The "winning" guess when played this game has been around 18.
What is rationality?The quality of being guided by or based on reasons is referred to as rationality. In this sense, a belief is rational if it is based on strong evidence, and an individual acts rationally if they have a good reason for doing so. This quality can be applied to a skill, like reasoning in a rational animal, a psychological process, like reasoning, mental states, like beliefs and intentions, or people who have these other kinds of rationality. If something does not meet the standards of rational evaluation, it is either irrational or rational, depending on where it falls outside the realm of rational evaluation.
Given you have to choose a number between 0 and 100. the winner of the game is the person whose number is closest to 2/3 times the average of all chosen numbers,
Theoretically, concur that the answer is 0. As stated, this is due to widespread rationality knowledge. You should guess 2/3*50=33 if everyone chose equally between 0 and 100. The average would be 50. Everyone ought to guess 33 if they are all aware of this. However, if I know that everyone correctly predicted 33, then 33 is the average, and should seek 2/3*33=22. This logic continues until everyone should correctly guess 0.
People actually don't think this way, but I disagree that you should guess between 5 and 10, which I believe is too low. depending on who you play with, as a group of game theorists might get a lower average than a group of people from the average population.) In a real-world setting, you need to consider how many iterations the others will make. You should guess 22 if you think they will only do one, which would result in everyone guessing 33. If you assume that they iterate the reasoning twice on average and guess 22, you should estimate 15 and so on.
The "winning" guess when played this game has been around 18.
Hence the guess should be 18.
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the complete question is,
Several people choose an integer between 0 and 100. The winner is the person whose integer is closest to 2/3 of the average of the numbers reported by all seven participants. What value would you choose?
find the quotient 8976 divided by 88
The quotient will be 102.
What is a quotient?
In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics and is commonly referred to as the integer part of a division, or as a fraction or a ratio.
The number 8976 is called the numerator or dividend, and the number 88 is called the denominator or divisor.
The quotient of 8976 and 88, the ratio of 8976 and 88, as well as the fraction of 8976 and 88 all, mean (almost) the same:
8976 divided by 88, often written as 8976/88.
= 8976/88
= 102
Hence, the quotient will be 102.
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Are the expressions 18+3.1 m+4.21 m-2 and 16+7.31 m equivalent
Answer:
Yes, they are equivalent.
Step-by-step explanation:
18 + 3.1m + 4.21m - 2
Combine like terms.
7.31m + 18 - 2
7.31m + 16 = 16 + 7.31m
Use the common ratio to find the next term of the geometric sequence. 120,60,30, ...
Answer:
15.
Step-by-step explanation:
The common ratio = 60/120 = 1/2
So the next term is 30 * 1/2 = 15.
Suppose you have the following three student loans: $12,000 with an APR of 7.5% for 17 years, $18,000 with an APR
of 8% for 22 years, and $13,500 with an APR of 9% for 12 years.
a. Calculate the monthly payment for each loan individually.
b. Calculate the total you'll pay in payments during the life of all three loans.
c. A bank offers to consolidate your three loans into a single loan with an APR of 8% and a loan term of 22 years.
What will your monthly payments be in that case? What will your total payments be over the 22 years?
a) The monthly payment for each loan individually is as follows:
Loan A Loan B Loan C Total
Monthly payments $133.82 $188.18 $195.00 $517.00
b) The total payments during the life of all three loans is $105,060, classified as follows:
Loan A Loan B Loan C Total
Total amount $27,300 $49,680 $28,080 $105,060
c) The monthly payments after consolidating the three loans are $350.69.
d) The total payments over the 22 years period will be $92,581.47.
How are the monthly payments determined?Note that student loans attract simple interest and not compound interest.
After the bank consolidates it, the interest system changes to compounding.
The total interest is computed by applying the annual percentage rate (APR) to the principal and the loan term. The result is added to the principal to obtain the total payment for each loan.
The total payment is divided by the loan period to get the monthly payments.
But for the consolidated loan by the bank, we can use an online finance calculator to determine the monthly payments and total payment over the loan term as follows.
Loan A Loan B Loan C Total
Student loan $12,000 $18,000 $13,500 $43,500
APR 7.5% 8% 9%
Loan term 17 years 22 years 12 years
Period in months 204 264 144
Total interests $15,300 $31,680 $14,580 $61,560
Total amount $27,300 $49,680 $28,080 $105,060
Monthly payments $133.82 $188.18 $195.00 $517.00
Consolidated Loans:Total amount of loans = $43,500 ($12,000 + $18,000 + $13,500)
Bank APR = 8%
Loan term = 22 years
N (# of periods) = 264 months (22 years x 12)
I/Y (Interest per year) = 8%
PV (Present Value) = $43,500
FV (Future Value) = $0
Results:
PMT = $350.69
Sum of all periodic payments = $92,581.47
Total Interest = $49,081.47
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When seven times a number is decreased by 8, the result is 13. What is the number?
Evan bought 12 postcards during 6 days of vacation. After 9 days of vacation,
how many total postcards will Evan have bought? Assume the relationship is
directly proportional.
Answer:
Step-by-step explanation:
Evan bought 18 postcards in 9 days
since he got 12 in 6 days that means he got 2 each day so 9x2=18
or if you add up by 2 9 times you would get 18 :)
Great job , you’ve already earned 1300 points, it there are 5 more days in this month. How many points a day do you need to reach your goal of 1580 points?
Answer:
56 points
Step-by-step explanation:
Let's say that for the next 5 days, you earn x points each day. This means that you will earn 5x more points before the month ends.
Since you already earned 1300 points, you will have earned 1300 + 5x points, which through the question we know is equal to 1580 points. Using this knowledge, we can set up the equation below:
1300 + 5x = 1580
5x = 280
x = 56
Therefore, you will need 56 points each day to reach your goal.
if this helps you, it would help me a lot if you could mark this as brainliest :)
NO LINKS!! Find the specified term of the geometric sequence
a8: 5/3, -1, 3/5, . . .
a8=
Answer:
[tex]a_8=-\dfrac{729}{15625}[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]
Given geometric sequence:
[tex]\dfrac{5}{3},\;-1,\;\dfrac{3}{5},\;...[/tex]
To find the common ratio, divide a term by the previous term:
[tex]\implies r=\dfrac{a_3}{a_2}=\dfrac{\frac{3}{5}}{-1}=-\dfrac{3}{5}[/tex]
Substitute the found common ratio and given first term into the formula to create an equation for the nth term:
[tex]a_n=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{n-1}[/tex]
To find the 8th term, substitute n = 8 into the equation:
[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{8-1}[/tex]
[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{3}{5}\right)^{7}[/tex]
[tex]\implies a_8=\dfrac{5}{3}\left(-\dfrac{2187}{78125}\right)[/tex]
[tex]\implies a_8=-\dfrac{729}{15625}[/tex]
convert the given rectangular coordinates to polar coordinates
(-2, 5)
So in this question, we have to convert the rectangular coordinates to polar coordinates.
Polar coordinates are (r, Q).
To convert rectangular coordinates to polar coordinates we use that x is equal to rcosQ and y is equal to rsinQ
In this question x is equal to -2 and y is equal to 5.
Hence we have -2 is equal to rcosQ and 5 is equal to rsinQ .
From here, we have tanQ equal to -2.5 . Hence, Q = tan⁻¹ (-2.5)
We have x²+y² =r² and also x²+y² = 5²+2²=29 .
Hence, r=√29 and Q = tan⁻¹ (-2.5) .
Therefore the polar coordinates of (-2,5) are ( √29, tan⁻¹ (-2.5)).
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What is the equation for the hyperbola shown?
The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.
Define hyperbola.A hyperbola is a particular kind of smooth curve that lies in a plane and is classified by its geometric characteristics or by equations for which it is the solution set. A hyperbola is made up of two mirror images of one another that resemble two infinite bows. These two sections are known as connected components or branches. A hyperbola is a geometric shape in mathematics where the difference between the lengths from any point on the figure to two fixed locations is a constant (the Greek letter o literally means "overshooting" or "excess"). We refer to the two fixed spots as foci (plural of focus).
Given
The equation for the hyperbola
The hyperbola's standard equation is [(x2/a2) - (y2/b2)] = 1, where X denotes the transverse axis and Y denotes the conjugate axis.
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Let f be the function given by f(x) = x+4/(x-1)(x+3) on the closed interval [-5, 5]. On which of the following closed intervals is the function f guaranteed by the extreme value theorem to have an absolute maximum and an absolute minimum?a. [-5, 5]b. [-3, 1]c. [-2, 0]d. [0, 5]
The function f is guaranteed by the extreme value theorem to have an absolute maximum and an absolute minimum on the closed interval a. [-5, 5].
The extreme value theorem states that if a function f is continuous on a closed interval [a, b], then it must have both an absolute maximum and an absolute minimum on that interval. Since the function f is given as being defined on the closed interval [-5, 5], it must have both an absolute maximum and an absolute minimum on this interval.
We can check that the function f is continuous on the interval [-5, 5] as follows:
First, we need to check that the function is defined at every point in the interval. The function f is defined as f(x) = x+4/(x-1)(x+3). The only points at which this function is not defined are x = 1 and x = -3, which are not within the interval [-5, 5]. Therefore, the function is defined at every point in the interval.
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Which fact represents the sum of (3 x 8) and (6 x 8)?
A. 9 x 8
B. 8 x8
C. 18 × 8
D 9x6
Answer:
Option A is the correct answer
Step-by-step explanation:
(3 × 8) + (6 × 8)
24 + 48 = 72
Thus, 9 x 8 = 72
Anna has 7,209 cans of soup that need to be boxed. If she puts 9 cans of soup in 1 box, how many boxes will she need.
Answer:
801 boxes
Step-by-step explanation:
[tex]\frac{7209}{9}[/tex] = 801
At what height does the ladder in Figure 1 rest on the wall?
Approximately how much higher up the wall does the ladder in Figure 2 rest compared to the ladder in Figure 1?
Answer:
The ladder rests against the wall at 6ft.
The ladder in Figure 2 rests about 3ft higher than the ladder in Figure 1.
Step-by-step explanation:
We can assume both of the figures display the ladder, ground, and wall forming a right triangle because the assignment is titled "apply the Pythagorean theorem", suggesting we should use the Pythagorean theorem and both figures are right triangles.
The Pythagorean theorem states that the sum of each leg squared is equal to the hypotenuse squared, or a^2 + b^2 = c^2, where "a" and "b" are legs and "c" is the hypotenuse.
At what height does the ladder in Figure 1 rest on the wall?
First, identify which sides are the legs and which is the hypotenuse. The hypotenuse is always the longest side and is the side opposite from the right angle. Thus, the hypotenuse in Figure 1 is the ladder length, and the two legs are the distance the ladder is from the wall and the height of the ladder against the wall. In other words, c = 10, a = 8, and b = the missing side length. Now, plug in the values to the Pythagorean theorem:
8^2 + b^2 = 10^2
64 + b^2 = 100
b^2 = 36
b = 6
The height at which the ladder in Figure 1 rests on the wall is 6 ft.
Approximately how much higher up the wall does the ladder in Figure 2 rest compared to the ladder in Figure 1?
We can set up the equation: Figure 2 ladder height - Figure 1 ladder height. We already know how where the ladder height of Figure 1 is, so we only need to solve for the ladder height in Figure 2 to get the result. Applying the same method as in the previous problem, we can input the values of Figure 2 into the Pythagorean theorem:
4^2 + b^2 = 10^2
16 + b^2 = 100
b^2 = 84
b = sqrt84
b ~ 9 ft
Now, we can find the difference between the Figure 2 ladder height and Figure 1 ladder height to get the answer to the question:
9 ft - 6 ft = 3 ft
REAL ANSWERS ONLY PLEASE! NEED THIS DONE ITS DUE SOON AND I DONT EVEN KNOW WHERE TO START! PLEASE SOMEONE!
Answer:
A.) Yes, because 0.064 is less than 0.4
Step-by-step explanation:
The equation [tex]0.4^3 = 0.064[/tex] is a counterexample to the given conjecture because 0.064 is less than 0.4. This shows that the cube of a number is not always greater than the number.
In general, the cube of a number is the result of multiplying the number by itself three times. For example, the cube of 2 is [tex]2 * 2 * 2 = 8[/tex], and the cube of 5 is [tex]5 * 5 * 5 = 125[/tex]. The cube of a number is always greater than the number when the number is positive, because a positive number multiplied by itself three times is always a larger number. However, when the number is negative or a decimal, this is not always the case.
In the given equation, 0.4 is a decimal number, and [tex]0.4^3[/tex] is the cube of 0.4. The result of this calculation is 0.064, which is less than 0.4. This means that the cube of 0.4 is not greater than 0.4, and therefore the given conjecture is false.
Sarah is setting up a garden plot in the shape of a right triangle. One leg off the triangle will be 7 feet longer than the other and the hypotenuse will be 1 foot longer than the longest leg. Find the length of each side of the garden plot. As a part of your work, include a sketch with the sides labeled. Show work algebraically.
The smallest leg of the triangular garden is 5 feet while the biggest leg is 12 feet.
What is a triangle?The three-sided shape known as a triangle is sometimes used to allude to it.
The area of a triangle is defined as (1/2) base height, and the sum of all three angles within a triangle will be 180°.
As the given triangular garden has been drawn below,
By Pythagoras' theorem,
x² + (x + 7)² = (x + 8)²
x² + x² + 49 + 14x = x² + 64 + 16x
x² - 2x - 15 = 0
x² - 5x + 3x - 15 = 0
x(x - 5) + 3(x - 5) = 0
(x + 3)(x - 5) = 0
x = -3,5
Since length can never be negative thus x = 5.
The smallest leg = 5 feet
Biggest leg = 5 + 7 = 12 feet
Hence "The triangular garden's smallest leg is 5 feet long, while its longest leg is 12 feet".
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mathematics needing some help
The principal that would need to be invested is $4202.
What is the principal?We know that the principal has to do with the amount of money that you have to invest so as to be able to obtain an interest. The interest in this case is a simple interest and it is charged on the principal. The amount is the sum of the principal and the interest.
We now have;
A = I + P
A = amount
I = interest
P = principal
But I = PRT/100
R = rate
T = time
We have;
A = PRT/100 + P
5000 = P * 9.5 * 2/100 + P
5000 = 19P/100 + P
5000 = 19P + 100P/100
5000 * 100 = 119P
P = 500000/119
P = $4202
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10) Find the missing angle measure. Round answers to the
nearest tenth.
37
24
Answer:
Θ = 49.6°
Step-by-step explanation:
cos Θ = adj/hyp
cos Θ = 24/37
cos Θ = 0.64864
Θ = cos^-1 0.64864
Θ = 49.6°
An airline offers daily flights between Chicago and Denver. The total monthly cost C (in millions of dollars) of these flights is C = â0.2x + 1
, where x is the number of passengers (in thousands). The total cost of the flights for June is 2.5 million dollars. How many passengers flew in June?
Total 26, 250 passengers are flew in June in an airline flights between Chicago and Denver .
What is Solution of Root Equations?An Equations with roots , such as the square root cannot be solved until its root is eliminated. Once we have isolated the square root on one side, we can eliminate it by raising both sides of the equation to powers equal to the square root. For example, if the square root is only one-sided, we can square both sides. We have , An airline offers daily flights between Chicago and Denver. Cost function of flight is,
C = 0.2x + 1
where x represents the number of passengers (in thousands) in flight.
To calculate how many passengers flew each day in June, we need to set the function to 2.5 (since the cost is in millions of dollars) and solve for the variables. This means taking the square root equal to 2.5. Therefore, to solve for the variables contained in its square root, we need to square both sides. This process and solution are described below,
√(0.2x + 1) = 2.5
Squaring both sides
=> (√(0.2x + 1) )² = ( 2.5 )²
=> 0.2x + 1 = 6.25
=> 0.2 x = 6.25 - 1 = 5.25
=> x = 5.25/0.2
=> x = 26.25
Now, we know that our variable i.e x measures passengers in thousands, so total passenger flew in June are 26.25 × 1000 = 26, 250 .
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Richard will increase the amount of time he studies each night from a period of 40 minutes to a period of 50 minutes. By what percentage will Richard increase the amount of time he studies each night?
Richard will increase the amount of time he studies each night by 25%.
What is the percentage?The percentage is defined as a ratio expressed as a fraction of 100.
To find the percentage increase, we need to first find the difference between the two amounts of time.
In this case, the difference is 50 minutes - 40 minutes = 10 minutes.
Next, we need to divide the difference by the original amount of time and multiply by 100% to express the answer as a percentage.
In this case, the percentage increase is :
⇒ (10 / 40) x 100% = 25%.
Therefore, he will increase the amount of time he studies each night by 25%.
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How can I turn a mixed fraction into a whole number?
Answer: Lets do 2 1/7
Step-by-step explanation: first you will start off by turning it to an improper fraction.
you multiply the denominator with the whole number (2x7) which you get 14 then add the numrinator with the number you got from multiplying (14+1= 15 then you get 15/7
Then after that divide the fraction 15/7 and you get 2 with a remander of 1
Here is a website you can use for fractions into whole numbers-https://www.cuemath.com
Mamadou leans a 18-foot ladder against a wall so that it forms an angle of 75° with
the ground. What's the horizontal distance between the base of the ladder and the
wall?
Could I get some help with this? Thanks! Here’s the question to go along with it.
For each diagram, calculate the value of x. Show your work and include an explanation of what you used (definitions and theorems) to solve the problem. If not possible, state why.
Based on the two (2) parallel lines m||n, the value of x is equal to 4.
What is the vertical angles theorem?The vertical angles theorem is also referred to as vertically opposite angles theorem and it states that two (2) opposite vertical angles that are formed whenever two (2) lines intersect each other are always congruent, which simply means being equal to each other.
In Geometry, when two (2) parallel lines are cut by a transversal, the pair of vertically opposite angles that are formed would always be equal to each other. In this context, we have the following;
m∠2 = m∠3
7x - 3 = x + 21
7x - x = 21 + 3
6x = 24
x = 24/6
x = 4.
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choose a student at random from a large statistics class. describe a sample space s for each of the following. (in some cases you may have some freedom in specifying s.) Step 1. Record the student's letter grade at the end of the course. S = {0,10,20,30,...,100} S = {0,1,2,...,100} S = {Fail, Passed, Good, Very Good, Excellent} S = {A, B, C, D, F} Step 2. Ask whether the student did or did not take a math class in each of the two previous years of school. S = {Yes, No} S = {Yes for each of the two years, No for each of the two years} S = {1st Year - Yes and 2nd Year - Yes, 1st Year - No and 2nd Year - Yes, 1st Year - Yes and 2nd Year - No, 1st Year - No and 2nd Year - No} S = {1st Year - Yes, 1st Year - No, 2nd Year Yes, 2nd Year - No}
The Sample of student at random from a large statistics class.
a) Sample space for Record the student's letter grade at the end of the course is , S = {A, B, C, D, F }. So, Correct option is option D.
b) Sample space for the student did or did not take a math class in each of the two previous years of school is S = {1st Year - Yes and 2nd Year - Yes, 1st Year - No and 2nd Year - Yes, 1st Year - Yes and 2nd Year - No, 1st Year - No and 2nd Year - No}. So, correct option is option C.
Sample Space: It is defined as the list of all possible outcome for given experiment.
a) record of Students grade are A, B, C, D, F.
These are all possible outcome for Student's letter grade at the end of course.
S= {A, B, C, D, F }
b) Possible outcome for whether student did or
did not take math class in each of the previous year of school is,
1st - Yes & 2nd - yes , 1st-yes & 2nd no
1st- NO & 2nd yes, 1st-NO & 2nd_NO
Hence, S = { 1st year - Yes & 2nd year - Yes,
1st year-NO & 2nd year- Yes, 1st year- yes & 2nd year - NO, 1st year - No & 2nd year - No}
Hence 3rd option is correct option.
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Complete question :
choose a student at random from a large statistics class. describe a sample space s for each of the following. (in some cases you may have some freedom in specifying s.)
a) Record the student's letter grade at the end of the course.
A) S = {0,10,20,30,...,100}
B) S = {0,1,2,...,100}
C) S = {Fail, Passed, Good, Very Good, Excellent}
D) S = {A, B, C, D, F}
b) Ask whether the student did or did not take a math class in each of the two previous years of school.
A) S = {Yes, No}
B)S = {Yes for each of the two years, No for each of the two years}
C) S = {1st Year - Yes and 2nd Year - Yes, 1st Year - No and 2nd Year - Yes, 1st Year - Yes and 2nd Year - No, 1st Year - No and 2nd Year - No}
D) S = {1st Year - Yes, 1st Year - No, 2nd Year Yes, 2nd Year - No}
Select the two answers that describe the unit rate at which he walks.
1.5 seconds per meter
90 meters per second
30 meters every second
1.5 meters per second
1 meters every second
30 seconds per meter
The unit rate at which Darrell walks is 1.5 meters per second.
What is the unit rate?
A ratio that compares two amounts of DIFFERENT types of UNITS is called a rate. When a rate is expressed as a fraction, the denominator is 1 unit. Divide the numerator and denominator of the rate by the denominator to represent a rate as a unit rate.
Here, we have
Given
Darrell is walking for a distance of 90 meters in time 60 seconds.
We have to find the unit rate at which he walks.
To find out the unit rate we must divide the distance traveled by Darrell by the time taken to travel that distance.
The unit rate is given by :
Unit rate = Distance traveled in meters / Time taken in seconds
Unit rate = 90 meters / 60 seconds
Unit rate = 1.5 meters per second.
Hence, the unit rate at which Darrell walks is 1.5 meters per second.
In 60 seconds, Darrell walks 90 meters. Select the two answers that describe the unit rate at which he walks.
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May I have help solving This I’ll give you brainliest
Answer:
1 1/6
Step-by-step explanation:
Hope it helps!
Which of the following is equivalent to
16^3/4
Answer:
B. 8
Step-by-step explanation:
I’m glad to help you
Question 2(Multiple Choice Worth 2 points)
(Multiplying Integers MC)
What is the product of (-25)(15)(-3)?
O 1,125
60
O-200
O-1,125
The product of the given values would be = 1,125. That is option A.
What is multiplication?Multiplication is defined as one of the major arithmetic operation along with others such as addition, subtraction and division which is used to to determine the product of two or more values.
The rule of multiplication is that
when a positive figure is multiplied by a positive figure, a positive value is obtained. When a negative figure multiplies a negative figure a positive value is obtained.Therefore;
= (-25)× (15) × (-3)
= -375 × -3
= 1,125.
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One tablecloth 48 inches wide & 72 inches long. Want 2nd tablecloth 30 inches wide how long should it be
The length (in inches) of the second tablecloth is 43.75 inches
How to determine the length of the second tablecloth?From the question, we have the following parameters that can be used in our computation:
Width = 48 inches. and Length = 72 inches ⇒ Small tablecloth
Width = 30 inches. and Length = ?? inches ⇒ Large tablecloth
Represent the unknown length with x
So, we have
Width = 48 inches. and Length = 72 inches ⇒ Small tablecloth
Width = 30 inches. and Length = x inches ⇒ Large tablecloth
This shows a direct variation
So, we cross multiply the equation
x * 48 = 70 * 30
So, we have
x = 70 * 30/48
Evaluate
x = 43.75
Hence, the length is 43.75 inches
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i will compare the calculated r-value against a critical value to deduce whether or not the value is statistically significant enough to act as a model for the entire world population.?
There is insufficient evidence to conclude that there is a significant linear relationship between
x and y because the correlation coefficient is not significantly different from zero.
The correlation coefficient, r, tells us about the strength and direction of the linear relationship between x and y.
However, the reliability of the linear model also depends on how many observed data points are in the sample.
We need to look at both the value of the correlation coefficient r and the sample size n, together.
We perform a hypothesis test of the “significance of the correlation coefficient” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.
The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
The symbol for the population correlation coefficient is ρ, the Greek letter “rho.”
The hypothesis test lets us decide whether the value of the population correlation coefficient
ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n.
If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”
There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero. What the conclusion means: There is a significant linear relationship between x and y. We can use the regression line to model the linear relationship between x and y in the population.
If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant.”
“There is insufficient evidence to conclude that there is a significant linear relationship between
x and y because the correlation coefficient is not significantly different from zero.” What the conclusion means: There is not a significant linear relationship between x and y. Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.
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We must consider the sample size n together with the correlation coefficient r's value.
To determine whether the linear relationship in the sample data is strong enough to utilize to model the relationship in the population, we conduct a hypothesis test of the "significance of the correlation coefficient."
The correlation coefficient for the sample, r, is calculated using the sample data. The population correlation coefficient could be discovered if we had data for the entire population. However, we are unable to determine the population correlation coefficient because we only have sample data. Our best guess for the unknown population correlation coefficient is the sample correlation coefficient, or r.
The Greek letter "rho" serves as the sign for the population correlation coefficient.
is "near zero" or "quite different from zero." Based on the sample correlation coefficient r and the sample size n, we make this determination.
We refer to the correlation coefficient as "significant" if the test results show that it differs considerably from zero.
The fact that the correlation coefficient differs sufficiently from zero is sufficient evidence that x and y are meaningfully related linearly. The result indicates that there is a significant linear relationship between x and y. The regression line can be used to model the population's linear relationship between x and y.
Because the correlation coefficient does not deviate considerably from zero, x and y. What the conclusion says is that x and y do not have a meaningful linear connection.