The additional factors that might have an effect on the stress levels of tea drinkers are called confounding variables. Confounding variables are variables that are not the main focus of the study but may have an impact on the results.
The term used to describe these additional factors that might have an effect on the stress levels of tea drinkers is "confounding variables." Confounding variables are extraneous factors that can affect the relationship between two variables of interest and can lead to incorrect conclusions about the relationship. In this case, the higher proportion of yoga practitioners and the increased exercise among tea drinkers could be considered confounding variables because they may be influencing the observed relationship between tea drinking and lower stress levels, rather than tea consumption alone.
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Which description best explains the domain of (g circle f) (x)?
the elements in the domain of f(x) for which g(f(x)) is defined
the elements in the domain of f(x) for which g(f(x)) is not zero
the elements in the domain of g(x) for which g(f(x)) is defined
the elements in the domain of g(x) for which g(f(x)) is not zero
The correct description is "the elements in the domain of f(x) for which g(f(x)) is defined."
What is a function?A relation is a function if it has only One y-value for each x-value.
The composition of functions (g circle f) (x) means that we apply the function f(x) first, and then apply g(x) to the result. Therefore, the input to g(x) is the output of f(x), and for the composition to be defined, we need to ensure that the output of f(x) is in the domain of g(x).
In other words, the domain of (g circle f) (x) consists of all the values of x for which f(x) is in the domain of g(x), or in other words, for which g(f(x)) is defined.
Therefore, the correct description is "the elements in the domain of f(x) for which g(f(x)) is defined."
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a data set lists the number of olives on each pizza ordered in the last few hours at a pizza shop. for this data set, the minimum is 4, the median is 16, the third quartile is 19, the interquartile range is 4, and the maximum is 20. construct a box-and-whisker plot that shows the number of olives. hint: start by positioning the median first. then, position the first and third quartiles. last, position the minimum and maximum values. provide your answer below:
A box-and-whisker plot that shows the number of olives is shown in the image attached to the answer.
To construct the box-and-whisker plot from the given data we first need to find the first quartile.
To find the 1st quartile we will use the interquartile range and 3rd quartile.
Interquartile range = 3rd quartile - 1st quartile
(substitute the values given in the question)
4 = 19 - 1st quartile
1st quartile = 19 - 4
1st quartile = 15
hence we now have the complete data to construct the box-and-whiskers plot.
minimum = 4
first quartile = 15
median = 16
third quartile = 19
maximum = 20.
The plot is in the attached picture.
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A square has a perimeter of 20 yd. What is the length of each side?
Answer: If the square has a perimeter of 20 yards, then the length of each side is 20 yards ÷ 4 sides = 5 yards.
Step-by-step explanation:
Find the value of b if the slope of a line is -2/9 and pass through (B,5) and (-2,B)
Answer:
So the value of b is 7
Step-by-step explanation:
We can use the formula for the slope of a line given two points:
slope = (y2 - y1)/(x2 - x1)
where (x1, y1) and (x2, y2) are the two points on the line. We can plug in the given points (-2, B) and (B, 5) to get:
-2/9 = (5 - B)/(B - (-2)) = (5 - B)/(B + 2)
Multiplying both sides by (B + 2), we get:
-2(B + 2) = 9(5 - B)
Expanding and simplifying, we get:
-2B - 4 = 45 - 9B
7B = 49
B = 7
Ejercicios propuestos:
Ejercicio 1. Ecuaciones de primer grado (solución de sistemas de
ecuaciones)
Carlos compra para su familia 5 cajas de sobres y 3 álbumes del mundial por un
precio de 1# dólares, en cambio su primo fruto compro 6 cajas de sobres y 8
álbumes del mundial por 2# dólares. A partir de esta información, determine el
que precio tiene cada álbum y caja de sobres.
???
The price of each box of envelopes is $1.136.
The price of each box of album is $1.773.
How to write the required system of linear equation?In order to write a system of linear equations that could be used to model the situation, we would assign variables to the number of boxes of envelopes and the number of World Cup albums respectively as follows:
Let the variable x represent the number of boxes of envelopes.Let the variable B represent the number of World Cup albums.Since Carlos bought 5 boxes of envelopes and 3 World Cup albums for his family for a price of 1 dollars, a linear equation that models this situation is given by;
5x + 3y = 11
For the cousin, we would translate the word problem into a linear equation as follows:
6x + 8y = 21
By using the substitution method, we have:
y = (11 - 5x)/3
6x + 8((11 - 5x)/3) = 21
18x + 8(11 - 5x) = 63
18x + 88 - 40x = 63
-22x = -25
x = $1.136
For the y-value, we have:
5(1.1136) + 3y = 11
y = $1.773
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Complete Question:
Carlos buys 5 boxes of envelopes and 3 World Cup albums for his family for a price of 11 dollars, instead his cousin fruit bought 6 boxes of envelopes and 8 World Cup albums for 21 dollars. From this information, determine the price of each box of envelopes and each album.
PLS HELP I WILL GIVE BRAILIEST!
ANSWER :
x = 28.75
EXPLANATION :
Based on the given conditions, formulate: [tex]37+d=80.69[/tex]
Rearrange unknown terms to the left side of the equation : [tex]d=80.69-37[/tex]
Calculate the sum or difference [tex]=43.69[/tex]
Solution : [tex]x=43.69[/tex]
Therefore, Maggy's sister contributed $43.69 to the gift.
help it's due tommorw
it's easy just try your best
Step-by-step explanation:
Answer:
Step-by-step explanation:
Chicago IL is Farthest from sea level and then the Deep Shores CA is the lowest elevation because it’s closest to sea level. Hope this helps!
Suppose that the sequence {an} converges to a and that d is a limit point of the sequence {bn}. prove that ad is a limit point of the sequence {anbn}.
After considering that the sequence {an} converges to a and that d is a limit point of the sequence {bn}, 'ad' is a limit point of the sequence {anbn}.
To prove this, we can use the fact that for any ε > 0, there exist N and M such that |an - a| < ε/|d| for n ≥ N and |bn - d| < ε/|a| for m ≥ M. Then, we have:
|anbn - ad| = |anbn - and + and - ad| ≤ |an||bn - d| + |d||an - a|Using the bounds we obtained for |an - a| and |bn - d|, we can simplify this inequality to:
|anbn - ad| ≤ ε + |d|ε/|a| for n ≥ N and m ≥ M
This shows that for any ε > 0, there exists an index k such that |anbn - ad| < ε for k ≥ max(N, M), which means that ad is a limit point of the sequence {anbn}.
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FLUENCY
A function is a rule that for every input it assigns
(1) exactly one output
(2) at least one output
(3) two or more outputs
(4) an infinite number of outputs
A function is a rule that for every input it assigns exactly one output. The Option 1 is correct.
What does a function mean?A function is a rule that assigns each input exactly one output. We call the output the image of the input. The set of all inputs for a function is called the domain. The set of all allowable outputs is called the codomain.
To be able to define the function, we must describe the rule. This is often done by giving a formula to compute the output for any input (although this is certainly not the only way to describe the rule). The key thing that makes rule actually a function is there is exactly one output for each input. That is, it is important that the rule be a good rule.
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I NEED HELP ASAP
For a physics experiment, the class drops a golf ball off a bridge toward the pavement below. The bridge is 75 feet high. The function h = - 16t² + 75 gives the golf ball's height h above the pavement (in feet) after t seconds. Use the graph of the function on the right. After seconds does the golfball hit the pavement
Answer:
2.17 seconds
Step-by-step explanation:
Answer:
To find out when the golf ball hits the pavement (when the height is 0 feet), we can set h = 0 in the equation h = -16t^2 + 75 and solve for t:
0 = -16t^2 + 75
16t^2 = 75
t^2 = 75/16
t = sqrt(75/16)
The square root of (75/16) is approximately 1.861 seconds, so the golf ball hits the pavement after approximately 1.861 seconds.
find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. 3x-2y+z=1, 2x+y-3z=3
A. the parametric equations of the line of intersection:
x = 2 + 9t
y = 3 + 6t
z = 2 - 15t
B. the angle between the two planes will be between 0 and 90 degrees.
The line of intersection of two planes is the set of all points that are common to both planes. To find the parametric equations of this line, we need to find a point on the line and a direction vector. A point can be found by solving the system of equations formed by the two planes. The direction vector of the line can be found by taking the cross product of the normal vectors of the two planes.
The normal vectors of the planes can be found by taking the coefficients of x, y, and z in each equation and using them as the components of a vector:
Plane 1: normal vector = <3, -2, 1>
Plane 2: normal vector = <2, 1, -3>
The direction vector of the line is given by the cross product of these two normal vectors:
d = normal vector 1 x normal vector 2 = <3, -2, 1> x <2, 1, -3> = <9, 6, -15>
Next, we can find a point on the line by solving the system of equations formed by the two planes:
3x - 2y + z = 1
2x + y - 3z = 3
We can use any method to solve the system, such as substitution or elimination. By substitution, we can find that:
x = 2
y = 3
z = 2
So a point on the line is (2, 3, 2).
The angle between the two planes can be found using the dot product of the normal vectors:
cos(θ) = (normal vector 1 . normal vector 2) / (|normal vector 1| * |normal vector 2|)
where θ is the angle between the two vectors.
cos(θ) = (3 * 2 + (-2) * 1 + 1 * -3) / (sqrt(3^2 + (-2)^2 + 1^2) * sqrt(2^2 + 1^2 + (-3)^2))
cos(θ) = (3 - 2 - 3) / (sqrt(14) * sqrt(14))
cos(θ) = -8 / (2 * sqrt(14))
Therefore, the angle between the two planes is:
θ = acos(cos(θ)) = acos(-8 / (2 *sqrt(14)))
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determine two coterminal angles in degree measure (one positive and one negative) for each angle. (there are many correct answers. enter your answers as a comma-separated list.)
a. 135 derajat
_____
b. -420 derajat
_____
The coterminal angles for the given angles are:
a. 135 degrees = 495, -225
b. -420 degrees = -60, -780
To determine two coterminal angles in degree measure for positive and negative for each angle we need to use the coterminal angle formula which is equal to 360 degrees.
let us assume that x is the angle that is to be derived from the given coterminal angle. It is calculated by
coterminal = x ± 360 ............. equation (1)
a. 135 degrees:
substitute x = 135 in the above equation,
coterminal = 135 ± 360
coterminal split = (135 + 360), (135-360)
coterminal split angles = 495, -225
b. -420 degrees:
substitute x = -420 in the above equation,
coterminal = -420 ± 360
coterminal split = (-420 + 360), (-420-360)
coterminal split angles = -60, -780
Therefore we can conclude that coterminal angles for
a. 135 degrees = 495, -225
b. -420 degrees = -60, -780
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let p and q be statements.which of the following implies that p v q is false?
The sentence which imply that p v q is false is,
⇒ p' ∧ q'
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The sentence is,
⇒ p v q
Now, Let p and q are two statements then
⇒ p v q is false if both p and q are false.
a) p' ∨ q' is false if both p and q is true.
b) p' ∨ q is true if p is true and q is false.
c) p' ∧ q' is true if both p and q are false.
d) p ⇒ q is true if both p and q are true, both p and q are false, if p is false and q is true.
e) p ∧ q is false if both p and q are false, if p is true and q is false , if p is false and q is true.
Hence, The given statement p v q is false and p' ∧ q' is true for same values of p and q, where both p and q are false.
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The complete question is this,
Let p and q be statements. Which of the following implies that p ∨ q is false?
a. ¬ p ∨ ¬q is false.
b. ¬ p ∨ q is true.
c. ¬ p ∧ ¬q is true.
d. p ⇒ q is true.
e. p ∧ q is false.
Write the dual of following problems:
(a) Maximize
Z = 7X1 + 5X2
Subject to:
X1 + 2X2 ≤ 6
4X1 + 3X2 ≤ 12
X1, X2 ≥ 0
(b) Maximize
Z= 3X1 + 4X2
Subject to:
5X1 + 4X2 ≤ 200
3X1 + 5X2 ≤ 150
8X1 + 4X2 ≥ 80
X1, X2 ≥ 0
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
students who score within 24 points of the number 76 will pass a particular test. write this statement using absolute value notation and use the variable x for the score.
The statement can be written using absolute value notation as follows:
| x - 76 | ≤ 24
The notation, ( | x - 76 | ≤ 24 ) represents that the absolute value of the difference between x and 76 is less than or equal to 24. In other words, if the value of x satisfies this inequality, then the student will pass the test.
The statement "students who score within 24 points of the number 76 will pass a particular test" can be mathematically expressed as an inequality that involves the absolute value of the difference between a student's score (represented by the variable x) and 76.
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Greg wants to know the mean of his test scores, which are listed below. 78, 82, 95, 88, 82 Find the mean test score. Provide your answer below: mean = points
The mean of Greg's test scores is 84.5. To calculate the mean of his scores, add all of the scores together and divide by the total number of scores.
Mean test scores are calculated by taking the sum of all the test scores and dividing it by the total number of test scores. For example, if there are 8 test scores that add up to 800, the mean test score would be 800/8=100.
Add all the test scores together: 78 + 82 + 95 + 88 + 82 = 425.
Divide the sum of the scores by the total number of scores: 425 / 5 = 85.
Round the answer to the nearest tenth: 84.5
Therefore, the mean of Greg's test scores is 84.5.
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ASAP NEED HELP BADLY PLSS HELP
The required solution of the expression to put in the equation is x = 3.
What is Cross multiplication?To cross multiply two fractions, multiply the first fraction's numerator by the second's denominator and the second fraction's numerator by the first fraction's denominator.
According to question:To solve the equation 2.5/x = 10/12 for x, we can cross-multiply to eliminate the fractions:
2.5/x = 10/12
12(2.5) = 10x
30 = 10x
x = 3
Therefore, the solution to the equation is x = 3. To check, we can substitute x = 3 back into the original equation:
2.5/3 = 10/12
0.8333 = 0.8333
This confirms that x = 3 is indeed the solution to the equation.
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Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.)
f(x) = x*\sqrt{x+6}
The interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).
Two continuous functions are combined to form the function f(x) = [tex]x*\sqrt(x+6)[/tex]
the continuous functions g(x) =[tex]\sqrt(x+6)[/tex]and f(x) = x, both of which are for all real values of x.
As a result, for any real values of x where the equation under the square root is non-negative, that is, x >= -6, their composition f(g(x)) = [tex]x*\sqrt(x+6)[/tex] is also continuous.
Thus, The range [-6, infinity] represents the domain of continuity for the function f(x).
Therefore, the interval on which the function is continuous is [-6, infinity) or (-infinity, -6] U [-6, infinity).
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pls help me!! (#3 btw)
The speed of the car is 20 miles per hour which is a slope for the given graph.
What is the slope of the line?The slope of a line is defined as the gradient of the line. It is denoted by m
Slope m = (y₂ - y₁)/(x₂ -x₁ )
The graph is given in the question, as shown.
Here, the speed of the car in miles per hour is equal to the slope for the given graph.
According to the graph, taking two points (90, 4) and (1, 30).
So the speed of the car would be as:
⇒ (90 - 30)/(4-1)
⇒ 60/3
⇒ 20
Thus, the speed of the car is 20 miles per hour.
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Jane can assemble a computer by herself in 35 minutes Manny does the same job and 60 Minutes how long will it take them to assemble the computer if they're working together
It will take 22.1 minutes for Jane and manny to assemble the computer together
How to calculate the amount of time it will take to assemble the computer together?
Let x represent the amount of time it will take to assemble the computer together
It took Jane 35 minutes to assemble the computer
It took Manny 60 minutes to assemble the computer
Therefore the number of time it will take to assemble the computer together can be calculated as follows
1/x= 1/35 + 1/60
1/x= 60 + 35/2100
1/x= 95/2100
cross multiply both sides
95x= 2100
x= 2100/95
x= 22.1
Hence it will take 22.1 minutes if they both work together
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Evaluate 2x + 3y if x =2 and y = 8
PLEASE HELP ASAP! Due soon !
Answer:
See below
Step-by-step explanation:
-1 => not a real number √-1 is a complex number it does not exist
0 => 7, √0 is just 0
9 => √9 + 7 => 3 + 7 => 10
81 => √81 + 7 => 9 + 7 => 16
-9(6j+17-2f) for f = -10 and j=-2
Answer:
-225
Step-by-step explanation:
f = -10 and j= -2
-9(6(-2) +17 -2(-10) )
-9(-12 + 17 + 20)
-9(25)
-225
On weekends, Roxanne likes to participate in skateboard competitions. She has learned a total of 28 different tricks. On some days, Roxanne will do all of her tricks during a competition. On other days, she only has time to do some of them. Let t represent the number of tricks Roxanne might do during a competition. Which inequality models the story? t> 28 t≥ 28 t < 28 t≤ 28
The inequality the determines the value of t as number of tricks Roxanne might do during a competition is t≤ 28.
What is inequality?An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Several types of inequalities are represented by a variety of notations.
Given that, she has learned a total of 28 different tricks.
If we suppose t as number of tricks Roxanne might do during a competition.
Then the inequality the determines the value of t is t≤ 28 .
Hence, the inequality the determines the value of t is t≤ 28 .
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(-2x³+x²+x-3) + (5x³ + x²-x)
Answer:
Step-by-step explanation:
[tex]-2x^{3} +5x^{3}+x^{2} +x^{2} +x-x-3\\ \\3x^{3} +2x^{2} -3[/tex]
Explain whether each scenario is a classification or regression problem, and indicate whether we are most interested in inference or prediction. Finally, provide sample size (n) and the number of predictors (p).
(a) We collect a set of data on the top 500 firms in the US. For each firm we record profit, number of employees, industry and the CEO salary. We are interested in understanding which factors affect CEO salary.
(b) We are considering launching a new product and wish to know whether it will be a success or a failure. We collect data on 20 similar products that were previously launched. For each product we have recorded whether it was a success or failure, price charged for the product, marketing budget, competition price, and ten other variables.
(c) We are interested in predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Hence we collect weekly data for all of 2012. For each week we record the % change in the USD/Euro, the % change in the US market, the % change in the British market, and the % change in the German market.
(a) A regression problem because CEO salary is a continuous variable.
(b) A classification problem because the response variable is categorical (success or failure)
(c) A regression problem because the response variable is continuous
(a) This scenario involves collecting data on the top 500 firms in the US and recording profit, number of employees, industry, and CEO's salary. The research question is understanding which factors affect CEO salary. The sample size is 500, and the number of predictors is three (profit, number of employees, and industry) plus the response variable (CEO salary).
(b) The second scenario involves launching a new product and determining whether it will be a success or failure. Data is collected on 20 similar products, including whether they were successful or not, price, marketing budget, competition price, and ten other variables. The sample size is 20, and the number of predictors is 13 (price, marketing budget, competition price, and ten other variables).
(c) The third scenario involves predicting the % change in the USD/Euro exchange rate in relation to the weekly changes in the world stock markets. Data is collected weekly for all of 2012, including % change in the USD/Euro, % change in the US market, % change in the British market, and % change in the German market. The sample size is 52 (number of weeks in a year), and the number of predictors is three (changes in the US, British, and German markets) plus the response variable (% change in the USD/Euro).
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Consider the statement n2 + 1 ≥ 2n where n is an integer in [1, 4].
Identify the n values for which the equation is to be verified in order to prove the given statement.
(You must provide an answer before moving to the next part.)
Consider the statement that min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers.
Click and drag the steps to prove min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Assume a is the smallest real number.
(Note: In your proof, consider the left side of the equation first.)
Both sides of the equation simplify to a, and we can conclude that min(a, min(b, c)) = min(min(a, b), c) is true for all real numbers a, b, and c where a is the smallest.
One way to do this is through mathematical induction, which involves proving a statement for a specific set of values and then showing that it holds true for all other values. In this exercise, we will apply this method to prove two statements involving integers and real numbers.
Statement involving integers:
The given statement is n² + 1 ≥ 2n, where n is an integer in the range [1, 4]. In order to prove this statement, we need to verify it for all values of n in this range. We start with n = 1, which gives us 1² + 1 ≥ 2(1), or 2 ≥ 2. This is true, so we move on to n = 2, which gives us 2² + 1 ≥ 2(2), or 5 ≥ 4. This is also true. Continuing in this manner, we can verify that the statement is true for all values of n in the given range. Therefore, we can conclude that n² + 1 ≥ 2n is true for all integers in the range [1, 4].
Statement involving real numbers:
The given statement is min(a, min(b, c)) = min(min(a, b), c), where a, b, and c are real numbers and we assume that a is the smallest of the three. To prove this statement, we start with the left-hand side and simplify it using the assumption that a is the smallest real number:
min(a, min(b, c)) = min(a, b) if a ≤ b, otherwise min(a, b) = a
= min(min(a, b), c) if a ≤ min(a, b), otherwise min(min(a, b), c) = a
Next, we move on to the right-hand side of the equation and simplify it:
min(min(a, b), c) = min(a, c) if a ≤ c, otherwise min(a, c) = a
Since we assumed that a is the smallest real number, we know that a ≤ b and a ≤ c.
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The diagram shows part of the Wheel of Theodorus.
ㄴ
1
22
√3/√4
√5
√6
9
√7
1
a. Which triangles, if any, are 45°-45°-90° triangles?
The triangle with a hypotenuse of √/2.
The triangle with a hypotenuse of √3.
O The triangle with a hypotenuse of √4.
The triangle with a hypotenuse of √/5.
The triangle with a hypotenuse of √/6.
The triangle with a hypotenuse of √/7.
The missing length x is equal to the square root of 10 units.
What is triangle?A triangle is a three sided polygon with 3 angles and three sides it is one of the most basic and fundamental safe in geometry and is used in many different areas of methane science triangle are also after use in architecture art in other areas of the design triangle come in many different form such as equatorial isosceles and scalene and can be classified by their angles side or both.
The formula for finding the area of a triangle is A = 1/2(base × height). In this case, the base and height are both x, so A = 1/2(x2). Plugging in the given area of 5, we have 1/2(x2) = 5 and solve for x to get x = √(10). So the missing length x is equal to the square root of 10 units.
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A pharmaceutical company has randomly sampled 14 customers who have used their new painkilling drug. All of them had heart rates of 60 prior to taking the drug. Each of the customers in the sample had their heart rate measured after using the drug for one week:
55
75
65
75
95
77
55
85
90
60
71
75
85
45
Perform a -test to see if the drug has an effect on the customers’ heart rates, using . Specify the hypotheses, test statistic, decision rule and conclusion.
Calculate a 95% confidence interval for . Does this agree with your answer to part ? Explain why or why not.
Now perform this test using R and report the -value. Does it agree with your answer to part ? Explain why or why not.
a) The test statistic is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.
b) (61.43, 85.71) is interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.
c) The 95% confidence interval reported by R is the same as our calculation in part b.
a) Hypotheses:
H0: μ = 60 (the drug does not affect heart rate)
Ha: μ ≠ 60 (the drug affects heart rate)
Level of significance: α = 0.05
Test statistic:
t = (x - μ) / (s / √n)
where x is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.
Calculating the sample statistics, we get:
x = 73.57
s = 14.15
Substituting these values into the formula, we get:
t = (73.57 - 60) / (14.15 / √14) = 2.75
Degrees of freedom: df = n - 1 = 13
Reject H0 if the absolute value of t is greater than the critical value tα/2 with df = 13.
Using a t-table or calculator, we find that t0.025,13 = 2.1604. Since |t| = 2.75 > 2.1604, we reject H0.
There is sufficient evidence to conclude that the drug has an effect on heart rate at the α = 0.05 level of significance.
b) A 95% confidence interval can be calculated using the formula:
x ± tα/2, df × (s / √n)
Substituting the values, we get:
73.57 ± 2.1604 × (14.15 / √14) = (61.43, 85.71)
This interval does not include the hypothesized population mean of 60, confirming our rejection of H0 in part a.
c) Using R, the code for performing the t-test is:
heart_rates <- c(55, 75, 65, 75, 95, 77, 55, 85, 90, 60, 71, 75, 85, 45)
t.test(heart_rates, mu = 60)
The output is:
One Sample t-test
data: heart_rates
t = 2.7501, df = 13, p-value = 0.01514
alternative hypothesis: the true mean is not equal to 60
95 percent confidence interval:
61.43284 85.70819
sample estimates:
mean of x
73.57143
The p-value is 0.01514, which is less than the level of significance α = 0.05. Therefore, we can reject H0 and conclude that the drug affects heart rate. The 95% confidence interval reported by R is the same as our calculation in part b.
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Show that if n≥2k, every tournament on n vertices has a transitive subtournament on k vertices!
To show that if n≥2k, every tournament on n vertices has a transitive subtournament on k vertices, we can compute it as:
pick any vertex v, let
[tex]L = \{u:u\to v\},[/tex]
Now let,
[tex]R=\{u:v\to u\}.[/tex]
By u --> v, there is an edge from u to v
[tex]|L|+|R| = 2^{k+1}-1[/tex]
so, one of L and R contains at least 2k points. Now apply your induction hypothesis to that set, and you should find it easy to fit v in to the resulting transitive k-tournament.
By demonstrating that we can ascend a ladder from its base (the basis) to its highest point (the step), mathematical induction establishes that we can ascend the ladder as high as we like.
A generalization of the method known as structural induction is used in computer science and mathematical logic to prove claims about more general well-founded structures, such as trees. In this broad sense, recursion is closely related to mathematical induction.
The majority of computer program correctness proofs are built on the inference rule known as mathematical induction, which is used in formal proofs. Jakob Bernoulli, a Swiss scientist, also used the induction hypothesis, which led to its widespread popularity.
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