the standard deviation for the numbers of sleepwalkers in groups of five is approximately 1.532.
To find the mean and standard deviation for the numbers of sleepwalkers in groups of five, we need to calculate the weighted average and variance using the given data.
Mean (Expected Value):
The mean is calculated by multiplying each value by its corresponding probability and summing up the results.
Mean = (0 * 0.147) + (1 * 0.367) + (2 * 0.319) + (3 * 0.133) + (4 * 0.031) + (5 * 0.003)
Mean = 0 + 0.367 + 0.638 + 0.399 + 0.124 + 0.015
Mean = 1.543
Therefore, the mean for the numbers of sleepwalkers in groups of five is 1.543.
Standard Deviation:
The standard deviation is calculated by first finding the variance and then taking the square root of the variance.
Variance =[tex](x^2 * P(x)) - (mean^2 * P(x))[/tex]
Variance =[tex](0^2 * 0.147) + (1^2 * 0.367) + (2^2 * 0.319) + (3^2 * 0.133) + (4^2 * 0.031) + (5^2 * 0.003) - (1.543^2 * 0.147)[/tex]
Variance = 0 + 0.367 + 1.278 + 0.532 + 0.496 + 0.015 - 0.343
Variance = 2.345
Standard Deviation = √Variance
Standard Deviation = √2.345
Standard Deviation ≈ 1.532 (rounded to three decimal places)
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kori categorized her spending for this month into four categories: rent, food, fun, and other. the percents she spent in each category are pictured here. if she spent a total of $2600 this month, how much did she spend on rent?
The amount Kori spent on rent this month if she spent a total of $2600 this month and 26% of her total budget is spent on rent is $676
How much did she spend on rent?Total amount Kori spent this month = $2600
Percentage spent on rent = 26%
Amount spent on rent = Percentage spent on rent × Total amount Kori spent this month
= 26% × $2600
= 0.26 × $2,600
= $676
Hence, Kori spent $676 on rent.
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Let E, F and G be three events in S with P(E) = 0.48, P(F) =
0.52, P(G) = 0.52, P(E ∩ F) = 0.32, P(E ∩ G) = 0.29, P(F ∩ G) =
0.26, and P(E ∩ F ∩ G) = 0.2.
Find P(EC ∪ FC ∪ GC).
The required probability of the union of the complements of events E, F, and G is 0.9631.
Given, the events E, F, and G in a sample space S are defined with their respective probabilities as follows: P(E) = 0.48, P(F) = 0.52, P(G) = 0.52, P(E ∩ F) = 0.32, P(E ∩ G) = 0.29, P(F ∩ G) = 0.26, and P(E ∩ F ∩ G) = 0.2. We need to calculate the probability of the union of their complements.
Let's first calculate the probabilities of the complements of E, F, and G.P(E') = 1 - P(E) = 1 - 0.48 = 0.52P(F') = 1 - P(F) = 1 - 0.52 = 0.48P(G') = 1 - P(G) = 1 - 0.52 = 0.48We know that P(E ∩ F) = 0.32. Hence, using the formula of probability of the union of events, we can find the probability of the intersection of the complements of E and F.P(E' ∩ F') = 1 - P(E ∪ F) = 1 - (P(E) + P(F) - P(E ∩ F))= 1 - (0.48 + 0.52 - 0.32) = 1 - 0.68 = 0.32We also know that P(E ∩ G) = 0.29. Similarly, we can find the probability of the intersection of the complements of E and G.P(E' ∩ G') = 1 - P(E ∪ G) = 1 - (P(E) + P(G) - P(E ∩ G))= 1 - (0.48 + 0.52 - 0.29) = 1 - 0.29 = 0.71We also know that P(F ∩ G) = 0.26.
Similarly, we can find the probability of the intersection of the complements of F and G.P(F' ∩ G') = 1 - P(F ∪ G) = 1 - (P(F) + P(G) - P(F ∩ G))= 1 - (0.52 + 0.52 - 0.26) = 1 - 0.76 = 0.24Now, we can calculate the probability of the union of the complements of E, F, and G as follows: P(E' ∪ F' ∪ G')= P((E' ∩ F' ∩ G')') {De Morgan's law}= 1 - P(E' ∩ F' ∩ G') {complement of a set}= 1 - P(E' ∩ F' ∩ G') {by definition of the intersection of sets}= 1 - P(E' ∩ F') ⋅ P(G') {product rule of probability}= 1 - 0.32 ⋅ 0.48 ⋅ 0.24= 1 - 0.0369= 0.9631.
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The Brady family received 27 pieces of mail on December 25 . The mail consisted of letters, magazines, bills, and ads. How many letters did they receive if they received three more magazines than bill
The Brady family received 12 letters on December 25th.
They received 9 magazines.
They received 3 bills.
They received 3 ads.
To solve this problem, we can use algebra. Let x be the number of bills the Brady family received. We know that they received three more magazines than bills, so the number of magazines they received is x + 3.
We also know that they received a total of 27 pieces of mail, so we can set up an equation:
x + (x + 3) + 12 + 3 = 27
Simplifying this equation, we get:
2x + 18 = 27
Subtracting 18 from both sides, we get:
2x = 9
Dividing by 2, we get:
x = 3
So the Brady family received 3 bills. Using x + 3, we know that they received 3 + 3 = 6 magazines. We also know that they received 12 letters and 3 ads. Therefore, the Brady family received 12 letters on December 25th.
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simplify the following expression 3 2/5 mulitply 3(-7/5)
Answer:
1/3
Step-by-step explanation:
I assume that 2/5 and -7/5 are exponents.
3^(2/5) × 3^(-7/5) = 3^(2/5 + (-7/5)) = 3^(-5/5) = 3^(-1) = 1/3
Answer: 136/5
Step-by-step explanation: First simplify the fraction
1) 3 2/5 = 17/5
3 multiply by 5 and add 5 into it.
2) 3(-7/5) = 8/5
3 multiply by 5 and add _7 in it.
By multiplication of 2 fractions,
17/5 multiply 8/5 = 136/5
=136/5
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n={n/2,3×n+1, if n is even if n is odd The conjecture states that when this algorithm is continually applied, all positive integers will eventually reach i. For example, if n=35, the secguence is 35, 106,53,160,60,40,20,10,5,16,4,4,2,1 Write a C program using the forki) systen call that generates this sequence in the child process. The starting number will be provided from the command line. For example, if 8 is passed as a parameter on the command line, the child process will output 8,4,2,1. Hecause the parent and child processes have their own copies of the data, it will be necessary for the child to outpat the sequence. Have the parent invoke the vaite() call to wait for the child process to complete before exiting the program. Perform necessary error checking to ensure that a positive integer is passed on the command line
The C program described generates a sequence of numbers based on a conjecture. The program takes a positive integer as input and uses the fork system call to create a child process.
The C program uses the fork system call to create a child process. The program takes a positive integer, the starting number, as a parameter from the command line. The child process then applies the given algorithm to generate a sequence of numbers.
The algorithm checks if the current number is even or odd. If it is even, the next number is obtained by dividing it by 2. If it is odd, the next number is obtained by multiplying it by 3 and adding 1.
The child process continues applying the algorithm to the current number until it reaches the value of 1. During each iteration, the sequence is printed.
Meanwhile, the parent process uses the wait() call to wait for the child process to complete before exiting the program.
To ensure that a positive integer is passed on the command line, the program performs necessary error checking. If an invalid input is provided, an error message is displayed, and the program terminates.
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Jackson rolls a fair 6-sided number cube. Then he spins a spinner that is divided into 4 equal sections numbered 1, 2, 3, and 4. What is the probability that at least one of the numbers is a 3? Enter your answer in the box.
Fill in the blank: When finding the difference between 74 and 112, a student might say, and then I added 2 more tens onto "First, I added 6 onto 74 to get a ______80 to get to 100 because that's another______
When finding the difference between 74 and 112, a student might say, "First, I added 6 onto 74 to get a number that ends in 0, specifically 80, to get to 100 because that's another ten."
To find the difference between 74 and 112, the student is using a strategy of breaking down the numbers into smaller parts and manipulating them to simplify the subtraction process. In this case, the student starts by adding 6 onto 74, resulting in 80. By doing so, the student is aiming to create a number that ends in 0, which is closer to 100 and represents another ten. This approach allows for an easier mental calculation when subtracting 80 from 112 since it involves subtracting whole tens instead of dealing with more complex digit-by-digit subtraction.
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Find the solution to initial value problem dt 2d2y−2dt dy+1y=0,y(0)=4,y ′(0)=1 Find the solution of y ′′−2y ′ +y=343e 8t with u(0)=8 and u ′(0)=6. y
Solution to initial value problem is u = (125/19)e^(20t) + (53/19)e^(-18t)
Given differential equation is
2d²y/dt² - 2dy/dt + y = 0;
y(0) = 4; y'(0) = 1.
And another differential equation is
y'' - 2y' + y = 343e^(8t);
u(0) = 8,
u'(0) = 6.
For the first differential equation,Let us find the characteristic equation by assuming
y = e^(mt).d²y/dt²
= m²e^(mt),
dy/dt = me^(mt)
Substituting these values in the given differential equation, we get
2m²e^(mt) - 2me^(mt) + e^(mt) = 0
Factorizing, we get
e^(mt)(2m - 1)² = 0
The characteristic equation is 2m - 1 = 0 or m = 1/2
Taking the first case 2m - 1 = 0
m = 1/2
Since this root is repeated twice, the general solution is
y = (c1 + c2t)e^(1/2t)
Differentiating the above equation, we get
dy/dt = c2e^(1/2t) + (c1/2 + c2/2)te^(1/2t)
Applying the initial conditions,
y(0) = 4c1 = 4c2 = 4
The solution is y = (4 + 4t)e^(1/2t)
For the second differential equation,
Let us find the characteristic equation by assuming
u = e^(mt).
u'' = m²e^(mt);
u' = me^(mt)
Substituting these values in the given differential equation, we get
m²e^(mt) - 2me^(mt) + e^(mt) = 343e^(8t)
We have e^(mt) commonm² - 2m + 1 = 343e^(8t - mt)
Dividing throughout by e^(8t), we get
m²e^(-8t) - 2me^(-8t) + e^(-8t) = 343e^(mt - 8t)
Setting t = 0, we get
m² - 2m + 1 = 343
Taking square roots, we get
(m - 1) = ±19
Taking first case m - 1 = 19 or m = 20
Taking the second case m - 1 = -19 or m = -18
Substituting the roots in the characteristic equation, we get
u1 = e^(20t); u2 = e^(-18t)
The general solution is
u = c1e^(20t) + c2e^(-18t)
Differentiating the above equation, we get
u' = 20c1e^(20t) - 18c2e^(-18t)
Applying the initial conditions,
u(0) = c1 + c2 = 8u'(0) = 20c1 - 18c2 = 6
Solving the above equations, we get
c1 = 125/19 and c2 = 53/19
Hence, the solution is
u = (125/19)e^(20t) + (53/19)e^(-18t)
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Consider the line y=-(1)/(5)x+3 (a) What is the slope of a line perpendicular to this line? (b) What is the slope of a line parallel to this line?
For a line to be parallel to the given line, it must have the same slope. The slope of the given line is -1/5, so a line parallel to it will also have a slope of -1/5. The slope of a line perpendicular to the given line is 5.
a) The slope of a line perpendicular to y=-(1)/(5)x+3 is 5. b) The slope of a line parallel to y=-(1)/(5)x+3 is -1/5.
The given equation is y = -(1/5)x + 3.
The slope of the given line is -1/5.
For a line to be perpendicular to the given line, the slope of the line must be the negative reciprocal of -1/5, which is 5.
Thus, the slope of a line perpendicular to the given line is 5.
For a line to be parallel to the given line, the slope of the line must be the same as the slope of the given line, which is -1/5.
Thus, the slope of a line parallel to the given line is -1/5.
To understand the concept of slope in detail, let us consider the equation of the line y = mx + c, where m is the slope of the line. In the given equation, y=-(1)/(5)x+3, the coefficient of x is the slope of the line, which is -1/5.
Now, let's find the slope of a line perpendicular to this line. To find the slope of a line perpendicular to the given line, we must take the negative reciprocal of the given slope. Therefore, the slope of a line perpendicular to y=-(1)/(5)x+3 is the negative reciprocal of -1/5, which is 5.
To find the slope of a line parallel to the given line, we must recognize that parallel lines have the same slope. Hence, the slope of a line parallel to y=-(1)/(5)x+3 is the same as the slope of the given line, which is -1/5. Therefore, the slope of a line parallel to y=-(1)/(5)x+3 is -1/5. Hence, the slope of a line perpendicular to the given line is 5, and the slope of a line parallel to the given line is -1/5.
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a. When we ADD two equations together (with the aim of solving a 2x2 system of equations), what do we need to happen?
b. What if it doesn’t happen?
When adding two equations together to solve a 2x2 system of equations, the aim is to eliminate one of the variables and create a new equation with only one variable, it can be done using elimination method However, if the elimination does not happen, it means that the equations do not have a unique solution or that the system is inconsistent.
a) When solving a 2x2 system of equations, one common approach is to add or subtract the equations to eliminate one of the variables. The objective is to create a new equation that contains only one variable, which simplifies the system and allows for finding the value of the remaining variable. This method is known as the method of elimination or addition/subtraction method.
If the addition of the equations successfully eliminates one variable, we end up with a simplified equation with only one variable. We can then solve this equation to find the value of that variable. Substituting this value back into one of the original equations will give us the value of the other variable, thus providing a unique solution to the system.
b) However, if the addition or subtraction of the equations does not result in the elimination of a variable, it means that the equations are not compatible or consistent. In such cases, the system either has no solution or an infinite number of solutions, indicating that the equations are dependent or the lines represented by the equations are parallel. It implies that the system is inconsistent and cannot be solved uniquely using the method of elimination.
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Let g(x)= x+2/(x^2 -5x - 14) Determine all values of x at which g is discontinuous, and for each of these values of x, define g in such a manner as to remove the discontinuity, if possible.
g(x) is discontinuous at x=______________(Use a comma to separate answers as needed.)
For each discontinuity in the previous step, explain how g can be defined so as to remove the discontinuity. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
A. g(x) has one discontinuity, and it cannot be removed.
B. g(x) has two discontinuities. The lesser discontinuity can be removed by defining g to beat that value. The greater discontinuity cannot be removed.
C. g(x) has two discontinuities. The lesser discontinuity cannot be removed. The greater discontinuity can be removed by setting g to be value.
at that
D. g(x) has two discontinuities. The lesser discontinuity can be removed by defining g to be at that value. The greater discontinuity can be removed by defining g to be
at that value.
E. g(x) has one discontinuity, and it can be removed by defining g to |
at that value.
F. g(x) has two discontinuities and neither can be removed.
The function g(x) is discontinuous at x = -2 and x = 7. The correct choice is B) g(x) has two discontinuities. The lesser discontinuity can be removed by defining g to beat that value. The greater discontinuity cannot be removed.
The function g(x) is discontinuous at x = -2 and x = 7.
x = -2
The denominator of g(x) is equal to 0 at x = -2. This means that g(x) is undefined at x = -2. The discontinuity at x = -2 cannot be removed.
x = 7
The numerator of g(x) is equal to 0 at x = 7. This means that g(x) approaches ∞ as x approaches 7. The discontinuity at x = 7 can be removed by defining g(7) to be 3.
Choice
The correct choice is B. The lesser discontinuity can be removed by defining g(-2) to be 3. The greater discontinuity cannot be removed.
Explanation
The function g(x) is defined as follows:
g(x) = x + 2 / ([tex]x^2[/tex] - 5x - 14) = x + 2 / ((x - 7)(x + 2))
The denominator of g(x) is equal to 0 at x = -2 and x = 7. This means that g(x) is undefined at x = -2 and x = 7.
The discontinuity at x = -2 cannot be removed because the denominator of g(x) is equal to 0 at x = -2. However, the discontinuity at x = 7 can be removed by defining g(7) to be 3. This is because the two branches of g(x) approach the same value, 3, as x approaches 7.
The following table summarizes the discontinuities of g(x) and how they can be removed:
x Value of g(x) Can the discontinuity be removed?
-2 undefined No
7 3 Yes
Therefore, the correct choice is B.
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a parallelogram has side lengths 2 and 5, and one diagonal measures 7. find the length of the other diagonal
The length of another diagonal will be 3 inches.
The formula for a parallelogram relationship between its sides and diagonals is
(D1)² + (D2)² = 2A² + 2B²
were
D1 represents one diagonal,
D2 represents the second diagonal,
A stand for one side and B stands for the adjacent side.
Putting the mentioned values in this formula will give -
= 7² +(D2)² = 2*2² + 2*5²
= 49 + (D2)² = 2*4 + 2*25
= 49 + (D2)² = 8 + 50
= 49 + (D2)² = 58
= D2 = 3 inch
So finally, the length of the other diagonal will be 3 inches.
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Suppose you are using the LCG xn+1 = (18xn + 53) mod 4913. The
value of x1 is 4600. What was x0?
xn+1 = (18xn + 53) mod 4913; x1 = 4600 We are given that the value of x1 is 4600 and we are to find the value of x0.Let's substitute the given value of x1 in the LCG equation and solve for x0. Thus,x2 = (18 * 4600 + 53) mod 4913x2 = 82853 mod 4913x2 = 1427... and so on.
Substituting x2 in the equation,
x3 = (18 * 1427 + 53) mod 4913x3 = 25751 mod 4913x3 = 2368...
and so on.Substituting x3 in the equation,
x4 = (18 * 2368 + 53) mod 4913x4 = 42657 mod 4913x4 = 1504...
and so on.This is a process of backward iteration of LCG. Since it is a backward iteration, x0 is the last generated random number before x1. So x0 is the random number generated after x4. Hence, x0 = 4600. We have been provided with a linear congruential generator (LCG), which is defined by the equation:xn+1 = (a xn + c) mod m...where xn is the nth random number, xn+1 is the (n+1)th random number, and a, c, and m are constants.Let's substitute the given values in the above equation,
xn+1 = (18 xn + 53) mod 4913; x1 = 4600
We can use backward iteration to solve for x0. In backward iteration, we start with the given value of xn and move backward in the sequence until we find the value of x0.Let's use the backward iteration to find the value of x0. Thus,
x2 = (18 * 4600 + 53) mod 4913x2 = 82853 mod 4913x2 = 1427...
and so on.Substituting x2 in the equation,
x3 = (18 * 1427 + 53) mod 4913x3 = 25751 mod 4913x3 = 2368...
and so on.Substituting x3 in the equation,
x4 = (18 * 2368 + 53) mod 4913x4 = 42657 mod 4913x4 = 1504...
and so on.The last generated random number before x1 is x0. Hence, x0 = 4600.Therefore, the value of x0 is 4600. This is the solution.
Thus, we can conclude that the value of x0 is 4600. We have solved this by backward iteration of LCG. This method involves moving backward in the sequence of random numbers until we find the value of x0.
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The weekly demand for Math Wars - Attack of the Limits video games is given by p=420/(x−6)+4000 where x is the number thousands of video games produced and sold, and p is in dollars. Using the Marginal Revenue function, R ′(x), approximate the marginal revenue when 12,000 video games have been produced and sold.
_____dollars
The marginal revenue when 12,000 video games have been produced and sold is 105 dollars.
Given function, p=420/(x-6)+4000
To find the marginal revenue function, R′(x)
As we know, Revenue, R = price x quantity
R = p * x (price, p and quantity, x are given in the function)
R = (420/(x-6) + 4000) x
Revenue function, R(x) = (420/(x-6) + 4000) x
Differentiating R(x) w.r.t x,
R′(x) = d(R(x))/dx
R′(x) = [d/dx] [(420/(x-6) + 4000) x]
On expanding and simplifying,
R′(x) = 420/(x-6)²
Now, to approximate the marginal revenue when 12,000 video games have been produced and sold, we need to put the value of x = 12
R′(12) = 420/(12-6)²
R′(12) = 105 dollars
Therefore, the marginal revenue when 12,000 video games have been produced and sold is 105 dollars.
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Assume the average selling price for houses in a certain county is $339,000 with a standard deviation of $60,000. a) Determine the coefficient of variation. b) Caculate the z-score for a house that sells for $329,000. c) Using the Empirical Rule, determine the range of prices that includes 68% of the homes around the mean. d) Using Chebychev's Theorem, determine the range of prices that includes at least 96% of the homes around the mear
a) The coefficient of variation is the ratio of the standard deviation to the mean. The formula for the coefficient of variation (CV) is given by:CV = (Standard deviation/Mean) × 100.
We are given the mean selling price of houses in a certain county, which is $339,000, and the standard deviation of the selling prices, which is $60,000.Substituting these values into the formula, we get:CV = (60,000/339,000) × 100= 17.69%Therefore, the coefficient of variation for the selling prices of houses in the county is 17.69%.
b) The z-score is a measure of how many standard deviations away from the mean a particular data point lies.
The formula for the z-score is given by:z = (x – μ) / σWe are given the selling price of a house, which is $329,000. The mean selling price of houses in the county is $339,000, and the standard deviation is $60,000.Substituting these values into the formula, we get:z = (329,000 – 339,000) / 60,000= -0.1667Therefore, the z-score for a house that sells for $329,000 is -0.1667.
c) The empirical rule states that for data that follows a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Therefore, the range of prices that includes 68% of the homes around the mean can be calculated as follows:Lower limit = Mean – Standard deviation= 339,000 – 60,000= 279,000Upper limit = Mean + Standard deviation= 339,000 + 60,000= 399,000Therefore, the range of prices that includes 68% of the homes around the mean is $279,000 to $399,000.
d) Chebychev's Theorem states that for any dataset, regardless of the distribution, at least (1 – 1/k²) of the data falls within k standard deviations of the mean. Therefore, to determine the range of prices that includes at least 96% of the homes around the mean, we need to find k such that (1 – 1/k²) = 0.96Solving for k, we get:k = 5Therefore, at least 96% of the data falls within 5 standard deviations of the mean. The range of prices that includes at least 96% of the homes around the mean can be calculated as follows:
Lower limit = Mean – (5 × Standard deviation)= 339,000 – (5 × 60,000)= 39,000Upper limit = Mean + (5 × Standard deviation)= 339,000 + (5 × 60,000)= 639,000Therefore, the range of prices that includes at least 96% of the homes around the mean is $39,000 to $639,000.
In statistics, the coefficient of variation (CV) is the ratio of the standard deviation to the mean. It is expressed as a percentage, and it is a measure of the relative variability of a dataset. In this question, we were given the mean selling price of houses in a certain county, which was $339,000, and the standard deviation of the selling prices, which was $60,000. Using the formula for the coefficient of variation, we calculated that the CV was 17.69%. This means that the standard deviation is about 17.69% of the mean selling price of houses in the county. A high CV indicates that the data has a high degree of variability, while a low CV indicates that the data has a low degree of variability.The z-score is a measure of how many standard deviations away from the mean a particular data point lies. In this question, we were asked to calculate the z-score for a house that sold for $329,000.
Using the formula for the z-score, we calculated that the z-score was -0.1667. This means that the selling price of the house was 0.1667 standard deviations below the mean selling price of houses in the county. A negative z-score indicates that the data point is below the mean. A positive z-score indicates that the data point is above the mean.The Empirical Rule is a statistical rule that states that for data that follows a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% of the data falls within two standard deviations of the mean, and approximately 99.7% of the data falls within three standard deviations of the mean.
In this question, we were asked to use the Empirical Rule to determine the range of prices that includes 68% of the homes around the mean. Using the formula for the range of prices, we calculated that the range was $279,000 to $399,000.
Chebychev's Theorem is a statistical theorem that can be used to determine the minimum percentage of data that falls within k standard deviations of the mean. In this question, we were asked to use Chebychev's Theorem to determine the range of prices that includes at least 96% of the homes around the mean.
Using the formula for Chebychev's Theorem, we calculated that the range was $39,000 to $639,000. Therefore, we can conclude that the range of selling prices of houses in the county is quite wide, with some houses selling for as low as $39,000 and others selling for as high as $639,000.
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Assume the random variable x is normally distributed with mean μ=90 and standard deviation σ=5. Find the indicated probability. P(x<85) P(x<85)= (Round to four decimal places as needed. )
The answer is P(x < 85) = 0.1587
Given that the random variable x is normally distributed with mean μ=90 and standard deviation σ=5. We need to find the probability P(x < 85).
Normal Distribution
The normal distribution refers to a continuous probability distribution that has a bell-shaped probability density curve. It is the most important probability distribution, particularly in the field of statistics, because it describes many natural phenomena.
P(x < 85)Using z-score:
When a dataset follows a normal distribution, we can transform the data using z-scores so that it follows a standard normal distribution, which has a mean of 0 and a standard deviation of 1, as shown below:z = (x - μ) / σ = (85 - 90) / 5 = -1P(x < 85) = P(z < -1)
We can find the area under the standard normal curve to the left of -1 using a z-table or a calculator.
Using a calculator, we can use the normalcdf function on the TI-84 calculator to find P(z < -1). The function takes in the lower bound, upper bound, mean, and standard deviation, and returns the probability of the z-score being between those bounds, as shown below:
normalcdf(-10, -1, 0, 1) = 0.1587
Therefore, P(x < 85) = P(z < -1) ≈ 0.1587 (to four decimal places).Hence, the answer is P(x < 85) = 0.1587 (rounded to four decimal places).
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The average number of misprints per page in a magazine is whixch follows a Poisson's Probability distribution. What is the probability that the number of misprints on a particular page of that magazine is 2?
The probability that a particular book is free from misprints is 0.2231. option D is correct.
The average number of misprints per page (λ) is given as 1.5.
The probability of having no misprints (k = 0) can be calculated using the Poisson probability mass function:
[tex]P(X = 0) = (e^{-\lambda}\times \lambda^k) / k![/tex]
Substituting the values:
P(X = 0) = [tex](e^{-1.5} \times 1.5^0) / 0![/tex]
Since 0! (zero factorial) is equal to 1, we have:
P(X = 0) = [tex]e^{-1.5}[/tex]
Calculating this value, we find:
P(X = 0) = 0.2231
Therefore, the probability that a particular book is free from misprints is approximately 0.2231.
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Question 13: The average number of misprints per page of a book is 1.5.Assuming the distribution of number of misprints to be Poisson. The probability that a particular book is free from misprints,is B. 0.435 D. 0.2231 A. 0.329 C. 0.549
A circle with radius 7 in. has circumference 43.96 in. Find the circumference of the circle if the radius changes to 13 in.
The circumference of the circle if the radius changes to 13 in. is 26π or approximately 81.64
Given that a circle with radius 7 in. has circumference 43.96 in. We need to find the circumference of the circle if the radius changes to 13 in.
The formula for the circumference of a circle is given by:
C = 2πr where C is the circumference, r is the radius and π is a constant equal to 3.14.
Applying the above formula we have:
Circumference of the circle with radius 7 in = 2π × 7= 14π
So, the circumference of the circle with radius 7 in. is 14π or approximately 43.96 in.
Given the radius of the circle changes to 13 in.
Now, the new circumference of the circle is:
Circumference of the circle with radius 13 in. = 2π × 13= 26π
Therefore, the circumference of the circle if the radius changes to 13 in. is 26π or approximately 81.64 in.
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Hudson and Knox are in a race. Hudson is running at a speed of 8. 8 feet per second. Knox got a 30-foot head start and is running at a speed of 6. 3 feet per second. How many seconds will it take until Hudson and Knox have run the same number of feet? Write the equation
It will take 12 seconds for Hudson and Knox to have run the same number of feet.
Let's first write the equation to represent the situation described in the problem.
Let's assume it takes t seconds for Hudson and Knox to run the same number of feet. In that time, Hudson will have run a distance of 8.8t feet, and Knox will have run a distance of 30 + 6.3t feet. Since they are running the same distance, we can set these two expressions equal to each other:
8.8t = 30 + 6.3t
Now we can solve for t:
8.8t - 6.3t = 30
2.5t = 30
t = 12
Therefore, it will take 12 seconds for Hudson and Knox to have run the same number of feet.
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in order to test a new drug for adverse reactions, the drug was administered to 1,000 test subjects with the following results: 60 subjects reported that their only adverse reaction was a loss of appetite, 90 subjects reported that their only adverse reaction was a loss of sleep, and 800 subjects reported no adverse reactions at all. if this drug is released for general use, what is the (empirical) probability that a person using the drug will suffer both a loss of appetite and a loss of sleep?
The empirical probability that a person will suffer both a loss of appetite and a loss of sleep is 5%.
What is the Empirical ProbabilityFirst step is to find the Number of subjects who reported both adverse reactions
Number of subjects who reported both adverse reactions = 1,000 - (60 + 90 + 800)
Number of subjects who reported both adverse reactions = 50
Now let find the Empirical Probability
Empirical Probability = Number of subjects who reported both adverse reactions / Total number of test subjects
Empirical Probability = 50 / 1,000
Empirical Probability = 0.05 or 5%
Therefore the empirical probability is 5%.
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a reporter bought hamburgers at randomly selected stores of two different restaurant chains, and had the number of calories in each hamburger measured. can the reporter conclude, at
Where the above conditions are given then the correct answer is -Yes, because the test value –3.90 is outside the noncritical region (Option C)
How is this so?To determine if the hamburgers from the two chains have a different number of calories, we can conduct an independent t-test.
Given -
Chain A -
- Sample size (n1) = 5
- Sample mean (x1) = 230 Cal
- Sample standard deviation (s1) = 23 Cal
Chain B -
- Sample size (n2) = 9
- Sample mean (x2) = 285 Cal
- Sample standard deviation (s2) = 29 Cal
The null hypothesis (H0) is that the two chains have the same number of calories, and the alternative hypothesis (Ha) is that they have a different number of calories.
Using an independent t-test, we calculate the test statistic -
t = (x1 - x2) / √((s1² / n1) + (s2² / n2))
Plugging in the values -
t = (230 - 285) / √((23² / 5) + (29² / 9))
t ≈ -3.90
To determine the critical region, we need to compare the test statistic to the critical value at a significance level of α = 0.05 with degrees of freedom df = smaller of (n1 - 1) or (n2 - 1).
The degrees of freedom in this case would be df = min(4, 8) = 4.
Looking up the critical value for a two-tailed t-test with df = 4 at α = 0.05, we find that it is approximately ±2.776.
Since the test statistic (-3.90) is outside the critical region (±2.776), we reject the null hypothesis.
Therefore, the reporter can conclude, at α = 0.05, that the hamburgers from the two chains have a different number of calories.
This means that the correct answer is -" Yes, because the test value –3.90 is outside the noncritical region" (Option C)
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Full Question:
Although part of your question is missing, you might be referring to this full question:
A reporter bought hamburgers at randomly selected stores of two different restaurant chains, and had the number of Calories in each hamburger measured. Can the reporter conclude, at α = 0.05, that the hamburgers from the two chains have a different number of Calories? Use an independent t-test. df = smaller of n1 - 1 or n2 - 1.
Chain A Chain B
Sample Size 5 9
Sample Mean 230 Cal 285 Cal
Sample SD 23 Cal 29 Cal
A) No, because the test value –0.28 is inside the noncritical region.
B) Yes, because the test value –0.28 is inside the noncritical region
C) Yes, because the test value –3.90 is outside the noncritical region
D) No, because the test value –1.26 is inside the noncritical region
Show that another approximation for log n! for large n is log n!=nlog(n)-n by expanding the log into a sum over the log of each term in the n! product and then approximating the resulting sum by an integral. What is the percentage error between log n! and your result when n=10?
The percentage error between log n! and the approximation when n = 10 is approximately 100%. This means that the approximation n log(n) - n is not very accurate for calculating log n! when n = 10.
The given approximation for log n! can be derived by expanding the logarithm of each term in the n! product and then approximating the resulting sum by an integral.
When we take the logarithm of each term in n!, we have log(n!) = log(1) + log(2) + log(3) + ... + log(n).
Using the properties of logarithms, this can be simplified to log(n!) = log(1 * 2 * 3 * ... * n) = log(1) + log(2) + log(3) + ... + log(n).
Next, we approximate this sum by an integral. We can rewrite the sum as an integral by considering that log(x) is approximately equal to the area under the curve y = log(x) between x and x+1. So, we approximate log(n!) by integrating the function log(x) from 1 to n.
∫(1 to n) log(x) dx ≈ ∫(1 to n) log(n) dx = n log(n) - n.
Therefore, the approximation for log n! is given by log(n!) ≈ n log(n) - n.
To calculate the percentage error between log n! and the approximation n log(n) - n when n = 10, we need to compare the values of these expressions and determine the difference.
Exact value of log(10!):
Using a calculator or logarithmic tables, we can find that log(10!) is approximately equal to 15.1044.
Approximation n log(n) - n:
Substituting n = 10 into the approximation, we have:
10 log(10) - 10 = 10(1) - 10 = 0.
Difference:
The difference between the exact value and the approximation is given by:
15.1044 - 0 = 15.1044.
Percentage Error:
To calculate the percentage error, we divide the difference by the exact value and multiply by 100:
(15.1044 / 15.1044) * 100 ≈ 100%.
Therefore, the percentage error between log n! and the approximation when n = 10 is approximately 100%. This means that the approximation n log(n) - n is not very accurate for calculating log n! when n = 10.
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Sugar consumption is a hot topic when it comes to good nutrition. Twelve-ounce case of soft drinks often contain 10 teaspoons of sugar in them. A random sample of 75 college students were asked how many cans of soda drinks they typically consume on a given day. That number was multiplied by 10 to give a daily amount of sugar from drinking soft drinks. The following statistics were calculated:
Min=8 max=62 Q1=25 Q3=38 n=75 mean=31.4 median=28 s=11.6
Dmitry says that there aren’t any outliers since
28-3(11.6)= -6.8 and 28-3(11.6) = 62.8
and the max and min fall within this range. Is Dmitry correct? Why or why not?
Dmitry is incorrect in his statement as his range is not comprehensive and adequate to determine if there is an outlier or not in the given data set.
The range he calculated is -6.8 to 62.8, but this range is not appropriate for the provided set of data as it is too wide. It is crucial to keep in mind that the formula for the range is Range = maximum – minimum, which is the absolute difference between the maximum and minimum values in a dataset. The range is not a good measure of variability because it is sensitive to outliers. Thus, it is not an adequate criterion for detecting outliers. It only focuses on the two extremes of the distribution rather than the entire dataset, so it is inadequate to determine if there is an outlier or not.
Dmitry is incorrect because the range he calculated is not appropriate for the given data set. Dmitry's argument is based on the incorrect assumption that a range of 3 standard deviations is sufficient to detect outliers. The rule that a range of 3 standard deviations is sufficient to detect outliers is based on the assumption that the data are normally distributed, but this is not the case for this particular data set.
The correct method to detect outliers, in this case, is to use the interquartile range (IQR), which is defined as the difference between the third quartile (Q3) and the first quartile (Q1). Outliers can be detected using the following formula: Outliers = Values < (Q1 - 1.5*IQR) or Values > (Q3 + 1.5*IQR)Therefore, in the case of the given data set, we can find the outliers by using the interquartile range (IQR), which is defined as follows:
IQR = Q3 – Q1= 38 – 25= 13Hence, the lower bound and upper bound of the data set will be Q1 – 1.5 × IQR and Q3 + 1.5 × IQR, respectively.
Lower bound = 25 – 1.5 × 13 = 5.5Upper bound = 38 + 1.5 × 13 = 57.5According to the above calculations, we can conclude that there are no outliers in the given data set since all the values lie within the range of 5.5 to 57.5.
Thus, Dmitry is incorrect in his statement. The range he calculated is not appropriate for the given data set. The correct method to detect outliers, in this case, is to use the interquartile range (IQR), which is defined as the difference between the third quartile (Q3) and the first quartile (Q1). All the values in the given data set lie within the range of 5.5 to 57.5, so there are no outliers in the data set.
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Find all the values of the following. (1) (−16) ^1/4Place all answers in the following blank, separated by commas: (2) 1 ^1/5 Place all answers in the followina blank. sebarated bv commas: (3) i ^1/4 Place all answers in the followina blank. sebarated bv commas:
The required roots of the given expressions are:
(1) (1/√2 + i/√2), (-1/√2 + i/√2), (-1/√2 - i/√2), (1/√2 - i/√2).
(2)1
(3) [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].
Formula used:For finding roots of a complex number `a+bi`,where `a` and `b` are real numbers and `i` is an imaginary unit with property `i^2=-1`.
If `r(cosθ + isinθ)` is the polar form of the complex number `a+bi`, then its roots are given by:r^(1/n) [cos(θ+2kπ)/n + isin(θ+2kπ)/n],where `n` is a positive integer and `k = 0,1,2,...,n-1.
Calculations:
(1) (-16)^(1/4)
This expression (-16)^(1/4) can be written as [16 × (-1)]^(1/4).
Therefore (-16)^(1/4) = [16 × (-1)]^(1/4) = 2^(1/4) × [(−1)^(1/4)] = 2^(1/4) × [cos((π + 2kπ)/4) + isin((π + 2kπ)/4)],where k = 0,1,2,3.
Therefore (-16)^(1/4) = 2^(1/4) × [(1/√2) + i(1/√2)], 2^(1/4) × [(−1/√2) + i(1/√2)],2^(1/4) × [(−1/√2) − i(1/√2)], 2^(1/4) × [(1/√2) − i(1/√2)].
Hence, the roots of (-16)^(1/4) are (1/√2 + i/√2), (-1/√2 + i/√2), (-1/√2 - i/√2), (1/√2 - i/√2).
(2) 1^(1/5)
This expression 1^(1/5) can be written as 1^[1/(2×5)] = 1^(1/10).
Now, 1^(1/10) = 1 because any number raised to power 0 equals 1.
Hence, the only root of 1^(1/5) is 1.
(3) i^(1/4).
Now, i^(1/4) can be written as (cos(π/2) + isin(π/2))^(1/4).Now, the modulus of i is 1 and its argument is π/2.
Therefore, its polar form is: 1(cosπ/2 + isinπ/2).
Therefore i^(1/4) = 1^(1/4)[cos(π/2 + 2kπ)/4 + isin(π/2 + 2kπ)/4], where k = 0, 1,2,3.
Therefore i^(1/4) = [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].
Therefore, the roots of i^(1/4) are [cos(π/8) + isin(π/8)], [cos(5π/8) + isin(5π/8)], [cos(9π/8) + isin(9π/8)], [cos(13π/8) + isin(13π/8)].
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Find all polynomial solutions p(t, x) of the wave equation utt=uzz with (a) deg p ≤ 2, (b) deg p = 3.
The polynomial solution for deg p = 3 is p(t, x) = At³ + Bx³ + Ct² + Dx² - 3At² - 2Ct - 3Bx² - 2Dx, where A, B, C, and D are constants.
(a) Case: deg p ≤ 2
Let's assume p(t, x) = At² + Bx² + Ct + Dx + E, where A, B, C, D, and E are constants.
Substituting p(t, x) into the wave equation, we have:
(p_tt) = 2A,
(p_zz) = 2B,
(p_t) = 2At + C,
(p_z) = 2Bx + D.
Therefore, the wave equation becomes:
2A = 2B.
This implies that A = B.
Next, we consider the terms involving t and x:
2At + C = 0,
2Bx + D = 0.
From the first equation, we get C = -2At. Substituting this into the second equation, we have D = -4Bx.
Finally, we have the constant term:
E = 0.
So, the polynomial solution for deg p ≤ 2 is p(t, x) = At² + Bx² - 2At - 4Bx, where A and B are constants.
(b) Case: deg p = 3
Let's assume p(t, x) = At³ + Bx³ + Ct² + Dx² + Et + Fx + G, where A, B, C, D, E, F, and G are constants.
Substituting p(t, x) into the wave equation, we have:
(p_tt) = 6At,
(p_zz) = 6Bx,
(p_t) = 3At² + 2Ct + E,
(p_z) = 3Bx² + 2Dx + F.
Therefore, the wave equation becomes:
6At = 6Bx.
This implies that A = Bx.
Next, we consider the terms involving t and x:
3At² + 2Ct + E = 0,
3Bx² + 2Dx + F = 0.
From the first equation, we get E = -3At² - 2Ct. Substituting this into the second equation, we have F = -3Bx² - 2Dx.
Finally, we have the constant term:
G = 0.
So, the polynomial solution for deg p = 3 is p(t, x) = At³ + Bx³ + Ct² + Dx² - 3At² - 2Ct - 3Bx² - 2Dx, where A, B, C, and D are constants.
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The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $356 to drive 380 mi and in June it cost her $404 to drive 620 mi. The function is C(d)=0.2+280 (b) Use part (a) to predict the cost of driving 1800 miles per month. (c) Draw a graph (d) What does the slope represent? What does the C-intercept represent? Why does a linear function give a suitable model in this situation?
(b) $640 (c) y-int of 280, positive slope (d) It represents the cost (in dollars) per mile. It represents the fixed cost (amount she pays even if she does not drive). A linear function is suitable because the monthly cost increases as the number of miles driven increases.
To predict the cost of driving 1800 miles per month, substitute 1800 in the given function C(d) = 0.2d + 280C(1800) = 0.2 (1800) + 280= $640 per month. Therefore, the cost of driving 1800 miles per month is $640.
(b) Graph is shown below:(c)The slope of the graph represents the rate of change of the cost of driving a car per mile. The slope is given by 0.2, which means that for every mile Lynn drives, the cost increases by $0.2.The y-intercept of the graph represents the fixed cost (amount she pays even if she does not drive).
The y-intercept is given by 280, which means that even if Lynn does not drive the car, she has to pay $280 per month.The linear function gives a suitable model in this situation because the monthly cost increases as the number of miles driven increases.
This is shown by the positive slope of the graph. The fixed cost is also included in the function, which is represented by the y-intercept. Therefore, a linear function is a suitable model in this situation.
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An empty shipping box weighs 250 grams. The box is then filled with T-shirts. Each T-shirt weighs 132. 5 grams. The equation W = 250 + 132. 5T represents the relationship between the quantities in this situation, where W is the weight, in grams, of the filled box and T the number of shirts in the box. Select two possible solutions to the equation W = 250 + 132. 5T.
Two possible solutions to the equation W = 250 + 132.5T are:
T = 2, W = 515
T = 5, W = 912.5
To find possible solutions to the equation W = 250 + 132.5T, we need to substitute values for T and calculate the corresponding value of W.
Let's consider two possible values for T:
Solution 1: T = 2 (indicating 2 T-shirts in the box)
W = 250 + 132.5 * 2
W = 250 + 265
W = 515
So, one possible solution is T = 2 and W = 515.
Solution 2: T = 5 (indicating 5 T-shirts in the box)
W = 250 + 132.5 * 5
W = 250 + 662.5
W = 912.5
Therefore, another possible solution is T = 5 and W = 912.5.
Hence, two possible solutions to the equation W = 250 + 132.5T are:
T = 2, W = 515
T = 5, W = 912.5
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Find the distance from the point (5,0,0) to the line
x=5+t, y=2t , z=12√5 +2t
The distance from the point (5,0,0) to the line x=5+t, y=2t, z=12√5 +2t is √55.
To find the distance between a point and a line in three-dimensional space, we can use the formula for the distance between a point and a line.
Given the point P(5,0,0) and the line L defined by the parametric equations x=5+t, y=2t, z=12√5 +2t.
We can calculate the distance by finding the perpendicular distance from the point P to the line L.
The vector representing the direction of the line L is d = <1, 2, 2>.
Let Q be the point on the line L closest to the point P. The vector from P to Q is given by PQ = <5+t-5, 2t-0, 12√5 +2t-0> = <t, 2t, 12√5 +2t>.
To find the distance between P and the line L, we need to find the length of the projection of PQ onto the direction vector d.
The projection of PQ onto d is given by (PQ · d) / |d|.
(PQ · d) = <t, 2t, 12√5 +2t> · <1, 2, 2> = t + 4t + 4(12√5 + 2t) = 25t + 48√5
|d| = |<1, 2, 2>| = √(1^2 + 2^2 + 2^2) = √9 = 3
Thus, the distance between P and the line L is |(PQ · d) / |d|| = |(25t + 48√5) / 3|
To find the minimum distance, we minimize the expression |(25t + 48√5) / 3|. This occurs when the numerator is minimized, which happens when t = -48√5 / 25.
Substituting this value of t back into the expression, we get |(25(-48√5 / 25) + 48√5) / 3| = |(-48√5 + 48√5) / 3| = |0 / 3| = 0.
Therefore, the minimum distance between the point (5,0,0) and the line x=5+t, y=2t, z=12√5 +2t is 0. This means that the point (5,0,0) lies on the line L.
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15, 6, 14, 7, 14, 5, 15, 14, 14, 12, 11, 10, 8, 13, 13, 14, 4, 13, 3, 11, 14, 14, 12
compute the standard deviation for both sample and population
The sample standard deviation of the given data is approximately 4.0 while the population standard deviation is approximately 3.94.
The formula for computing standard deviation is as follows:
[tex]\[\large\sigma = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\mu)^2}{n-1}}\][/tex]
where:x is the individual value.μ is the mean (average).n is the number of values.[tex]\(\sigma\)[/tex] is the standard deviation.
A standard deviation is the difference between the average and the square root of the variance of a set of data. Standard deviation measures the amount of variability or dispersion for a subject set of data. We will compute both the sample standard deviation and the population standard deviation.
To calculate the sample standard deviation, we can use the same formula as we did in the population standard deviation, but we must divide by n - 1 instead of n. Thus:
[tex]\[\large s = \sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}\][/tex]
where:[tex]\(\sigma\)[/tex] is the standard deviation.x is the individual value.μ is the mean (average).n is the number of values. [tex]\(\sigma\)[/tex] is the standard deviation.
For the given data 15, 6, 14, 7, 14, 5, 15, 14, 14, 12, 11, 10, 8, 13, 13, 14, 4, 13, 3, 11, 14, 14, 12
we first calculate the mean.
µ = (15+6+14+7+14+5+15+14+14+12+11+10+8+13+13+14+4+13+3+11+14+14+12) / 23=10.6
After that, we compute the standard deviation (sample).
s = √ [ (15-10.6)² + (6-10.6)² + (14-10.6)² + (7-10.6)² + (14-10.6)² + (5-10.6)² + (15-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² + (11-10.6)² + (10-10.6)² + (8-10.6)² + (13-10.6)² + (13-10.6)² + (14-10.6)² + (4-10.6)² + (13-10.6)² + (3-10.6)² + (11-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² ] / 22
s = 4.0
The sample standard deviation is approximately 4.0.
For the population standard deviation, we should replace n-1 by n in the above formula. Thus:
σ = √ [ (15-10.6)² + (6-10.6)² + (14-10.6)² + (7-10.6)² + (14-10.6)² + (5-10.6)² + (15-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² + (11-10.6)² + (10-10.6)² + (8-10.6)² + (13-10.6)² + (13-10.6)² + (14-10.6)² + (4-10.6)² + (13-10.6)² + (3-10.6)² + (11-10.6)² + (14-10.6)² + (14-10.6)² + (12-10.6)² ] / 23
σ = 3.94 (approximately)
Therefore, the population standard deviation is approximately 3.94.
The sample standard deviation of the given data is approximately 4.0 while the population standard deviation is approximately 3.94.
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the point is located six units to the right of the y-axis and 8 units above the x-axis (x,y)
The point is located at (6,8). In the coordinate plane, the point is defined by an ordered pair of numbers, one for the x-coordinate and one for the y-coordinate. The first number represents the x-coordinate, and it specifies the horizontal position of the point, while the second number represents the y-coordinate and it specifies the vertical position of the point.
In this particular case, the point is located six units to the right of the y-axis and 8 units above the x-axis. This means that the x-coordinate is 6, and the y-coordinate is 8. In other words, the point is 6 units to the right of the y-axis, which means that it is on the positive x-axis, and it is 8 units above the x-axis, which means that it is in the positive y-direction.
Therefore, the point is at (6,8) which means that it is six units to the right of the y-axis and 8 units above the x-axis. This point is in the first quadrant of the coordinate plane, which is where both the x- and y-coordinates are positive.The coordinate plane is an essential tool in algebra that helps graphically represent functions and equations. It is divided into four quadrants by two perpendicular lines, the x-axis, and the y-axis. These axes intersect at the origin, which has the coordinates (0,0).
The location of a point in the coordinate plane is determined by its ordered pair of x- and y-coordinates. By plotting these points on the coordinate plane, we can graph lines, functions, and other mathematical concepts. The coordinate plane is also helpful in finding solutions to equations by identifying the points that satisfy the equation or inequality.
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