7
2 points
Which is greater- 9/20 or -0.5?

The surface area of the regular pyramid is 178.3 mm².

Here, we need to find the surface area of the regular pyramid.

This regular pyramid consists of three equal triangular faces.

The base of the triangle is 10 mm and height is 9 mm.

Using formula of the area of triangle, the area of a triangle would be,

A = (1/2) × base × height

A = (1/2) × 10 × 9

A = 45 sq. mm.

So, the surface area of the three sides would be,

B = 3A

B = 3 × 45

B = 135 sq. mm.

Here, the area of the base is 43.3 sq.mm.

so, the total surface area of regular pyramid would be,

S = B + 43.3

S = 135 + 43.3

S = 178.3 sq.mm.

This is the required surface area.

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The age of a movie attendee with a z-score of 0.89 is approximately 41 years.

Normal distribution problem

Let's use the standard normal distribution table or a calculator to find the proportion/probability corresponding to the given z-score of 0.89, and then use the inverse z-score formula to find the corresponding age value.

Using a standard normal distribution table, the proportion/probability corresponding to a z-score of 0.89 is 0.8133.

Using the inverse z-score formula:

Rearranging the formula to solve for x, we get:

Therefore, the age of a movie attendee with a z-score of 0.89 is approximately 41 years rounded to the nearest whole number.

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Answer:

1414 tickets, in explanation

Hope this helps!

Step-by-step explanation:

1 ticket = $9.50

? tickets = $13,433

13,433 ÷ 9.50 = 1414

9.50 × 1414 = 13,433

1 ticket × 1414 = ? tickets

? tickets = 1414 tickets

From the graph, the complex number with the greatest modulus is z1

From the question, we have the following parameters that can be used in our computation:

The complex numbers z1, z2, z3 and z4

The general rule of modulus of complex numbers is that

The complex number that has the greatest modulus is the complex number that is at the farthest distance from the origin

Using the above as a guide, we have the following:

From the graph, the complex number that is at the farthest distance from the origin is the complex number z1

Hence, the complex number with the greatest modulus is z1

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Since we know that the heptagon is convex, all of its interior angles are less than 180 degrees. Let's call the smallest angle x degrees.

According to the problem, the other six angles are each 1 degree larger than the angle just smaller than it. This means the second angle is x+1, the third angle is x+2, and so on, until we get to the seventh angle which is x+6.

We know that the sum of the interior angles of a heptagon is (7-2) * 180 = 900 degrees. So we can set up an equation:

x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) = 900

Simplifying this equation, we get:

7x + 21 = 900

Subtracting 21 from both sides:

7x = 879

Dividing both sides by 7:

x = 125.57

So the smallest angle is approximately 125.57 degrees.

To find the fourth largest angle, we need to find the value of x+3.

x+3 = 125.57 + 3 = 128.57

So the fourth largest angle is approximately 128.57 degrees.

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The degree of the polynomial 2 + 2w⁶ + 15y²w + 64u² - uy⁶ is found to be 7 as the term with highest power is 7.

A degree of the polynomial is the highest power to which any of its term is expressed as. For finding the degree we have to find the term with the highest degree in the polynomial. The given polynomial is,

2 + 2w⁶ + 15y²w + 64u² - uy⁶,

The term with the highest degree is uy⁶, which has a degree of 7 (the sum of the exponents of u and y ). Therefore, the degree of the polynomial is 7.

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Complete question - Give the degree of the polynomial. 2 + 2w⁶ + 15y²w + 64u² - uy⁶.

The number of distinct solutions of the given congruences and find the solutions.

a) 7x = 9 (mod 14) has no solution

b) 8x = 9 mod (mod 11) [tex]x\equiv 8 \hspace{0.1cm}(mod \hspace{0.1cm}11)[/tex]

c) 16x = 20 (mod 36) [tex]8, 17, 26, 35 \hspace{0.2cm}mod(36)[/tex]

(a) 7x = 9mod(14) 20

Here, gcd(7,14) =7 , and we know that 7 does not divide 9.

Thus, from Theorem 1, we can say that it has no solution.

(b)8x =  9 mod(11)

Here, gcd(8,11) = 1, so using theorem 2, we can say that it has a unique solution.

For that we need to find [tex]\phi (11)[/tex],  Since 11 is an prime number, therefore the gcd of 11 with any positive integer smaller than 11 will be 1. So,

[tex]\phi (11)[/tex] = 10  = |{1,2,3,..., 10}| ,

So, the solution for the congruence is given by using theorem 2:

[tex]x\equiv a^{\phi (m)-1}b \hspace{0.1cm}(mod \hspace{0.1cm}m)[/tex]

x = 810-19 (mod 11) (

x = 88*9*8 (mod 11)

[tex]x\equiv 64^{4}*72 \hspace{0.1cm}(mod \hspace{0.1cm}11)x\equiv 9^{4}*6 \hspace{0.1cm}(mod \hspace{0.1cm}11)x\equiv 81^{2}*6 \hspace{0.1cm}(mod \hspace{0.1cm}11)[/tex]

x = 16 * 6 (mod 11)

2 = 5*6 (mod 11

[tex]x\equiv 8 \hspace{0.1cm}(mod \hspace{0.1cm}11)[/tex]

which is the final solution.

(c) [tex]16x\equiv 20 \hspace{0.1cm}(mod \hspace{0.1cm}36)[/tex]

Here, d=gcd(16,36) =4 and 4 divides 20, so it has 4 unique solutions.

So, we will use theorem 3.

Divide by 4 whole congruence:

[tex]16x/4\equiv 20/4 \hspace{0.1cm}(mod \hspace{0.1cm}36/4)[/tex]

[tex]4x\equiv 5 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

[tex]So, \phi (9)=\left | \left \{ 1,2,4,5,7,8 \right \} \right |=6[/tex]

[tex]So, x\equiv 4^{\phi (9)-1}*5 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

[tex]x\equiv 4^{5}*5 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

[tex]x\equiv 4^{4}*20 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

[tex]x\equiv 16^{2}*20 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

x = 72 * 2 (mod 9)

[tex]x\equiv 8 \hspace{0.1cm}(mod \hspace{0.1cm}9)[/tex]

Thus, the 5 unique solutions using theorem3 are given as follows:

[tex]t,t+\frac{m}{d}, t+\frac{2m}{d},. . ., t+\frac{(d-1)m}{d} \hspace{0.2cm} mod(m)[/tex]

[tex]8, 17, 26, 35 \hspace{0.2cm}mod(36)[/tex].

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The standard deviation of the total sales per day for the shoe store is approximately $1743.28.

To find the standard deviation of the total sales per day for the shoe store, we need to first find the mean and variance of the total sales.

The number of customers visiting the store follows a Poisson distribution with a rate of 70 per day. Let X be the number of customers per day, then X ~ Poisson(70).

The proportion of customers that make a purchase is 45%. Let Y be the number of customers that make a purchase, then Y ~ Binomial(X, 0.45).

The amount spent by a paying customer follows an exponential distribution with a mean of $120. Let Z be the amount spent by a paying customer, then Z ~ Exp(1/120).

Now, let's find the mean and variance of the total sales per day.

E[XY] = E[E[XY|X]] = E[X * 0.45] = 70 * 0.45 = 31.5

E[Z] = 120

E[XYZ] = E[E[XYZ|XY]] = E[XY * 120] = 31.5 * 120 = 3780

Var(XY) = E[Var(XY|X)] + Var(E[XY|X]) = E[X * 0.45 * 0.55] + Var(X * 0.45) = 70 * 0.45 * 0.55 + 70 * 0.45 * 0.55 = 17.325

Var(Z) = 120^2

Var(XYZ) = E[Var(XYZ|XY)] + Var(E[XYZ|XY]) = E[XY * 120^2] + Var(XY * 120) = 31.5 * 120^2 + 17.325 * 120^2 = 3035250

Therefore, the variance of the total sales per day is Var(XYZ) = 3035250.

The standard deviation of the total sales per day is the square root of the variance:

SD(XYZ) = sqrt(3035250) = 1743.28

Therefore, the standard deviation of the total sales per day for the shoe store is approximately $1743.28.

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The value of the unknown angle is 68°

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

Sin(tetha) = opp/hyp

cos(tetha) = adj/hyp

tan(tetha) = opp/adj

In the triangle, 51 is the opposite and 55 is the hypotenuse.

therefore;

sin(tetha) = 51/55

sin(tetha) = 0.927

tetha = sin^-1( 0.927)

tetha = 67.97

approximately to 68°

therefore the value of the unknown angle is 68°

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Answer:

The mean weight of six boys as per the incorrect data is 59.1667 kilograms.

The actual mean weight of the boy's group is 58.5 kilograms.

To find the mean weight of the six boys as per the incorrect data, we add up all the weights and divide by 6:

(55 + 58 + 57 + 60 + 59 + 65)/6 = 354/6 = 59.1667 kilograms

To find the actual mean weight of the boy's group, we add up the weights of the first five boys and the corrected weight of the sixth boy, and divide by 6:

(55 + 58 + 57 + 60 + 59 + 56)/6 = 345/6 = 58.5 kilograms

Step-by-step explanation:

Answer: 10x - 1 and 20x - 12

Step-by-step explanation:

perimeter is defined as the distance around a figure:

as such, the perimeter of this triangle is: (2x -3) + (3x + 5) + (5x - 3)

which simplifies to 10x - 1

Area is a little bit tougher. the area of a triangle is defined as 0.5 base x height.

both are given, so we simply plug in: 0.5 x (5x-3) x 8 = 4(5x-3) = 20x - 12

And thats it!

the Perimeter is 10x - 1 units.

the Area is 20x - 12 square units.

Answer: red hot= $4

gummies=$9

Step-by-step explanation:

3r+1g=21

-(3r+3g=39)

3r+1g=21

-3r-3g=-39

-2g=-18

g=9

3r+1(9)=21

3r=12

r=4

Armando was charged approximately $5.08 in interest for the billing cycle.

To calculate the interest charged for the billing cycle, we need to find the average daily balance (ADB) and then multiply it by the daily periodic rate (DPR) and the number of days in the billing cycle. For a credit card that uses the adjusted balance method, the ADB is calculated as the sum of the balances on each day in the billing cycle divided by the number of days in the cycle.

To find the balance on each day in the billing cycle, we need to divide the cycle into three periods: the first 10 days, the next 10 days, and the last 10 days.

During the first 10 days, the balance was $2500, so the total balance for this period was:

10 * $2500 = $25000

During the next 10 days, the balance was $900, so the total balance for this period was:

10 * $900 = $9000

During the last 10 days, the balance was $2200, so the total balance for this period was:

10 * $2200 = $22000

The total balance for the entire billing cycle was:

$25000 + $9000 + $22000 = $56000

The number of days in the billing cycle is 30, so the ADB is:

ADB = $56000 / 30 = $1866.67

The DPR can be calculated by dividing the APR by the number of days in the year:

DPR = 0.33 / 365 = 0.00090411

Finally, we can calculate the interest charged for the billing cycle by multiplying the ADB by the DPR and the number of days in the billing cycle:

Interest = ADB * DPR * Days

Interest = $1866.67 * 0.00090411 * 30

Interest ≈ $5.08

Therefore, Armando was charged approximately $5.08 in interest for the billing cycle.

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The solution to the partial differential equation is u(x,t) = [tex]kx^{a/10[/tex], where k is a constant.

A differential equation is a mathematical formula that includes one or more terms as well as the derivatives of one variable with respect to another.

To find the Laplace transform of f(t) = te^(-4t) cosh(5t), we use the formula:

[tex]L{f(t)} = \int_0 f(t) e^{(-st)} dt[/tex]

= ∫₀^∞ [tex]te^{(-4t)} cosh(5t) e^{(-st)} dt[/tex]

= ∫₀^∞ t cosh(5t) [tex]e^{(-(4+s)t)[/tex] dt

Using integration by parts with u = t and dv = cosh(5t) [tex]e^{(-(4+s)t)[/tex] dt, we get:

L{f(t)} = [-t/(4+s) cosh(5t) [tex]e^{(-(4+s)t)[/tex]]₀^∞ + ∫₀^∞ (1/(4+s)) cosh(5t) [tex]e^{(-(4+s)t)[/tex] dt

Simplifying the boundary term, we get:

L{f(t)} = (1/(4+s)) ∫₀^∞ cosh(5t) [tex]e^{(-(4+s)t)}[/tex] dt

= (1/(4+s)) ∫₀^∞ (1/2) [[tex]e^{(5t)[/tex] + [tex]e^{(-5t)[/tex]] [tex]e^{(-(4+s)t)[/tex] dt

= (1/2(4+s)) ∫₀^∞ [[tex]e^{((1-s)t)[/tex] + [tex]e^{(-(9+s)t)[/tex]] dt

Using the Laplace transform of [tex]e^{(at)[/tex], we get:

L{f(t)} = (1/2(4+s)) [(1/(s-1)) + (1/(s+9))]

= (1/2) [(1/(4+s-4)) + (1/(4+s+36))]

= (1/2) [(1/(s+1)) + (1/(s+40))]

To solve the partial differential equation au/ax = 10 au/at by variable separable method, we can write:

(1/u) du/dt = 10/a dx/dt

Integrating both sides with respect to t and x, we get:

ln|u| = 10ax + C₁

Taking the exponential of both sides, we get:

|u| = [tex]e^{(10ax+C_1)[/tex]

= [tex]e^{(10ax)[/tex] [tex]e^{(C_1)[/tex]

= [tex]ke^{(10ax)[/tex]  (where k is a constant)

Since u is positive, we can drop the absolute value and write:

[tex]u = ke^{(10ax)[/tex]

Taking the partial derivative of u with respect to x, we get:

au/ax = [tex]10ke^{(10ax)[/tex]

Substituting this into the given partial differential equation, we get:

[tex]10ke^{(10ax)[/tex] = 10 au/at

Dividing both sides by 10u, we get:

(1/u) du/dt = a/(10x)

Integrating both sides with respect to t and x, we get:

ln|u| = (a/10) ln|x| + C₂

Taking the exponential of both sides, we get:

|u| = [tex]e^{(a/10 ln|x|+C_2)[/tex]

= [tex]e^{(ln|x|^{a/10)[/tex] [tex]e^{(C_2)[/tex]

= [tex]kx^{a/10[/tex]  (where k is a constant)

Since u is positive, we can drop the absolute value and write:

[tex]u = kx^{a/10[/tex]

The solution to the partial differential equation is u(x,t) = [tex]kx^{a/10[/tex], where k is a constant.

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Given a ray passing through a line at an angle of 29 degrees, the angle opposite to it (angle R) can be found by subtracting 29 degrees from 180 degrees. Therefore, the value of angle R is 151 degrees.

We are given that a ray passes through a line, making an angle of 29 degrees with the line. Let us represent this situation as follows

The angle R represents the angle opposite to the angle of 29 degrees. Since the ray and the line form a straight line, their angles add up to 180 degrees. Therefore, we can write

angle R + 29 degrees = 180 degrees

To solve for angle R, we can subtract 29 degrees from both sides of the equation

angle R = 180 degrees - 29 degrees

Simplifying the expression, we get

angle R = 151 degrees

Therefore, the value of angle R is 151 degrees.

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The probability of randomly drawing a vowel is 2/8

From the question, we have the following parameters that can be used in our computation:

P, E, R, C, E, N, T, S,

Using the above as a guide, we have the following:

Vowels = 2

Total = 8

So, we have

P(Vowel) = Vowel/Total

Substitute the known values in the above equation, so, we have the following representation

P(Vowel) = 2/8

Hence, the solution is 2/8

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In an English literature course, the professor asks students to read three books by selecting one memoire, one book of poetry, and one novel to read. The students can select these books from a list of 8 memoires, 9 poetry books, and 4 novels.

To determine how many different ways a student can select their reading assignment of three books, we will use the multiplication principle.

1. Choose one memoire:  There are 8 memoires to choose from, so there are 8 ways to make this choice.
2. Choose one book of poetry:  There are 9 poetry books to choose from, so there are 9 ways to make this choice.
3. Choose one novel:  There are 4 novels to choose from, so there are 4 ways to make this choice.

Now, multiply the number of choices for each step together to find the total number of ways to select the reading assignment:

8 (memoires) x 9 (poetry books) x 4 (novels) = 288 different ways to select the reading assignment of three books.

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Answer:

24.997 cm

Step-by-step explanation:

         r = 7 cm

[tex]\sf Circumference \ of \ quarter \ circle = \dfrac{1}{2}*\pi *r[/tex]

                                               [tex]\sf =\dfrac{1}{2}*3.142*7\\\\= 10.997 \ cm[/tex]

        Perimeter of quarter circle  = r + r + circumference of quarter circle

                                                      = 7 + 7 + 10.997

                                                      = 24.997 cm

Using the considered ordered pairs:

The domain is {1,3,5}

The range is {2,4,5,6}

It is not a function

We have,

To determine the domain and range of a function, we need to know the set of possible input values (domain) and the set of possible output values (range).

To determine if a relation is a function, we need to check if every input has a unique output.

In other words, if there are no two distinct ordered pairs with the same first element.

This means,

A function can be one-to-one or onto.

For example,

Let's consider the relation given by the set of ordered pairs:

{(1,2), (3,4), (1,5), (5,6)}

To determine if this is a function, we first need to check if there are any two distinct ordered pairs with the same first element.

In this case, we see that both (1,2) and (1,5) have a first element of 1, so this relation is not a function.

The domain of this relation is the set of all first elements of the ordered pairs, which is {1,3,5}.

The range of this relation is the set of all second elements of the ordered pairs, which is {2,4,5,6}.

Thus,

Domain: {1,3,5}

Range: {2,4,5,6}

Not a function

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Andrés will be 34 years old and Juan will be 13 years old.

From the question, we shall make  Juan's age as J as well as Andrés' age as A.

According to the question, Juan is 21 years younger than Andrés, so we can write it as:

J = A - 21  --------Equation 1

The sum of their ages is 47 will be:

J + A = 47  ----------Equation 2

Then we substitute the sum of J from Equation 1 into Equation 2 to remove J and look for A:

(A - 21) + A = 47

2A - 21 = 47

2A = 47 + 21

2A = 68

A = 68 / 2

A = 34

Hence Andrés' age (A) is 34 years.

So we also need to substitute the value of A back into Equation 1 to know Juan's age (J):

J = A - 21

J = 34 - 21

J = 13

Hence, Juan's age (J) is 13 years.

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Juan is 21 years younger than Andrés and we know that the sum of their ages is 47. How old is each of them?

The best measure of variability for the data and the player which was more consistent include the following: B. Player B is the most consistent, with a range of 9.

In Mathematics and Statistics, interquartile range (IQR) of a data set and it is typically calculated as the difference between the first quartile (Q₁) and third quartile (Q₃):

Interquartile range (IQR) of Player A = Q₃ - Q₁

Interquartile range (IQR) of Player A = 4 - 1.5

Interquartile range (IQR) of Player A = 2.5.

Range of Player A = Highest number - Lowest number

Range of Player A = 8 - 1

Range of Player A = 7

Interquartile range (IQR) of Player B = Q₃ - Q₁

Interquartile range (IQR) of Player B = 5 - 1.5

Interquartile range (IQR) of Player B = 4.5.

Range of Player B = Highest number - Lowest number

Range of Player B = 10 - 1

Range of Player B = 9

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Answer: You can convert it by dividing it by 12 which would give you 3.5 and since half of 12 is 6 the answer is 3ft and 6inches

Step-by-step explanation:

The midpoint of the two points of P (-3,-7) and Q (3,-5) is (0, -6).

When you have the vertices of two points, you can find the midpoint by the formula :

= ( ( x 1 + x 2 ) / 2 , ( y 1 + y 2 ) / 2 )

Solving for the midpoint therefore gives:

= ( ( - 3 + 3 ) / 2 , ( - 7 + ( - 5 ) ) / 2 )

=  ( 0 / 2 , ( - 12 ) / 2 )

= (0, -6)

In conclusion, the midpoint is (0, -6).

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Both statements are true. When events A and B are mutually exclusive, meaning they cannot occur simultaneously, you can use the special rule of addition.

If events A and B are not mutually exclusive, meaning they can occur together, you should use the general rule of addition. The specific rule of addition can only be used when dealing with mutually exclusive events, while the general rule of addition can be used for any two events, whether they are mutually exclusive or not. The specific rule of addition states that the probability of either event A or event B occurring is equal to the sum of their individual probabilities, while the general rule of addition states that the probability of event A or event B occurring is equal to the sum of their individual probabilities minus the probability of their intersection (if they are not mutually exclusive).

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True statement is Player 3 is more likely to hit the ball than Player 2 because P(Player 3) > P(Player 2)

Convert to decimals first

Player 1: 7/11 ≈ 0.636

Player 2: 6/9 = 2/3 ≈ 0.667

Player 3: 5/7 ≈ 0.714

From here we have to compare the options

1. False. Player 1 is more likely to hit the ball than Player 2 because P(Player 1) >P(Player 2)

Reason

(0.636 < 0.667)

2. False. Player 2 is more likely to hit the ball than Player 3 because P(Player 2) > P(Player 3)

Reason

(0.667 < 0.714)

3. False. Player 1 is more likely to hit the ball than Player 3 because P(Player 1) > P(Player 3)

Reason

(0.636 < 0.714)

4. True. Player 3 is more likely to hit the ball than Player 2 because P(Player 3) > P(Player 2)

Reason

(0.714 > 0.667)

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The number of textbooks of each type sold is found by solving the system of equations and got as,

Number of chemistry textbooks = 210

Number of history textbooks = 158

Given that,

A textbook store sold a combined total of 368 chemistry and history textbooks in a week.

let c be the number of chemistry textbooks sold and h be the number of history textbooks sold.

c + h = 368

The number of history textbooks sold was 52 less than the number of chemistry textbooks sold.

h = c - 52

Substituting the second equation in first,

c + (c - 52) = 368

2c = 420

c = 210

h = 210 - 52 = 158

Hence the number of each textbooks is 210 and 158.

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The number of hours that it will take Lindsey to rake the lawn alone will also be 3 hours just like Camilla.

The total number of hours it takes two people to rake the lawn = 2 hours.

The more people the less number of hours it will take to take the lawn.

That is;

If 2 people = 2 hours

Camilla = 3 hours

1 person (Lindsey) = 3 hours.

Therefore, for either Lindsey or Camilla, they will rake separately for 2 hours when working alone.

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The value of the iterated integral  ∫85∫√x12ye−xdydx is

-[tex]4e^(-5) + 7e^(-8)[/tex] where the inner integral is first integrated with respect to y.

We are inquiring to assess the iterated integral:

[tex]∫85∫√x12ye−xdydx[/tex]

We are able to coordinate the internal integral, to begin with regard to y:

[tex]∫√x12ye−xdy = (-1/2)e^(-x) y√x1/2 | from y = to y = √x^1/2[/tex]

[tex]= (-1/2)e^(-x) (√x^1/2)^2 - (-1/2)e^(-x) (0)[/tex]

[tex]= (-1/2)x e^(-x)[/tex]

Substituting this into the first necessity, we get:

[tex]∫85∫√x12ye−xdydx = ∫85(-1/2)x e^(-x)dx[/tex]

To assess this necessarily, we utilize integration by parts with u = x and [tex]dv = e^(-x) dx, so that du/dx = 1 and v = -e^(-x):[/tex]

[tex]∫85(-1/2)x e^(-x)dx = (-1/2)xe^(-x) + ∫85(1/2)e^(-x)dx[/tex]

[tex]= (-1/2)xe^(-x) - (1/2)e^(-x) | from x = 8 to x = 5[/tex]

[tex]= (-1/2)(8e^(-8) - 5e^(-5)) - (1/2)(e^(-8) - e^(-5))[/tex]

[tex]= -4e^(-5) + 7e^(-8)[/tex]

therefore, the value of the iterated integral is [tex]-4e^(-5) + 7e^(-8).[/tex]

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Applying the Friedman test, we conclude that there is evidence that at least two population locations differ at a significance level of 10%, since our calculated [tex]$\chi^2$[/tex] value (979.5) is greater than the critical value (7.81).

To apply the Friedman test, we need to first rank the data within each block (column) and calculate the average ranks for each treatment (row). The ranks are calculated by assigning a rank of 1 to the smallest value, 2 to the second-smallest value, and so on. Ties are given the average rank of the tied values.

Treatment Block 1 Block 2 Block 3 Block 4 Ranks

1 2 1.5 3 3.5 10

2 1 3 5 5 14

3 3 2.5 4 2 11.5

4 2 4 2 1 9

5 1 4.5 1 4.5 11

The Friedman test statistic is calculated as:

[tex]$ \chi^2 = \frac{12}{n(k-1)} \left[ \sum_{j=1}^k \left( \sum_{i=1}^n R_{ij}^2 - \frac{n(n+1)^2}{4} \right) \right] $[/tex]

where [tex]$n$[/tex] is the number of blocks, [tex]$k$[/tex] is the number of treatments, and [tex]$R_{ij}$[/tex] is the rank of the [tex]$j^t^h[/tex] treatment in the [tex]$i^t^h[/tex] block.

In this case, [tex]$n=4$[/tex] and [tex]$k=5$[/tex], so:

[tex]$ \chi^2 = \frac{12}{4(5-1)} \left[ \sum_{j=1}^5 \left( \sum_{i=1}^4 R_{ij}^2 - \frac{4(4+1)^2}{4} \right) \right] $[/tex]

[tex]$ \chi^2 = \frac{3}{2} \left[ (10^2 + 14^2 + 11.5^2 + 9^2 + 11^2) - \frac{4(5^2)}{4} \right] $[/tex]

[tex]$ \chi^2 = \frac{3}{2} \left[ 727 - 50 \right] = 979.5 $[/tex]

The critical value for the Friedman test with [tex]$k=5$[/tex] treatments and [tex]$n=4$[/tex]blocks, at a significance level of [tex]\alpha = 0.1$,[/tex] is:

[tex]$ \chi_{0.1}^2 = 7.81 $[/tex]

Since our calculated [tex]$\chi^2$[/tex] value (979.5) is greater than the critical value (7.81), we reject the null hypothesis that there is no difference between the population locations, and conclude that there is evidence that at least two population locations differ at a significance level of 10%.

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