To model the division of -24 by -4 on a number line, we can start at -24 and move to the left by -4 units at a time, counting the number of times we move until we reach 0.
How to explain the number line?A number line is a diagram of a graduated straight line used to represent real numbers in elementary mathematics. It is assumed that each real number and each point on a number line correspond to one another.
Since -24 divided by -4 is 6, we would move to the left 6 times and end at 0 on the number line. The negative signs of the dividend and divisor indicate that the quotient will be a negative number.
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Solve using Cramer's rule.
6x + 5y = 25
7x − 9y = 44
x = 7, y = −1
x = 5, y = −1
x = 5, y = −2
x = 7, y = −2
The solution to the simultaneous equations using Cramer's rule is;
x = 5, y = −1
How to use Cramer's rule to solve simultaneous equations?The simultaneous equations are;
6x + 5y = 25
7x − 9y = 44
We will first find the determinant of the left hand side;
[tex]\left[\begin{array}{ccc}6&&5\\7&&-9\\\end{array}\right][/tex]
|A| = (6 * -9) - (7 * 5)
|A| = -89
Now, will replace the x part of the matrix with the right side values to get;
[tex]\left[\begin{array}{ccc}25&&5\\44&&-9\\\end{array}\right][/tex]
The determinant is;
|A_x| = (25 * -9) - (44 * 5)
|A_x| = -445
Now, will replace the y part of the matrix with the right side values to get;
[tex]\left[\begin{array}{ccc}6&&25\\7&&44\\\end{array}\right][/tex]
The determinant is;
|A_x| = (6 * 44) - (25 * 7)
|A_x| = 89
Thus;
x = |A_x|/|A|
x = -445/-89
x = 5
y = |A_y|/|A|
y = 89/-89
y = -1
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An experiment was conducted to investigate the relationship between the dose of a puin medication and the number of hours of pain relief; Twenty individuals with chronic pain wcre randomly assigned t0 one of five doses 0.0,0.5,1.0,1.5.2.0 in milligrams (mg) of medication. The results are shown in the scatterplot below. The data were used to fit least-squares regression line to predict the number of hours of- pain relief for given dose. Which of the following would be revealed plot of the residuals of the regression versus the dose" (A)The SUm of the residuals is less than 0, (B) The sum of the residuals is greater than 0_ (C) There are outliers associated with the lower doses (D) The variation in the hours of pain relief is not the same across the doses. (E) There is positive linear relationship between the residuals and the dose _
The least-squares regression line can be used to predict the number of hours of pain relief given a certain dose.
This is done by taking the sum of the squares of the residuals and finding the minimum value. The residuals are the differences between the observed values and the predicted values. The plot of the residuals versus the dose would reveal the variation in the hours of pain relief across the doses.
For example, the formula for finding the least-squares regression line is
Y = b0 + b1X
where Y is the predicted value, b0 is the intercept, b1 is the slope, and X is the dose.
To calculate the residuals, we take the difference between the observed values and the predicted values for each dose.
residual = Y (observed) - Y (predicted)
The plot of the residuals versus the dose would reveal if the variation in the hours of pain relief is not the same across the doses. This is because if the variation is not the same, then the residuals would not be evenly distributed. If the residuals were evenly distributed, then the sum of the residuals would be equal to zero.
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A file that is 237 megabytes is being downloaded. If the download is 17.6% complete, how many megabytes have been downloaded?
Answer:
Step-by-step explanation:
The answer is 42 MB.If it's round to nearest tenth than it's 40 MB
For Problems 13 and 14, draw a bar model to represent each expression.
13. y - 2
14. 4x
50 POINTS REWARD + BRAINLEST!! PLEASE HEL
Bar models can be used to represent the quotient and the product of numbers or expressions that are; y = x + 2 and x = m/4
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Given the expression as;
y - 2
LEt the expression Represent y - 2 with x.
y - 2 = x
Add 2 on both sides we get;
y = x + 2
Similalry,
Represent the 4x with m.
4x = m
Divide by 4 on both sides we get;
x = m/4
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g components of force f with respect to x and y axes can be calculated as follow. please note that x and y axes are not perpendicular to each others. the angle between f and x is denote as (theta). the angle between f and y is denote as (gamma).
The force with respect to the x and y axes can be calculated as follows:
F_x = 100N cos(30) = 86.6N
F_y = 100N sin(60) = 50N
Let F be the magnitude of the force, (theta) be the angle between the force and the x-axis, and (gamma) be the angle between the force and the y-axis. The components of the force with respect to the x and y axes can be calculated using the following formulas:
F_x = F cos(theta)
F_y = F sin(gamma)
For example, if the magnitude of the force is 100N and (theta) = 30 degrees and (gamma) = 60 degrees, then the components of the force with respect to the x and y axes can be calculated as follows:
F_x = 100N cos(30) = 86.6N
F_y = 100N sin(60) = 50N
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at a certain bakery, the price of each doughnut is $1.50. let the random variable d represent the number of doughnuts a typical customer purchases each day. the expected value and variance of the probability distribution of d are 2.6 doughnuts and 3.6 (doughnuts)2, respectively. let the random variable p represent the price of the doughnuts that a typical customer purchases each day. which of the following is the standard deviation, in dollars, of the probability distribution of p?
The standard deviation, in dollars, of the probability distribution of p is √(1.50 × 3.6) dollars.
We are given the following parameters -
Expected value = 2.6
Variance = 3.6
Price per doughnut = $1.50
Now we can write the price for the variance of the distribution as -
Price per doughnut × Variance
Variance = $1.50 × 3.6
The formula for the standard deviation of the distribution D in relation to the variance is given by -
Standard deviation = √variance
Standard deviation = √(1.50 × 3.6)
Therefore, the standard deviation of P in dollars will be √(1.50 × 3.6).
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Find the length of the midsegment of the trapezoid with the vertices S(-2,4) ,T(-2,-4) ,U(3,-2) ,V(13,10)
11.69 is the exact length needed for the mid-segment of the trapezoid with STUV vertices.
What is trapezoid?The Greek roots of the term trapezoid are trapeza, which means "table," and -oeides, which means "formed." A trapezoid is shaped like a table. It has two parallel sides, which are often referred to as the figure's bases. Take the average of the bases and multiply it by the height to determine the area of a trapezoid.
According to question:We have,
vertices S(-2,4) ,T(-2,-4) ,U(3,-2) ,V(13,10).
Half of the lengths of the two parallel sides, ST and UV, make up the length of the trapezoid's middle section.
The trapezoid's vertices are S(2, 4), T(2, 4), U(3, 2), and V. (13, 10).
The distance formula is then used to calculate the length of the ST and UV sides;
Length = [tex]\sqrt{x_{2}-x_{1}+y_{2}-y_{1}}[/tex]
Side ST's length is S(-2, 4), T(-2, -4).
ST = [tex]\sqrt{x_{2}-x_{1}+y_{2}-y_{1}}[/tex]
ST = [tex]\sqrt{(-2-(-2))^2+(-4-(4))^2[/tex]
ST = [tex]\sqrt{0 + 64}[/tex]
ST √64
ST = 8
Side UV's length is U(3, 2), V. (13, 10).
UV = [tex]\sqrt{(13-3)^2+(10-(-2))^2[/tex]
UV = [tex]\sqrt{(10)^2+(12)^2[/tex]
UV = √244
UV = 15.62
Therefore,
The mid-segment STUV's length is,
Length of mid-segment = (ST + UV)/2
Length of mid-segment = (8 + 15.62)/2
Length of mid-segment = (23.62)/2
Length of mid-segment = 11.81
Therefore, 11.69 is the exact length needed for the mid-segment of the trapezoid with STUV vertices.
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Which icon points out the Essential Question that you should be able to answer by the end of the lesson?
A.
A question mark icon is shown.
B.
An icon of a computer mouse is shown.
C.
A hand icon is shown.
D.
A book icon is shown.
The icon that points out the Essential Question that you should be able to answer by the end of the lesson is this:
C. A hand icon is shown.
What is the function of a hand icon in a text?A hand icon is used for the purpose of identifying the important questions that should be answered at the end of the lesson. It shows that the reader should pay careful attention to the indicated question as they may have to provide an answer to it by the end of the lesson.
Icons in reading have different meanings and understanding their purposes will lead to a more immersive reading.
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Select all the equations that are NOT equivalent to p-7=22
The equivalent expressions are B. p-7-p = 22 -, C. p - 7 + 22 = 22 - 7, D. p - 7 + 7 = 22 - 22
What are algebraic expressions?Algebraic expressions are simply defined as expressions that comprised of terms. coefficients, factors, constants and variables.
They are also made up of some operations which are arithmetic. They include;
AdditionMultiplicationBracketFloor divisionParenthesesSubtractionDivisionFrom the information given, we have the equation as;
p- 7 = 22
To solve the equation, collect like terms
p = 22 + 7
Add the like terms
p = 29
Hence, the value is 29
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The complete question:
Which equation is NOT equivalent to p- 7 = 22? Select all that apply.
A. p-p-7 = 22-p
B. p-7-p = 22 -
C. p - 7 + 22 = 22 - 7
O
D. p - 7 + 7 = 22 - 22
Ep-7 + 7 = 22 +7
What is the value of x in this proportion?
56=−4x+2
Responses
x=−2 4/5
x equals negative 2 and 4 over 5
x=−4 2/5
x equals negative 4 and 2 over 5
x=−5 1/5
x equals negative 5 and 1 over 5
x=−6 4/5
x equals negative 6 and 4 over 5
The value of x or the expression 56 = -4x + 2 is -27/2.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
56 = -4x + 2
Solve for x.
56 = -4x + 2
Subtract 2 on both sides.
56 - 2 = -4x
54 = -4x
Divide -4 into both sides.
-54/4 = x
x = -54/4
x = -27/2
x = -13(1/2)
Thus,
The value of x is -27/2.
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Emma is buying a gift for her mother's birthday. She purchases flowers and a gift certificate that cost a total of $68, not including tax. If the cost of the gift certificate is $14 more than 2 times the cost of the flowers, what is the cost of the flowers? Set up a system of equations, then solve using a matrix.
The cost of the flowers she bought for her mum alone would be = $18
How to calculate the cost of flowers?The items purchased by Emma = flowers and a gift certificate.
The total cost of the flowers and a gift certificate = $68
But, cost of gift certificate= $14 +2x
cost of flower = X
That is 14 + 2x + X = 68
2x + X = 68 - 14
3X = 54
X = 54/3
X= $18
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There are 5
green apples and 7
red apples in a basket.
(a) What is the ratio of all
apples in the basket to green apples?
0
(b) What is the ratio of all apples in the basket to red apples?
0
a.
5+7=13
total amount of apples = 13
green apples = 5
5:13
b.
total amount of apples = 13
red apples = 7
7:13
When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate
As per the elimination method, in order to solve a system of equations when multiplication is necessary to eliminate a variable.
The term equation is known as algebra is a statement of equality that contains one or more unknown quantities or variables.
Here we have given that When using the multiplication method, it is important to multiply all the terms on both sides of the equation, then here we do not just the one term you are trying to eliminate.
According to the rule of elimination method we all know while solving systems of linear equations uses the addition property of equality. You can add the same value to each side of an equation.
So in order to multiplying to solve linear systems is to find a number to multiply to one or both of the equations so that the x or y terms in one of the equations will have opposite coefficients from the x or y in the other equation.
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Quinn deposit $3000 into a savings account her freshman year of college. She will earn simple interest on the account at 2.56%. If she makes no contributions or withdrawals, what will the accrued value of the account be when she graduates four years later 
[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$3000\\ r=rate\to 2.56\%\to \frac{2.56}{100}\dotfill &0.0256\\ t=years\dotfill &4 \end{cases} \\\\\\ A = 3000[1+(0.0256)(4)] \implies A=3000(1.1024)\implies A = 3307.2[/tex]
Determine how (if possible) the triangles can be proved similar.
Select one:
A. SAS~
B. SSS~
C. Not Similar
D. AA~
The triangles are proved similar by; D. AA similarity.
What are similar angles of triangles?When on comparing the sides and measures of internal angles of two or more triangles, if there is a common properties among them, then the triangles are similar. Note that being similar does not imply that they are congruent.
So that on comparing the properties of the two triangles given, it can be inferred that:
the two traingles are similar by Angle-Angle theorem which states that; when two angles of two triangles are the same, then the triangles are similar.
Therefore, the statement to prove that the triangles are similar is giben in option D. AA similarity.
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#8 i need help with this problems.
ΔABD and ΔBCD are congruent by AAS congruence theorem.
The proof is given below.
What is a triangle?It is a two-dimensional figure which has three sides and the sum of the three angles is equal to 180 degrees.
We have,
From the figure below,
ΔABD and ΔBCD are congruent.
∠BCD = ∠BAD [ 90 degrees ]
∠CDB = ∠ABD [ alternate angles ]
BD = BD [ Common ]
ΔABD ≈ ΔBCD [ AAS ]
Thus,
ΔABD and ΔBCD are congruent by AAS congruence theorem.
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In circle G the length of arc HI = 2 pi and measures angle HGI = 60 degrees. Find the area shaded below
The area of the circular sector is equal to 6π square units.
How to find the area of a circular sector
In this question we must determine the area of the circular sector between points H, G and I from given information of the central angle (θ), in degrees, and arc (p). The perimeter and area of the circular sector are described by the following formulae:
Area
A = (θ / 360) · π · r²
Perimeter
p = (θ / 360) · 2π · r
Where r is the radius of the circle.
If we know that p = 2π and θ = 60°, then the area of the circular sector is:
r = (360 / θ) · (p / 2π)
r = (360 / 60) · (2π / 2π)
r = 6
A = (60 / 360) · π · 6²
A = 6π
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billy makes $20 a week in allowance plus $5 for each lawn that he mows. this week billy wants to make over $50. select the inequality for the number of lawns he needs to mow.
The inequality for the number of lawns he needs to mow is: $50 < $20 + $5n
The correct answer is an option (b)
Let n represents the number of lawns he needs to mow.
Here, Billy needs to make more than $50 between his allowance and the lawns that he mows. This means that an inequality should include 50 <
As Billy will make $5 for each lawn that he mows, for 'n' number of lawns the cost would be = 5n
Based on the mentioned conditions we formulate an inequality,
50 < 20 + 5n
20 + 5n > 50
20 + 5n - 20 > 50 - 20 ...........(subtract 20 from each side)
5n/5 > 30/5 .............(divide each side by 5)
n > 6
This means, he needs to mow more that 6 lawns.
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The complete question is:
Billy makes $20 a week in allowance plus $5 for each lawn that he mows. this week billy wants to make over $50. select the inequality for the number of lawns he needs to mow.
a) $50 < $20+$5
b) $50 > $20+$5n
c) $50 < $20n+$5
d) $50 > $20n+$5
e) $50 < $20+$5n
Sally has a leaking faucet in her bathroom. When she first noticed the leak, there was a puddle that was 2 inches in diameter. Each hour, the diameter will triple in size. If Sally doesn’t do anything to stop the leak, how large will the puddle be after 10 hours?
The required size of the hole after 10 hours will be 118098 inches.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
here,
Sally has a leaking faucet in her bathroom. When she first noticed the leak, there was a puddle that was 2 inches in diameter.
According to the given condition
let Y be the size of the hole after 10 hours
Y=2×3¹⁰
Y=118098
Thus, the required size of the hole after 10 hours will be 118098 inches.
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Paul and $10 for mowing 2 lawns. What is the rate she earned per per lawn mowed  
Answer:
5$ Per Lawn
Step-by-step explanation:
10$/2=5$
2/2=1
MARK ME BRAINLISIST
Answer: the rate she is earning is 5$ per lawn mowed
Step-by-step explanation:
if she earns 10$ after mowing 2 lawns then if you divide 10 and 2 you get 5
Please make me brainllest
How many terms are there in the sequence 10, 13, 16, 19, …, 238
According to the information there are 76 numbers in the sequence between 10 and 238.
How to calculate how many terms are there in the sequence?To calculate how many terms there are in the sequence we have to perform the following mathematical operation:
238 - 10 = 228228 / 3 = 76According to the above, there are 76 numbers in the sequence between 10 and 238. Also, we can prove this calculating each number:
10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100...
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Given the midsegment below:
41°
14
Z
z=[?]
62⁰
y
Enter
8. Stick A is 120 cm shorter than stick B. Stick C is 84 cm shorter than stick B. Sticks A, B, and C have a total length of 360 cm. What is the total length (in meters and centimeters) of sticks A and C?
The total length of A+C will be 172 cm(unit).
What is a unit?
Units are the tools to measure and compare different things. Comparison becomes easy when all the units for the measurement are the same. Different units can be classified depending on their use. A few of the units can be classified as:
The seven SI base units (The International System of Units (SI), commonly known as the metric system, is the international standard for measurement) which are comprised of:
Length - meter (m)
Time - second (s)
Amount of substance - mole (mole)
Electric current - ampere (A)
Temperature - kelvin (K)
Luminous intensity - candela (cd)
Mass - kilogram (kg)
Now,
Given B-A=120 -->1
B-C=84 -->2
A+B+C=360 -->3
Therefore, from equations 1,2 and 3
B-120+B+B-84=360
B=188
By putting B=188 in 1 and 2
A=68
C=104
A+C=172cm
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24(20) this is my question please answer
Answer:
480
Step-by-step explanation:
Lets break this down to make it even easier
(24x2)x10
=(48)x10
=480
Answer:
If you are looking for the simplest form of 24/20 it is basically 6/5.
because if you multiply 6/5 by 4/4 you could get the answer which is 24/20.
and the same thing if you divide
I hope that I helped
the directions are : The table shows a proportional relationship between x and y. What is the unit rate of y per x.
Since the table shows a proportional relationship between x and y, the unit rate of y per x is equal to 4.
What is a proportional relationship?In order to have a proportional relationship and equivalent ratios, the variable (x) and the variable (y) must have the same constant of proportionality.
Next, we would calculate the constant of proportionality (k) as follows:
Constant of proportionality (k) = y/x
Constant of proportionality (k) = 12/3 = 20/5 = 24/6 = 32/8
Constant of proportionality (k) = 4.
In this context, we can reasonably infer and logically deduce that the unit rate for this proportional relationship is equal to 4.
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Complete Question:
The table shows a proportional relationship between x and y what is the unit rate of y per x (3, 5, 6, 8) (12, 20, 24, 32)?
Jamie answered 16 questions in 4/5 minutes. How many questions were answered per minute?
The number of questions that were answered per minute 20 questions
How to determine the number of questions?the question reads: Jamie answered 16 questions in 4/5 minutes. How many questions were answered per minute?
You should number that a minute is made up of 60 seconds
This implies that 4/5 of a minute is given as 4/5 * 60
This is equal to 48 seconds
So every 48 seconds, Jamie answers 16 questions
This implies that (16*60)/48 = 20
Therefore in one minute Jamie would answer 20 questions
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Help me ASAP math question.
The client should invest $8889 in AAA bonds, $4444 in A bonds, and $5667 in B bonds.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example: 2x + 4 = 8 is an equation.
We have,
Amount invested in AAA bond = 2x
Amount invested in A bond = y
Amount invested in B bond = x
Now,
Average yield:
AAA bond = 4%
A bond = 5%
B bond = 8%
Now,
Total amount = $19,000
Total yield = $990
Now,
Two equations can be made:
2x + y + x = 19000
3x + y = 19000 _____(1)
0.04 x (2x) + 0.05 x y + 0.08x = 990
0.08x + 0.05y + 0.08x = 990
0.16x + 0.05y = 990 ______(2)
From (1) and (2),
3x = 19000 - y
x = (19000 - y)/3
Substituting in (2)
0.16 x (19000 - y)/3 + 0.05y = 990
0.053 x 19000 - 0.053y + 0.05y = 990
1007 - 0.003y = 990
1007 - 990 = 0.003y
y = 17/0.003
y = 5666.6
y = 5667
Now,
x = (19000 - y)/3
x = (19000 - 5667)/3
x = 4444.333
x = 4444
Now,
x = $4444
y = $5667
2x = $8889
Thus,
Amount invested in AAA bond = $8889
Amount invested in A bond = $5667
Amount invested in B bond = $4444
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according to government data, 22 percent of children in the united states under the age of 6 years live in households with incomes that are classified at a particular income level. a simple random sample of 300 children in the united states under the age of 6 years was selected for a study of learning in early childhood. if the government data are correct, which of the following best approximates the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level? (note: z represents a standard normal random variable.) A. P( z>(0.27-0.22)/root((0.22)(0.73))/300 B. P( z>(0.27-0.22)/root((0.27)(0.73))/300 C. P( z>(0.22-0.27)/root((0.22)(0.78))/300 D. P( z>(0.22-0.27)/root((0.27)(0.73))/300
The probability that at least 27 percent of the children in the sample live in households that are classified at a particular income level is [tex]P(z > (0.22-0.27)/ \sqrt{(0.27)(0.73)/300}.[/tex] Hence, the correct answer is option D.
This probability can be calculated using the normal approximation to the binomial distribution, with the sample proportion (p-hat) being 0.27, the population proportion (p) being 0.22, and the sample size (n) being 300.
The formula for the standard error of a sample proportion is -
[tex]\sqrt{p(1-p)/n}[/tex]
substituting the values, we get,
[tex]\sqrt{(0.27)(0.73)/300}.[/tex]
Then, we use the formula for a standard normal probability -
P(z > (p-hat - p)/standard error),
substituting the values, we get,
[tex]P(z > (0.22-0.27)/ \sqrt{(0.27)(0.73)/300}.[/tex]
Hence, the probability that at least 27 percent of the children in the sample live in households that are classified at a particular income level is [tex]P(z > (0.22-0.27)/ \sqrt{(0.27)(0.73)/300}.[/tex]
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In the following situations, the sampling frame does not match the population, resulting in undercoverage. Give examples of population members that might have been omitted.
The population consists of all 250 students in your large statistics class. You plan to obtain a simple random sample of 30 students by using the sampling frame of students present next Monday. (Select all that apply.)
A. Students who are skipping class cannot be sampled.
B. Home-schooled students cannot be sampled.
C. Students who are not taking statistics cannot be sampled.
D. Dropouts cannot be sampled.
E. Students who are on a school trip cannot be sampled.
F. Students who are out sick cannot be sampled.
Answer:
c
Step-by-step explanation:
NO LINKS!!!
Find the solution set for each inequality.
Answer:
41. (-∞, -1] ∪ [2, 3)
42. (2, 5) ∪ (5, ∞)
Step-by-step explanation:
Question 41Given rational inequality:
[tex]\dfrac{x^2-x-2}{x-3} \leq 0[/tex]
Factor the numerator:
[tex]\dfrac{(x-2)(x+1)}{x-3} \leq 0[/tex]
Find the roots by solving f(x) = 0 (set the numerator to zero):
[tex]\implies (x-2)(x+1)=0[/tex]
[tex]\implies x-2=0 \implies x=2[/tex]
[tex]\implies x+1=0 \implies x=-1[/tex]
Find the restrictions by solving f(x) = undefined (set the denominator to zero):
[tex]\implies x-3=0 \implies x=3[/tex]
Create a sign chart, using closed dots for the roots and open dots for the restrictions (see attachment 1).
Choose a test value for each region, including one to the left of all the critical values and one to the right of all the critical values.
Test values: -2, 0, 2.5, 4For each test value, determine if the function is positive or negative:
[tex]f(-2)=\dfrac{(-2)^2-(-2)-2}{(-2)-3} =\dfrac{+}{-}=-[/tex]
[tex]f(0)=\dfrac{(0)^2-(0)-2}{(0)-3} =\dfrac{-}{-}=+[/tex]
[tex]f(2.5)=\dfrac{(2.5)^2-(2.5)-2}{(2.5)-3} =\dfrac{+}{-}=-[/tex]
[tex]f(4)=\dfrac{(4)^2-(4)-2}{(4)-3} =\dfrac{+}{+}=+[/tex]
Record the results on the sign chart for each region (see attachment 1).
As we need to find the values for which f(x) ≤ 0, shade the appropriate regions (zero or negative) on the sign chart (see attached).
Therefore, the solution set is:
x ≤ -1 or 2 ≤ x < 3As interval notation:
(-∞, -1] ∪ [2, 3)Question 42Given rational inequality:
[tex]\dfrac{\sqrt{x}}{(x-2)|x-5|} > 0[/tex]
Find the roots by solving f(x) = 0 (set the numerator to zero):
[tex]\implies \sqrt{x}=0 \implies x=0[/tex]
Find the restrictions by solving f(x) = undefined (set the denominator to zero):
[tex]\implies (x-2)|x-5|=0[/tex]
[tex]\implies x-2=0 \implies x=2[/tex]
[tex]\implies |x-5|=0 \implies x=5[/tex]
Create a sign chart, using closed dots for the roots and open dots for the restrictions (see attachment 1).
Choose a test value for each region, including one to the left of all the critical values and one to the right of all the critical values.
Test values: -1, 1, 3, 6For each test value, determine if the function is positive or negative:
[tex]f(-1)=\dfrac{\sqrt{-1}}{(-1-2)|-1-5|}=\dfrac{\sf unde\:\!fined}{-}=\sf unde\:\!fined[/tex]
[tex]f(1)=\dfrac{\sqrt{1}}{(1-2)|1-5|}=\dfrac{+}{-}=-[/tex]
[tex]f(3)=\dfrac{\sqrt{3}}{(3-2)|3-5|}=\dfrac{+}{+}=+[/tex]
[tex]f(6)=\dfrac{\sqrt{6}}{(6-2)|6-5|}=\dfrac{+}{+}=+[/tex]
Record the results on the sign chart for each region (see attachment 2).
As we need to find the values for which f(x) > 0, shade the appropriate regions (positive) on the sign chart (see attached).
Therefore, the solution set is:
2 < x < 5 or x > 5As interval notation:
(2, 5) ∪ (5, ∞)