Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose six integers are chosen from A. Must there be two integers whose sum is 11
Answers
When six integers are chosen from the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there must be at least one pair of integers that adds up to 11. This can be proven using a proof by contradiction.
To solve this problem, we can use a proof by contradiction. We assume that there are six integers chosen from A such that no two integers add up to 11.
If there is no pair of integers in the six chosen that sum up to 11, then we can consider the pairs of integers (1,10), (2,9), (3,8), (4,7), and (5,6). These are the only possible pairs of integers in A that add up to 11.
However, since there are five pairs and we can choose at most one integer from each pair, we can choose at most five integers in total. This is a contradiction since we were asked to choose six integers from A. Therefore, our assumption that there are no pairs of integers that add up to 11 is false.
Hence, we conclude that there must be at least one pair of integers among the six chosen that adds up to 11.
To know more about integer here
https://brainly.com/question/4196546
#SPJ4
Related Questions
Help pls
Mitchell went on a shopping trip to Kensington. He purchased a pair of headphones originally
priced at $50 but discounted 60%. If sales tax in Kensington is 15%, what was the total cost?
Answers
= 30
$50-$30=$20
20x(15/100)
= $3
$20+$3 = $23
The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Required percentage value = a
total value = b
Percentage = a/b x 100
The total cost of the pair of headphones is $23.
What is a percentage?The percentage is calculated by dividing the required value by the total value and multiplying by 100.
Example:
Required percentage value = a
total value = b
Percentage = a/b x 100
Example:
50% = 50/200 = 1/2
25% = 25/100 = 1/4
20% = 20/100 = 1/5
105 = 10/100 = 1/10
We have,
Original price of the headphones = $50
60% of $50
= 60/100 x 50
= $30
The price of the headphones after the discount.
= $50 - $20
= $20
The amount of sales tax.
= 15% of $20
= 15/100 x 20
= $3
The total cost.
= $20 + $3
= $23
Thus,
The total cost of the pair of headphones is $23.
Learn more about percentages here:
https://brainly.com/question/11403063
#SPJ2
Zahara mixed 5 quarts of dry ingredients and 2 quarts of water and a batch of concrete .She ran out of the mixture before finishing her project so she made another batch 10 quarts of dry ingredients . How many quarts of water does zahara need to add to keep the consistency the same? Complete the ratio square
Answers
Answer: 4 quarts.
Step-by-step explanation:
Given data:
Amount of dry ingredients = 5 quarts.
Amount of water in the mixture = 2 quarts.
Solution:
Let’s a represent the amount of dry ingredients.
Let b represent the amount of water
For first Batch
a/b = 5/2.
b = 2/5a ....... equation 1
For second Batch
a= 10
substitute a into equation 1 to solve for b
b = 2/5(10)
b = 4.
The ratio for batch b = 10/4.
Zahara needs to add 4 quarts of water to the mixture.
Answer:4 quarts!
Step-by-step explanation:
if a and b are sets, |a| = 10 and |b| = 5, then |a × b| = ?
Answers
If a and b set, then the cardinality of the Cartesian product of the two sets, |a × b|, is equal to the product of the cardinalities of the individual sets, |a| and |b|.
So, if |a| = 10 and |b| = 5, then |a × b| = 10 × 5 = 50.
Therefore, the cardinality of the Cartesian product of the two sets is 50.
In mathematical notation, this can be written as:
|a × b| = |a| × |b|
|a × b| = 10 × 5
|a × b| = 50
So the answer is 50.
Learn more about sets:
https://brainly.com/question/28365114
#SPJ11
Date
Jazzlyn is applying for a job online. She has completed 12 questions. Throughout the application, a
bar shows up to represent what percentage of questions she's completed. The progress bar
currently shows 25%. How many questions are there for the job application? Use a table, diagram,
or equation to justify your answer.
Answers
Answer:
so what
Step-by-step explanation:
Focus groups are not appropriate for generating statistics and making scientific generalizations. Select one: True False
Answers
Focus groups are not appropriate for generating statistics and making scientific generalizations.
The given statement is True.
Focus groups are qualitative research methods used to gather in-depth insights and subjective opinions from a small group of participants.
They typically involve a moderator guiding a discussion among a selected group of individuals to explore their perspectives, attitudes, and experiences on a specific topic.
The main objective of focus groups is to obtain rich, contextual information rather than to produce statistical data.
Focus groups usually consist of a small number of participants, typically ranging from 6 to 12 individuals, which makes it difficult to achieve statistical significance or representativeness.
The participants are not randomly selected, but rather purposively chosen based on specific criteria relevant to the research focus.
While focus groups offer valuable qualitative insights, they are not designed to generate statistically reliable data or make generalizations to a larger population.
The results obtained from focus groups are specific to the participants involved and their unique perspectives.
Therefore, it is not appropriate to use focus group findings as a basis for making scientific generalizations or drawing statistically significant conclusions about a broader population.
To make scientific generalizations and produce statistical data, researchers typically employ quantitative research methods, such as surveys or experiments, with larger and more diverse samples that allow for statistical analysis and inference.
For similar question on generating statistics.
https://brainly.com/question/32204197
#SPJ11
Concept 9.5: Find x Assume that segments that appear to be tangent are tangent.
Answers
Answer:
x = 2
Step-by-step explanation:
Here's your solution
=> we know that tangents from a point to a common circle are equal
=> so, ST = SU
=> 7x - 3 = 5x + 1
=> 7x - 5x = 1 + 3
=> 2x = 4
=> x = 4/2
=> x = 2
hope it helps
Answer:
X will be 2
Step-by-step explanation:
ST=SU
7x-3=5x+1
2x=3+1
2x=4
x=4/2
x=2
hope it helps....
have a great day!!!
y=3x-1
y=2x+7
solve by substitution
Answers
Therefore, the solution to the system of equations is x = 8 and y = 23.
What is equation?An equation is a statement that shows the equality between two mathematical expressions using symbols, numbers, and/or variables. Equations are used to describe relationships between different quantities, and they are commonly used in many fields, including mathematics, physics, chemistry, engineering, and economics.
by the question.
Let's solve the first equation for y:
y = 3x - 1
Now we substitute this expression for y in the second equation:
2x + 7 = y
We can substitute 3x - 1 for y in this equation:
2x + 7 = 3x - 1
Now we can solve for x:
2x - 3x = -1 - 7
-x = -8
x = 8
Now we can substitute x = 8 into either equation to solve for y. Let's use the first equation:
y = 3x - 1
y = 3(8) - 1
y = 24 - 1
y = 23
To learn more about equations:
https://brainly.com/question/10413253
#SPJ1
O
An escalator runs from the first floor of a shopping mall to
Answers
Answer: D.) the hypotenuse
__________________________________________________________
Have a good day!
pls consider marking me brainliest!
Question:
What is the longest side of a right triangle?
Answer:
The answer would be D. hypotenuse
Step-by-step explanation:
Leg A and B would be the legs remember pythagorean theorem. Pythagorean theorem would be a2 + b2 = c2, a is a side of right triangle b is also a side of right triangle and c is the hypotenuse.
I hope it helps you!
~XxBells is a cute girlxX~
#Learn with Brainly
Mike is a sales person in a retail carpet store he paid 500 base salary per month plus 5% commission on sales over 10,000. His sales this month were 23,750 his total deductions were 152.75 What is the total amount on which mike could earn commission? What was the total amount of mikes commission? What was mikes base salary plus commission? What's mike net pay?
Answers
Answer:
Total amount on which Mike could earn commission:
Mike can earn amounts on sales over $10,000 so the commission amount is:
= 23,750 - 10,000
= $13,750
Total amount of Mike's compensation:
= 5% * 13,750
= $687.50
Mikes base salary plus commission:
= 500 + 687.50
= $1,187.50
Mike's net pay:
= Gross pay - deductions
= 1,187.50 - 152.75
= $1,034.75
What are two of the ways in which the digestive system increases its ability to absorb nutrients?
Answers
Passive diffusion and active transport absorption are two of the ways in which the digestive system increases its ability to absorb nutrients.
How can the digestive system increase its ability to absorb nutrients?On the opposite side, the epithelial cells of these structures release the nutrients that were absorbed from the digested meal into the bloodstream. With their numerous folds, the villi and microvilli enhance the surface area of the small intestine and improve nutritional absorption.
Catabolism, or the breakdown of things, is a process that includes two distinct steps: mechanical digestion and chemical digestion.
The small intestinal mucosa's lining is extremely well-trained to maximise nutrition absorption and digestion. Villi, which enhance the surface area to aid in absorption, are minute finger-like projections made from the highly folded lining.
To learn more about catabolism refer to :
https://brainly.com/question/15198053
#SPJ4
2. Consider the data points (−1,−1),(2,2) and (1,0). (a) Find the least squares line of best fit for the data. (b) Compute the least-squares error.
Answers
(a) the equation of the least squares line is y = (3/5)x - 1/5. b) Hence, the least-squares error for the given data points is 26/5.
To find the slope, we use the formula:
m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)
Using the given data points:
Σx = -1 + 2 + 1 = 2
Σy = -1 + 2 + 0 = 1
Σxy = (-1)(-1) + (2)(2) + (1)(0) = 5
Σx^2 = (-1)^2 + 2^2 + 1^2 = 6
n = 3
Substituting these values into the formula, we get:
m = (3(5) - (2)(1)) / (3(6) - (2)^2) = 3/5
To find the y-intercept, we use the formula:
b = (Σy - mΣx) / n
Substituting the values, we get:
b = (1 - (3/5)(2)) / 3 = -1/5
Therefore, the equation of the least squares line is y = (3/5)x - 1/5.
(b) To compute the least-squares error, we need to calculate the sum of the squared differences between the observed y-values and the predicted y-values using the least squares line.
Using the given data points and the equation of the least squares line, we calculate the squared differences and sum them up:
(-1 - (-1/5))^2 + (2 - (6/5))^2 + (0 - (2/5))^2 = 26/5
Hence, the least-squares error for the given data points is 26/5.
Learn more about squares here:
https://brainly.com/question/28776767
#SPJ11
what is the correct set up to solve for x?
Answers
Equiangular (same angles) triangles are also equilateral (same sides), so both sides are equal. Therefore, in the equation, you have to set them equal to each other.
In a chemistry lab, you measure the same sample of carbon 6 times and get the following measurements: 9.91g. 10.03g, 10.01g. 9.92g. 9.99g, 10.02g. If you measure the sample a seventh time, by how much would you expect your answer to be off? Round your answer to two decimal places. 0.02
Answers
The expected error of the mean would be ± 0.032g or ± 0.03g rounded to two decimal places. Hence, the answer is 0.03.
The mean of all measurements would be,Mean = (9.91g + 10.03g + 10.01g + 9.92g + 9.99g + 10.02g) / 6= 59.88 / 6= 9.98 g
Therefore, the expected value or the seventh measurement should be 9.98 g, as it is based on the previous measurements.
Now, let's calculate the variance and the standard deviation to estimate the expected error,Variance,σ² = ∑ (xᵢ - μ)² / Nσ² = (9.91g - 9.98g)² + (10.03g - 9.98g)² + (10.01g - 9.98g)² + (9.92g - 9.98g)² + (9.99g - 9.98g)² + (10.02g - 9.98g)² / 6σ² = 0.00617g ²
Standard Deviation,σ = √σ²σ = √0.00617g²σ = 0.078g
Thus, by one standard deviation (68.26% confidence), the expected error would be ± 0.078g.
However, we want to estimate the error of the mean, which has a larger sample size.
Hence, we need to adjust the standard deviation for the sample size using the following equation,σᵢ = σ / √NIgnoring the subscript,σ = 0.078g / √6σ = 0.032g
Therefore, the expected error of the mean would be ± 0.032g or ± 0.03g rounded to two decimal places. Hence, the answer is 0.03.
To know more about decimal visit :-
https://brainly.com/question/28393353
#SPJ11
Find the perimeter of stu
Answers
Answer:
32
Step-by-step explanation:
we have to find the side of the bigger triangle by making a ratio
5/8=7/x
cross multiply
5x=7(8)
5x=56
/5. /5
x=11.2
now for the other side
5/8=8/x
5x=8(8)
5x=64
/5. /5
x=12.8
we know all three sides 11.2, 12.8 and 8 so we add them
11.2+12.8+8
24+8
32
hopes this helps
What is the distance between points A and B? Round your answer to the nearest tenth
A (22)
1
D
(21)
3
1
2
Answers
Answer:
AB = 5 units
Step-by-step explanation:
Calculate the distance d using the distance formula
d = \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\)
with (x₁, y₁ ) = A (2, 2 ) and (x₂, y₂ ) = B (- 2, - 1 )
AB = \(\sqrt{(-2-2)^2+(-1-2)^2}\)
= \(\sqrt{(-4)^2+(-3)^2}\)
= \(\sqrt{16+9}\)
= \(\sqrt{25}\)
= 5
Given \( v=\left[\begin{array}{l}21 \\ 98 \\ 28 \\ 37\end{array}\right] \) find the coordinates for \( v \) in the subspace \( W \) spanned by \[ u_{1}=\left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{ar
Answers
To find the coordinates of vector v in the subspace W spanned by u1, u2, and u3, we first need to check if the vectors u1, u2, and u3 are linearly independent. If they are linearly dependent, we can reduce them to obtain a basis for W. If they are linearly independent, we can directly use them as the basis for W.
To check for linear independence, we can form a matrix A by arranging u1, u2, and u3 as columns:
A = [u1 u2 u3] = [[4 2 1] [-2 1 -1] [1 1 0] [0 1 2]]
Next, we can reduce this matrix to row echelon form using Gaussian elimination:
[4 2 1]
[-2 1 -1]
[1 1 0]
[0 1 2]
Performing row operations:
R2 ← R2 + (1/2)R1
R3 ← R3 - (1/4)R1
R4 ← R4 - (1/8)R1
Resulting matrix:
[4 2 1]
[0 5/2 1/2]
[0 3/2 -1/4]
[0 1/2 7/8]
Performing further row operations:
R3 ← R3 - (3/5)R2
R4 ← R4 - (1/5)R2
Resulting matrix:
[4 2 1]
[0 5/2 1/2]
[0 0 -11/20]
[0 0 5/4]
Lastly, perform the final row operation:
R4 ← R4 + (11/25)R3
Resulting matrix:
[4 2 1]
[0 5/2 1/2]
[0 0 -11/20]
[0 0 0]
Since the reduced row echelon form has a row of zeros, the system of equations x1u1 + x2u2 + x3u3 = 0 only has the trivial solution. Therefore, u1, u2, and u3 are linearly independent and form a basis for the subspace W.To find the coordinates of v in subspace W, we solve the system of equations x1u1 + x2u2 + x3u3 = v using Gaussian elimination. Substituting the values of v and the basis vectors, we get the following matrix equation:
[4 2 1] [x1] [21]
[-2 1 -1] [x2] = [98]
[1 1 0] [x3] [28]
[0 1 2] [37]
Simplifying the matrix equation using row operations, we get:
[4 2 1 21]
[0 5/2 1/2 70]
[0 0 -11/20 -22]
[0 0 0 69/11]
From the last row of the matrix, we can determine that x3 = 69/11. Substituting this value into the third row of the matrix, we find x2 = 10. Finally, substituting the values of x2 and x3 into the second row of the matrix, we find x1 = -39/4.
Therefore, the coordinates of v in the subspace W are:
vW = [-39/4, 10, 69/11]
The given solution is correct, and it provides a detailed explanation of the process to find the coordinates of vector v in the subspace W spanned by u1, u2, and u3.
To know more about coordinates visit:
https://brainly.com/question/32836021
#SPJ11
example 2 major premise: no dogmatists are scholars who encourage free thinking. minor premise: some theologians are scholars who encourage free thinking. conclusion: some theologians are not dogmatists. the major premise in example 2 is an proposition. the minor premise in example 2 is an proposition. the conclusion in example 2 is an proposition. therefore, the mood of the categorical syllogism in example 2 is .
Answers
The mood of the categorical syllogism in example 2 is AIO.
In your example, we have the following premises and conclusion:
1. Major Premise: No dogmatists are scholars who encourage free thinking.
2. Minor Premise: Some theologians are scholars who encourage free thinking.
3. Conclusion: Some theologians are not dogmatists.
The major premise in example 2 is an A proposition (All S are not P). The minor premise in example 2 is an I proposition (Some S are P). The conclusion in example 2 is an O proposition (Some S are not P).
To learn more about premises, refer here:
https://brainly.com/question/29699382#
#SPJ11
What’s the answer ????
Answers
Answer:
A-1
Step-by-step explanation:
It perfectly represents the scenario.
Answer:
A - 2
Step-by-step explanation:
younger = -
A - 2
Hope this helped! Have a nice day! Plz mark as brainliest!!! :D
-XxDeathshotxX
Scott rolled a number cube several times and counted how many times he rolled a 3.
If Scott rolls the number cube 300 times, how many times should he predict to roll a 3?
Answers
Answer:
if it is a 6 sided cube, then 50
Step-by-step explanation:
300/6=50
Jing has $800 to deposit into two different savings account. She will deposit $600 into an account that earns simple interest annually at a rate of 2.75%. She will deposit $200 into an account that earns 2.5% interest compounded annually. Jing will not make any additional withdrawals or deposits. Which amount is closest to the total balance of the two accounts at the end of 3 years?
Answers
Answer:
the question answer is A
Step-by-step explanation:
Fill in the missing proportional value.
I just need to figure out what divided by 10/3 gets me 2.5
If someone could tell me how to find it out that would be really helpful too thanks
Answers
Answer:
8 1/3
Step-by-step explanation:
We want to find a number, x, that when divided by 10/3 equals 2.5. We can write that as:
x/(10/3) = 2.5
x = (2.5)*(10/3) [MULTIPLY BOTH SIDES BY (10/3)
x = 25/3
x = 8 1/3 [
1. Use the K-map to determine the prime implicants, essential prime implicants, a minimum sum of products, prime implicates, essential prime implicates, and a minimum product of sums for each of the following Boolean functions. Also, for each one compute a minimum product of sums and a minimum sum of products of its complements.
a. f(a,b,c,d)= Π M(0,1,8,11,12,14)
b. g(a,b,c,d)= Σ m(0,1,3,5,6,8,11,13,15)
c. h(a,b,c)= Σ m(1,4,5,6)
2. Write the decimal representation of SSOP and SPOS for each of the above functions and its complement.
Answers
The questions pertain to Boolean functions and involve using Karnaugh maps (K-maps) to determine prime implicants, essential prime implicants, minimum sum of products, prime implicates, essential prime implicates, minimum product of sums, and decimal representations of SSOP and SPOS forms for the given Boolean functions and their complements.
For Boolean function f(a, b, c, d) = ΠM(0, 1, 8, 11, 12, 14):
Using the K-map, we can determine the prime implicants and essential prime implicants.
The minimum sum of products can be derived from the prime implicants.
The prime implicates and essential prime implicates can also be determined.
To find the minimum product of sums of its complements, we can use the prime implicants and essential prime implicants of the complement function.
For Boolean function g(a, b, c, d) = Σm(0, 1, 3, 5, 6, 8, 11, 13, 15):
Similar to the first question, we can use the K-map to determine the prime implicants, essential prime implicants, minimum sum of products, prime implicates, essential prime implicates, and minimum product of sums of its complements.
The decimal representation of the SSOP (Sum of Sum of Products) and SPOS (Sum of Product of Sums) forms can be obtained for the given Boolean function and its complement.
For Boolean function h(a, b, c) = Σm(1, 4, 5, 6):
Follow a similar process using the K-map to find the prime implicants, essential prime implicants, minimum sum of products, prime implicates, essential prime implicates, minimum product of sums of its complements, and the decimal representation of SSOP and SPOS forms for the given Boolean function and its complement.
The process involves using K-maps and Boolean algebra techniques to determine the required values for each given Boolean function and its complement. The specific steps and calculations can be performed based on the provided Boolean functions and their respective minterms.
To learn more about decimal representations visit:
brainly.com/question/29220229
#SPJ11
A student always catches his train if class ends on time. However, 30% of classes run late and then there's a 45% chance he'll miss it. What is the probability that he misses the train today?
Answers
The probability that he misses the train today is 0.135 or 13.5%.
P(Class runs late)=30%=0.30
P(Miss train ∣ Class runs late)=45%=0.45
General multiplication rule:
P(A and B)=P(A)×P(B ∣ A)
Using the general multiplication rule, we then obtain (and using that the student always catches his train if the class ends on time) :-
P( Miss train)=P( Miss train and Class runs late)=P(Class runs late)×P(Miss train ∣ Class runs late)
= 0.30×0.45
= 0.135
= or 13.5%.
So, the the probability that he misses the train today is 0.135 or 13.5%.
Learn more about Probability and multiplication rule here :-
https://brainly.com/question/11234923
#SPJ4
The equation �=112�y=1\frac{1}{2}xy=1
2
1
x represents the number of cups of dried fruit, y, needed to make x pounds of granola. Determine whether each point would be on the graph of this proportional relationship.
Choose Yes or No for each point.
Answers
The coordinates (2,1) will be on graph but (1,3) is not on graph.
What is a coordinate?
A coordinate is a set of two or more numbers or variables that identify the position of a point, line, or plane in a space of a given dimension. Coordinates are used to pinpoint a particular location, such as a specific point on a map or a specific point in a mathematical equation.
This means that for every 1.5 cups of dried fruit, there is 1 pound of granola. The graph of this proportional relationship would be a line that goes through the origin and has a slope of 1.5. For the point (2,1), the x-coordinate (2) is exactly 1.5 times the y-coordinate (1). This means that if you used 2 cups of dried fruit, you would get 1 pound of granola. Therefore, this point would be on the graph of the proportional relationship, so the answer is Yes. However, for the point (1,3), the x-coordinate (1) is not 1.5 times the y-coordinate (3). This means that if you used 1 cup of dried fruit, you would not get 3 pounds of granola.
For more questions related to variable
brainly.com/question/25223322
#SPJ9
The cost to hire a tent consists of two parts.$c + $d per daythe total cost for 4 days is $27.10 and for 7 days is $34.30.write down two equations in c and d and solve them.
Answers
The value of c is 17.50 and the value of d is 2.40
How to solve the value of C and D in equation?A finite collection of symbols that are properly created in line with context-dependent criteria is referred to as an expression, sometimes known as a mathematical expression.
Given
Equation for 4 days
C + 4d=27.10
Equation for 7 days
C + 7d=34.30
Solving simultaneously equation
C + 4d=27.10
- C + 7d=34.30
by simplifying,
0- 3d = - 7.20
d= -7.30/-3
=2.40
Replace the d 2.40 in either side
C+4d = 27.10
C=27.10 - 4d
=27.10 - 4(2.40)
=17.50
Therefore the value of c is 17.50 and the value of d is 2.40
To learn more about expression refer to :
https://brainly.com/question/1859113
#SPJ4
find the rate of change of the area of a square when the side of the square is 2 meters and the side is growing at a rate of 3 meters per second.
Answers
The rate of change of the area of a square is 12m^2/sec.
What do you mean by rate of change?
The term "rate of change" refers to the rate at which one quantity changes in respect to another. Consequently, if y value is the dependent variable and x value is the independent variable.
Rate of Change is equal to \(\frac{Change in y}{Change in x}\).
Change rates may be positive or negative. This reflects a change in the y-value between the two data points, either up or down. Zero rate of change is the state where a quantity does not change over time.
Solution Explained:
Given,
S = 2m and ds/dt = 3m/s
So, we use
Area = s^2
dA/dt = 2s ds/dt
= 2 X 2 X 3
Therefore, the rate of change of the area is 12m^2/s.
To learn more about the rate of change, use the link given
https://brainly.com/question/25184007
#SPJ4
i need help answer it correctly
Answers
Answer:
(1) when x is 0 y=5/2 and when y is 0 x=5
(2)Graph the points : (0 , 5/2) and (5 , 0)
(3)the slope is -1/2
Step-by-step explanation:
Simultaneous Equation
(x+5)(x+2)=65 , xy=24
Plssss help me ✌
Answers
Answer:
(-11.70, -2.05), (4.70, 5.10)Step-by-step explanation:
Given equations
(x+5)(x+2)=65xy=24Solving the first equation for x:
(x+5)(x+2)=65x^2 + 7x + 10 = 65x^2 + 7x - 55 = 0x = (-7 ± √(49+4*55))/2x = -11.70, x = 4.70Then solving for y:
y= 24/(-11.7) = -2.05y = 24/4.7 = 5.10Solution set:
(-11.70, -2.05), (4.70, 5.10)Create a frequency distribution, find the midpoint, relative frequency, cumulative frequency and create a histogram for the following:
Data Set: Highway fuel consumptions (in miles per gallon) for a sample of cars
Number of Classes: 5 classes
32, 35, 28, 40, 30, 42, 55, 40, 45, 24, 28, 34, 40, 36, 34, 40, 30, 25, 28, 32, 40, 35, 25, 44, 26, 39, 38, 42, 45, 32
Answers
First sort the data:
24, 25, 25, 26, 28, 28, 28, 30, 30, 32, 32, 32, 34, 34, 35, 35, 36, 38, 39, 40, 40, 40, 40, 40, 42, 42, 44, 45, 45, 55
Not the size of this data (30) and the range, 55 - 24 = 31.
Now, 31/5 = 6.2, so split up the interval from 24 to 55 into the following 5 classes with equal size:
[1] 24 - 30.2
[2] 30.2 - 36.4
[3] 36.4 - 42.6
[4] 42.6 - 48.8
[5] 48.8 - 55
If a given data has a value in one of these sub-ranges, you place it in that group.
You would end up with the following frequency distribution:
[1] 24, 25, 25, 26, 28, 28, 28, 30, 30
[2] 32, 32, 32, 34, 34, 35, 35, 36
[3] 38, 39, 40, 40, 40, 40, 40, 42, 42
[4] 44, 45, 45
[5] 55
To get the relative frequencies for each class, count the number of data in each class and divide it by the total size of the dataset.
The relative frequencies in this case would be
[1] 9/30 = 3/10
[2] 8/30 = 4/15
[3] 9/30 = 3/10
[4] 3/30 = 1/10
[5] 1/30
The cumulative frequencies are the cumulative sums of the frequencies as you move up from one class to another:
• For any data less than 24, the cumulative frequency is 0
• Up to class [1], the cumulative frequency is 3/10
• Up to class [2], the c.f. is 3/10 + 4/15 = 17/30
• Up to class [3], the c.f. is 17/30 + 3/10 = 26/30 = 13/15
• Up to class [4], the c.f. is 13/15 + 1/10 = 29/30
• Up to class [5] and beyond, the c.f. is 29/30 + 1/30 = 1
The c.f.'s always range from 0 to 1 in this way.
The histogram should roughly look like the one I've attached.
3 2/3 divided by 3/4
Answers
Answer:
\( \frac{44}{9} \)