90 POINTS!!!!! Given: KLIJ inscr. in k(O),m∠K = 64°, measure of arc LI = 69°, measure of arc IJ = 59°, measure of arc KJ =97°
Find: All angles of KLIJ
Answers
In any cyclic quadrilateral, angles opposite one another are supplementary, meaning
[tex]m\angle K+m\angle I=m\angle L+m\angle J=180^\circ[/tex]
and given that [tex]\boxed{m\angle K=64^\circ}[/tex], we have [tex]\boxed{m\angle I=116^\circ}[/tex].
By the inscribed angle theorem,
[tex]m\angle JLK=\dfrac12m\widehat{KJ}[/tex]
[tex]m\angle ILJ=\dfrac12m\widehat{IJ}[/tex]
and since
[tex]m\angle L=m\angle JLK+m\angle ILJ[/tex]
we have
[tex]m\angle L=\dfrac{97^\circ+59^\circ}2\implies\boxed{m\angle L=78^\circ}[/tex]
and it follows that
[tex]m\angle J=180^\circ-m\angle L\implies\boxed{m\angle J=102^\circ}[/tex]
Related Questions
Find the volume of the rectangular prism. Write answer as a fraction. Length=2/3 mm. Width=1/3mm. Height=2/3 mm.
Answers
put simply, the volume of a rectangular prism is just the product of all its 3 dimensions.
[tex]\bf V=\cfrac{2}{3}\cdot \cfrac{1}{3}\cdot \cfrac{2}{3}\implies V=\cfrac{4}{27}[/tex]
circle O is shown below. the diagram is not drawn to scale. 73°
Answers
53.5
Explanation:
Since triangle AOB is an isosceles triangle (AO = OB), the base angles of the triangle are 53.5. (180-73 = 107, 107/2 = 53.5)
Answer:
Option A
53.5
Step-by-step explanation:
A scatter plot containing the point (5, 29) has the regression equation yˆ=5x+2 . What is the residual e when x = 5? Enter your answer in the box.
e = ?
Answers
Answer:
The answer is below
Step-by-step explanation:
Answer:
The residual e=2 when x = 5.
Step-by-step explanation:
A scatter plot containing the point (5, 29), it means the observed value at x=5 it 29.
The given regression equation is
[tex]\hat{y}=5x+2[/tex]
Substitute x=5 in the above regression equation, to find the predicted value at x=-5.
[tex]\hat{y}=5(5)+2=25+2=27[/tex]
The formula to find the residual value e is
e = Observed value - Predicted value
[tex]e=29-27[/tex]
[tex]e=2[/tex]
Therefore the residual e=2 when x = 5.
(98 POINTS) (PLEASE HURRY!)
[1] x^2 - xy - x + y =
[2] 1 - x - x^2 + x^3 =
[3] xy + 1 + x + y =
[4] x^2y + y^2x + x^2x + y^3 =
Answers
Answer:
1. no like terms
2. x^ 3 − x^ 2 − x + 1
3.x y + x + y + 1
4.x^ 3 + x^ 2 y + x y^ 2 + y^ 3
Lol
A number by which another number is to be divided
Answers
Answer:
Dividend
Step-by-step explanation:
If you take a simple division problem like A = B / C
A is the quotient (result)
B is the dividend
C is the divisor
So, a number by which another number is to be divided is called a dividend, like A in the example above.
If C doesn't divide B in an exact manner (like in the case of 7 / 2), there's a remainder for the operation.
A divisor is a number by which another number, the dividend, is divided. In scientific notation, division involves dividing the coefficients and subtracting the exponents of the divisor from the dividend. Division is fundamentally related to multiplication, as it can be represented by multiplying by a reciprocal.
What does the relationship between the mean and median reveal about the shape of the data? The mean is less than the median, so the data is skewed left. The mean is more than the median, so the data is skewed right. The mean is equal to the median, so the data is symmetrical. The mean is equal to the median, so the data is linear.
Answers
Answer:
The mean is equal to the median, so the data is symmetrical
Step-by-step explanation:
Here is the data.
10 5 8 10 12 6
8 10 15 6 12 18
The given data: 10 5 8 10 12 6 8 10 15 6 12 18
For finding the Mean, we will have to add all numbers together and divide it by total number. i.e sum of terms divided by number of terms
Mean= 10+5+8+10+12+6+8+10+15+6+12+18 ÷ 12
Mean = 120 ÷ 12 = 10
For finding the Median, first we need to rearrange the data in ascending order
5 6 6 8 8 10 10 10 12 12 15 18
We can see that the middle values are 10 and 10. So, the median will be the average of those two middle values.
Median = 10+10 ÷ 2
Median = 20 ÷ 2 = 10
From the calculation, we can see that both the median and mean are equal so, the data is symmetrical
NEED HELP ASAP PRETTY PLEASE WITH A CHERRY ON TOP WILL GIVE A FOOT RUB IF REQUESTED ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Solve the equation check for extraneous solutions 4|5-5x|=7x+6
Answers
Answer:
[tex]\large\boxed{b.\ x=\dfrac{14}{27}\ \text{and}\ c.\ x=2}[/tex]
Step-by-step explanation:
[tex]4|5-5x|=7x+6\\4|-5(x-1)|=7x+6\\4|-5||x-1|=7x+6\\(4)(5)|x-1|=7x+6\\20|x-1|=7x+6\\\\\text{First step:}\\\text{Based on the de}\text{finition of the absolute value}\\\\|x-1|=\left\{\begin{array}{ccc}x-1&\text{for}\ x\geq1\\1-x&\text{for}\ x<1\end{array}\right\\\\\text{Let}\ x<1\to x\in(-\infty,\ 1).\ \text{Then}\ |x-1|=1-x:\\\\20(1-x)=7x+6\qquad\text{use the distributive property}\\20-20x=7x+6\qquad\text{subtract 20 from both sides}\\-20x=7x-14\qquad\text{subtract}\ 7x\ \text{from both sides}\\-27x=-14\qquad\text{divide both sides by (-27)}\\x=\dfrac{14}{27}<1\qquad \bold{:)}[/tex]
[tex]\text{Let}\ x\geq0\to x\in\left<1,\ \infty\right).\ \text{Then}\ |x-1|=x-1:\\\\20(x-1)=7x+6\qquad\text{use the distributive property}\\20x-20=7x+6\qquad\text{add 20 to both sides}\\20x=7x+26\qquad\text{subtract}\ 7x\ \text{from both sides}\\13x=26\qquad\text{divide both sides by 13}\\x=2\geq1\qquad \bold{:)}[/tex]
You invest $3000 in an account at 3.5% per year simple interest. How much will you have in the account at the beginning of the 7th year? Round your answer to the nearest whole dollar.
Answers
Answer:
$3735
Step-by-step explanation:
The formula for simple interest is I = Prt, where I is the interest earned, P is the initial investment, r is the interest rate in decimal form, and t is the time in years. We have everything we need to find the interest, which is the amount your investment earned while it sat there for 7 years. Once we find that interest amount, we will add it to the intial investment to find the total amount after 7 years that your money has grown to.
I = 3000(.035)(7) so
I = 735
3000 + 735 = 3735
Answer:
$3,630
Step-by-step explanation:
You invest $3,000 in an account at 3.5% per year simple interest.
We have to calculate the amount in the account at the beginning of the 7th year. This means we have to calculate the interest for completed 6 years.
Formula for simple interest
A = P(1+rt)
A = Amount after maturity
P = Principal amount ( 3,000)
r = rate of interest in decimal ( 0.035)
t = time in years ( 6 )
Now we put the values in to formula
A = 3,000(1 + 0.035 × 6)
A = 3,000 ( 1 + 0.021 )
A = 3,000 × 1.21
A = $3,630
The amount would be $3,630 at the beginning of the 7th year.
please help me and plz explain thoroughly i need help asap!
Given the function f(x) = 2(x + 8), find x if f(x) = 12.
2
40
−2
14
Answers
f(x) = 2(x+8)
replace f(x) with 12 and then solve for x:
12 = 2(x +8)
Use the distributive Property on the right side:
12 = 2x +16
Subtract 16 from each side:
-4 - 2x
Divide both sides by 2:
x = -4 / 2
x = -2
Answer:
−2
Step-by-step explanation:
f(x) = 2(x+8)
f(x) = 12
12 = 2(x+8)
Divide each side by 2
12/2 = 2/2(x+8)
6 = x+8
Subtract 8 from each side
6-8 =x+8-8
-2 = x
You play a board game with your little brother. You pick a card that sends you back five spaces. On your next turn, you must move back three more spaces. On your third turn, you get to move forward three spaces. Which expression best shows how you could find your new spot on the game board?
Answers
Answer:
Step-by-step explanation:
- 5 + (-3) + 3 is the correct option.
Answer:
-5 + (-3) + 3
Step-by-step explanation:
Which trigonometric function requires a domain restriction of [-pi/2, pi/2] to make it invertable?
Answers
Answer:
[tex]y=\tan x[/tex]
Step-by-step explanation:
The trigonometric function that needs a domain restriction of [tex][-\frac{\pi}{2},\frac{\pi}{2} ][/tex] to make it invertible is [tex]y=\tan x[/tex].
The function [tex]y=\tan x[/tex] will pass the horizontal line test on this interval therefore making it an invertible function on this interval.
This explains why the inverse tangent function, [tex]y=\tan^{-1} x[/tex] has range [tex][-\frac{\pi}{2},\frac{\pi}{2} ][/tex].
Answer:
A) [tex]f(x)=sin x[/tex]
Step-by-step explanation:
I just did the test and this was the correct answer.
25 pts Maureen tracks the range of outdoor temperatures over three days. She records the following information.
(picture attached)
Which answer below expresses the intersection of the three days as an inequality in terms of temperature, t. (The Intersection would be the temperatures they have in common.)
0 < t < 40
0 ≤ t ≤ 40
-23 ≤ t ≤ 50
-23 < t < 50
Answers
Answer: 0 ≤ t ≤ 40
Step-by-step explanation:
0 and 40 are included in all 3 number lines.
Given that g(x)=3x^2 -4x+3, find each of the following.
PICTURE DOWN BELOW!!!
Answers
Answer:
see explanation
Step-by-step explanation:
To evaluate the expression for the given values.
Substitute the given values for x into g(x)
a
g(0) = 3(0)² - 4(0) + 3 = 0 - 0 + 3 = 3
b
g(- 1) = 3(- 1)² - 4(- 1) + 3 = 3 + 4 + 3 = 10
c
g(2) = 3(2)² - 4(2) + 3 = 12 - 8 + 3 = 7
d
g(- x) = 3(- x)² - 4(- x) + 3 = 3x² + 4x + 3
e
g(1 - t)
= 3(1 - t)² - 4(1 - t) + 3
= 3(1 - 2t + t²) - 4 + 4t + 3
= 3 - 6t + 3t² - 4 + 4t + 3
= 3t² - 2t + 2
Can someone help me with these questions please?
1. Which function pass through the points (1, 4), (2, 9), and (3, 16)?
y = (x + 1)2
y = (x + 3)2
y = 7x - 5
2. Suppose that after swimming 1 lap in a pool, you can swim 10 laps in an hour. What is an equation that represents the total number of laps you have swam y in terms of the number of hours x?
y(x) = 10 + x
y(x) = 10(x + 1)
y(x) = 10x + 1
3. Is the following an arithmetic sequence? If so, what is the common difference? {4, 2, 0, -2, -4, -6, …}
Yes, d = -2
No
Yes, d = 2
4. Which function has the following graph?
y = -2x + 4 b. y = 2x + 4 c. y =12x + 4
5. What is the slope of the line containing (3, 5) and (2, 7)?
m = 2 b. m = -2 c. m = 125
6. The graph of a line passes through the point (-3, 2) and has a slope of 4. What is an equation of the line?
y = 4x - 10
y = 4x - 11
y = 4x+ 14
7. Which line is parallel to y = 4x - 2?
y = -14x + 3
y = -4x + 5
y = 4x + 5
8. Which system of equations has (2, 3) as a solution?
y = 2x - 1 b. y = 2x + 1 c. y = 4x - 5
y = x + 1 y = x - 1 y = 2x
Answers
Answer:
the first function, y = (x + 1)^2, does pass through the points (1, 4), (2, 9), and (3, 16).
Step-by-step explanation:
Note that 4, 9 and 16 are squares of 2, 3 and 4. So it's likely that the parent function is of the form f(x) = x^2. Note that f(1) = 1, f(2) = 4, f(3) = 9 and f(4) = 16.
Let's now determine whether these three points satisfy the first given equation, y = (x + 1)^2: Does 4 = (1 + 1)^2? Does 4 = 2^2? YES.
Does 9 = (2 + 1)^2? Does 9 = 3^2? YES
Does 16 = (3 + 1)^2? YES
So the first function, y = (x + 1)^2, does pass through the points (1, 4), (2, 9), and (3, 16).
Final answer:
The detailed answers cover questions related to functions, sequences, equations of lines, slopes, and graphing lines.
Explanation:
Question 1:
The function that passes through the points (1, 4), (2, 9), and (3, 16) is y = (x + 1)^2.
Question 2:
The equation representing the total number of laps swum y in terms of hours x is y(x) = 10x + 1.
Question 3:
The sequence {4, 2, 0, -2, -4, -6, …} is an arithmetic sequence with a common difference of -2.
Question 5:
The slope of the line containing (3, 5) and (2, 7) is m = -2.
Question 6:
For a line passing through (-3, 2) with a slope of 4, the equation of the line is y = 4x + 14.
The following binomials are multiplied together as shown below. Which of the following is a true statement?
(x+y)(a+b)
= xa+xb+ya+yb
A. There is no error; the binomials were correctly multiplied together.
B. Binomials with different variables cannot be multiplied together.
C. The product was not simplified correctly.
D. There should only be three terms when multiplying two binomials.
Answers
For this case we have that by definition:
[tex](a + b) (c + d)[/tex] is equal to:
[tex]ac + ad + bc + bd[/tex]
Applying the distributive property.
Then, it can be seen that the given binomials were multiplied correctly.
ANswer:
There is no error; the binomials were correctly multiplied together.
The length of a rectangle is 3 meters more than the width. The perimeter of the rectangle is 26 meters. If the width is b, which equation represents the situation? How many solutions will this equation have?
Answers
Answer:
Thehe answer is 75
Step-by-step explanation:
Answer with Step-by-step explanation:
The length of a rectangle is 3 meters more than the width.
If the width is b.
⇒ Length=b+3
The perimeter of the rectangle is 26 meters.
Since, Perimeter=2(Length+width)
⇒ 2(b+3+b)=26
dividing both sides by 2
⇒ b+3+b=13
⇒ 2b=13-3
⇒ 2b=10
dividing both sides by 2
⇒ b=5
⇒ Width=5 meter
and Length=5+3=8 meter
which equation represents the situation: 2(b+3+b)=26How many solutions will this equation have: UniqueThe function f(t) = 4 cos(pi over 3t) + 15 represents the tide in Bright Sea. It has a maximum of 19 feet when time (t) is 0 and a minimum of 11 feet. The sea repeats this cycle every 6 hours. After five hours, how high is the tide?
13.5 feet
16 feet
18.5 feet
17 feet
Answers
Answer:
17 ft
Step-by-step explanation:
I don't think you meant to put pi over 3t ... I think you meant to put pi just over 3
Just plug in 5 assuming t is time in hours.
Evaluate 4 *cos(pi/3 *5)+15
which is 17 ft
The height of tower after 5 hours is 17.68 feet.
What is cosine range?
The graph of the cosine function looks like this: The domain of the function y=cos(x) is all real numbers (cosine is defined for any angle measure), the range is −1≤y≤1 .
The Cosine function : f(t) = 4 cos(π/3t) + 15
and, f(t) = 4 cos(2kπ + π/3t) + 15
where k= 0,1,2,3....
The range of cosine function is : Maximum= +1 and minimum= -1
At t=0,
For maximum, f(t)= 4 x1 +15
= 19 feet
For minimum, f(t) = 4 x(-1)+15
= 11 feet
After , 6 hours ,the tide function is: 6 n=5
n= 6/5
f(t) = 4 cos ( 2* [tex]\frac{5* \pi}{6}[/tex] + [tex]\frac{\pi}{3*5}[/tex] ) +15
= 4 cos (26 π/15) + 15
= 4 cos[tex]312^{0}[/tex] + 15
= [tex]4 cos 48^{0}+ 15[/tex]
= 4 x 0.6691 +15
= 17.68 feet.
Thus, the height of tower after 5 hours is 17.68 feet
Learn more about concept here:
https://brainly.com/question/3714457
#SPJ2
Please help me please !!
Answers
Answer:
Acute
Step-by-step explanation:
7² + 9² ? 11²
49 + 81 ? 121
130 is greater than 121, so it is an acute triangle
A circle has a diameter of 8 m. What is the approximate area of the circle? 200.96 m2 12.56 m2 25.12 m2 50.24 m2
Answers
Answer:
50.24
Step-by-step explanation:
area of a circle =pi r^2
diameter = 2 * radius
radius=8/2=4
area=pi 4^2
=pi 16
=50.26
Please answer this multiple choice question CORRECTLY for 30 points and brainliest!!
Answers
Answer:
[tex]\text{D.}\quad d=\dfrac{206-8(10)}{7}[/tex]
Step-by-step explanation:
The total length of the space between rungs is the overall length less the width of 8 rungs, so is 206 -8(10). That space is divided into 7 equal parts, as shown by the equation in choice D.
_____
Choice A looks similar, but is not. In that equation, only the term 8(10) is divided by 7. You want the difference to be divided by 7, so must have a grouping symbol of some kind. Choice D uses the division bar to group the terms of the numerator. Parentheses would work, too, as in ...
d = (206 -8(10))÷7
but without them, the equation is incorrect.
Find the length of each leg. Leave answer in simplest radical form.
Question 28 options:
16√2
8
8√2
4√2
Answers
Answer:
8√2
Step-by-step explanation:
The hypotenuse of a right triangle is √2 times the leg length, so you have ...
[tex]x\sqrt{2}=16[/tex]
Dividing by the coefficient of x gives ...
[tex]x=\dfrac{16}{\sqrt{2}}=\dfrac{16\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}}=8\sqrt{2}[/tex]
Each leg has length 8√2.
Triangle TUV has been dilated to form triangle TꞌUꞌVꞌ.
What is the scale factor?
Answers
Answer:
4.4, 4.4 : 1, 22 : 5
Step-by-step explanation:
There are many ways to solve this, but I'll do it by finding the ratio between V'T' and VT.
That's easy, we're given the lengths of these two. The ratio would be 8.8/2. 8.8/2 = 4.4, or (8.8*5)/(2*5) = 22/5
Identify the coefficient of 13m
Answers
Answer: the coefficient of 13m is 13.
Answer:
The coefficient of 13 m is 13.
Step-by-step explanation:
Consider the provided expression.
[tex]13m[/tex]
Numbers appear with variables are called coefficients.
Alphabets in an expression are called variables.
Numbers appear without variables are called Constant terms.
Here we need to identity the coefficient.
From the above definition we know the number appear with the variables are coefficients.
Since m is variable, thus the number 13 must be the coefficient.
Hence, the coefficient of 13 m is 13.
A fair coin is tossed 3 times. What is the probability that the coin with land showing heads on all three tosses.
Answers
Answer:2/3
Step-by-step explanation:
Answer:
1/8
Step-by-step explanation:
one way we can solve this is by making a tree using H for heads and T for tails
since it can land H or T, we put them down like this. then we can get H or T again because its a coin, so we put that under each letter. finally we do the same thing again to get 8 possibilities. now we can count how many of them was HHH, which is one
H T
/ \ / \
H T H T
/\ /\ /\ /\
H T H T H T H T
^ this one
or do 2 x 2 x 2 because there are 2 possibilities each three times
The question is in the picture PLEASE HELP ME!!! idk how to do this
Answers
Answer:
35
Step-by-step explanation:
Put the numbers in the formula and do the arithmetic.
nCk = n!/(k!(n -k)!)
7C3 = 7!/(3!(7-3)!) = 7·6·5/(3·2·1) = 7·5 = 35
_____
It is convenient to use the largest of the factorials in the denominator to cancel as many factors as you can from the numerator, then cancel factors from the remaining numbers. Here after canceling 4! = 4·3·2·1 from the numerator, we are left with 7·6·5 divided by 3! = 3·2·1 = 6. Obviously, this will cancel the 6 in the numerator product, leaving only 7·5 = 35.
Some graphing and/or scientific calculators will have this function built in.
Maryanne began a job that gives her 3 days of vacation each year she is employed up to a maximum of 24 days of vacation time. How many years will she be at the company before she reaches the maximum amount of vacation time?
Answers
Answer:
8 years
Step-by-step explanation:
Answer:
she will be at the company for 8 years.
Step-by-step explanation:
The number of days of vacation for each year = 3
Maryanne is employed up to a maximum of 24 days of vacation time.
We need to find the number of years she will be in the company before she reaches the maximum amount of vacation of time.
The number of years she will be in the company = The number of maximum days of vacation time divided by the number of days of vacation for each year
= [tex]\frac{24}{3}[/tex]
= 8 years
Therefore, she will be at the company for 8 years.
Please Help!
When Mario has to leave the house for a while, he tethers his mischievous puppy to the corner of a 12 ft-by-8 ft shed in the middle of his large backyard. The tether is 18 feet long. Which description fits the boundary of the locus of points in the yard that the puppy can reach?
A. semicircles of radii 18 ft, 10 ft, and 6 ft
B. a three-quarter circle of radius 18 ft, quarter circles of radii 10 ft and 6 ft
C. semicircles of radii 18 ft, 12 ft, and 8 ft
D. a three-quarter circle of radius 18 ft, quarter circles of radii 12 ft and 8 ft
Answers
Answer:
B. a three-quarter circle of radius 18 ft, quarter circles of radii 10 ft and 6 ft
Step-by-step explanation:
A diagram can be a useful aid to answering this question. At one corner of the shed, 8 ft of rope is no longer available, so the quarter circle has a radius of 18-8 = 10 ft. (That side of the shed is 12 ft, so the area is only a quarter circle.)
At the other corner of the shed, 12 ft of rope is no longer available, so the quarter circle has a radius of 18-12 = 6 ft. (That side of the shed is 8 ft, so the area is only a quarter circle.)
In the attached diagram, the 3/4 circle with radius 18 ft is shown red, and the quarter circles are shown in green.
You are installing a brick sidewalk. The brick portion of the sidewalk will occupy an area of 100 feet long by 4 feet wide. Each brick will occupy an area 8 inches long by 4 inches wide. What is the minimum number of bricks you will need to build the sidewalk?
Answers
Answer:
1,800 bricks are needed
Step-by-step explanation:
First convert the Length and width of the sidewalk into inches
Entire Sidewalk: L= 100x12 =1200in and W= 4 x 12 = 48in
Then we know Area = LxW, so we will do this for both the sidewalk and the brick.
Area of sidewalk: 1200 x 48 = 57600
Bricks ( no need to convert since the measurements are already in inches): 8 x 4 = 32
Now we will divide the area of the entire sidewalk by the area of a single brick to find out how many bricks you need to complete the whole sidewalk:
57600/32= 1,800 bricks
Thelma and Laura start a lawn-mowing business and buy a lawnmower for $225. They plan to charge $15 to mow one lawn. They would like to earn a profit of at least $750
Part A: Define a variable to represent the unknown quantity in this situation.
Part B: Write an inequality to represent the amount of money charged per lawn, the cost of the lawnmower and the profit.
Answers
Answer:
I got 65
Step-by-step explanation:
The profit will be given as:
P=$15x−$225
(where we subtracted the cost of the mower) so that P=$750
x will be the required number of needed lawns.
We need to solve:
$750=$15x−$225
rearranging:
x=$750+$225 / $15 =65 lawns
Define a variable and write an inequality to represent earning a specific profit in a lawn-mowing business.
Define variable:
Let x represent the number of lawns mowed.
Write inequality:
The inequality is 15x - 225 ≥ 750, where 15x is the amount earned, 225 is the cost of the lawnmower, and 750 is the desired profit.
In a simple random sample of 90 patients who saw a certain dentist, 8 patients had their teeth whitened. Which interval is the 95% confidence interval for the percent of all the dentists patients who had their teeth whitened?
Answers
Answer:
(3.01%, 14.77%)
Step-by-step explanation:
The confidence interval of a proportion is:
CI = p ± SE × CV,
where p is the proportion, SE is the standard error, and CV is the critical value (either a t-score or a z-score).
We already know the proportion: 8/90. But we need to find the standard error and the critical value.
The standard error is:
SE = √(p (1-p) / n)
SE = √((8/90) * (82/90) / 90)
SE = 0.03
To find the critical value, we must first find the alpha level and the degrees of freedom.
The alpha level for a 95% confidence interval is:
α = (1 - 0.95) / 2 = 0.025
The degrees of freedom is one less than the sample size:
df = n - 1 = 90 - 1 = 89
Since df > 30, we can approximate this with a normal distribution.
If we look up the alpha level in a z score table, we find the z-score is 1.96. That's our critical value. CV = 1.96.
Now we can find the confidence interval:
CI = 8/90 ± 0.03 * 1.96
CI = 0.0889 ± 0.0588
CI = (0.0301, 0.1477)
So we are 95% confident that the percent of patients who had their teeth whitened is between 3.01% and 14.77%.