacks of pancakes.

Cost of 1 pancake = $5.50

Cost of S number of stacks of pancakes = S*5.50

=5.50S

Total money spent =$3.25+5.50S

The total money spent should either be lesser than or equal to $30

The inequality is

3.25 + 5.50S ≤ 30

Largest number of pancakes Benjamin can afford

3.25 + 5.50S ≤ 30

5.50S ≤ 30-3.25

5.50S ≤ 26.75

Divide both sides by 5.50

S ≤ 4.86

1 stack=44 pancakes

4.86 stacks= 4.86 *44

=213.84 pancakes

Answer:

3.25+5.50S≤30

and 16 pancakes

Step-by-step explanation:

Answer:

[tex] d = 7 + 3\sqrt{3} [/tex] and

[tex] d = 7 - 3\sqrt{3} [/tex]

Step-by-step explanation:

To solve the equation, [tex] d^2 - 14d - 22 = 0 [/tex], using the quadratic formula,

Recall: quadratic formula = [tex] \frac{-b ± \sqrt{b^2 - 4ac}}{2a} [/tex]

Where,

a = 1

b = -14

c = 22

Plug in your values into the formula and solve:

[tex] \frac{-(-14) ± \sqrt{(-14)^2 - 4(1)(22)}}{2(1)} [/tex]

[tex] \frac{14 ± \sqrt{196 - 88}}{2} [/tex]

[tex] \frac{14 ± \sqrt{108}}{2} [/tex]

[tex] d = \frac{14 + \sqrt{108}}{2} [/tex]

[tex] d = \frac{14 + 6\sqrt{3}}{2} [/tex]

[tex] d = (\frac{2(7 + 3\sqrt{3})}{2} [/tex]

[tex] d = 7 + 3\sqrt{3} [/tex]

And

[tex] d = \frac{14 - \sqrt{108}}{2} [/tex]

[tex] d = \frac{14 - 6\sqrt{3}}{2} [/tex]

[tex] d = (\frac{2(7 - 3\sqrt{3})}{2} [/tex]

[tex] d = 7 - 3\sqrt{3} [/tex]

Answer:

Yes, it is invertible

Step-by-step explanation:

We need to find in the matrix determinant is different from zero, since iif it is, that the matrix is invertible.

Let's use co-factor expansion to find the determinant of this 4x4 matrix, using the column that has more zeroes in it as the co-factor, so we reduce the number of determinant calculations for the obtained sub-matrices.We pick the first column for that since it has three zeros!

Then the determinant of this matrix becomes:

[tex]4\,*Det\left[\begin{array}{ccc}1&4&6\\0&3&8\\0&0&1\end{array}\right] +0+0+0[/tex]

And the determinant of these 3x3 matrix is very simple because most of the cross multiplications render zero:

[tex]Det\left[\begin{array}{ccc}1&4&6\\0&3&8\\0&0&1\end{array}\right] =1 \,(3\,*\,1-0)+4\,(0-0)+6\,(0-0)=3[/tex]

Therefore, the Det of the initial matrix is : 4 * 3 = 12

and then the matrix is invertible

[tex]a=\dfrac{2}{3}\\r=\dfrac{1}{2}[/tex]

The sum exists if [tex]|r|<1[/tex]

[tex]\left|\dfrac{1}{2}\right|<1[/tex] therefore the sum exists

[tex]\displaystyle\\\sum_{k=0}^{\infty}ar^k=\dfrac{a}{1-r}[/tex]

[tex]\dfrac{2}{3}+\dfrac{1}{3}+\dfrac{1}{6}+\ldots=\dfrac{\dfrac{2}{3}}{1-\dfrac{1}{2}}=\dfrac{\dfrac{2}{3}}{\dfrac{1}{2}}=\dfrac{2}{3}\cdot 2=\dfrac{4}{3}[/tex]

Answer:

(A) A 95% confidence for the population mean is [$332.16, $447.84] .

(B) If the confidence level in part (a) changed from 95% to 99%, then the margin of error for the confidence interval would increase.

(C) If the sample size in part (a) changed from 19 to 22, then the margin of error for the confidence interval would decrease.

(D) A 99% confidence interval for the proportion of students who purchase used textbooks is [0.363, 0.477]  .

Step-by-step explanation:

We are given that 19 students are randomly selected the sample mean was $390 and the standard deviation was $120.

Firstly, the pivotal quantity for finding the confidence interval for the population mean is given by;

                             P.Q.  =  [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex]  ~ [tex]t_n_-_1[/tex]

where, [tex]\bar X[/tex] = sample mean = $390

            s = sample standard deviation = $120

            n = sample of students = 19

            [tex]\mu[/tex] = population mean

Here for constructing a 95% confidence interval we have used a One-sample t-test statistics because we don't know about population standard deviation.

So, 95% confidence interval for the population mean, [tex]\mu[/tex] is ;

P(-2.101 < [tex]t_1_8[/tex] < 2.101) = 0.95  {As the critical value of t at 18 degrees of

                                               freedom are -2.101 & 2.101 with P = 2.5%}  

P(-2.101 < [tex]\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }[/tex] < 2.101) = 0.95

P( [tex]-2.101 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]{\bar X-\mu}[/tex] < [tex]2.101 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95

P( [tex]\bar X-2.101 \times {\frac{s}{\sqrt{n} } }[/tex] < [tex]\mu[/tex] < [tex]\bar X+2.101 \times {\frac{s}{\sqrt{n} } }[/tex] ) = 0.95

95% confidence interval for [tex]\mu[/tex] = [ [tex]\bar X-2.101 \times {\frac{s}{\sqrt{n} } }[/tex] , [tex]\bar X+2.101 \times {\frac{s}{\sqrt{n} } }[/tex] ]

                        = [ [tex]\$390-2.101 \times {\frac{\$120}{\sqrt{19} } }[/tex] , [tex]\$390+2.101 \times {\frac{\$120}{\sqrt{19} } }[/tex] ]

                        = [$332.16, $447.84]

(A)  Therefore, a 95% confidence for the population mean is [$332.16, $447.84] .

(B) If the confidence level in part (a) changed from 95% to 99%, then the margin of error for the confidence interval which is [tex]Z_(_\frac{\alpha}{2}_) \times \frac{s}{\sqrt{n} }[/tex] would increase because of an increase in the z value.

(C) If the sample size in part (a) changed from 19 to 22, then the margin of error for the confidence interval which is [tex]Z_(_\frac{\alpha}{2}_) \times \frac{s}{\sqrt{n} }[/tex]  would decrease because as denominator increases; the whole fraction decreases.

(D) We are given that to estimate the proportion of students who purchase their textbooks used, 500 students were sampled. 210 of these students purchased used textbooks.

Firstly, the pivotal quantity for finding the confidence interval for the population proportion is given by;

                             P.Q.  =  [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex]  ~ N(0,1)

where, [tex]\hat p[/tex] = sample proportion students who purchase their used textbooks = [tex]\frac{210}{500}[/tex] = 0.42    

            n = sample of students = 500

            p = population proportion

Here for constructing a 99% confidence interval we have used a One-sample z-test statistics for proportions

So, 99% confidence interval for the population proportion, p is ;

P(-2.58 < N(0,1) < 2.58) = 0.99  {As the critical value of z at 0.5%

                                               level of significance are -2.58 & 2.58}  

P(-2.58 < [tex]\frac{\hat p-p}{\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < 2.58) = 0.99

P( [tex]-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < [tex]{\hat p-p}[/tex] < [tex]2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.99

P( [tex]\hat p-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] < p < [tex]\hat p+2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ) = 0.99

99% confidence interval for p = [ [tex]\hat p-2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] , [tex]\hat p+2.58 \times {\sqrt{\frac{\hat p(1-\hat p)}{n} } }[/tex] ]

= [ [tex]0.42 -2.58 \times {\sqrt{\frac{0.42(1-0.42)}{500} } }[/tex] , [tex]0.42 +2.58 \times {\sqrt{\frac{0.42(1-0.42)}{500} } }[/tex] ]

= [0.363, 0.477]

Therefore, a 99% confidence interval for the proportion of students who purchase used textbooks is [0.363, 0.477]  .