The three intervals do you have the most confidence that the interval captures the population mean (x) is (2.4, 4.5). option (b).
An interval is a range of values that is used to estimate a population parameter, such as the mean.
To answer your question, let's look at the three intervals you provided: (1) (1.3, 4), (2) (2.5, 4.5), and (3) (2, 5). We want to determine which interval has the highest level of confidence in capturing the population mean.
Interval (1) has a range of 2.7 units, which is relatively large. This means that it is possible that the true population mean could fall anywhere within this range.
Interval (2) has a smaller range of 2 units, which means that it is more likely to capture the true population mean. Interval (3) has the largest range of 3 units, which makes it less likely to capture the true population mean compared to interval (2).
Hence the correct option is (b).
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Complete Question:
For which of the three intervals do you have the most confidence that the interval captures the population mean (x)?
a) (13,4)
(b) (2.5, 4.5)
(c) (2,5)
She begins at sea level, which is an elevation of 0 feet.
She travels down 14.3 feet.
She then travels directly up 4.3 feet.
Next, she travels down a second time, 57.9 feet.
If the scientist used the submarine to study about "ocean-life", then the "total-distance" that she need to ascend to get back at "sea-level" is 67.9 feet.
The scientist begins her journey at sea-level, which is an elevation of 0 feet.
She then descends 14.3 feet, which means her current elevation is -14.3 feet.
Next, she travels up by 4.3 feet, which brings her current elevation to -10 feet.
She then descends a second time by 57.9 feet, which brings her current elevation to -67.9 feet.
Therefore, To return to sea level from here, she needs to ascend by a 67.9 feet.
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The given question is incomplete, the complete question is
A scientist uses a submarine to study ocean life,
She begins at sea level, which is an elevation of 0 feet.
She travels down 14.3 feet.
She then travels directly up 4.3 feet.
Next, she travels down a second time, 57.9 feet.
How much distance (in feet) must she now ascend to get back at sea-level?
The triangle above has the following measures.
m∠C = 45°
a = 1.5 in
Use the 45-45-90 Triangle Theorem to find the length of the hypotenuse. Include correct units.
Show all your work.
The length of the hypotenuse is 1.5√2 inches
Finding the length of the hypotenuse.From the question, we have the following parameters that can be used in our computation:
m∠C = 45°
a = 1.5 in
The length of the hypotenuse in a 45-45-90 Triangle is
Hypotenuse = Leg * √2
In this case, we have
Leg = a = 1.5
Hypotenuse = b
So, we have
b = 1.5 * √2
Evaluate
b = 1.5√2
Including the correct units, we have
b = 1.5√2 inches
Hence, the length of the hypotenuse is 1.5√2 inches
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Can someone help me please!!!
Answer:
Option A
4.59
Step-by-step explanation:
The given triangle is a right triangle
The known angle measures are 74° and 90°
The other angle opposite side x = 180 - (74 + 90) = 16°
(Or we could simply use 90° - 74° since the sum of the non-right angles must be 90°)
The relationship between the side opposite the angle 16°, namely x and the side adjacent to the angle 16° namely the side of length 16 is given by
tan(16°) = x/16
tan(16°) = 0.286745
=> 0.286745 = x/16
=> x = 16 x 0.286745
=> x = 4.5879
Rounded to two decimal places that would be 4.59 which is Option A
what is the distance from X to Y. X (0,6) Y (9,-6)
A. 225 units
B. 21 units
C. 15 units
D. 3 units
1 unit is 2 if that makes sense
The distance between X and Y is 15 units. So, the correct answer is option C.
The distance between two locations on a plane is the length of the line segment connecting them.
The formula to find the distance between the two points is usually given by d=√{(x₂-x₁)² + (y₂-y₁)²}
Here, X(0, 6) and Y(9, -6), so
d =√((9 - 0)² + (-6 - 6)²)
= √(81 + 144)
= √(225)
= 15
Therefore, the required distance is 15 units.
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What is the value of nine 2/3-4 1/5
let's firstly convert the mixed fraction to an improper fraction.
[tex]\stackrel{mixed}{4\frac{1}{5}}\implies \cfrac{4\cdot 5+1}{5}\implies \stackrel{improper}{\cfrac{21}{5}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{2}{3}-\cfrac{21}{5}\implies \cfrac{(5)2~~ - ~~(3)21}{\underset{\textit{using this LCD}}{15}}\implies \cfrac{10-63}{15}\implies \cfrac{-53}{15}\implies -3\frac{8}{15}[/tex]
The hypotenuse of a right triangle measures 16 cm and one of its legs measures 5 cm. Find the measure of the other leg. If necessary, round to the nearest tenth.
Answer:
15.2 (rounded)
Step-by-step explanation:
To solve we need to use the pythagrom theorem. This theory states that [tex]a^{2} + b^2 = c^2[/tex]. In which a and b are the legs of the triangle and c is the hypotenuse.
Because we have these numbers we can plug them into our equation:
[tex]5^2 + b^2 = 16^2[/tex]. [tex]5^2 = 25[/tex] and [tex]16^2 = 256[/tex] so our final equation is [tex]25 + b^2 = 256[/tex]. To find b (our missing leg) we need to subtract 25 from both sides. This leaves us with the problem [tex]b^2=231[/tex]. The final step would be to square root both sides and we get our answer of about 15.2
Use graph to work out x and y
Answer: x = 1, y = 4
Step-by-step explanation:
First, you would need to set the equations equal to each other.
y = 3x + 1 and y = -2x + 6
They can be set equal to each other because they are equal to the same y.
So, you would get the equation:
3x + 1 = -2x + 6
From there, solve for x.
3x + 2x = 6 - 1
5x = 5
x = 1
Now, plug that x-value into the original equations to get y.
y = 3(1) +1
= 3 + 1
= 4
Plug the x-value into the other equation and if you get the same answer, you're correct!
y = -2(1) + 6
= -2 + 6
= 4
So, as an ordered pair (in the form [x, y]), the answer would be (1, 4).
The number line shown can be used to model the expression 7/10-3/10 To model the expression, how many parts should the segment of the number line between 0 and 1 be divided into? Explain your answer. Describe each of the remaining steps needed to model the expression. What is the value of the expression?
The number of parts is 10 and the value of the expression is 4/10
Given that
7/10 - 3/10
The above expressions are fractions and the denominator is 10
This means that the number of parts between 0 and 1 is 10
What is the value of the expression?Recall that
7/10 - 3/10
When evaluated, we have
7/10 - 3/10 = 4/10
Hence, the solution si 4/10
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a survey asked how many colleges undergraduate students applied to, with 206 students responding to this question. this sample yielded an average of 9.7 college applications with a standard deviation of 7. the college board website states that counselors recommend students apply to roughly 8 colleges. how would you test if the data provide convincing evidence that the average number of colleges students apply to is higher than recommended? what would be your hypotheses?
The hypothesis regarding the convincing evidence that the average number of colleges students apply to is accepted because p-value is more than the chosen significance level.
The null hypothesis will be that the average number of colleges students apply to is equal to or less than 8. The alternative hypothesis would be that the average number of colleges students apply to is greater than 8.
We can apply a one-sample t-test to test this hypothesis. By doing so we can evaluate the t-statistic
t = (x'- μ) / (s / √n)
here
x' = sample mean,
μ = hypothesized population mean,
s = sample standard deviation,
n = sample size
(x'- μ) / (s / √n)
Staging values
(9.7 - 8) /(7) /√206)
=( 0.3) /( 7/ 14.35)
= (0.3) /(0.4)
= 0.75
Now, the p-value associated with this t-statistic using a t-distribution . The p-value in this case is greater than the chosen significance level (usually 0.05), we will accept the null hypothesis.
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g a finite well always has at least one bound state. why does the argument of exercise 38 fail in the case of a finite well?
The wavelength of a particle in a finite well can be very large, allowing it to extend into the classically forbidden region outside the barriers. This means even a small potential barrier can hold the particle, resulting in at least one bound state.
A finite well always has at least one bound state, meaning that even a small potential barrier can hold a particle. This is because the wavelength of the wave extends into the classically forbidden region outside the barriers on both sides of the well, leading to a very large wavelength.
According to the formula λ = hp, a larger wavelength corresponds to a smaller momentum. As kinetic energy is directly proportional to momentum, a smaller momentum means a larger kinetic energy. Thus, even a small potential barrier can hold the particle. The argument of fails in the case of a finite well because of the wave's extended wavelength.
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--The given question is incomplete, the complete question is given
" A finite well always has at least one bound state. Why does the argument of fail in the case of a finite well?
In the case of the finite well, the wave extends into the classically forbidden region on the outside of the barriers on both sides of the well, so its wavelength can be very large."--
100 POINTS PLEASE HELP ANSWER ALL QUESTIONS!!!!!!
Answer: for number 1 it is 4 and 5
For number 2 it is 7.6 and 7.7
Number 3 is c
Step-by-step explanation: (I don’t know the last one)
In the above figure, m∠A = 34° and m∠B = (2x + 22)°. If angles A and B are complementary angles, what are the value of x and the measure of angle B?
Answer:
Step-by-step explanation:
[tex]m\angle(A) + m\angle(B)=90\\34+(2x+22)=90\\\textrm{Solve for x.}\\\\34+2x+22=90\\2x+56=90\\2x=90-56\\2x=34\\x=17\\m\angle(B)=2x+22=2(17)+22=34+22=56\degree[/tex]
Colin leaves school to go home. He walks 3 blocks south and then 9 blocks west. If Colin could walk in a straight line to the school, what is the exact distance between Colin and the school?
3√10 blocks
2√45 blocks
6√2 blocks
2√3 blocks
Answer:
The answer would be A. 3√10 blocks
Step-by-step explanation:
Hope this helps :))
Please let me know if im wrong :(
Answer:
3√10 blocks
Step-by-step explanation:
It's A I took the test
Which of these values is a possible solution
to the inequality?
52+3<5
The inequality equation which represents Three less than twice a number is equal to at least 52 is 10 - 2x ≥ 52 and x ≤ -21
What is inequality in mathematics?In mathematics, an inequality is described as a relation which makes a non-equal comparison between two numbers or other mathematical expressions.
Let
The unknown number = x
Ten less than twice a number is equal to at least 52;
10 - 2x ≥ 52
we Subtract 10 from both sides
- 2x ≥ 52 - 10
- 2x ≥ 42
we then divide both sides by - 2
x ≤ 42/-2
x ≤ -21
Therefore , x ≤ -21 is the solution to the inequality 10 - 2x ≥ 52 if x comes first.
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A vehicle purchased for $20,700 depreciates at a constant rate of 6%. Determine the approximate value of the vehicle
13
years after purchase. Round to the nearest whole dollar.
$
After 13 years the depreciation value of the vehicle will be $9253 if the vehicle is purchased for $20,700.
A percentage is a number that indicates how many out of 100 it is, and it can also be expressed as a decimal or a fraction. Put the percentage value in the numerator and 100 in the denominator to convert a percentage to a fraction.
V(n) = v₀ × (1-rate)ⁿ
V(n) is the value at year n (we want n =13 years)
V₀ is the starting value = $20,700
Rate is the depreciation rate expressed as a decimal = 0.06
n = years = 13
v(10) = 20700 × (1-0.06)¹³
= 20700×(0.94)¹³
= 20700×(0.447)
= 9252.9
= 9253 (approximately)
Therefore after 13 years the vehicle will cost $9253
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pls I need help I ve doing 2 hours and 25 mins and 43 seconds of ixl and I cant figure out this question I already answered 69 questions help
Mrs. Gregory runs a house cleaning business. To save money, she bought a 1-liter container of concentrated cleaner. She mixed the cleaner with 10 liters of water. To make sure it was strong enough, she used 200 milliliters to clean her kitchen. It worked well, so she filled as many 600-milliliter spray bottles as she could. How many spray bottles did she fill?
Answer:
Mrs Gregory started with 1 litre of concentrated cleaner, which is equal to 1000 millilitres of cleaner. (we have to convert litre to millilitres because " she used 200 millilitres to clean her kitchen)
She mixed the 1000 millilitres of cleaner with 10 litres of water, which is equal to 10,000 millilitres of water. This gives a total of:
1000 + 10,000 = 11,000 millilitres of cleaner solution
She used 200 millilitres of the solution to clean her kitchen, which leaves:
11,000 - 200 = 10,800 millilitres of cleaner solution
To fill the 600-millilitre spray bottles, she needs to divide the total amount of cleaner solution by the amount of solution in each bottle:
10,800 / 600 = 18
Therefore, Mrs Gregory filled 18 spray bottles.
Elsie is going to flip a coin and spin a spinner. The coin has a heads and a tails. The spinner has four equal parts that are yellow, blue, red, and green.
What are the chances that Elsie gets heads and red on the spinner?
A.1/8
B.1/6
C.1/4
D.1/2
The probability that Elsie gets heads and red on the spinner is 1/8
Given data ,
Elsie is going to flip a coin and spin a spinner
The coin has a heads and a tails
The spinner has four equal parts that are yellow, blue, red, and green
Now , The chance of getting heads on the coin is 1/2, and the chance of getting red on the spinner is 1/4, since there are four equal parts.
To find the probability of getting both events together, we need to multiply the probabilities:
P(heads and red) = P(heads) × P(red) = (1/2) × (1/4) = 1/8
Hence , the probability of Elsie getting heads and red on the spinner are 1/8
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Sketch a graph of the first two cycles of s (0) = sin 0. Then label your graph to show the following positions of a passenger on The Screamer.
a. The passenger gets on initially.
b. The passenger reaches the bottom of the water pit.
c. The passenger is halfway between the highest point of The Screamer and the ground level.
The point where the passenger gets on initially cannot be determined as it is undefined
Sketching the graph and interpreting the statementsThe graph of the function s(θ) = sin(θ) is added as an attachment
a. The passenger gets on initially.
From the graph, this is when θ = 0
However, s(θ) = sin(θ) is undefined at θ = 0
This means that when the passenger gets on initially cannot be determined
b. The passenger reaches the bottom of the water pit.
From the graph, this is when s(θ) = sin(θ) equals -1
The values of θ at this point according to the graph are
θ = 3π/2 and θ = 7π/2
c. The passenger is halfway between the highest point of The Screamer and the ground level.
From the graph, this is when s(θ) = sin(θ) equals 0.5
The values of θ at this point according to the graph are
θ = π/6, θ = 5π/6, θ = 13π/6, and θ = 17π/6
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each letter of the word probable is written on a separate card. the cards are placed face down and mixed up. what is the probability that randomly selected card has a consonant?
Thus, the probability that a randomly selected card has a consonant is 62.5%. This means that if you were to randomly select a card from the pile of cards, there is a 62.5% chance that the card would be a consonant.
The number of consonants in the word "probable" and the total number of cards. There are 5 consonants in "probable" (p, r, b, l, and b) and a total of 8 cards.
In the word "probable," there are 5 consonants (p, r, b, b, and l) and 3 vowels (o, a, and e). To calculate the probability of choosing a consonant, divide the number of consonants by the total number of letters:
Probability = (Number of consonants) / (Total number of letters)
The probability of selecting a consonant can be calculated by dividing the number of consonants by the total number of cards:
5 consonants / 8 cards = 0.625 or 62.5%
Therefore, the probability that a randomly selected card has a consonant is 62.5%. This means that if you were to randomly select a card from the pile of cards, there is a 62.5% chance that the card would be a consonant.
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F(x) = x squared + 2x + 1
show F(x+2) - F(x) = 4x+8
full working out
The given statement F(x + 2) - F(x) = 4x + 8 is proved where the function is F(x) = x² + 2x + 1.
Given the function is,
F(x) = x² + 2x + 1
And the given statement which we have to prove is: F(x + 2) - F(x) = 4x + 8.
Now, to find F(x + 2) we have to replace x with (x + 2).
F(x + 2) = (x + 2)² + 2(x + 2) +1
F(x + 2) = x² + 2² + 2*2*x + 2x + 2*2 + 1 [Since, (a +b)² = a² + b² + 2ab]
F(x + 2) = x² + 4 + 4x + 2x + 4 + 1
F(x + 2) = x² + 6x + 9
So now calculating left hand side of expression which have to be proved,
F(x + 2) - F(x)
= x² + 6x + 9 - (x² + 2x + 1)
= x² + 6x + 9 - x² - 2x - 1
= (x² - x²) + (6 - 2)x +(9 - 1)
= 0 + 4x + 8
= 4x + 8
Hence the given statement is proved.
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1.) For a ∆PQR, find the value of x where S and T are points on sides PQ and PR respectively and ST is parallel to QR.
A. PS=5 cm, SQ=(3x+1) cm, PT=6 cm, and TR=(4x) cm
B. PS=(3x+4) cm, SQ= 12 cm, PT = (4x+3) cm, and TR= 15 cm
2.) The distance from Sam’s house to school is (3x+17)m and from school to library is (5x-3)m. Find the distance from house to library and school to house, if the school is in the middle of the house and the library.
3.) Find the value of x, y, and z in the figure.
The distance from the house to the library is 8x + 14
Finding the value of x if S and T are points on sidesWe use the following ratio of corresponding sides:
PS/SQ = PT/TR
So, we have
5/(3x + 1) = 6/4x
Cross multiiply
20x = 18x + 6
2x = 6
So, we have
x = 3
Next, we have
(3x + 4)/12 = (4x + 3)/15
Cross multiiply
45x + 60 = 48x + 36
3x = 24
So, we have
x = 8
Calculating the distance from the house to the libraryHere, we have
House to school = 3x + 17School to library = 5x - 3So, we have
Distance = 3x + 17 + 5x - 3
Evaluate
Distance = 8x + 14
Finding x, y and z in the figureWe use the following ratio of corresponding sides:
x = 5
y = 4
So, we have
x/z = (x + 4)/6.5
This gives
5/z = (5 + 4)/6.5
Rewrite as
z/5 = 6.5/(5 + 4)
This gives
z = 5 * 6.5/(5 + 4)
Evaluate
z = 3.6
Hence, the values are x = 5, y = 4 and z = 3.6
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a random sample of 22 people, the mean commute time to work was 34.2 minutes and the standard deviation was 7.1 minutes. assume the population is normally distributed and use a t-distribution to construct a 95% confidence interval for the population mean . what is the margin of error of y? interpret the results.
We're 95% confident that the true population mean is between 31.361 and 37.039 minutes, with a margin of error of 2.839 minutes.
To find the t-value from the t-distribution table, we need to know the degrees of freedom. Since we have a sample size of 22, the degrees of freedom are 22-1=21. Looking up the t-value with 21 degrees of freedom and a 95% confidence level, we get 2.080.
Plugging in this value, we get:
CI = 34.2 ± 2.080*(7.1/√22)
CI = 34.2 ± 2.839
Therefore, the 95% confidence interval for the population mean commute time to work is (31.361, 37.039). This means that if we were to take many random samples of size 22 from the population and construct a 95% confidence interval for each sample, 95% of those intervals would contain the true population mean.
The margin of error of y is the amount added and subtracted from the sample mean to get the upper and lower bounds of the confidence interval. In this case, the margin of error is 2.839 minutes.
This means that we're 95% confident that the true population mean commute time to work is between 31.361 and 37.039 minutes, with a margin of error of 2.839 minutes.
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HELP PLEASE
On this document you will notice there are no sides lengths provided. However, the rooftop length AD is 300 ft, CD is 330 ft, AB is 210 ft; Based on the math you learned at the beginning of this unit you should be able to complete the table located on the “Rooftop Artist” document to prove you understand distance and midpoint given two points.
Answer:
Step-by-step explanation:
To find the distance between two points, you can use the distance formula:
d = √[(x2 - x1)^2 + (y2 - y1)^2]
where (x1, y1) and (x2, y2) are the coordinates of the two points.
To find the midpoint between two points, you can use the midpoint formula:
[(x1 + x2)/2, (y1 + y2)/2]
where (x1, y1) and (x2, y2) are the coordinates of the two points.
Using the given lengths, you should be able to find the coordinates of each point and then use the formulas above to fill out the table in the "Rooftop Artist" document.
Which expression represents the product of 3 and (5/4+1.8)(5/4n+1.8) ?
An expression represents the product of 3 and ((5/4)n+1.8) is 3.75n + 5.4
The correct answer is an option (D)
In the expression ((5/4)n + 1.8), first we write the fraction 5/4 in the decimal form.
5/4 = 1.25
So, the expression ((5/4)n + 1.8) becomes,
((5/4)n + 1.8)
= (1.25 n + 1.8) ..........(1)
Now consider the product of 3 and ((5/4)n + 1.8)
3 × ((5/4)n + 1.8)
= 3 × ((1.25)n + 1.8) ............(from (1))
= (3 × (1.25 n)) + (3 × 1.8)
= (3 × 1.25)n + 5.4
= 3.75n + 5.4
Thus, 3 × ((5/4)n + 1.8) = 3.75n + 5.4
The correct answer is an option (D)
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Which expression represents the product of 3 and (5/4n + 1.8)? (SHOW WORK PLEASE)
A) 5.55n
B) 9.15n
C) 3.75n + 1.8
D) 3.75n + 5.4
Please help me with this homework
Answer:
42 Hope this is what you were looking for :)
Step-by-step explanation:
180=46+92+x
add
180=138+x
Subtract from both sides
180-138=138-138+x
Answer
42
Triangular prism A and triangular prism B have bases of
the same area. The height of prism A is 18 inches and
the height of prism B is 6 inches. The volume of prism B
can be found by multiplying the volume of prism A by a
scale factor of -
A
B
с
D
A. 1/3
B. 3
C. 1/6
D. 6
The victim was not shredded by paper cuts
The victim was not attacked by a lizard.
The victim did not have a fatal overdose of caffeine.
The victim was not electrocuted by a cell phone.
A) The victim was not shredded by paper cuts.
B) The victim was not attacked by a lizard.
C) The victim did not have a fatal overdose of caffeine.
D) The victim was not electrocuted by a cell phone.
This is a required question
The volume of prism B can be found by multiplying the volume of A by a factor of 1/3
What is scale factor?A scale factor is when you enlarge a shape and each side is multiplied by the same number. The value of scale factor is expressed as;
scale factor = new dimension / original dimension
Since the two triangular prisms have same base areas , this means that the volume will only be affected by the height. The height will be the determinant of the volume.
Volume = base area × height.
The volume of prism A = y × 18
= 18y units³
The volume of Prism B
= y × 6
= 6y³
Therefore the volume of prism of B can be found by multiplying the volume of a by a scale factor of 3.
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can someone help with 1 and 2 please
1. The standard form of each quadratic function is given as follows:
a) (x + 4)(x - 1) = x² + 3x - 4.
b) (2x - 1)(3x - 1) = 6x² - 5x + 1.
2. The expression 8 - 6x + x² is not in standard form, as the coefficients are not in descending order.
What is the standard format of a quadratic function?The standard format of a quadratic function is given with the coefficients ordered in descending format as follows:
y = ax² + bx + c.
To transform from the factored form into the standard form, we multiply the terms then combine the like terms, hence:
a) (x + 4)(x - 1) = x² + 4x - x - 1 = x² + 3x - 4.
b) (2x - 1)(3x - 1) = 6x² - 2x - 3x + 1 = 6x² - 5x + 1.
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Which of the following is not a solution to the following inequality
-2-12y<-1
A:(-3,4)
B:(3,-2)
C;(-1,3)
D:(0,1)
Only option B: (3, -2) is not a solution to this inequality out of the available answer options. This is due to the fact that this point's y-coordinate is smaller than -1/12, which defies the inequality y > -1/12.
We can use the steps listed below to solve the inequality -2-12y -1:
To both sides, add 2. -12y < 1
Divide the two sides by -12, remembering to invert the inequality sign when doing so: y > -1/12
Since y > -1/12, or any value greater than -1/12, is the answer to the inequality, y can have any value.
Because they all have y values of more than -1/12, options A, C, and D are all viable solutions to the inequality. All y values larger than -1/12 are included in option A, (-3,4); all y values between -1/12 and 3 are included in option C, (-1,3); and all y values between -1/12 and 1 are included in option D, (0,1).
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You are building a triangular patio from a blueprint with base 8 in and height 11 in The base of the patio is going to be 16 ftWhat will be the height of the patio, h?
The calculated height of the patio, h will be 22 ft
What will be the height of the patio, h?From the question, we have the following parameters that can be used in our computation:
Blueprint with base 8 in and height 11
This means that
Ratio = 8 in : 11 in
So, we have
16 ft : height = 8 in : 11 in
Multiply by 2
So, we have the following representation
16 ft : height = 16 in : 22 in
By comparison, we have
height = 22 ft
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X1 and S 2 1 are the sample mean and sample variance from a population with mean µ1 and variance σ 2 1. Similarly, X2 and S 2 2 are the sample mean and sample variance from a second independent population with mean µ2 and variance σ 2 2. The sample sizes are n1 and n2, respectively. A) Show that X1 − X2 is an unbiased estimator of µ1 − µ2. B) Find the standard error of X1 − X2. How could you estimate the standard error?
a. X1 - X2 is an unbiased estimator of µ1 - µ2.
b. The standard error of X1 - X2 is √(S1²/n1 + S2²/n2).
A) To show that X1 − X2 is an unbiased estimator of µ1 − µ2, we need to show that the expected value of X1 − X2 equals µ1 − µ2.
E[X1 − X2] = E[X1] − E[X2] (since expectation is linear)
= µ1 − µ2 (since X1 and X2 are unbiased estimators of µ1 and µ2, respectively)
Therefore, X1 − X2 is an unbiased estimator of µ1 − µ2.
B) The variance of X1 − X2 can be calculated as:
Var[X1 − X2] = Var[X1] + Var[X2] (since X1 and X2 are independent)
= σ1²/n1 + σ2²/n2
The standard error of X1 − X2 is the square root of the variance:
SE(X1 − X2) = √(σ1²/n1 + σ2²/n2)
To estimate the standard error, we can use the sample standard deviations S1 and S2 as estimators of the population standard deviations σ1 and σ2, respectively. Therefore, the estimated standard error of X1 − X2 is:
SE(X1 − X2) ≈ √(S1²/n1 + S2²/n2)
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