The proof that AE = EF is done from the congruency theorem of triangles and the Pythagorean identity.
How can it be proved that AE = EF?In order to prove that AE = EF, note that angle CEF is a right angle since EF is perpendicular to the diameter CD and intersects it at E.
Since AE = EB, triangles AEB and CED are congruent (SAS theorem)
Therefore, angle EAF is congruent to angle ECF.
Also, angle EAF is a right angle, since it is inscribed in a semicircle with a diameter CD.
Therefore, triangles EAF and ECF are similar by the Angle-Angle (AA) theorem.
Since they are similar, the ratio of corresponding side lengths is the same.
Since AE = EB, we have that:
AE / EF = EC / CF
But we also know that EC = ED + DC = ED + 2EF, since CD is the diameter of the semicircle and EF meets it at E perpendicularly.
Similarly, we have CF = CD + DF
CF = 2EF + DF, where DF is the other leg of the right triangle CDF.
Substituting these expressions for EC and CF in the above equation, we get:
AE / EF = (ED + 2EF) / (2EF + DF)
Simplifying this equation, we get:
AE / EF = (ED/EF) + 2 / (2 + DF/EF)
But ED/EF = sin(DEF) and DF/EF = sin(EFD) by the definition of sine in right triangle DEF. Since the angles DEF and EFD add up to 90 degrees, their sines are complementary.
Therefore, we have that:
ED/EF = cos(EFD) and DF/EF = sin(DEF)
Substituting these expressions in the above equation, we get:
AE / EF = cos(EFD) + 2 / (2 + sin(DEF))
But cos(EFD) + 2 = sin(DEF) (Pythagorean identity).
Therefore, we have:
AE / EF = sin(DEF) / (2 + sin(DEF))
Multiplying both sides by (2 + sin(DEF)), we get:
AE = EF
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solve the systems of equations using elimination
2x-7y=-13
8x-7y=11
Answer:
( [tex]-\frac{1}{3}[/tex] , [tex]-\frac{41}{21}[/tex] )
Step-by-step explanation:
#5
CONNECTING CONCEPTS You use 94 inches of plastic to frame the perimeter of a kite. One side of the kite has a
length of 18 inches. Find the length of each of the three remaining sides in order from least to greatest.
Length: in.,in., in
The length of the tree remaining sides are 18 inches, 29 inches and 29 inches.
What are the lengths of the remaining sides?A kite is an object with four sides. It length of the two adjacent sides are of equal length. No pair of sides in a kite are parallel. The perimeter of a kite is the sum of the length of the four sides of the kite.
Perimeter of a kite = 2( a + b)
94 = 2(18 + b)
94 / 2 = 18 + b
47 = 18 + b
b = 47 - 18
b = 29 inches
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Which expression is NOT equal to 125?
5(5^3/2/5)^2
(5^3/5^4)^-3
5^-2/5^-5
5(5^5/5^3)
help me solve this equation
The area of the shaded portion is 6x^2
How to determine this?
The area of the shaded portion = Area of big rectangle - Area of small rectangle.
To get the area of big rectangle = L * B
A = 2x * 4x
A = 8x^2
To get area of small rectangle = L * B
A = 2x * x
A = 2x^2
To get the area of shaded portion ,
= 8x^2 - 2x^2
= 6X^2
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Let f(x) = x^2 − 2x + 1. Find the inverse function of f by identifying an appropriate restriction of its domain.
The inverse of f(x) is restricted to the domain x ≥ 1, since this is when[tex]x^2 − 2x + 1[/tex] is always greater than 0.
To find the inverse function of f(x), we must first determine the domain of f(x). We can do this by finding the values of x where f(x) is greater than or equal to 0.
We can start by setting f(x) to 0 and solving for x:
[tex]x^2 − 2x + 1 = 0[/tex]
[tex]x^2 − 2x + 1 − 1 = 0 − 1[/tex]
[tex]x^2 − 2x = -1[/tex]
(x − 1)(x − 1) = -1
x = 1
Therefore, the inverse of f(x) is restricted to the domain x ≥ 1, since this is when [tex]x^2[/tex] − 2x + 1 is always greater than 0.
The inverse of a function f(x) is a function that "undoes" f(x). In order to find the inverse of a function, the domain of the function must first be identified. This is done by solving f(x) = 0 and determining which values of x make the equation equal to 0.In the case of f(x) =[tex]x^2[/tex] − 2x + 1, the equation is equal to 0 when x = 1. This means that the inverse of f(x) is restricted to the domain x ≥ 1, since this is when [tex]x^2[/tex] − 2x + 1 is always greater than 0.Once the domain of the function is restricted, the inverse function can be found by switching the x and y values, and solving for the new equation. This process can be used to find the inverse of any function, as long as the domain is appropriately restricted.
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write the equations from word form.
The transformed functions is
a) The absolute value function reflected across the x axis and shifted up 2 units is y₁ = -| x₁ | + 2
b) The absolute value function that is shifted left 3 units and is compressed by a third is y₂ = ( 1/3 ) | x₂ + 3 |
c) The absolute value function reflected across the x axis stretched vertically by 5 units shifted left 4 units and up 6 units is y₃ = -5 | x₃ + 4 | + 6
How does the transformation of a function happen?The transformation of a function may involve any change.
Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs), etc.
If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:
Horizontal shift (also called phase shift):
Left shift by c units: y=f(x+c) (same output, but c units earlier)
Right shift by c units: y=f(x-c)(same output, but c units late)
Vertical shift:
Up by d units: y = f(x) + d
Down by d units: y = f(x) - d
Stretching:
Vertical stretch by a factor k: y = k × f(x)
Horizontal stretch by a factor k: y = f(x/k)
Given data ,
a) The absolute value function reflected across the x axis and shifted up 2 units is y₁
The absolute value function is y₁ = | x₁ |
when reflected over x axis , the new function is
y₁ = - | x |
when shifted up 2 units ,
y₁ = -| x₁ | + 2
b)The absolute value function that is shifted left 3 units and is compressed by a third is y₂
The absolute value function is y₂ = | x₂ |
when shifted left 3 units ,
y₂ = | x₂ + 3 |
when compressed by a third , the new function is
y₂ = ( 1/3 ) | x₂ + 3 |
c)The absolute value function reflected across the x axis stretched vertically by 5 units shifted left 4 units and up 6 units is y₃
The absolute value function is y₃ = | x₃ |
when reflected across the x axis ,
y₃ = - | x₃ |
when stretched vertically by 5 units ,
y₃ = - 5 | x₃ |
when shifted 4 units left ,
y₃ = - 5 | x₃ + 4 |
when shifted 6 units up ,
y₃ = -5 | x₃ + 4 | + 6
Hence , the transformation of functions are solved
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Three times the quantity of a number increased by 4 is -9. Write the equation and solve
The value of the number is -13/3
What are algebraic expressions?Algebraic expressions are defined as expressions that are composed of arithmetic operations such as subtraction, multiplication, division, addition, bracket and parentheses.
These expressions are also made up of variables, terms, coefficients, constants and factors.
From the information given, we have that;
The number = x
3x + 4 = -9
collect like terms
3x = -9 - 4
subtract the like terms
3x = -13
Make 'x' the subject
x = -13/3
Hence, the value is -13/3
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a university is conducting research on ways to improve study habits and help students with lower grade point averages. it sends out a questionnaire to undergraduate students asking how they study and what they feel is the most productive way to study. what data collection method is the university using?
As the university is sending out questionnaires, the data collection method the university is using is a survey
The term "survey" describes the method of gathering data on a certain subject generally by employing questionnaires. In order to understand more about a certain topic, a survey is a method of data collection that involves asking a sample of respondents a number of questions. In the given case, the university is using a survey as a tool for gathering data.
Here, the institution is distributing a survey to undergraduate students to learn more about their study habits and what they believe to be the most effective method of studying. In order to efficiently gather data from a large number of participants quickly and effectively, surveys are frequently employed in research investigations.
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Use the Pythagorean Theorem to find the distance between points F and C.
A. 2 √3
B. √41
C. 4 √3
D. 3 √5
find the value of x . round to the nearest tenth 15° -2- 70°
The value of x, given the triangle, the length of the side and the angle, would be
How to find the value x ?The angle given is 23 degrees and the sides given are 15 and x. These two sides are the hypotenuse and the opposite sides which means that the operation to be used is Sin.
The value of x would therefore be :
Sin ( angle ) = Opposite / Hypotenuse
Sin ( 23 ) = x / 15
x = 15 x Sin ( 23 )
The value of x is:
x = 15 x Sin ( 23 )
x = 5. 86
x = 5. 9 units
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Find the approximate values of the five number summary for the data set represented by this box plot,
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
The Approximate values of the five number summary are ;
(i) the Minimum is 10 ,
(ii) the First quartile is 30 ,
(iii) the Third Quartile is 80 ,
(iv) the Median is 55 ,
(v) the Maximum is 100 .
The values of the data set are : {10 ,15 ,20 ,25 ,30 ,35 ,40 ,45 ,50 ,55 ,60 ,65 ,70 ,75 ,80 ,85 ,90 ,95 ,100 } ;
The Five number summary consists of the minimum , maximum , first quartile , third quartile and the median .
(i) the Minimum of the data set is = 10 ;
(ii) we split the data set into , 2 halves ;
that means ; {10 ,15 ,20 ,25 ,30 ,35 ,40 ,45 ,50 } and {60 ,65 ,70 ,75 ,80 ,85 ,90 ,95 ,100 } ;
the first quartile lies in the lower half : {10 ,15 ,20 ,25 ,30 ,35 ,40 ,45 ,50 } ;
So , first quartile (Q₁) = 30 ;
(iii) the third quartile (Q₃) lies in the upper half : {60 ,65 ,70 ,75 ,80 ,85 ,90 ,95 ,100 } ;
So , Q₃ = 80 ;
(iv) the total number of terms in the set is = 19 ;
So , the median is = tenth term = 55 ;
(v) the maximum of the data set is = 100 .
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Determine if events A and B are independent.
P(A)=1/5, P(B)=7/20, P(A and B)= 7/100. The events A and B are independent.
What is independent probability?Two events are independent if the occurrence of one event does not affect the chances of the occurrence of the other event. The mathematical formulation of the independence of events A and B is the probability of the occurrence of both A and B being equal to the product of the probabilities of A and B (i.e., P(A and B).
1) P(A)=1/5, P(B)=7/20, P(A and B)= 7/100
Here, P(A and B)= P(A)×P(B)
= 1/5 × 7/20
= 7/100
So, the events A and B are independent.
2) P(A)=13/20, P(B)=13/20, P(A and B)= 169/400
Here, P(A and B)= P(A)×P(B)
= 13/20 × 13/20
= 169/400
So, the events A and B are independent.
3) P(A)=7/10, P(B)=2/5, P(A and B)= 21/100
Here, P(A and B)= P(A)×P(B)
= 7/10 × 2/5
= 14/50
So, the events A and B are not independent.
2) P(A)=1/5, P(B)=1/5, P(A and B)= 1/25
Here, P(A and B)= P(A)×P(B)
= 1/5 × 1/5
= 1/25
So, the events A and B are independent.
Therefore, options 1, 2 and 4 are correct answers.
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what is the probability of picking a red ball from the bag and getting heads upon flipping a fair coin? a.) 3 over 16 b.) 7 over 8 c.) 3 over 4 d.) 1 over 8
the probability of picking a red ball from the bag and getting heads upon flipping a fair coin is 1/2 * 1/2 = 1/8.
There are two independent events, and therefore the probability of both occurring is calculated by multiplying the probability of each event. The probability of picking a red ball from the bag is 1/2, and the probability of getting heads upon flipping a fair coin is also 1/2. Therefore, the probability of picking a red ball from the bag and getting heads upon flipping a fair coin is 1/2 * 1/2 = 1/8.
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The Martinez family just bought 6 crates of eggs, and each crate had 12 eggs. The family already had 9eggs in their refrigerator. How many eggs do they have now?
The Martinez family just bought 6 crates of eggs, and each crate had 12 eggs. The family already had 9 eggs in their refrigerator. Total number of eggs they have now is 81.
The problem states that the Martinez family bought 6 crates of eggs, and each crate had 12 eggs. So, we can start by multiplying the number of crates by the number of eggs per crate to find out how many eggs they bought in total:
6 crates x 12 eggs/crate = 72 eggs
This tells us that the family now has 72 eggs. However, the problem also states that the family already had 9 eggs in their refrigerator. To find out how many eggs they have now in total, we need to add the number of eggs they bought to the number of eggs they already had:
72 eggs + 9 eggs = 81 eggs
So the answer is that the Martinez family now has a total of 81 eggs.
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X-1
2x + 4
6
X
Determine the value of x in the
given diagram
The value of x in the diagram is given as follows:
x = 10.
What are similar triangles?Similar triangles share these two features:
Congruent angles, that is, angles that have the same measure.Proportional side lengths.Considering the similar triangles, the proportional relationship for the side lengths is given as follows:
6/(6 + x) = (x - 1)/(2x + 4).
Applying cross multiplication, the value of x is obtained as follows:
(x - 1)(x + 6) = 6(2x + 4)
x² + 5x - 6 = 12x + 24
x² - 7x - 30 = 0.
Factoring, we obtain the solutions are follows:
(x - 10)(x + 3) = 0.
x - 10 = 0 -> x = 10 -> value of x.x + 3 = 0 -> x = -3 -> extraneous, as a length cannot be negative.More can be learned about similar triangles at brainly.com/question/11920446
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Please help I’ll give brainliest!!!!!!!!
The solution set of the system of equations is {(2,2), (4,-2)}
What is the solution to an equation?
In order to make the equation's equality true, the unknown variables must be given values as a solution. In other words, the definition of a solution is a value or set of values (one for each unknown) that, when used as a replacement for the unknowns, transforms the equation into equality.
4x+2y =12------>(1)
Take some random values for x. Find the corresponding y values
Example: when x=0
(1) => 4(0)+2y = 12
=> 0+2y = 12
=> 2y =12
=> y= 12/2 = 6
The point obtained is (x,y) = (0,6)
This way, we get a line when we graph the equation 4x+2y =12.
Similarly, when we graph the equation y= -x² + 4x -2 , we get a parabola
Graph is attached:
In the graph, both graphs intersect at the points (2,2) and (4,-2)
Thus, the solution set of the system of equations is {(2,2), (4,-2)}
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Help me please
Choose all that are correct
A.Random samples from the same population will vary from sample to sample
B.Values of a sample statistic of different random samples of the same size from the same population will be the same
C.As the sample size increases, the sample distribution more closely resembles the population distribution.
D.if a random sample is chosen from a population with a large cluster of points at the maximum, the sample is likely to have at least one element near the maximum.
The solution is, Option a
Stratified random sample
What is random sample?In statistics, a simple random sample is a subset of individuals chosen from a larger set in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way.
here, we have,
Explanation:
(a) In stratified random sampling, a division of the population occurs in such a way that a population is stratified in different small groups or strata with common traits and same proportion both in population and sample. this type of sample so formed is known as stratified random sample.
(b) Simple random Sample:
This sample is formed by the random and unbiased selection of individual that represents the population.
(c) Population:
The entire set of individuals with certain specific characteristics represents a population whereas a sample is drawn out of the population and can be referred to as the subset of the population.
(d) Cluster Sample:
In this, separate sample groups are formed known as clusters, here sampling of random clusters take place.
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For some value of b, the expression 3(5x - 1) + b(2x - 7) is a constant. What is the constant?
The expression is constant for b = -7.5, and the constant is 49.5.
What is a mathematical Expression?A mathematical expression is a phrase that contains at least two terms or numbers and one action. It's possible to multiply, divide, add, or subtract with this mathematical operation.
Given that, The expression 3(5x - 1) + b(2x - 7) is constant
It means the expression doesn't consist of a variable. So, we need to find the 'b' such that the variable 'x' need to be eliminated.
The co-efficient of the 'x' should be '0'.
3(5x - 1) + b(2x - 7)
= 15x - 3 + 2bx - 7b
= (15 + 2b)x - 3 - 7b
Now, 15 + 2b = 0
2b = -15
b = -7.5
On substituting the value of 'b' in the expression,
= -3 - 7 * (-7.5)
= 52.5 - 3
= 49.5
Therefore, The expression is constant for b = -7.5, and the constant is 49.5
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g(n) = -1 +1 n f(n) = - 1 n-1
Please help i’ll give brainliest
Answer: 4.5
Step-by-step explanation:
what do i do if there is a word problem with x but there isnt a number for x
You might need to utilise algebra to answer for the variable x in a word problem if there isn't a particular number provided for it.
Create an equation involving x using the information provided.
For instance, if the question is, "What is the value of x if x is 5 more than twice y and y is 3?" you may write up an equation like this:
x = 5 + 2y
x = 5 + 2(3) (3) (Replace y = 3)
x = 11
By changing the supplied value of y, you can answer for x in this situation by first solving the ensuing equation.
You might need to utilise many equations and algebraic techniques like substitution, elimination, or graphing to solve for x if the issue is more complicated. It could be beneficial to divide the problem into smaller steps and work through each step methodically if you are unclear of how to proceed.
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If 3 garbage trucks can collect the trash of 24 homes in a day, how many trucks are needed to collect in 72 houses?
Answer:
9
Step-by-step explanation:
72/24 = 3
72 is 3 times 24, so you need 3 times as many garbage trucks.
Answer: 9
There are 342 boys and girls at the canteen. After 51 boys and an equal number of girls as boys returned to their classroom, there were three times the number of boys than girls in the canteen. How many boys were there in the canteen at first?
Let b be the initial number of boys, and let g be the initial number of girls. We know that the total number of students at the canteen is 342, so:
b + g = 342
After 51 boys and an equal number of girls returned to their classroom, there were three times the number of boys than girls in the canteen. This means that the number of girls remaining in the canteen is:
g - 51
And the number of boys remaining in the canteen is:
b - 51
Since there were three times as many boys as girls, we can write:
b - 51 = 3(g - 51)
Now we have two equations and two unknowns. We can solve for one variable in terms of the other in the first equation, and substitute into the second equation:
b + g = 342
g = 342 - b
b - 51 = 3(g - 51)
b - 51 = 3(342 - b - 51)
b - 51 = 3(291 - b)
b - 51 = 873 - 3b
4b = 924
b = 231
So there were 231 boys in the canteen at first. To check our answer, we can substitute into the first equation:
b + g = 342
231 + g = 342
g = 111
Therefore, there were 231 boys and 111 girls in the canteen at first.
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In the country of Enigmia the postage charge for sending a letter is calculated according to the
following formula:
20 cents per 50 square centimetres of envelope area plus 32 cents per 40 grams of weight.
All measurements of area and weight are rounded up to the next multiples of 50 square
centimetres and 40 grams before calculation.
Jennifer has a letter weighing 125 grams in a 20 cm by 12 cm rectangular envelope that she
wants to send.
What will the postage charge be for Jennifer's letter?
I
Answer:
$2.12
Step-by-step explanation:
First, let's find the area of Jennifer's envelope: 20 cm x 12 cm = 240 square cm.
Since the area needs to be rounded up to the next multiple of 50 square cm, we will round up 240 square cm to 250 square cm.
Next, let's round up the weight of the letter to the next multiple of 40 grams: 125 grams rounded up to 140 grams.
Now that we have the rounded up values for both weight and area, we can use the formula to find the postage charge:
20 cents per 50 square cm x (250 square cm ÷ 50) = 100 cents
32 cents per 40 grams x (140 grams ÷ 40) = 112 cents
So the total postage charge will be 100 cents + 112 cents = 212 cents = $2.12.
Therefore, the postage charge for Jennifer's letter will be $2.12.
Aɳʂɯҽɾҽԃ Ⴆყ ɠσԃKEY ꦿ
How would you check values for x and y after solving linear equations in x and y?
Answers:
1. substitute the x and y values into the original equations
2. divide the x and y values into the original equations
3. cross-multiply the x and y values by the original equations
4. subtract the x and y values from the original equations
a Which equation represents a line that passes through (-2, 4) and has a slope of 2/5?
y = 2/5x +24/5, is the equation which represents a line that passes through (-2, 4) and has a slope of 2/5.
What is a straight line?A straight line is an endless one-dimensional figure that has no width. It is a combination of endless points joined both sides of a point and has no curve.
here, we have,
The slope intercept form of a line is
y = mx+b
where m is the slope and b is the y intercept
y = 2/5 x +b
Substitute the point (-2,4) into the equation and solve for b
4 = 2/5(-2)+b
4 = -4/5 +b
Add 4/5 to each side
20/5 +4/5 = b
24/5 = b
Hence, y = 2/5x +24/5, is the equation which represents a line that passes through (-2, 4) and has a slope of 2/5.
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A.seqn T5 = 20, T₂² = T₁ T4 Find the order of the 1st term whose value is greater than 106.
The order of the first term whose value is greater than 106 is the smallest integer greater than 1 + [log(0.5)(T₁) - log(0.5)(212)].
How did we get the value?Using the given information, we can find the common ratio of the geometric sequence A.
First, we can use the equation T₂² = T₁ T4 to find T4. Substituting n = 4, we get:
T₂² = T₁ T4
20² = T₁ T4
T4 = 400/T₁
Next, we can use the equation T5 = 20 to find T₁r⁴:
T5 = T₁r⁴
20 = T₁r⁴
We can now substitute T4 in terms of T₁ in the equation above:
20 = T₁r⁴
20 = T₁(400/T₁)r⁴
20 = 400r⁴
r⁴ = 1/20
Therefore, the common ratio r = (1/20)^(1/4) = 0.5.
To find the order of the first term whose value is greater than 106, we can use the formula for the nth term of a geometric sequence:
Tn = T₁r^(n-1)
We want to find the smallest value of n such that Tn > 106. Substituting r = 0.5 and rearranging, we get:
T₁(0.5)^(n-1) > 106
T₁ > 212(0.5)^(1-n)
Since T₁ > 0, we can take the logarithm base 0.5 of both sides:
log(0.5)(T₁) < log(0.5)(212) + (n-1)
Solving for n, we get:
n > 1 + [log(0.5)(T₁) - log(0.5)(212)]
Therefore, the order of the first term whose value is greater than 106 is the smallest integer greater than 1 + [log(0.5)(T₁) - log(0.5)(212)].
Note that the exact value of this order depends on the value of T₁, which was not given in the original problem statement.
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In a normal distribution the mean and the __ are equal
In a normal distribution, the mean and the 'median' are equal.
In a normal distribution, the mean represents the central tendency of the data, while the median represents the middle value. In a perfectly symmetrical normal distribution, the mean and median are exactly the same value. This is because the normal distribution is a bell-shaped curve that is symmetric around the mean. However, in skewed distributions, the mean and median can differ, with the mean being pulled towards the tail of the distribution. Overall, the equality of the mean and median in a normal distribution is a reflection of the symmetry of the data.
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One hundred people were asked about their favorite sport.
The statement that best describes the two - way frequency table is b. More females like soccer.
How to find the correct statement ?The number of females who like soccer are 31 females out of a total of fifty females who were interviewed.
This means that the number of females who like basketball are :
= 50 - 31 females for soccer
= 19 females
31 females like soccer while 19 females like basketball which shows that more females prefer soccer to basketball.
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The blood pressure in millimeters was measured for a large sample of people. The average pressure is 140 mm, and the SD of the measurements is 20 mm. The histogram looks reasonably like a normal curve. Use the normal curve to estimate the following percentages. Choose the answer that is closest to being correct.
a. 10.6%
b. 89.4%
c. 39.4%
d. 78.8%
e. 68.27%
