What is the solution to the equation below?

log 7+log(x-4)= 1

Answer:

The temperature that separates the lowest​ 59.5% of temperatures from the​ rest is 83.66 degrees Fahrenheit.

Step-by-step explanation:

Mean temperature = u = 77.9

Standard Deviation = [tex]\sigma[/tex] = 2.4

Since the distribution is normal and we have the value of population standard deviation, we will use the concept of z-score to find the desired value.

We have to find the value that separates lowest 59.5% of the temperature from the rest. This means our desired value is above 59.5% of the data values.

From the z-table we can find a z-score which is above 59.5% of the values and then convert it to its equivalent temperature using the formula.

From the z-table, the z-score which is above 59.5% of the value is:

z = 0.24

The formula for z-score is:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Using the values we have:

[tex]0.24=\frac{x-77.9}{2.4}\\\\ 5.76=x-77.9\\\\ x=83.66[/tex]

This is our data value equivalent to a z-score of 0.24. Just like 59.5% of z values were below 0.24, in the same manner, 59.5% of temperatures are below 83.66 degrees Fahrenheit.

Therefore, the temperature that separates the lowest​ 59.5% of temperatures from the​ rest is 83.66 degrees Fahrenheit.

Answer:

5/8

Step-by-step explanation:

8 children in class

3 have pink on

8-3 = 5

That means 5 do not have pink

Fraction that do not have pink

those that do not have pink/total

5/8

Answer:

LMN is not a right triangle.

Step-by-step explanation:

To verify that the lengths of a triangle for a right triangle, all we need to do is to check it they satisfy the Pythagoras Theorem.

Pythagoras Theorem:[tex]Hypotenuse^2=Opposite^2+Adjacent^2[/tex]

The Hypotenuse is always the longest side.

Since no angle is given, the two other sides can be alternated.

Given lengths of triangle LMN: 18.5 inches, 10, inches, and 15.5 inches

[tex]LHS=18.5^2=342.25\\RHS=10^2+15.5^2=100+240.25=340.25\\Since \: LHS\neq RHS,\\$The triangle is not a right triangle.[/tex]

Answer:

Jamal needs to save $254.17 each month to cover his expenses.

Step-by-step explanation:

We know that there are 12 months in a year, and Jamal already has $1,000 towards college. Jamal still needs to save $3,050 more dollars because: 4,050 - 1,000 = 3,050. Next, we take how much he has yet to pay ($3,050) divided by how many months in 1 year (12), and that gives us 254.17 per month.

For his first year of college expenses, Jamal needs to save $254.17 each month.

Jamal needs to contribute $4,050 towards his first year of college. He has 1 year to save and $1,000 in his savings account already.

Calculate the remaining amount needed: $4,050 - $1,000 = $3,050.

Divide the remaining amount by 12 months:

[tex]\frac{3050}{12} = 254.17[/tex].

Jamal needs to save $254.17 each month to cover his college expenses.

Answer:

(a) 1.6197

(b) 1.3803

Step-by-step explanation:

(a) Let X denote the number of games the Reds win. If Reds win exactly one it means that in the other two games Reds lost i.e Cubs won. Now,  the probability of the Reds losing is determined by [tex] 1 -\text{probability of winning} = 1-0.54 = 0.46 [/tex]. So, if X wins exactly [tex]i[/tex] times, then we should multiply the probabilities associated with

[tex]P(X=0) = \binom{3}{0} \times 0.46 \times 0.46 \times 0.46 = 0.0973[/tex]

[tex] P(X=1) = \binom{3}{1} 0.54 \times 0.46 \times 0.46 = 0.3429[/tex]

[tex] P(X=2) = \binom{3}{2} 0.54 \times 0.54 \times 0.46 = 0.4023[/tex]

[tex] P(X=3) = \binom{3}{3} 0.54 \times 0.54 \times 0.54 = 0.1574[/tex]

Now,

[tex] E(X) = \sum_{i=0}^{3} i \times P(X=i) = 1.6197[/tex]

(b) Let Y denote the number of games won by Cubs. Then by the similar logic as above,

[tex] P(Y=3) = \binom{3}{0} 0.46 \times 0.46 \times 0.46 = 0.0973[/tex]

[tex] P(Y=2) = \binom{3}{1} 0.54 \times 0.46 \times 0.46 = 0.3429[/tex]

[tex] P(Y=1) = \binom{3}{2} 0.54 \times 0.54 \times 0.46 = 0.4023[/tex]

[tex] P(Y=0) = \binom{3}{3} 0.54 \times 0.54 \times 0.54 = 0.1574[/tex]

Now

[tex] E(Y) = \sum_{j=0}^{3} j \times P(Y=j) = 1.3803[/tex]

Answer:

import pandas as pd

import import statsmodels.api as sm

dataframe = pd.read_csv(Your model)

mod = sm.OLS(formula = ABSENT ~ COMPLX + SENING+FS1+FS2+FS3+FS4)

res = mod.fit()

print(res.summary())

Step-by-step explanation:

Using python you can load the dataframe using pandas library. Once you have your pandas library imported to the system you can also import the statsmodels.api  module. What you do is this. You fit the model using the variables mentioned "COMPLX" ,"SENINV","FS1","FS2","FS3", once you fit the model you use .summary() and that will give you a summary of each coefficient and the level of significance,  the level of significance must be less than 5% in order to be significant. The code would look like this.

import pandas as pd

import import statsmodels.api as sm

dataframe = pd.read_csv(Your model)

mod = sm.OLS(formula = ABSENT ~ COMPLX + SENING+FS1+FS2+FS3+FS4)

res = mod.fit()

print(res.summary())

The question involves performing a regression analysis on absenteeism data from the ABX Company, with a particular focus on the variable of supervisor satisfaction. Hypotheses are formulated to test if there is a significant difference in absenteeism among different supervisor satisfaction groups. The result of the F-test in the regression analysis will provide the answer.

The problem involves the analysis of absenteeism data collected from the employees of the ABX Company with the goal of determining the effects of various variables on absenteeism. These variables include supervisor satisfaction (SATIS), which is transformed into indicator variables (FS1, FS2, FS3, FS4), as well as the reciprocal of the seniority variable (SENINV), and COMPLX.

The regression equation to be fitted is: ABSENT = β0 + β1COMPLX + β2SENINV + β3FS1 + β4FS2 + β5FS3 + β6FS4.

To test the hypothesis that there is a difference in average absenteeism for employees in different supervisor satisfaction groups, the null and alternative hypotheses are defined as follows:

These hypotheses are tested using the F-test in the regression analysis. The F-statistic will tell you if the variations in the SATIS groups are significantly different. If the p-value associated with this F-statistic is less than your significance level (0.05 in this case), you would reject the null hypothesis and conclude that there is a difference in average absenteeism among the different satisfaction groups.

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Answer:

option D.

Step-by-step explanation:

Given,

Area of a figure = 13 in²

All dimension is multiplied by 4.

Area of the figure will be multiplied by = ?

We know that,

[tex]Area\ \alpha \ (Length)^2[/tex]

If length is multiplied by 4.

[tex]Area\ \alpha \ 4^2 (L)^2[/tex]

Where L is the dimension of the figure.

[tex]Area\ \alpha \ 16 (L)^2[/tex]

Area of the figure will be multiplied by 16.

Hence, correct answer is option D.

Answer:

The z statistic for this sample is -2.04.

Step-by-step explanation:

The null hypothesis is:

[tex]H_{0} = 7.44[/tex]

The alternate hypotesis is:

[tex]H_{1} \neq 7.44[/tex]

The z-statistic is:

[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

In which X is the sample mean, [tex]\mu[/tex] is the population mean(the hypothesis tested), [tex]\sigma[/tex] is the standard deviation and n is the size of the sample.

In this problem:

[tex]X = 3.9, \mu = 7.44, \sigma = 10.98, n = 40[/tex]. So

[tex]z = \frac{X - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]

[tex]z = \frac{3.9 - 7.44}{\frac{10.98}{\sqrt{40}}}[/tex]

[tex]z = -2.04[/tex]

The z statistic for this sample is -2.04.

Answer:

x = 5, 15 y = 30

Step-by-step explanation:

y=3x is the rule, thus you can use it to find all values.

15 = 3x  Divide 15 by 3 in order to isolate the variable

5 = x

y = 3(10)  Multiply 3 by 10 in order to find y

y = 30

45 = 3x  Divide 45 by 3 in order to isolate the variable

15 = x

Answer:

D. mn

Step-by-step explanation:

The amount of elements that are in an m x n matrix is mn.

An m x n matrix has m*n elements since the elements are the product of the number of rows (m) and the number of columns (n).

An m x n matrix is a rectangular grid containing elements that are arranged in m rows and n columns.

So, the elements in a matrix are represented by the multiplication of the number of rows (m) and the number of columns (n). If, for instance, you have a matrix that is 3 x 4, this matrix contains 3 (rows) multiplied by 4 (columns) which equals to 12 elements total in the matrix.

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Answer:

700 nanometers

Step-by-step explanation:

8 wavelengths had a combined length of 5,600 nanometers.

We want to find the wavelength (in nm) of 1 of those wavelengths.

We will assume all of them are same wavelength. So, we can basically divide the total of 8 wavelengths by 8 to get 1 wavelength. Thus:

5600/8 = 700

Hence,

1 wavelength distance is 700 nanometers (or 700 nm)

Answer:

C. How many movies do the students in my class watch in a month

Step-by-step explanation:

Answer:

c

Step-by-step explanation:

Step-by-step explanation:

2πr^/4

the formula of volume of a circle

Answer:

the beach ball will hold 36,000pi  

Step-by-step explanation:

The volume of a sphere which is V=4/3 x pi x r^3

The beach ball has a radius of 30 cm

V= 4/3 x pi x 30^3

= 4/3 x pi x 27,000

= 36,000pi

Answer:

a)[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

[tex]\bar X = 112.75[/tex]

[tex]s^2 = \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]

[tex] s^2 = 540.5[/tex]

b) [tex] t = \frac{\bar X -\mu}{\frac{s}{\sqrt{n}}}[/tex]

And if we replace we got:

[tex] t = \frac{112.75-112}{\frac{23.249}{\sqrt{8}}}= 0.0912[/tex]

Step-by-step explanation:

Part a

For this case we have the following data: 111 113 145 105 90 100 150 88

We can calculate the mean with the following formula:

[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

And replacing we got:

[tex]\bar X = 112.75[/tex]

And the sample variance can be calculated with this formula:

[tex]s^2 = \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]

And replacing we got:

[tex] s^2 = 540.5[/tex]

Part b

For this case we want to check is the true mean is equal to 102 or no, the t statistic is given by:

[tex] t = \frac{\bar X -\mu}{\frac{s}{\sqrt{n}}}[/tex]

And if we replace we got:

[tex] t = \frac{112.75-112}{\frac{23.249}{\sqrt{8}}}= 0.0912[/tex]

To answer the student's question, the sample mean of electrical energy usage is calculated to be 112.75 kwH. The sample variance requires computing the squared differences of each data point from the mean, summing them up and dividing by n-1. The t-statistic can then be calculated using the historical mean, the sample mean, and the sample standard deviation.

To calculate the mean of the sample data, we sum all the sample values and divide by the number of observations. The sample data for electrical energy usage in kwH per month are: 111, 113, 145, 105, 90, 100, 150, and 88. The sum of these values is 902 kwH, and with 8 residents in the sample, the mean (μ) is 902 kwH / 8 = 112.75 kwH.

The sample variance (s²) is calculated by taking the sum of squared differences from the mean and dividing by the sample size minus one. First, we find the squared differences for each data point: (111-112.75)², (113-112.75)², (145-112.75)², (105-112.75)², (90-112.75)², (100-112.75)², (150-112.75)², and (88-112.75)², which then summed up give us the total sum of squares. Dividing this total by 7 (n-1), we obtain the sample variance.

For the t statistic, we would need the hypothesized population mean to compare our sample mean against. However, we can calculate the t-value if we assume the historical mean provided (102 kwH) is the population mean we are testing against. The t-statistic is calculated using the formula: t = (sample mean - population mean) / (sample standard deviation / sqrt(n)). In this scenario, n=8 and we would need the sample standard deviation, which is the square root of the variance we calculated earlier. Once these values are obtained, the t-statistic can be computed.

The appropriate domain for this situation is all non-negative real numbers or the interval [0, ∞).

The appropriate domain for this situation can be determined by considering the values that the independent variable, t, can take in the given function. In this case, the function is h(t) = -16t^2 + 24t, where t represents the time since the ball was thrown.

Since time cannot be negative in this context, the domain of the function is t ≥ 0. This means that the appropriate domain for this situation is all non-negative real numbers or the interval [0, ∞).

Answer:

$280 y $120

Step-by-step explanation:

Primero haces una ecuación. Entonces puedes adivinar y verificar o usar la ecuación. Solía ​​adivinar y comprobar.

¡Espero que esto haya ayudado!

Answer:

t = $4400 +/- $450

mean = $4400

standard deviation = $450

Step-by-step explanation:

Given;

Base salary b = $2000

Sales s = $8000 +/- $1500

Commission c = 30% sales = 0.3 × s

Total Salary t = base salary + commission = b + c

Commission c = 0.3×s = 0.3×($8000 +/-$1500)

c = $2400 +/- $450

Total salary t = b + c

Substituting the values;

t = $2000 + ($2400 +/- $450)

t = $4400 +/- $450

mean = $4400

standard deviation = $450

Answer:

DOGS is a parallelogram.

Step-by-step explanation:

Given the quadrilateral DOGS with coordinates D(1, 1), O(2, 4),  G(5, 6), and S(4,3).

To prove that it is a parallelogram, we need to show that the opposite lengths are equal. That is:

Using the Distance Formula

[tex]Distance=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

For D(1, 1) and O(2, 4)

[tex]|DO|=\sqrt{(2-1)^2+(4-1)^2}=\sqrt{1^2+3^2}=\sqrt{10} \:Units[/tex]

For G(5, 6), and S(4,3).

[tex]|GS|=\sqrt{(4-5)^2+(3-6)^2}=\sqrt{(-1)^2+(-3)^2}=\sqrt{10}\:Units[/tex]

For O(2, 4) and G(5, 6)

[tex]|OG|=\sqrt{(5-2)^2+(6-4)^2}=\sqrt{(3)^2+(2)^2}=\sqrt{13}\:Units[/tex]

For S(4,3) and D(1, 1)

[tex]|SD|=\sqrt{(1-4)^2+(1-3)^2}=\sqrt{(-3)^2+(-2)^2}=\sqrt{13}\:Units[/tex]

Since:

Then, quadrilateral DOGS is a parallelogram.

Answer:

4 13/9 - 1 7/9

3 6/9

3 2/3

Step-by-step explanation:

The answer would be 3.6 because I simplified my answer into decimal. Hope this helps! If you have any other questions, please ask me. Stay safe! And have a good day!

Answer:

a-area-22

a-perimeter-19

b-area-98

b-perimeter-42

Step-by-step explanation:

for the sides just add the numbers 2x

for the area multiply the numbers

hope this helps

Answer:

The probability that exactly one of these mortgages is delinquent is 0.357.

Step-by-step explanation:

We are given that according to the Mortgage Bankers Association, 8% of U.S. mortgages were delinquent in 2011. A delinquent mortgage is one that has missed at least one payment but has not yet gone to foreclosure.

A random sample of eight mortgages was selected.

The above situation can be represented through Binomial distribution;

[tex]P(X=r) = \binom{n}{r}p^{r} (1-p)^{n-r} ; x = 0,1,2,3,.....[/tex]

where, n = number of trials (samples) taken = 8 mortgages

            r = number of success = exactly one

           p = probability of success which in our question is % of U.S.

                  mortgages those were delinquent in 2011, i.e; 8%

LET X = Number of U.S. mortgages those were delinquent in 2011

So, it means X ~ [tex]Binom(n=8, p=0.08)[/tex]

Now, Probability that exactly one of these mortgages is delinquent is given by = P(X = 1)

                 P(X = 1)  = [tex]\binom{8}{1}\times 0.08^{1} \times (1-0.08)^{8-1}[/tex]

                               = [tex]8 \times 0.08 \times 0.92^{7}[/tex]

                               = 0.357

Hence, the probability that exactly one of these mortgages is delinquent is 0.357.

Answer:

a. reflection

b. rotation

c. translation

d. translation

e. reflection

f. rotation

27. MD

28. QP

29. NM

30. BA

31. ∠FGK

32. CS

33. ∠V

34. DC

Step-by-step explanation:

There really isn't any work to show...

Congruent means that a letter is equal to the letter on the other side of the equation that is in the same location

Answer:

13

Step-by-step explanation:

use pythagorean theorem

Answer:

13

Step-by-step explanation:

Hi! Use the height and length of the line to form the line into the hypotenuse of a triangle. The Pythagorean Theorem can be used to find the length of the line.

Side a is 5 squares high

Side b is 12 squares long

Side c (the line) is x squares long

a^2+b^2=c^2

5^2+12^2=x^2

x can equal 13 or -13 (because of square rules) but in this case it's thirteen because length can't be negative

Please mark brainliest and have a great day!

and if you have anymore questions please let me know

Answer:

Lyla used 1.4-1 = 0.4 = 40% more brown sugar than white sugar

Step-by-step explanation:

To find how much brown sugar she used, we divide the amount she used of brown sugar divided by the amount she used of white sugar.

7/10 of a scoop of brown sugar

1/2 of a scoop of white sugar.

So

[tex]\frac{\frac{7}{10}}{\frac{1}{2}} = \frac{7*2}{10*1} = \frac{14}{10} = 1.4[/tex]

Lyla used 1.4-1 = 0.4 = 40% more brown sugar than white sugar

Lyla used 2/10 of a scoop more brown sugar than white sugar in her baking, a comparison found by subtracting the fractions representing the amounts of sugar used.

To solve this, we need to find the difference in the amounts of the two types of sugar Lyla used. She used 7/10 of a scoop of brown sugar and 1/2 (or 5/10) of a scoop of white sugar. Converting these to equivalent fractions with a common denominator allows for an easy comparison. We then subtract the smaller fraction (white sugar) from the larger fraction (brown sugar) to determine how much more brown sugar was used:

7/10 - 5/10 = 2/10

So, Lyla used 2/10 of a scoop more brown sugar than white sugar.

Answer:

Step-by-step explanation:

a) The null hypothesis is the hypothesis that is assumed to be true. It is an expression that is the opposite of what the researcher predicts.

The alternative hypothesis is what the researcher expects or predicts. It is the statement that is believed to be true if the null hypothesis is rejected.

From the given situation,

Carpetland salespersons average $8000 per week in sales. This is the null hypothesis.

H0: µ = 8000

Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per salesperson. This is the alternative hypothesis.

Ha: µ > 8000

b) A type I error occurs when a true null hypothesis is rejected.

In this situation, a Type I error would occur if it was concluded that the new compensation plan provides a population mean weekly sales greater than 8000 (correct) when in fact it does not.

c) a Type II error would occur if it was concluded that the new compensation plan does not provide a population mean weekly sales greater than 8000 when in fact, it does.

To test Carpetland's new compensation plan, the null hypothesis is that weekly sales per salesperson is $8000, and the alternative is that it's greater than $8000. A Type I error means falsely concluding the new plan increases sales, while a Type II error refers to failing to detect an actual increase in sales.

In the case of Carpetland's new compensation plan, the appropriate null hypothesis (H0) and alternative hypothesis (H1) would be:

H0: The population mean weekly sales per salesperson is $8000. (μ = $8000)

H1: The population mean weekly sales per salesperson is greater than $8000. (μ > $8000)

A Type I error would occur if we reject the null hypothesis when it is actually true, meaning we conclude that the new compensation plan increases average sales when in reality, it does not. A Type II error would occur if we fail to reject the null hypothesis when the alternative hypothesis is true, meaning we conclude that the new compensation plan does not increase average sales when in fact, it does.

It is crucial to design a study properly to reduce the chances of these errors, weighing the consequences that each type of error may have on the business decisions.

Answer:

Step-by-step explanation:

Hello!

You have two regression models:

The multiple regression model that was estimated is y=40.7+ 8.63x1+ 2.71x2

The Simple regression model that was estimated is y=42.0+9.01x1

a.

MRmodel the 8.63 represents the modification in the estimated mean of Y when X₁ increases one unit and X₂ remains constant.

SRmodel the 9.01 represents the modification in the estimated average of Y when X₁ increases one unit.

b.

Yes, since both variables X₁ and X₂ are correlated, the effect that X₁ has over Y is directly affected by the precence of X₂

I hope you have a nice day!

Answer:

x=-4,x= -14

Step-by-step explanation:

Step 1: Simplify both sides of the equation.

x2+18x+81=25

Step 2: Subtract 25 from both sides.

x2+18x+81−25=25−25

x2+18x+56=0

Step 3: Factor left side of equation.

(x+4)(x+14)=0

Step 4: Set factors equal to 0.

x+4=0 or x+14=0

x=−4 or x=−14

========================================================

Explanation:

Apply the square root to both sides

(x+9)^2 = 25 leads to x+9 = 5 or x+9 = -5

This is because (-5)^2 = 25 and (5)^2 = 25. This is the plus/minus you often see with quadratics.

Solve x+9 = 5 to get x = -4. We subtract 9 from both sides to get this.

Solve x+9 = -5 to get x = -14

That's why the two answers are x = -4 and x = -14.

------

To check our answers, we plug those x values into the original equation and simplify. We should get the same thing on both sides

(x+9)^2 = 25

(-4+9)^2 = 25 ... replace x with -4

(5)^2 = 25

25 = 25 ... So x = -4 has been confirmed

and,

(x+9)^2 = 25

(-14+9)^2 = 25 ... replace x with -14

(-5)^2 = 25

25 = 25 ... and x = -14 has been confirmed

Answer:

12  29 /48  in mixed number form,

≈ 12.6 as a rounded decimal

Step-by-step explanation:

Multiply all three numbers together for the volume.

Answer:

The absolute value is   [tex]|z| = 1.92[/tex]

Step-by-step explanation:

From the question we are told that

            The mean is  [tex]\mu = 20.1[/tex]

            The  standard deviation is  [tex]\sigma = 1.4[/tex]

For the null hypothesis [tex]H_0[/tex]

                 The mean remains  [tex]\mu = 20.1[/tex]

For the alternative hypothesis [tex]H_a[/tex]

                 The mean is [tex]\mu < 20[/tex]

This mean that the claim drops

   The test statistic(the calculated value ) (z) is mathematically obtained with the following formula

                        [tex]z = \frac{\= x -\mu}{\frac{\sigma}{\sqrt{n} } }[/tex]

Where [tex]\= x[/tex] is the mean for the the alternative hypothesis

         substituting values

                    [tex]z = \frac{18.9 - 20.1}{\frac{1.4}{\sqrt{5} } }[/tex]

                       [tex]= -1.92[/tex]

the absolute value of the calculated value is  

                [tex]|z| = 1.92[/tex]

Answer:

[tex]\mid z \mid = 1.92[/tex]

Step-by-step explanation:

The null hypothesis is that the average number of complaints per week is 20.1

If H₀ =  Null hypothesis

H₀:  μ = 20.1

After the training, the complaints drop to 18.9 < 20.1. This means that the alternative hypothesis, [tex]H_{a} : \mu < 20.1[/tex]

[tex]\bar{x} = 18.9[/tex]

To know if the alternative hypothesis is true, we need to calculate the absolute z value using the test statistic

number of days, n = 5

Standard deviation, [tex]\sigma = 1.4[/tex]

[tex]z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex]

[tex]z = \frac{18.9 - 20.1}{\frac{1.4}{\sqrt{5} } }[/tex]

z = -1.92

[tex]\mid z \mid = 1.92[/tex]

Final answer:

Fitting a polynomial regression model involves determining coefficients that minimize the sum of squared differences between observed and predicted values. After plotting the data, the cubic model is fitted using statistical software. The coefficients, r-squared, and standard error of the estimate are then calculated to assess the model's accuracy.

Explanation:

The process of fitting a polynomial regression model involves finding the coefficients that minimize the sum of the squares of the differences between the observed values and the values predicted by the model. For a cubic equation, we will be fitting a model in the form y = a + bx + cx2 + dx3, where a, b, c, and d are the coefficients that need to be determined from the data.

To fit a cubic equation to the data and assess the goodness-of-fit, we usually perform the following steps:

The calculation of r2 (r-squared) is important because it tells us the strength of the relationship between the independent and dependent variables. An r2 value of 0.72, for instance, would indicate that 72% of the variability in the dependent variable can be explained by the model. The r2 value cannot be negative because it represents the proportion of variance explained, which is inherently non-negative.

Coefficient of determination is the square of the correlation coefficient (r), and since the correlation coefficient ranges between -1 and +1, its square will always be non-negative as well.