Answers
total length of the line is:
J = -10
K = 8
for a total of 18 units long
midpoint would be 18/2 =9
-10+9 = -1
so need the point that is located at negative 1, which is point N
so N is the midpoint
Related Questions
A salesperson sold a total of $6,400.00.If her rate of commission is 6%, what is her commission?
Answers
multiply 6400 x 6%
6% = 0.06
6400 x 0.06 = 384
her commission was $384
Answer:
The commission amount of the salesperson is $384.
Step-by-step explanation:
A salesperson sold a total of $6,400.00.
The rate of commission is 6% or 0.06. Commissions are based on sales. These are some percentage of the sales amount.
So, here the amount will be = [tex]0.06\times6400=384[/tex] dollars
So, the commission amount of the salesperson is $384.
The quotient of (x4 + 5x3 – 3x – 15) and a polynomial is (x3 – 3). What is the polynomial?
Answers
Answer:
(x +5)
Step-by-step explanation:
The problem statement is telling you that one factor of (x⁴ +5x³ -3x -15) is (x³ -3). It is asking for the other factor. Clearly, you can find the other factor by dividing the polynomial by the given factor.
That is ...
(x⁴ +5x³ -3x -15) / (x³ -3) = (x +5)
so ...
(x⁴ +5x³ -3x -15) / (x +5) = (x³ -3)
The divisor of interest is (x +5).
Answer:
(x+5)
The answer is c.
the quadratic formula gives which roots for the equation 2x^2+7x+-2
Answers
[tex]a=2[/tex]
[tex]b=7[/tex]
[tex]c=-2[/tex]
Quadratic formula to find the roots is given as
x₁,₂ = [-b plus minus √(b)²-4ac)] ÷ 2a
Substitute a, b, and c from our expression we have
x₁,₂ = [-7 plus minus √(7)²-(4×2×-2)] ÷ 2(2)
x₁,₂ = [-7 plus minus√65] ÷ 4
from here we'll work out x₁ and x₂ separately
x₁ = (-7+√65) ÷ 4 = 0.266 (round to 3 dp)
x₂ = (-7-√65) ÷ 4 = -3.766 (round to 3 dp)
The roots for the equation [tex]\(2x^2 + 7x = -2\)[/tex] are [tex]\(x = \frac{{-7 \pm \sqrt{65}}}{{4}}\).[/tex] So, option D is correct.
To find the roots of the quadratic equation [tex]\(2x^2 + 7x = -2\),[/tex] we can use the quadratic formula:
[tex]\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\][/tex]
Here, [tex]\(a = 2\), \(b = 7\), and \(c = -2\).[/tex]
Substituting these values into the formula:
[tex]\[x = \frac{{-7 \pm \sqrt{{7^2 - 4 \cdot 2 \cdot (-2)}}}}{{2 \cdot 2}}\][/tex]
[tex]\[x = \frac{{-7 \pm \sqrt{{49 + 16}}}}{{4}}\][/tex]
[tex]\[x = \frac{{-7 \pm \sqrt{{65}}}}{{4}}\][/tex]
So, the correct answer is option D:
[tex]\[x = \frac{{-7 \pm \sqrt{{65}}}}{{4}}\][/tex]
Complete Question:
Find a rational zero of the polynomial function and use it to find all the zeros of the function. f(x) = x4 + 3x3 - 5x2 - 9x - 2
Answers
Graph the equation by plotting arbitrary points. The graph looks like that in the figure. The points at which the curve passes the x-axis are the solution which are encircled in red.In approximation, the rational roots or zero's are -3.73, -1, -0.28 and 2.
The rational zero -1 is a root of f(x). Synthetic division yields [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Further factorization or testing other rational roots finds the remaining zeros.
To find a rational zero of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\)[/tex], we can use the Rational Root Theorem. According to this theorem, any rational zero of the polynomial function must be of the form ±p/q, where p is a factor of the constant term (-2 in this case) and q is a factor of the leading coefficient (1 in this case).
The factors of -2 are ±1, ±2, and the factors of 1 are ±1. Therefore, the possible rational zeros are:
±1, ±2
We can try these values to see if they are roots of the polynomial.
Let's start by trying x = 1:
[tex]\[f(1) = (1)^4 + 3(1)^3 - 5(1)^2 - 9(1) - 2\]\[= 1 + 3 - 5 - 9 - 2\]\[= -12\][/tex]
So, x = 1 is not a root.
Next, let's try x = -1:
[tex]\[f(-1) = (-1)^4 + 3(-1)^3 - 5(-1)^2 - 9(-1) - 2\]\[= 1 - 3 - 5 + 9 - 2\]\[= 0\][/tex]
Therefore, x = -1 is a root of the polynomial.
To find the other zeros, we can perform polynomial division or synthetic division by dividing f(x) by (x + 1). Let's use synthetic division:
-1 1 3 -5 -9 -2
1 2 -7 -2 ↓
The result is [tex]\(x^3 + 2x^2 - 7x - 2\)[/tex]. Now, we can factor this cubic polynomial or continue using the Rational Root Theorem to find additional roots. Let's try x = 1 again:
[tex]\[f(1) = (1)^3 + 2(1)^2 - 7(1) - 2\]\[= 1 + 2 - 7 - 2\]\[= -6\][/tex]
x = 1 is not a root, so we continue to try the other possible rational zeros. However, to save time, let's check if any of the values of [tex]\(x = \pm 2\)[/tex] are roots using synthetic division:
For x = 2:
2 1 2 -7 -2
1 4 1 ↓
For \(x = -2\):
-2 1 2 -7 -2
1 0 -7 ↓
Since none of these values result in a remainder of 0, [tex]\(x = \pm 2\)[/tex] are not roots.
Therefore, the zeros of the polynomial function [tex]\(f(x) = x^4 + 3x^3 - 5x^2 - 9x - 2\) are \(x = -1\),[/tex] and the other zeros can be found by further factoring the reduced cubic polynomial.
A supply company manufactures copy machines. The unit cost C (the cost in dollars to make each copy machine) depends on the number of machines made. If x machines are made, then the unit cost is given by the function
C (x) = 0.8x ^ 2 - 256x +25,939 . How many machines must be made to minimize the unit cost? Do not round your answer.
Number of copy machines:
Answers
To minimize unit cost, 160 machines must be made.
Step-by-step explanation:
Find the first derivative of the cost function: C'(x) = 1.6x - 256.
Set the derivative equal to 0 and solve for x to find the critical point: 1.6x - 256 = 0. x = 160.
Check the nature of the critical point using the second derivative test to confirm that x = 160 gives the minimum cost.
Therefore, the number of machines that must be made to minimize the unit cost is 160 machines.
The number of machines that must be made to minimize the unit cost is 160.
To find the number of machines that must be made to minimize the unit cost, we need to find the minimum point of the function [tex]\(C(x) = 0.8x^2 - 256x + 25,939\).[/tex]
The function \(C(x)\) represents a quadratic equation, and the vertex of a quadratic equation represents its minimum or maximum point. The x-coordinate of the vertex of a quadratic function in the form [tex]\(ax^2 + bx + c\) is given by \(-\frac{b}{2a}\).[/tex]
[tex]For the function \(C(x) = 0.8x^2 - 256x + 25,939\), we have \(a = 0.8\) and \(b = -256\).Now, let's calculate the x-coordinate of the vertex:\[ x_{\text{vertex}} = -\frac{b}{2a} = -\frac{-256}{2 \times 0.8} = -\frac{-256}{1.6} = 160 \][/tex]
So, the number of machines that must be made to minimize the unit cost is 160.
If thewronskian w of f and g is 3e4t,and if f(t) = e2t,find g(t).
Answers
We have [tex]f(t)=e^{2t}\implies f'(t)=2e^{2t}[/tex], so
[tex]W(f(t),g(t))=e^{2t}g'(t)-2e^{2t}g(t)=3e^{4t}[/tex]
[tex]\implies e^{-2t}g'(t)-2e^{-2t}g(t)=3[/tex]
[tex]\implies\dfrac{\mathrm d}{\mathrm dt}[e^{-2t}g(t)]=3[/tex]
[tex]\implies e^{-2t}g(t)=\displaystyle\int3\,\mathrm dt[/tex]
[tex]\implies e^{-2t}g(t)=3t+C[/tex]
[tex]\implies g(t)=3te^{2t}+Ce^{2t}[/tex]
where [tex]C[/tex] is any arbitrary constant.
The graph below shows the fine that a college student pays to the library based on the number of minutes a loaner laptop is overdue:
A graph is shown. The values on the x axis are 0, 2, 4, 6, 8. The values on the y axis are 0, 0.70, 1.40, 2.10, 2.80. Points are shown on ordered pairs 0, 0 and 2, 0.70 and 4, 1.40 and 6, 2.10. These points are joined by a line. The label on the x axis is Minutes Overdue. The title on the y axis is Fine.
Which statement best describes the point (0, 0) on the graph?
Answers
This graph shows an increasing linear trend. It shows that the fine is directly proportional with time. At time 0 minutes, the college student does not have to pay a fine because he hasn't even used the laptop yet. Therefore, the origin (0,0) signifies the starting point of the observation. But when he used 2 minutes of the time, he would pay $0.7. The trend goes on until he used up 6 minutes and paid a total of $2.1
Answer:
So the answer would be No fine is paid if the laptop is returned exactly at the time at which it is due
1) On average, Donna's Cafe has 42 customers, which represents 20% of the total approved occupancy by the fire department.
a) According to the fire department's occupancy approval, what percentage of the cafe is still available for customers?
b)According to the fire department's occupancy approval, how many seats are still available for customers?
Answers
a.) According to the fire department's occupancy approval, the percentage of the cafe still available for customers is 100% - 20% = 80%
b.) Given that 42 customers represent 20% of the total approved occupancy by the fire department.
Let the total approved occupancy be x, then 20% of x is 42
i.e.
[tex]0.2x=42 \\ \\ x= \frac{42}{0.2} =210[/tex]
i.e. the total approved occupancy is 210.
According to the fire department, the number of seats that are still available for customers is given by
[tex]80\% \ of \ x = 0.8 \times 210 = 168[/tex]
Write an algebraic expression which represents the volume of a box whose width is 4y, height is 6y and length is 3y + 1.
Answers
please help me right now will give brainliest and offering 50 points
Above are two different models of the same hexagon. If the side length of the model on the left is in, what is the corresponding side length of the model on the right?
A. 10 1/4 in
B. 4 in
C. 5 in
D. 3 3/4 in
Answers
Answer:
The answer is 5 in.
Step-by-step explanation:
Since the scale for the model on the left is 1 in = 12 ft, and the scale for the model on the right is 1 in = 3 ft, the model on the right is 4 times larger than the model on the left.
Multiply the side length of the model on the left by 4 to find the side length of the model on the right.
A 31-m tall building casts a shadow. The distance from the top of the building to the tip of the shadow is 37 m. Find the length of the shadow. If necessary, round your answer to the nearest tenth.
Answers
if the divisor is 40 what is the least 3 digit number dividend that would give a remainder of 4
Answers
If the divisor is 40 what is the least 3 digit number dividend that would give a remainder of 4"124/40"
Hope this help, Lukifer677
Your Welcome :)
A new bank account is opened on week 1 with a $200 deposit. After that first week, weekly deposits of $55 are made in the account. If y represents the total deposited into the account and x represents the number of weeks, which function rule describes this situation?
Answers
We know y is total amount of money (deposits) in the bank account.
We also know x represents the number of weeks.
On week 1, we would have $200 in the back account.
On week 2, we would have $200 + $55 or $255 in the bank account.
So, we could build an expression of 55(x - 1) + 200 which is equal to y.
So, y = 55(x - 1) + 200 is the answer.
What is the value of x?
16
50
130
164
Please hurry !!!
Answers
Answer:
x = 16.
Step-by-step explanation:
Given : Transverse line b and parallel line e and f.
To find : What is the value of x.
Solution : We have given Transverse line b and parallel line e and f.
Corresponding angles : When two lines are crossed by another line the angles in matching corners are called corresponding angles.
corresponding angles are always equal.
2x + 18 = 4x - 14.
On subtracting both sides by 4x
2x -4x + 18 = -14.
- 2x + 18 = - 14 .
On subtracting both sides by 18
- 2x = - 14 -18 .
- 2x = - 32 .
On dividing both sides by -2 .
x = 16.
Therefore, x = 16.
What does the value of the LCM represent
Answers
The circumference of a coin is 8π What is the radius? What is the diameter?
Answers
Circumference of a circle:
[tex]=2 \pi r[/tex]
[tex]8 \pi =2 \pi r[/tex][tex] \frac{8 \pi }{2 \pi } = r[/tex]
r=4
diameter = radius x 2
4x2 = 8
πd=C divide both sides by π
d=C/π, if C=8π then
d=8π/π
d=8
So the diameter is 8 units.
David wishes to accumulate $1 million by the end of 20 years by making equal annual end-of-year deposits over the next 20 years. if david can earn 10 percent on his investments, how much must he deposit at the end of each year? $50,000 $17,460 $14,900 $117,453
Answers
Fv=pmt [(1+r)^(n)-1)÷r]
Fv accumulated amount 1000000
PMT annual payment ?
R interest rate. 0.1
N time 20 years
Solve the formula for PMT
PMT=FV÷[(1+r)^(n)-1)÷r]
PMT=1,000,000÷(((1+0.1)^(20)−1)÷(0.1))
PMT=17,459.62 round your answer
PMT=17460
David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.
To calculate the equal annual end-of-year deposits that David must make to accumulate $1 million in 20 years at a 10 percent interest rate, we can use the formula for the future value of an ordinary annuity.
The formula for the future value of an ordinary annuity is given by:
[tex]FV = P * ((1 + r)^n - 1) / r[/tex]
where:
FV is the future value of the annuity (the desired $1 million in this case)
P is the annual deposit (what we need to find)
r is the annual interest rate (10% or 0.10 as a decimal)
n is the number of years (20 years in this case)
Substituting the known values:
[tex]$1,000,000 = P * ((1 + 0.10)^{20} - 1) / 0.10[/tex]
Now, we can solve for P:
$1,000,000 = P * (6.1917364224) / 0.10
$1,000,000 = P * 61.917364224
P = $1,000,000 / 61.917364224
P ≈ $16,150.01
So, David must deposit approximately $16,150.01 at the end of each year to accumulate $1 million by the end of 20 years at a 10 percent interest rate.
To know more about deposit:
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which statements describe the function f(x)=2(x-4)^4
A) The left end of the graph of the function goes up, and the right end goes down
B) It has 3 zeros and at most 4 relative maximums or minimums
C) It has 4 zeros and at most 3 relative maximums or minimums
D) It is a translation of the parent function 4 units to the right
E) It is a translation of the parent function 4 units to the left
F) Both ends of the graph of the function go up
Answers
There was 3 answers.
Answer one is It has 4 zeros and at most 3 relative maximums or minimums.
Answer two is It is a translation of the percent function 4 units to the right.
Answer three is Both ends of the graph of the function go up.
:)
it is a transition of the parent function 4 units to the right, it has 4 zeros and at most 3 relative maximums and minimums, both ends of the graph of the function go up this is for apex
What is the slope of a line that is perpendicular to the line whose equation is 0.5x−5y=9 0.5 x − 5 y = 9
Answers
y=mx+k
where m is the slope of the line.
ii)
given lines y=mx+k and y=nx+t, which are perpendicular to each other.
Then, m*n=-1
Now consider the line 0.5x−5y=9:
0.5x−5y=9
take -5y to the right side and 9 to the left side
0.5x-9=5y
switch sides
5y=0.5x-9
divide by 5
y=0.1x-9/5
so the slope of the line is m=0.1
let the slope of the line perpendicular to 0.5x−5y=9 be n, then
0.1*n=-1
n=-1/(0.1)=-10
Answer: -10
Israel claims that all 45degree right triangles are similar. Is he correct? Explain.
Answers
Please hurry !!!
Which is an x-intercept of the graphed function?
A) 0,4
B)-1,0
C)4,0
D)0,-1
Answers
we know that
The x-intercept is the value of the coordinate x when the value of the function is equal to zero
so
In this problem we have that the x-intercepts of the graphs are the points
[tex](-2,0)\\(-1,0)\\(1,0)\\(2,0)[/tex]
therefore
the answer is the option
B)-1,0
Given the following sequence, find the 23rd term: 10.5, 11, 11.5, 12, 12.5, . . .
Answers
an = a1 + (n - 1) * d
n = term to find = 23
a1 = first term = 10.5
d = common difference = 0.5
sub and solve
a(23) = 10.5 + (23 - 1) * 0.5
a(23) = 10.5 + 22 * 0.5
a(23) = 10.5 + 11
a(23) = 21.5 <===
If a boatman rows his boat 35km up stream and 55km downstream in 12 hours and he can row 30km upstream and 44 km downstream in 10hr , then the speed of the stream and that of the boat in still water
Answers
To answer this item, we let x be the speed of the boat in still water. The speed of the current, we represent as y.
When the boat travels upstream or against the current, the speed is equal to x – y and x + y if it travels downstream or along with the current.
The time it takes for the an object to travel a certain distance is calculated by dividing the distance by the speed.
First Travel: 35 / (x – y) + 55 / (x + y) = 12
Second travel: 30 / (x – y) + 44 / (x + y) = 10
Let us multiply the two equations with the (x-y)(x+y)
This will give us,
35(x + y) + 55(x – y) = 12(x-y)(x+y)
30(x + y) + 44(x – y) = 10(x-y)(x+y)
Using dummy variables:
Let a = x + y and b be x – y
35a + 55b = 12ab
30a + 44b = 10ab
From the first equation,
b = 35a/(12a – 55)
Substituting to the second equation,
30a + 44(35a/(12a – 55)) = 10a(35a/(12a-55))
The value of a is 11.
b = 35(11)/(12(11) – 55))
b = 5
Putting back the equations,
x + y = 11
x – y = 5
Adding up the equations give us,
2x = 16
x = 8 km/hr
The value of x, the speed of the boat in still water, is 8 km/hr.
speed of the stream = 3 km/hr
and speed of boat in still water= 8 km/hr
Step-by-step explanation:Let s be the speed of the boat upstream
and s' be the speed of the boat downstream.
We know that:
[tex]Time=\dfrac{distance}{speed}[/tex]
Hence, we get:
[tex]\dfrac{35}{s}+\dfrac{55}{s'}=12[/tex]
and
[tex]\dfrac{30}{s}+\dfrac{44}{s'}=10[/tex]
Now, let
[tex]\dfrac{1}{s}=a\ and\ \dfrac{1}{s'}=b[/tex]
Hence, we have:
[tex]35a+55b=12--------------(1)\\\\\\and\\\\\\30a+44b=10--------------(2)[/tex]
on multiplying equation (1) by 4 and equation (2) by 5 and subtract equation (1) from (2) we get:
[tex]a=\dfrac{1}{5}[/tex]
and by putting value of a in (2) we get:
[tex]b=\dfrac{1}{11}[/tex]
Hence, speed of boat in upstream= 5 km/hr
and speed of boat in downstream= 11 km/hr
and we know that:
speed of boat in upstream=speed of boat in still water(x)-speed of stream(y)
and speed of boat in downstream=speed of boat in still water(x)+speed of stream(y)
Hence, we get:
[tex]x-y=5\\\\\\and\\\\\\x+y=11[/tex]
Hence, on solving the equation we get:
[tex]x=8[/tex]
and y=3
Hence, we get:
speed of the stream = 3 km/hr
and speed of boat in still water= 8 km/hr
A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same. The cone is tilted at an angle so its peak touches the edge of the cylinder’s base. What is the volume of the space remaining in the cylinder after the cone is placed inside it?
Answers
Answer:
Step-by-step explanation:
Given that A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same
Whatever position cone is placed, the space remaining will have volume as
volume of the cylinder - volume of the cone
Let radius of cylinder be r and height be h
Then volume of cylinder = [tex]\pi r^2 h[/tex]
The cone has height as h and radius as r/2
So volume of cone = [tex]\frac{1}{3} \pi (\frac{r}{2} )^2h\\=(\pi r^2 h)\frac{1}{24}[/tex]
the volume of the space remaining in the cylinder after the cone is placed inside it
=[tex]\pi r^2 h (1-\frac{1}{24} )\\=\frac{23 \pi r^2 h}{24}[/tex]
Answer:
11/12 pie r^2 h
Step-by-step explanation:
Which algebraic expression shows the average melting points of helium, hydrogen, and neon if h represents the melting point of helium, j represents the melting point of hydrogen, and k represents the melting point of neon?
Answers
h = melting point of helium
j = melting point of hydrogen
k = melting point of neon
Number of values = 3
Therefore, the average is
(Sum of values)/(Number of values) = (h+j+k)/3
Answer:
The average melting point of helium, hydrogen, and neon is
(h+j+k)/3
Final answer:
The algebraic expression for finding the average melting points of helium, hydrogen, and neon, using variables h, j, and k as their respective melting points, is (h + j + k) / 3.
Explanation:
The question asks for the algebraic expression that represents the average melting points of helium, hydrogen, and neon. The variables h, j, and k denote the individual melting points of these elements, respectively. To calculate the average melting point, you would add the melting points of each element and divide by the number of elements.
The algebraic expression for the average melting point is:
(h + j + k) / 3
What is the value of 3×5+6×2+1
Answers
[tex]Parentheses \\ Equations \\ Multiplication \\ Addition \\ Subtraction[/tex]
Looks like we're doing: [tex]multiplication [/tex] and [tex]addition [/tex]
[tex]3(5) + 6(2) + 1 \\ 3(5)= 15 \\ 15 + 6(2) = \\ 6(2) =12 \\ 15 + 12 = 27 \\ 27 + 1 = 28 \\ Answer: 28 [/tex]
~ [tex]MeIsKaitlyn :) [/tex]
Good luck on your assignment and enjoy your day!
what is 1/3m-1-1/2n when m=21 and n=12
Answers
Answer:
Value of the expression is 0
Step-by-step explanation:
[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]
Given the value of m and n
m= 21 and n= 12
We plug in the value of m and n in the given expression
[tex]\frac{1}{3} m -1-\frac{1}{2} n[/tex]
[tex]\frac{1}{3}(21) -1-\frac{1}{2}(12)[/tex]
[tex]\frac{21}{3} -1-\frac{12}{2}[/tex]
[tex]7-1-6= 0[/tex]
So the value of given expression is 0 when we plug in the values of m and n
A spring is oscillating so that its length is a sinusoidal function of time. Its length varies from a minimum of 10 cm to a maximum of 14 cm. At t=0 seconds, the length of the spring was 12 cm, and it was decreasing in length. It then reached a minimum length at time t= 1.2 seconds. Between time t=0 and t=8 seconds, how much of the time was the spring longer than 13.5 cm?
Answers
The length of the oscillating spring varies from a minimum of 10 cm to a maximum of 14 cm.
Therefore its amplitude is A = (14 - 10)/2 = 2.
When t = 0 s, x = 12 cm.
Therefore the function is of the form
x(t) = 2 sin(bt) + 12
At t=0, x(t) is decreasing, and it reaches its minimum value when t = 1.2 s.
Therefore, a quarter of the period is 1.2 s.
The period is given by
T/4 = 1.2
T = 4.8 s
That is,
b = (2π)/T = (2π)/4.8 = π/2.4 = 1.309
The function is
x(t) = 2 sin(1.309t) + 12
A plot of x(t) is shown below.
When x(t) = 13.5, obtain
2 sin(1.309t) + 12 = 13.5
sin(1.309t) = (13.5 - 12)/2 = 0.75
1.309t = sin⁻¹ 0.75 = 0.8481 or π - 0.8481
t = 0.8481/1.309 or t = (π - 0.8481)/1.309
= 0.649 or 1.751
The difference in t is 1.751 - 0.649 = 1.1026.
This difference occurs twice between t=0 and t=8 s.
Therefore the spring length is greater than 13.5 cm for 2*1.1026 = 2.205 s.
Answer:
Between t=0 and t=8, the spring is longer than 13.5 cm for 2.205 s.
Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p. n = 195, x = 162; 95% confidence
Answers
To construct a 95% confidence interval for the population proportion, calculate the sample proportion p' and its complement q', determine the Z-score for 95% confidence, calculate the margin of error using the formula E = Z*sqrt((p'q')/n), and add/subtract E from p' to get the lower and upper bounds.
Explanation:To construct a 95 percent confidence interval for the population proportion p using the given sample data, we must first calculate the sample proportion (p') and its complement, the estimated proportion of failures (q'). Using the formula p' = x/n, we find that p' = 162/195. Next, we determine q' by calculating q' = 1 - p'.
With the sample proportion and its complement, we can use the standard formula for a confidence interval for a population proportion: p' ± Z*sqrt((p'q')/n), where Z* is the Z-score corresponding to the given degree of confidence. For a 95% confidence level, the Z-score is approximately 1.96.
By substituting the values of p', q', n, and the Z-score into the formula, we calculate the margin of error (E) and then the lower and upper bounds of the 95 percent confidence interval.
Suppose p' is 0.83 and q' is 0.17 for n = 195 and the Z-score for a 95% confidence interval is 1.96. The margin of error (E) would then be 1.96 * sqrt((0.83*0.17)/195), and the confidence interval would be p' ± E, resulting in a specific numerical range which would constitute our 95% confidence interval for the true population proportion.
The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is [tex]\( (0.7783, 0.8833) \)[/tex].
To construct a confidence interval for the population proportion [tex]\( p \),[/tex] we will use the given information: sample size [tex]\( n = 195 \)[/tex], number of successes [tex]\( x = 162 \),[/tex] and a confidence level of 95%.
The formula for the confidence interval for a population proportion [tex]\( p \)[/tex] is:
[tex]\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]
where:
- [tex]\( \hat{p} \)[/tex] is the sample proportion [tex](\( \frac{x}{n} \)),[/tex]
- [tex]\( z^* \)[/tex] is the critical value from the standard normal distribution corresponding to the desired confidence level.
Calculate the sample proportion [tex]\( \hat{p} \):[/tex]
[tex]\[ \hat{p} = \frac{x}{n} = \frac{162}{195} \][/tex]
[tex]\[ \hat{p} \approx 0.8308 \][/tex]
For a 95% confidence level, the critical value [tex]\( z^* \)[/tex] can be found using the standard normal distribution table or a calculator. It corresponds to the middle 95% of the distribution, which leaves 2.5% in each tail.
The critical value [tex]\( z^* \)[/tex] for a 95% confidence level is approximately 1.96.
Calculate the standard error [tex]\( SE \):[/tex]
[tex]\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ SE = \sqrt{\frac{0.8308 \cdot (1-0.8308)}{195}} \\ SE \approx \sqrt{\frac{0.8308 \cdot 0.1692}{195}} \\ SE \approx \sqrt{\frac{0.1405}{195}} \\ SE \approx \sqrt{0.0007205} \\ SE \approx 0.0268 \][/tex]
Now, we can construct the 95% confidence interval for [tex]\( p \):[/tex]
[tex]\[ \hat{p} \pm z^* \cdot SE \][/tex]
[tex]\[ 0.8308 \pm 1.96 \cdot 0.0268 \][/tex]
Calculate the margin of error:
[tex]\[ 1.96 \cdot 0.0268 \approx 0.0525 \][/tex]
So, the confidence interval is:
[tex]\[ 0.8308 \pm 0.0525 \][/tex]
Finalize the interval: [tex]\[ (0.7783, 0.8833) \][/tex]
The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is approximately [tex]\( (0.7783, 0.8833) \)[/tex]. This means we are 95% confident that the true population proportion [tex]\( p \)[/tex] lies between 0.7783 and 0.8833.
John’s gross pay for the week is $500. He pays 1.45 percent in Medicare tax, 6.2 percent in Social Security tax, 2 percent in state tax, 20 percent in federal income tax, and $20 as an insurance deduction. He does not have any voluntary deductions. What is John’s net pay for the week?
Answers
deductions :
medicare tax : 0.0145(500) = 7.25
S.S tax : 0.062(500) = 31.00
sales tax : 0.02(500) = 10.00
income tax : 0.2(500) = 100
insurance = 20
total deductions : 7.25 + 31 + 10 + 100 + 20 = 168.25
gross pay - deductions = net pay
500 - 168.25 = net pay
331.75 = net pay <===
John's net pay is calculated by subtracting deductions for Medicare, Social Security, state and federal taxes, and insurance from his gross pay of $500. The total deductions amount to $168.25, resulting in a net pay of $331.75.
Explanation:Calculation of John's Net Pay
To calculate John's net pay, we need to subtract all the deductions from his gross pay. Since his gross pay is $500, we will apply the following deductions:
Medicare tax: 1.45% of $500 = $7.25
Social Security tax: 6.2% of $500 = $31.00
State tax: 2% of $500 = $10.00
Federal income tax: 20% of $500 = $100.00
Insurance deduction: $20.00
Add up all deductions: $7.25 (Medicare) + $31.00 (Social Security) + $10.00 (State Tax) + $100.00 (Federal Tax) + $20.00 (Insurance) = $168.25
John's net pay is therefore calculated by subtracting the total deductions from his gross pay: $500.00 - $168.25 = $331.75.