coefficient - a numerical or constant quantity placed before and multiplying the variable in an algebraic expression

2 times a number added to another number is 25. 3 times the first number minus the other number is 2

2 times a number added to another number is 25. 3 times the first number minus the other number is 20.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]2x + y = 25
[*]3x - y = 20
[/LIST]
Since we have matching opposite coefficients for y (1 and -1), we can add both equations together and eliminate a variable.
(2 + 3)x + (1 - 1)y = 25 + 20
Simplifying, we get:
5x = 45
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%3D45&pl=Solve']Typing this equation into the search engine[/URL], we get:
[B]x = 9[/B]
To find y, we plug in x = 9 into equation (1) or (2). Let's choose equation (1):
2(9) + y = 25
y + 18 = 25
[URL='https://www.mathcelebrity.com/1unk.php?num=y%2B18%3D25&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]y = 7[/B]
So we have (x, y) = (9, 7)
Let's check our work for equation (2) to make sure this system works:
3(9) - 7 ? 20
27 - 7 ? 20
20 = 20 <-- Good, we match!

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7.
Write this out:
3^2 * 3^x = 3^7
When we multiply matching coefficients, we add exponents, so we have:
3^(2 + x) = 3^7
Therefore, 2 + x = 7. To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2%2Bx%3D7&pl=Solve']type it into our search engine[/URL] and we get:
x = [B]5[/B]

admission to the school fair is $2.50 for students and $3.75 for others. if 2848 admissions were col

admission to the school fair is $2.50 for students and $3.75 for others. if 2848 admissions were collected for a total of 10,078.75, how many students attended the fair
Let the number of students be s and the others be o. We're given two equations:
[LIST=1]
[*]o + s = 2848
[*]3.75o + 2.50s = 10078.75
[/LIST]
Since we have no coefficients for equation 1, let's solve this the fast way using substitution. Rearrange equation 1 by subtracting o from each side to isolate s
[LIST=1]
[*]o = 2848 - s
[*]3.75o + 2.50s = 10078.75
[/LIST]
Now substitute equation 1 into equation 2:
3.75(2848 - s) + 2.50s =10078.75
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=3.75%282848-s%29%2B2.50s%3D10078.75&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]481[/B]

Basic Statistics

Given a number set, and an optional probability set, this calculates the following statistical items:

Expected Value

Mean = μ

Variance = σ^{2}

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Expected Value

Mean = μ

Variance = σ

Standard Deviation = σ

Standard Error of the Mean

Skewness

Mid-Range

Average Deviation (Mean Absolute Deviation)

Median

Mode

Range

Pearsons Skewness Coefficients

Entropy

Upper Quartile (hinge) (75th Percentile)

Lower Quartile (hinge) (25th Percentile)

InnerQuartile Range

Inner Fences (Lower Inner Fence and Upper Inner Fence)

Outer Fences (Lower Outer Fence and Upper Outer Fence)

Suspect Outliers

Highly Suspect Outliers

Stem and Leaf Plot

Ranked Data Set

Central Tendency Items such as Harmonic Mean and Geometric Mean and Mid-Range

Root Mean Square

Weighted Average (Weighted Mean)

Frequency Distribution

Successive Ratio

Big John weighs 300 pounds and is going on a diet where he'll lose 3 pounds per week. Write an equat

Big John weighs 300 pounds and is going on a diet where he'll lose 3 pounds per week. Write an equation in slope-intercept form to represent this situation.
[LIST]
[*]The slope intercept form is y = mx + b
[*]y is John's weight
[*]x is the number of weeks
[*]A 3 pound per week weight loss means -3 as the coefficient m
[*]b = 300, John's starting weight
[/LIST]
[B]y = -3x + 300[/B]

Bob bought 10 note books and 4 pens for 18$. Bill bought 6 notebooks and 4 pens for 12$. Find the pr

Bob bought 10 note books and 4 pens for 18$. Bill bought 6 notebooks and 4 pens for 12$. Find the price of one note book and one pen.
Let the price of each notebook be n. Let the price of each pen be p. We're given two equations:
[LIST=1]
[*]10n + 4p = 18
[*]6n + 4p = 12
[/LIST]
Since we have matching coefficients for p, we subtract equation 1 from equation 2:
(10 - 6)n + (4 - 4)p = 18 - 12
Simplifying and cancelling, we get:
4n = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=4n%3D6&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]n = 1.5[/B]
Now, substitute this value for n into equation (2):
6(1.5) + 4p = 12
Multiply through:
9 + 4p = 12
[URL='https://www.mathcelebrity.com/1unk.php?num=9%2B4p%3D12&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]p = 0.75[/B]

Can a coefficient of determination be negative? Why or why not?

Can a coefficient of determination be negative? Why or why not?
[B]Yes, reasons below[/B]
[LIST]
[*] predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data
[*] where linear regression is conducted without including an intercept
[*] Yes, negative values of R2 may occur when fitting non-linear functions to data
[/LIST]

Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit

Given two distributions X and Y, this calculates the following:

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r^{2}

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

DeAndre is a spelunker (someone who explores caves). One day DeAndre is exploring a cave that has a

DeAndre is a spelunker (someone who explores caves). One day DeAndre is exploring a cave that has a series of ladders going down into the depths. Every ladder is exactly 10 feet tall, and there is no other way to descend or ascend (the other paths in the cave are flat). DeAndre starts at 186 feet in altitude, and reaches a maximum depth of 86 feet in altitude.Write an equation for DeAndre's altitude, using x to represent the number of ladders DeAndre used (hint: a ladder takes DeAndre down in altitude, so the coefficient should be negative).
Set up a function A(x) for altitude, where x is the number of ladders used. Each ladder takes DeAndre down 10 feet, so this would be -10x. And DeAndre starts at 186 feet, so we'd have:
[B]A(x) = 186 - 10x[/B]

Fisher Transformation and Fisher Inverse

Given a correlation coefficient (r), this calculates the Fisher Transformation (z).

Given a Fisher Transformation (r), this calculates the Fisher Inverse (r)

Given a Fisher Transformation (r), this calculates the Fisher Inverse (r)

If 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners an

If 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners and 2 walkers?
[U]Set up a joint variation equation, for the 100 runners, 4 bicyclists, and 5 walkers:[/U]
100 = 4 * 5 * k
100 = 20k
[U]Divide each side by 20[/U]
k = 5 <-- Coefficient of Variation
[U]Now, take scenario 2 to determine the bicyclists with 20 runners and 2 walkers[/U]
20 = 2 * 5 * b
20 = 10b
[U]Divide each side by 10[/U]
[B]b = 2[/B]

If the correlation between two variables is close to minus one, the association is: Strong Moderate

If the correlation between two variables is close to minus one, the association is:
Strong
Moderate
Weak
None
[B]Strong[/B] - Coefficient near +1 or -1 indicate a strong correlation

If you take a Uber and they charge $5 just to show up and $1.57 per mile, how much will it cost you

If you take a Uber and they charge $5 just to show up and $1.57 per mile, how much will it cost you to go 12 miles? (Assume no tip.)
a. Create an equation from the information above.
b. Identify the slope in the equation?
c. Calculate the total cost of the ride?
2. With the same charges as #1, how many miles could you go with $50, if you also gave a $7.50 tip? (Challenge Question! Hint, you only have a $50, exactly, with you
a. Cost Equation C(m) for m miles is as follows:
[B]C(m) = 1.57m + 5
[/B]
b.
Slope of the equation is the coefficient for m, which is [B]1.57
[/B]
c.
Total cost of the ride for m = 12 miles is:
C(12) = 1.57(12) + 5
C(12) = 18.84 + 5
C(12) = [B]23.84
[/B]
d.
If you give a 7.50 tip, we subtract the tip from the $50 to start with a reduced amount:
50 - 7.50 = 42.50
So C(m) = 42.50
1.57m + 5 = 42.50
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=1.57m%2B5%3D42.50&pl=Solve']type it in our search engine[/URL] and we get:
m = 23.89
Since we deal in full miles, we round our answer down to m = [B]23[/B]

In simple linear regression the slope and the correlation coefficient will have the same signs True

In simple linear regression the slope and the correlation coefficient will have the same signs
True
False
[B]FALSE[/B] - Only if they are normalized

Justin is older than Martina. The difference in their ages is 22 and the sum of their ages is 54. Wh

Justin is older than Martina. The difference in their ages is 22 and the sum of their ages is 54. What age is Martina?
[U]Assumptions and givens:[/U]
[LIST]
[*]Let Justin's age be j
[*]Let Martina's age be m
[*]j > m ([I]since Justin is older than Martina[/I])
[/LIST]
We're given the following equations :
[LIST=1]
[*]j - m = 22
[*]j + m = 54
[/LIST]
Since the coefficients of m are opposites, we can take a shortcut using the [I]elimination method[/I] and add equation (1) to equation (2)
(j + j) + (m - m) = 22 + 54
2j = 76
To solve for j, we [URL='https://www.mathcelebrity.com/1unk.php?num=2j%3D76&pl=Solve']type this equation into our math engine[/URL] and we get:
j = 38
The question asks for Martina's age (m), so we can pick equation (1) or equation (2). Let's use equation (1):
38 - m = 22
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=38-m%3D22&pl=Solve']type it in our math engine[/URL] and we get:
m = [B]16[/B]

Set Notation

Given two number sets A and B, this determines the following:

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J_{σ}(A,B)

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

* Union of A and B, denoted A U B

* Intersection of A and B, denoted A ∩ B

* Elements in A not in B, denoted A - B

* Elements in B not in A, denoted B - A

* Symmetric Difference A Δ B

* The Concatenation A · B

* The Cartesian Product A x B

* Cardinality of A = |A|

* Cardinality of B = |B|

* Jaccard Index J(A,B)

* Jaccard Distance J

* Dice's Coefficient

* If A is a subset of B

* If B is a subset of A

The coefficient of determination is found by taking the square root of the coefficient of correlatio

The coefficient of determination is found by taking the square root of the coefficient of correlation. True or False
[B]FALSE[/B] - It is found by squaring the coefficient of correlation

The sum of Mr. Adams and Mrs. Benson's age is 55. The difference is 3. What are their ages?

The sum of Mr. Adams and Mrs. Benson's age is 55. The difference is 3. What are their ages?
[U]Givens[/U]
[LIST]
[*]Let Mr. Adam's age be a
[*]Let Mrs. Benson's age be b
[*]We're given two equations where [I]sum[/I] means we add and [I]difference[/I] means we subtract:
[/LIST]
[LIST=1]
[*]a + b = 55
[*]a - b = 3
[/LIST]
Since we have opposite coefficients for b, we can take a shortcut and add equation 1 to equation 2:
(a + a) + (b - b) = 55 + 3
Combining like terms and simplifying, we get:
2a = 58
To solve this equation for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=2a%3D58&pl=Solve']type it in our search engine[/URL] and we get:
a = [B]29[/B]

Two numbers total 12, and their differences is 20. Find the two numbers.

Two numbers total 12, and their differences is 20. Find the two numbers.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]x + y = 12
[*]x - y = 20
[/LIST]
Since we have y coefficients of (-1 and 1) that cancel, we add the two equations together:
(x + x) + (y - y) = 12 + 20
The y terms cancel, so we have:
2x = 32
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%3D32&pl=Solve']Type this equation into our search engine[/URL] and we get:
x = [B]16[/B]
Substitute this value of x = 16 back into equation 1:
16 + y = 12
[URL='https://www.mathcelebrity.com/1unk.php?num=16%2By%3D12&pl=Solve']Typing this equation into our search engine[/URL], we get:
y = [B]-4
[/B]
Now, let's check our work for both equations:
[LIST=1]
[*]16 - 4 = 12
[*]16 - -4 --> 16 + 4 = 20
[/LIST]
So these both check out.
(x, y) = ([B]16, -4)[/B]

What can we conclude if the coefficient of determination is 0.94?

What can we conclude if the coefficient of determination is 0.94?
[LIST]
[*]Strength of relationship is 0.94
[*]Direction of relationship is positive
[*]94% of total variation of one variable(y) is explained by variation in the other variable(x).
[*]All of the above are correct
[/LIST]
[B]94% of total variation of one variable(y) is explained by variation in the other variable(x)[/B]. The coefficient of determination explains ratio of explained variation to the total variation.

What is the range of possible values for a coefficient of correlation?

What is the range of possible values for a coefficient of correlation?
[B]The range is -1.0 to +1.0[/B]