Answer:
Step-by-step explanation:
Let
[tex]f(x,y,z) = z-x^2-y^4-e^{xy}[/tex].
The partial derivatives of this function are
[tex]\frac{\partial f}{\partial x} (x,y,z) = ye^{xy} - 2x \\ \frac{\partial f}{\partial y} (x,y,z) = xe^{xy} - 4y^3 \\\frac{\partial f}{\partial z} (x,y,z) = 1[/tex]
The tangent plane equation through a point [tex]A(x_1,y_1,z_1)[/tex] is given by [tex]f'_x (x_1,y_1,z_1)(x-x_1) + f'_y(x_1,y_1,z_1)(y-y_1) + f'_z(x_1,y_1,z_1)(z-z_1) = 0[/tex]
In this case, we have
[tex]x_1 = 1, y_1 = 0, z_1 = 2.[/tex]
The values of the partial derivatives in this point are
[tex]\frac{\partial f}{\partial x} (1,0,2) = 0 \cdot e^{0} - 2\cdot 1 = -2 \\ \frac{\partial f}{\partial y} (1,0,2) = 1 \cdot e^{0} - 0 = 1 \\ \frac{\partial f}{\partial y} (1,0,2) = 1[/tex]
So, the equation is
[tex]-2(x-1) + 1 \cdot (y-0) + 1\cdot (z-2) = 0[/tex]
Therefore, the equation for the plane tangent to the surface at the point [tex](1,0,2)[/tex] is given by
[tex]2x-y-z = 0[/tex]
evaluate the expression 6÷3+17=
Answer:
=19
Step-by-step explanation:
=6÷3+17
=2+17
=19
What is the probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5
Answer:
Step-by-step explanation:
The probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19,
What is probability?It is defined as the ratio of the number of favorable outcomes to the total number of outcomes, in other words, the probability is the number that shows the happening of the event.
It is given that:
The randomly chosen number between 1 and 100 is divisible by 3
Applying conditional probability:
Let A is the event: the numbers divisible by 3
Let B is the event: At least one digit equal to 5
P(A|B) = n(A∩B)/n(B)
P(A|B) = 6/19
Thus, the probability that a randomly chosen number between 1 and 100 is divisible by 3, given that the number has at least one digit equal to 5 is 6/19.
Learn more about the probability here:
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1. Explain or show how you could find 5/ 1/3
by using the value of 5x3
Find 12/ 3/5
Answer:
20
Step-by-step explanation:
You could find 5/⅓
by using 5 × 3
Knowing that:
i. Any number multiplied by 1, gives the number itself.
ii. Dividing any number by itself gives 1.
You would agree with me that
i. (5×3)/(5×3) = 1
ii. Writing 5/⅓ as 5/⅓ × 1 doesn't change the value.
Then I can write 5/⅓ as
5/⅓ × (5×3)/(5×3) = 1
This can become
[5×(5×3)] / [(⅓) × (5×3)]
= 75/(15/3)
= 75/5
= 15
In a similar way,
12/ 3/5
= [12/ (3/5)] × [(5×3)/(5×3)]
= 12×(5×3) / (3/5)×(5×3)
= (12×5×3) / [(3×5×3)/5]
= 180 / (45/5)
= 180 / 9
= 20
Prove that among 502 positive integers, there are always two integers so that either their sum or their difference is divisible by 1000.
Final answer:
Using the pigeonhole principle, we can prove that among 502 positive integers, at least two will have the same remainder when divided by 1000, implying their difference is divisible by 1000.
Explanation:
The question asks to prove that among 502 positive integers, there are always two integers so that either their sum or their difference is divisible by 1000. This statement can be understood through the pigeonhole principle, which in basic terms means if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon.
In this case, consider the remainders when these integers are divided by 1000. Since there are only 1000 possible remainders (from 0 to 999), and we have 502 numbers, at least two of them must have the same remainder when divided by 1000, according to the pigeonhole principle.
Let these two numbers be a and b, where without loss of generality, a ≥ b. If a and b have the same remainder when divided by 1000, then a - b is divisible by 1000. Alternatively, if we had a case where the sum is considered, assuming complementary pairs mod 1000, a similar argument involving the pigeonhole principle can conclude that there must be at least one pair whose sum or difference gives a number divisible by 1000, satisfying the initial claim.
Let x and y be any numbers at all with x ≤ y. Show that the number of integers between x and y is [y] - [x] +1.That is show that the number of integers between x and y is = (the floor of y) - (the ceiling of x) +1
Answer:
the explanation is given below.
Step-by-step explanation:
Here what is applied is assumption of range of values of number from say 1 - 100In total, i stopped at 100 on the dot.from this, the lowest number is 1 and the highest number is 100
hence the range of the numbers = Difference between Highest and Lowestrange = 100 - 1 = 99, the 99 gotten as the range is indicative that a number has been missing.In order to make up the 100, an integer is added to the difference = 99, i.e 99 is added to 1 to make up the 100.Furthermore, if 0 is exclusively out when numbers are counted up 100 with 0 inclusive, in such case, the first and last number are excluded from the counting. as such the integers will be {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20.........., 99} since both 0 and 100 are not included.Here, if we try to get the range = highest - lowest = 99 - 1 = 98, it implies that to make up the 99, an integer is added to the result of the difference = 98+1 = 99As such, the number of integers between two numbers is the difference between the highest and the lowest number plus 1 i.e highest - lowest + 1 = y - x +1 = (the floor of y) - ( the ceiling of x) + 1Using the laws of logic to prove tautologies.Use the laws of propositional logic to prove that each statement is a tautology.a. ¬r ∨ (¬r → p)b. ¬(p → q) → ¬q
Answer: The proofs are given below.
Step-by-step explanation: We are given to prove that the following statements are tautologies using truth table :
(a) ¬r ∨ (¬r → p) b. ¬(p → q) → ¬q
We know that a statement is a TAUTOLOGY is its value is always TRUE.
(a) The truth table is as follows :
r p ¬r ¬r→p ¬r ∨ (¬r → p)
T T F T T
T F F T T
F T T T T
F F T F T
So, the statement (a) is a tautology.
(b) The truth table is as follows :
p q ¬q p→q ¬(p→q) ¬(p→q)→q
T T F T F T
T F T F T T
F T F T F T
F F T T F T
So, the statement (B) is a tautology.
Hence proved.
Final answer:
To prove that a statement is a tautology using propositional logic, we need to show that the statement is true under all possible truth values of its variables. By applying the laws of implication, disjunction, and contradiction, we can prove that the given statements are tautologies.
Explanation:
To prove that a statement is a tautology using the laws of propositional logic, we need to show that the statement is true under all possible truth value assignments of its variables. Let's consider each statement:
a. ¬r ∨ (¬r → p)
We can use the law of implication, which states that ¬p ∨ q is equivalent to p → q, to rewrite the statement as ¬r ∨ (r → p). By applying the law of disjunction, which states that p ∨ (q ∧ r) is equivalent to (p ∨ q) ∧ (p ∨ r), we can further rewrite the statement as (¬r ∧ r) ∨ (r ∨ p). Using the law of contradiction, which states that p ∧ ¬p is always false, we can simplify the statement to r ∨ p, which is a tautology.
b. ¬(p → q) → ¬q
We can use the law of implication to rewrite the statement as ¬(¬p ∨ q) → ¬q. By applying De Morgan's law, which states that ¬(p ∨ q) is equivalent to ¬p ∧ ¬q, we can simplify the statement to (p ∧ ¬q) → ¬q. Using the law of contradiction, we know that p ∧ ¬p is always false, so the statement simplifies to false → ¬q, which is always true. Therefore, it is a tautology.
A large university will begin a 13-day period during which students may register for that semester’s courses. Of those 13 days, the number of elapsed days before a randomly selected student registers has a continuous distribution with density function f (t) that is symmetric about t = 6.5 and proportional to 1/(t + 1) between days 0 and 6.5.A student registers at the 60th percentile of this distribution.Calculate the number of elapsed days in the registration period for this student.(A) 4.01
(B) 7.80
(C) 8.99
(D) 10.22
(E) 10.51
Answer:
8.99 days elapsed. Option (C) is correct
Step-by-step explanation:
The distribution has density function k/t+1 for a constant k and t between 0 and 6.5 . Since the distribution is symmetrical in 6.5, the area it forms between 0 and 6.5 should be 1/2, thus
[tex]\frac{1}{2} = \int\limits_0^{6.5} \frac{k}{t+1} \, dt = k *(ln(t+1) \, |_0^{6.5}) = k * (ln(7.5)-ln(1)) = k*ln(7.5)[/tex]
Hence k = 1/(2ln(7.5)), approx 1/4.
We need to find the percentil 0.6, since the integral of the random variable is 1/2 over the first half, we need to find t such that the integral of the random variable between o and 6.5 + t is 0.6. This is equivalent to find t such that the integral between 6.5 and 6.5+t is 0.1. Due to the over 6.5, this t should satisfy that the integral between 6.5-t and 6.5 is also 0.1. Lets compute the integral and find t
[tex]\int\limits^{6.5}_{6.5-t} {\frac{k}{t+1}} \, dx = \frac{1}{2ln(7.5)}*(ln(t+1) \, |_{6.5-t}^{6.5} \, ) = \frac{1}{2ln(7.5)} * (ln(7.5)-ln(7.5-t)) = \\\frac{1}{2} - \frac{ln(7.5-t)}{2ln(7.5)} = 0.1[/tex]
Therefore,
[tex]\frac{ln(7.5-t)}{2ln(7.5)} = 0.4\\\\ln(7.5-t) = 0.8*ln(7.5)\\\\7.5-t = e^{0.8*ln(7.5)}\\\\t = 7.5-e^{0.8*ln(7.5)} = 2.49[/tex]
As a result, the student sould have registered 2.49 days after the day 6.5, thus it should have registeredd at day 8.99. Option (C) is correct.
Solve the system of equations by row-reduction. At each step, show clearly the symbol of the linear combinations that allow you to clear the entries below each pivot or of the operations that allow you to swap two rows or to scale a row.
1) 3x₂ - 5x₃ = 89
6x₁ + x₃ = 17
x₁ - x₂ + 8x₃ = -107
2) 4x₁ - x₂ + 3x₃ = 12
2x₁ + 9x₃ = -5
x₁ + 4x₂ + 6x₃ = -32
Answer:
1) The solution of the system is
[tex]\left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]
2) The solution of the system is
[tex]\left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]
Step-by-step explanation:
1) To solve the system of equations
[tex]\left\begin{array}{ccccccc}&3x_2&-5x_3&=&89\\6x_1&&+x_3&=&17\\x_1&-x_2&+8x_3&=&-107\end{array}\right[/tex]
using the row reduction method you must:
Step 1: Write the augmented matrix of the system
[tex]\left[ \begin{array}{ccc|c} 0 & 3 & -5 & 89 \\\\ 6 & 0 & 1 & 17 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]
Step 2: Swap rows 1 and 2
[tex]\left[ \begin{array}{ccc|c} 6 & 0 & 1 & 17 \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]
Step 3: [tex]\left(R_1=\frac{R_1}{6}\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 1 & -1 & 8 & -107 \end{array} \right][/tex]
Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 3 & -5 & 89 \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]
Step 5: [tex]\left(R_2=\frac{R_2}{3}\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & -1 & \frac{47}{6} & - \frac{659}{6} \end{array} \right][/tex]
Step 6: [tex]\left(R_3=R_3+R_2\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & \frac{37}{6} & - \frac{481}{6} \end{array} \right][/tex]
Step 7: [tex]\left(R_3=\left(\frac{6}{37}\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{1}{6} & \frac{17}{6} \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]
Step 8: [tex]\left(R_1=R_1-\left(\frac{1}{6}\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & - \frac{5}{3} & \frac{89}{3} \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]
Step 9: [tex]\left(R_2=R_2+\left(\frac{5}{3}\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right][/tex]
Step 10: Rewrite the system using the row reduced matrix:
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 5 \\\\ 0 & 1 & 0 & 8 \\\\ 0 & 0 & 1 & -13 \end{array} \right] \rightarrow \left\begin{array}{ccc}x_1&=&5\\x_2&=&8\\x_3&=&-13\end{array}\right[/tex]
2) To solve the system of equations
[tex]\left\begin{array}{ccccccc}4x_1&-x_2&+3x_3&=&12\\2x_1&&+9x_3&=&-5\\x_1&+4x_2&+6x_3&=&-32\end{array}\right[/tex]
using the row reduction method you must:
Step 1:
[tex]\left[ \begin{array}{ccc|c} 4 & -1 & 3 & 12 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]
Step 2: [tex]\left(R_1=\frac{R_1}{4}\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 2 & 0 & 9 & -5 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]
Step 3: [tex]\left(R_2=R_2-\left(2\right)R_1\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 1 & 4 & 6 & -32 \end{array} \right][/tex]
Step 4: [tex]\left(R_3=R_3-R_1\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & \frac{1}{2} & \frac{15}{2} & -11 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]
Step 5: [tex]\left(R_2=\left(2\right)R_2\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & - \frac{1}{4} & \frac{3}{4} & 3 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]
Step 6: [tex]\left(R_1=R_1+\left(\frac{1}{4}\right)R_2\right)[/tex]
[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & \frac{17}{4} & \frac{21}{4} & -35 \end{array} \right][/tex]
Step 7: [tex]\left(R_3=R_3-\left(\frac{17}{4}\right)R_2\right)[/tex]
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & - \frac{117}{2} & \frac{117}{2} \end{array} \right][/tex]
Step 8: [tex]\left(R_3=\left(- \frac{2}{117}\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{cccc} 1 & 0 & \frac{9}{2} & - \frac{5}{2} \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]
Step 9: [tex]\left(R_1=R_1-\left(\frac{9}{2}\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 15 & -22 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]
Step 10: [tex]\left(R_2=R_2-\left(15\right)R_3\right)[/tex]
[tex]\left[ \begin{array}{cccc} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right][/tex]
Step 11:
[tex]\left[ \begin{array}{ccc|c} 1 & 0 & 0 & 2 \\\\ 0 & 1 & 0 & -7 \\\\ 0 & 0 & 1 & -1 \end{array} \right]\rightarrow \left\begin{array}{ccc}x_1&=&2\\x_2&=&-7\\x_3&=&-1\end{array}\right[/tex]
A(n) _________ is a person or object that is a member of the population being studied.
Answer: individual
Step-by-step explanation:
An individual is a person or object that is a member of the population being studied. A population is defined as a group of individuals with a common characteristic living and interbreeding within a given area, in statistics, population is a collection of individuals to be studied. Individuals can also be referred to as the objects/person described by a set of data. For example: when studying the height of students in a school, the students attending that school are individuals.
The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target. What are the null and alternative hypotheses (H0 = null hypothesis and Ha = alternative hypothesis)?
(A) H0: Mean= 5 and Ha: Mean is not equal to 5
(B) H0: Mean = 5 and Ha: Mean <5
(C) H0: Mean < 5 and Ha: Mean > 5
(D) H0: Mean = 5 and Ha: Mean > 5
Answer:
a) H0: mean =5 and Ha: mean≠ 5
Step-by-step explanation:
In hypothesis testing procedure the trait of null hypothesis is that it always contain an equality sign. We are known that diameter of spindle is known to be 5mm. This our null value. Hence the null hypothesis is
H0:μ=5.
Now for alternative hypothesis we are given that the mean diameter has moved away from the target. This means that mean diameter could be increases or decreases from 5mm. Hence the alternative hypothesis is
Ha:μ≠5
From the information given, it is found that the correct option is:
(A) H0: Mean= 5 and Ha: Mean is not equal to 5.
At the null hypothesis, it is tested if the motor works properly, that is, the spindle has diameter significantly close to 5 mm, hence:
[tex]H_0: \mu = 5[/tex]
At the alternative hypothesis, it is tested if the motor does not work properly, that is, the spindle has diameter different from 5 mm, either too high or too low, hence:
[tex]H_a: \mu \neq 5[/tex]
Thus, a is the correct option.
A similar problem is given at https://brainly.com/question/15704285
Business Weekly conducted a survey of graduates from 30 top MBA programs. On the basis of the survey, assume the mean annual salary for graduates 10 years after graduation is 133000 dollars. Assume the population standard deviation is 31000 dollars. Suppose you take a simple random sample of 87 graduates.
1. Find the probability that a single randomly selected salary is at least 134000 dollars.
The probability that a single randomly selected salary is at least $134,000 is approximately 0.5120 or 51.20%.
To find the probability that a single randomly selected salary is at least $134,000, we need to calculate the z-score and use the standard normal distribution.
1. Calculate the z-score:
z = (x - μ) / σ
where x is the value we want to find the probability for, μ is the mean, and σ is the standard deviation.
In this case, x = $134,000, μ = $133,000, and σ = $31,000.
z = (134000 - 133000) / 31000
z = 1 / 31000
z ≈ 0.0323
2. Find the probability associated with the z-score:
We can use a z-table or a calculator to find the probability.
From the z-table, we find that the probability corresponding to a z-score of 0.0323 is approximately 0.5120.
Therefore, the probability that a single randomly selected salary is at least $134,000 is 0.5120 or 51.20%.
Solve the following system of equations using Gaussian elimination method. If there are no solutions, type "N" for both xx, yy and zz. If there are infinitely many solutions, type "z" for zz, and expressions in terms of zz for xx and yy.-5x-7y-4z=-66x+2y+3z=-2-1x+2y-7z=0
Answer:
x=-224/229,
y=296/229,
z=118/229
Step-by-step explanation:
-5x-7y-4z=-6
6x+2y+3z=-2
-x+2y-7z=0...........( multiple with (-5) and sum with 1st equation, mult with 6 and sum with 2nd equation)
______________
-x+2y-7z=0
-17y+31z=-6.....(mult with 14)
14y-39z=-2.....(mult with -17) then sum
___________
-x+2y-7z=0
-229z=-118, so here we have z=118/229.
14y-39*(118/229)=-2, from here we have y=296/229
-x+2*(296/229)-7*(118/229)=0, we get that x=-234/229
In the same way you can do this in the matrix form>>
Let X and Y be independent binomial random variables having parameters(N,p) and (M,p), respectively. Let Z = X+Y;(a) Argue that Z has a binomial distribution with parameters(N+M,P) bywriting X and Y as appropriate sums of Bernoulli random variables.(b) Validate the result in (a) by evaluating the necessary convolution.
Answer:
See explanation below.
Step-by-step explanation:
Part a
For this case we can use the moment generating function for the bernoulli distribution for n trials, given by:
[tex] p(s)^n = (q+ps)^n =\sum_{r=0}^n (nCr) (ps)^r q^{n-r}[/tex]
Wehre p is the probability of success and [tex] P(s_ = q+ps[/tex]
Using this property we see that;
If we multiply the two generating functions we got:
[tex] p(s)^N = (q+ps)^N =\sum_{r=0}^N (NCr) (ps)^r q^{N-r}[/tex]
[tex] p(s)^M = (q+ps)^M =\sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]
[tex] p(s)^M p(s)^N = \sum_{r=0}^N (NCr) (ps)^r q^{N-r} \sum_{r=0}^M (MCr) (ps)^r q^{M-r}[/tex]
And the mass function would be given by:
[tex]= (N+M C r) p^{r} (1-p)^{N+M-r}[/tex]
So we see that follows a binomial random variable with parameters (N+M, p)
Part b
For this case we are assuming that [tex] X \sim Bin (N,p) , Y\sim Bin (M,p)[/tex] and for this case we can assume that [tex] 0 \leq k \leq N+M[/tex] for the proof.
We are interested on the random variable [tex] Z= X+Y[/tex] since the two random variables are independent we can write the probability mass function for Z like this:
[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i , Y=k-i)[/tex]
[tex] P(Z = X+Y = k) =\sum_{i=0}^k P(X=i) P(Y=k-i)[/tex]
And we can replace the mass function for X and Y
[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} \sum_{i=0}^k (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]
And we can rewrite this like that:
[tex] P(Z = X+Y = k) = \sum_{i=0}^k (NCi) p^i (1-p)^{N-i} (M C i-1) p^{k-i} (1-p)^{M-K+i}[/tex]
[We can take out the constant p:
[tex] P(Z = X+Y = k) = p^k (1-p)^{N+M-k} \sum_{i=0}^k (NCi)(M C k-i)[/tex]
And using properties of the binomial formula we can write this like that:
[tex] P(Z = X+Y = k) = (N+M Ck) p^k (1-p)^{N+M-k} [/tex]
So then we see that [tex] Z= X+Y \sim Bin(N+M ,p)[/tex]
Find the tangent line approximation for 10+x−−−−−√ near x=0. Do not approximate any of the values in your formula when entering your answer below.
The tangent line approximation near x=0 for the function f(x) = \\sqrt{10 + x} is found by first calculating its derivative, then using that derivative to construct the equation of the tangent line at x=0, resulting in the linear approximation y = (1/2)(10)^{-1/2}x + \\sqrt{10}.
Explanation:Finding the tangent line approximation for a function near a point involves using the function's derivative at that point. For the function f(x) = \\sqrt{10 + x}, the derivative at x = 0, denoted as f'(0), will provide the slope of the tangent. To find this, let's differentiate f(x) using the chain rule. The derivative of f(x) with respect to x is (1/2)(10 + x)^{-1/2}. At x = 0, this simplifies to 1/2(\\sqrt{10}), which is the slope of the tangent line at that point. Hence, the tangent line equation is y - f(0) = f'(0)(x - 0), which simplifies to y = (1/2)(10)^{-1/2}x + \\sqrt{10}. This form equation is the linear approximation of f(x) near x = 0.
Determine whether the data set could represent a linear function.
Answer:
yes
Step-by-step explanation:
the rate of change is constant
This might be hard to explain, but could you try to explain where I should put the points. I’m really confused?
Answer:
im pretty sure you have to find out how far Joel walked and then double joels distance to get Brents
Step-by-step explanation:
Answer:
Step-by-step explanation:
5 small size companies, 9 median and 41 large ones are randomly selected for a financial auditing from 31 small. 59 median and 43 large companies. Identify which sampling technique is used Stratified Convenience Random Cluster Systematic
Answer:
Stratified
Step-by-step explanation:
Stratified sampling is a type of sampling method in which the observer divides the overall population into different groups.
These groups are known as strata.
Within the each group, a probability sample is selected.
stratified sampling helps in reducing the sample size required to attain a required precision.
Consider a square whose size varies. Let s s represent the side length of the square (in cm) and let P P represent the perimeter of the square (in cm).
Write a formula that expresses P in terms of s.
Answer:
P = 4s
Step-by-step explanation:
The perimeter of a geometric shape is simply the sum of all its sides length. Since the shape in question is a square, which means that all of the four sides have the same length 's', the perimeter can be expressed by:
[tex]P = s+s+s+s\\P=4s[/tex]
For any value of 's', the formula above expresses the perimeter 'P' as a function of 's'
Determine what type of observational study is described. Explain. Researchers wanted to determine whether there was an association between high blood pressure and the suppression of emotions. The researchers looked at 1800 adults enrolled in a health initiative observational study. Each person was interviewed and asked about their response to emotions. In particular they were asked whether their current tendency was to express or to hold in anger and other emotions. The degree of suppression of emotions was rated on a scale of 1 to 10. Each person's blood pressure was also measured. The researchers analyzed the results to determine whether there was an association between high blood pressure and the suppression of emotions. A. The observational study is a retrospective study because individuals are asked to look back in time. B. The observational study is a cross-sectional study because information is collected at a specific point in time. C. The observational study is a cohort study because individuals are observed over a long period of time.
Final answer:
The described observational study is a cross-sectional study, as it involves collecting data once at a particular point in time, without any follow-up observations. So the correct option is B.
Explanation:
The type of observational study described in the scenario is a cross-sectional study. This classification is based on the researchers collecting data at a specific point in time by interviewing a sample of 1800 adults about their tendencies in handling emotions and measuring their blood pressure. Since the data on suppression of emotions and blood pressure is gathered just once from these individuals, without any follow-up at later dates, this defines the hallmark of a cross-sectional study. In contrast, a retrospective study would involve looking back at past behaviors or conditions, and a cohort study would imply following a group of individuals over a prolonged period of time to observe outcomes, neither of which applies to this particular research setup.
Final answer:
The study in the question is a cross-sectional study because it gathers data from subjects at a specific point in time to assess the association between emotional suppression and high blood pressure.
Explanation:
The study described in the question can be classified as a cross-sectional study because it involved collecting data from the individuals at a specific point in time. Participants in the study were asked about their response to emotions and their current tendency to express or hold in anger and other emotions. Additionally, their blood pressure was measured. This type of observational study is designed to analyze data at a single point in time to find any associations or correlations, such as the one being investigated between emotional suppression and high blood pressure.
Fireworks on July4th.A local news outlet reported that 56% of 600 randomly sampled Kansasresidents planned to set off fireworks on July 4th. Determine the margin of error for the 56% point estimateusing a 95% confidence level.1
Answer:
The margin of error is 3.97 percentage points.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
[tex]n = 600, p = 0.56[/tex]
95% confidence interval
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.56 - 1.96\sqrt{\frac{0.56*0.44}{600}} = 0.5203[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.56 + 1.96\sqrt{\frac{0.56*0.44}{600}} = 0.5997[/tex]
The margin of error is the upper limit subtracted by the proportion, or the proportion subtracted by the lower limit. They are the same values.
So the margin of error is 0.5997 - 0.56 = 0.56 - 0.5203 = 0.0397 = 3.97 percentage points.
Use the given information to find the length of a circular arc. Round to two decimal places.the arc of a circle of radius 11 inches subtended by the central angle of pie/4
Answer is in inches(in)
Answer:
The length of the circular arc is 8.64 inches
Step-by-step explanation:
Length of circular arc (L) = central angle/360° × 2πr
central angle = pie/4 = 45°, r (radius) = 11 inches
L = 45°/360° × 2 × 3.142 × 11 = 8.64 inches (to two decimal places)
Answer:
Step-by-step explanation:
The formula for determining the length of an arc is expressed as
Length of arc = θ/360 × 2πr
Where
θ represents the central angle.
r represents the radius of the circle.
π is a constant whose value is 3.14
From the information given,
Radius, r = 11 inches
θ = pi/4
2π = 360 degrees
π = 360/2 = 180
Therefore,
θ = 180/4 = 45 degrees
Therefore,
Length of arc = 45/360 × 2 × 3.14 × 11
Length of arc = 8.64 inches rounded up to 2 decimal places
Match each differential equation to a function which is a solution
FUNCTIONS
A. y = 3x + x^2,
B. y = e^{-3 x},
C. y = \sin(x),
D. y = x^{\,\frac{1}{2}},
E. y = 5 \exp(5 x),
DIFFERENTIAL EQUATIONS
1. y'' + 8 y' + 15 y = 0
2. y'' + y = 0
3. y' = 5 y
4. 2x^2y'' + 3xy' = y
Best Answer
Answer:
1. First equation is option B
2. Second equation is option C
3. Third equation is option E
4. Fourth equation no best option.
explanation:
Check the attachment for solution
Following are the calculation to the differential equation:
For point 1)
[tex]y'' + 8 y' + 15 y = 0\\[/tex]
B
[tex]Y = e^{-3x}[/tex] be the solution of this equation
[tex]Y' = -3 e^{-3x}\\\\ y''= 9 e^{-3x} \\\\\therefore \\\\y'' +8y' + 15 y= 9e^{-3x} + 8(-3e^{-3x})+ 15 e^{-3x} \\\\e^{-3x}( 9-24+15)=0[/tex]
For point 2)
[tex]y'' + y = 0[/tex]
C
[tex]y = \sin x[/tex] be the solution of above equation
[tex]y'= -\cos x \\\\y''= -\sin x = -y \\\\y''+y=0\\\\[/tex]
For point 3)
[tex]y' = 5 y[/tex]
[tex]y'=e^{5x}[/tex] be the solution of equation 3
[tex]y'= 5 e^{5y}= 5y =y'=5y[/tex]
For point 4)
[tex]2x^2 y'' + 3xy' = y[/tex]
Let [tex]y=\sqrt{x}[/tex] be the solution of equation (4)
[tex]y'=\frac{1}{2 \sqrt{x} }\\\\y''=- \frac{1}{2} \times \frac{1}{2} \times {x^{- \frac{3}{2}}} ==- \frac{1}{4} \times {x^{- \frac{3}{2}}} \\\\-2x^2 \times =- \frac{1}{4} {x^{- \frac{3}{2}}}+ 3x \times =- \frac{1}{2 \sqrt{x}}\\\\- \frac{1}{2} {x^{ \frac{1}{2}}}+ \frac{3}{2} x^{\frac{1}{2}} =\sqrt{x} =y\\\\[/tex]
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Mariel thinks the tens digit goes up by 1 in these numbers. Do you agree? Explain. 864,865,866,867,868,869
Answer:
No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.
Step-by-step explanation:
It is important to know the concepts of units, tenths and cents.
For example
1 = 1 unit
10 = 1*10 + 0 = The tens digit is one the unit digit is 0
21 = 2*10 + 1 = The tens digit is two and the unit digit is 1.
120 = 1*100 + 2*10 + 0 = The cents digit is 1, the tens digit is two and the unit digit is 0.
So
Adding 1 is the same as the unit digit going up by 1.
Adding 10 is the same as the tens digit going up by 1.
Adding 100 is the same as the cents digit going up by 1.
In this problem, we have that:
864,865,866,867,868,869
Each value is the 1 added to the previous value, that is, the unit digit goes up by 1.
Mariel thinks the tens digit goes up by 1 in these numbers. Do you agree?
No, because it is not the tens digit that goes up by 1 in these numbers, it is the unit digit.
Answer:
disagree, its the unit number that goes up by 1
Step-by-step explanation:
Claim: High School teachers have incomes with a standard deviation that is more than $22,500. A recent study of 126 high school teacher incomes shower a standard deviation of $24,500.
A. Express the original claim in symbolic form.
B. Identify the null and the alternative hypotheses that should be used to arrive at the conclusion that supports the claim.
Answer:
A.
sigma > 22500
B.
Null hypothesis:sigma = 22500
Alternative hypothesis:sigma > 22500
Step-by-step explanation:
A.
The claim states that the standard deviation of high school teachers income is more than 22,500. This can be represented in the symbolic form as sigma > 22500.
B.
The null hypothesis and alternative hypothesis for the given scenario can be written as
Null hypothesis: Standard deviation of income of high school teachers is 22,500.
The standard deviation is represented as sigma.Symbolically it can be written as
Null hypothesis: sigma = 22500
Alternative hypothesis: Standard deviation of income of high school teachers is more than 22,500.
Symbolically it can be written as
Alternative hypothesis: sigma > 22500
An experiment consists of four outcomes with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4. The probability of outcome E4 is__________.
Answer: 0.1
Step-by-step explanation:
WE know that the total probability in an experiments = 1.
i.e. Sum of the probabilities of occurring each event is 1.
i.e. If there are n outcomes in any experiment., then the total probability will be:
[tex]P(E_1)+P(E_2)+P(E_3)+...........+P(E_n)=1[/tex]
Given : An experiment consists of four outcomes ,with P(E1) = 0.2, P(E2) = 0.3, and P(E3) = 0.4.
Then , [tex]P(E_1)+P(E_2)+P(E_3)+P(E_4)=1[/tex]
Substitute corresponding values , we get
[tex]0.2+0.3+0.4+P(E_4)=1[/tex]
[tex]0.9+P(E_4)=1[/tex]
[tex]P(E_4)=1-0.9=0.1[/tex]
Hence , the probability of outcome [tex]E_4[/tex] is 0.1.
The probability of event E4 in the given experiment is 0.1, because the sum of probabilities of all outcomes should equal 1.
Explanation:The problem falls under the subject of
Probability Theory
within Mathematics. In probability, the total probability of all possible outcomes is always 1. So, for the given problem where you have four events E1, E2, E3, E4, the total probability P(E1)+P(E2)+P(E3)+P(E4) should equal 1. Given that P(E1) = 0.2, P(E2) = 0.3 and P(E3) = 0.4, we can find P(E4) using the equation
1 - P(E1) - P(E2) - P(E3) = P(E4)
. By substituting the given values into this equation, we find that P(E4) = 1 - 0.2 - 0.3 - 0.4 =
0.1
. Therefore, the probability of outcome E4 is 0.1.
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HELP PLEASE!
When two functions have an inverse relationship, the line of reflection will be which of the following equations?
Select the correct response(s):
x = 0
y = x
y = 0
y = -x
Thank you!
Answer:
Option C: The line y = x
Step-by-step explanation:
Let us take an ordered pair (x, y) of a function. Then the ordered pair of its inverse function would be (y, x).
That is to say, when we reflect a point (x, y) across the line y = x we get the point (y, x).
Note that since, this function is invertible, it is both 'one - one' and 'onto'.
The line of reflection for two functions with an inverse relationship is the line y = x, as this line shows the symmetry of the original function and its inverse on a graph.
Explanation:When two functions have an inverse relationship, the line of reflection across which their graphs are mirrored is the line y = x. This is because for an inverse relationship, each x value in the first function corresponds to a y value in the second function, and vice versa. Thus, when graphing the original function and its inverse, you will notice that they are symmetric with respect to the line y = x.
A random sample of 11 days were selected from last year's records maintained by the maternity ward in a local hospital, and the number of babies born each day of the days is given below: 3 7710 0 712 530 Find the five number summary (minimum, first quartile, second quartile, third quartile, maximum) of the data.
Answer: ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )
Step-by-step explanation:
Given : A random sample of 11 days were selected from last year's records maintained by the maternity ward in a local hospital, and the number of babies born each day of the days is given below:
3 7 7 10 0 7 1 2 5 3 0
We first arrange them in increasing order , we get
0 0 1 2 3 3 5 7 7 7 10
Here , N= 11
Now , we can see that
Minimum value = 0
Maximum value = 10
First quartile [tex]Q_1[/tex]= [tex](\dfrac{N+1}{4})^{th}\ term=(\dfrac{12}{4})^{th}\ term = 3^{rd} term =1[/tex]
Second quartile [tex]Q_2[/tex]= Median = Middlemost number = 3
Third quartile [tex]Q_3[/tex] = [tex](\dfrac{3(N+1)}{4})^{th}\ term=(\dfrac{36}{4})^{th}\ term[/tex]
[tex]= 9^{th} term =7[/tex]
∴ The required five number summary : ( Min = 0 , [tex]Q_1=1[/tex] , [tex]Q_2=3[/tex] , [tex]Q_3=7[/tex] , Max = 10 )
Find the general indefinite integral. (Use C for the constant of integration. Remember to use absolute values where appropriate.) integral 9 + Squareroot x + x/x dx x + Squareroot x + 9 log (x) +
Answer:
[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]
Step-by-step explanation:
We have been an integral [tex]\int \frac{9+\sqrt{x}+x}{x}dx[/tex]. We are asked to find the general solution for the given indefinite integral.
We can rewrite our given integral as:
[tex]\int \frac{9}{x}+\frac{\sqrt{x}}{x}+\frac{x}{x}dx[/tex]
[tex]\int \frac{9}{x}+\frac{1}{\sqrt{x}}+1dx[/tex]
Now, we will apply the sum rule of integrals as:
[tex]\int \frac{9}{x}dx+\int \frac{1}{\sqrt{x}}dx+\int 1dx[/tex]
[tex]9\int \frac{1}{x}dx+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]
Using common integral [tex]\int \frac{1}{x}dx=\text{ln}|x|[/tex], we will get:
[tex]9\text{ln}|x|+\int x^{-\frac{1}{2}}dx+\int 1dx[/tex]
Now, we will use power rule of integrals as:
[tex]9\text{ln}|x|+\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+\int 1dx[/tex]
[tex]9\text{ln}|x|+\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+\int 1dx[/tex]
[tex]9\text{ln}|x|+2x^{\frac{1}{2}}+\int 1dx[/tex]
[tex]9\text{ln}|x|+2\sqrt{x}+\int 1dx[/tex]
We know that integral of a constant is equal to constant times x, so integral of 1 would be x.
[tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex]
Therefore, our required integral would be [tex]9\text{ln}|x|+2\sqrt{x}+x+C[/tex].
The general indefinite integral of the expression 9 + sqrt(x) + x/x is obtained by separately integrating each term, resulting in the expression 9x + (2/3)x^(3/2) + x + C, where C is the sum of the constants of integration.
Explanation:The question asks to compute the indefinite integral of an algebraic function. Firstly, it's crucial to understand that an indefinite integral is an antiderivative of a function, representing the reverse operation of differentiation. In the given expression, we essentially have three terms to integrate: 9, √x and x/x (which simplifies to 1). To find the integral of these expressions we can use the power rule of integration: ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration.
When we integrate 9, treating it as 9x^0, it becomes 9x + C1. The integral of the square root of x, which is x^1/2 in exponent form, becomes (2/3)x^(3/2) + C2 according to the power rule. Finally, x/x simplifies to 1, and its integral is simply x + C3.
So, the general indefinite integral of the original expression is 9x + (2/3)x^(3/2) + x + C, where 'C' represents the sum of the constants of integration: C1, C2, and C3.
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Design a sine function with the given properties:
It has a period of 24 hr with a minimum value of 10 at t=4 hr and a maximum value of 16 at t=16 hr.
please show all the steps
To design a sine function with a period of 24 and minima and maxima at given times, we scale the function using a coefficient of 2π/24, shift the function to make the peak occur at x=16, and stretch it by a factor of 3 to make it go from 10 to 16.
Explanation:To design a sine function with a period of 24 and minima and maxima at given times, we first need to understand a few concepts about sine functions. The standard sine function, sine(x), has a period of 2π. Therefore, to stretch it to a period of 24 hours, we would scale the function using a coefficient of 2π/24 or π/12. Thus, our function becomes sine((π/12) x).
Next, we want to shift the function so its maximum occurs at t=16. Normally, the sine function peaks at π/2, so we need to shift the function to the right by an amount that makes the peak occur at x=16. This would be 16 - π/2, which gives us the function sine(π/12 x - 16 + π/2).
Finally, to stretch the function vertically to accommodate the minimum and maximum values of 10 and 16, we note that the amplitude of the sine function is usually 1 (from -1 to 1), so we need to stretch it by a factor of (16-10)/2 = 3 to make it go from 10 to 16. This gives us the function y= 3sin((π/12)x - 16 + π/2)+13.
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A government watchdog association claims that 70% of people in the U.S. agree that the government is inefficient and wasteful. You work for a government agency and asked to test this claim to determine if the true proportion differs from 70%. You find that in a random sample of 1165 people in the U.S., 746 agreed with this view. Test the claim at 0.02 level of significance and determine which one of the following is a correct conclusion?
A.There is not sufficient evidence that the true population proportion is not equal to 70%.
B.There is sufficient evidence that the true population proportion is greater than 70%.
C.There is sufficient evidence that the true population proportion is less than 70%.
D.There is sufficient evidence that the true population proportion is not equal to 70%.
Answer:
Option D) There is sufficient evidence that the true population proportion is not equal to 70%.
Step-by-step explanation:
We are given the following in the question:
Sample size, n = 1165
p = 70% = 0.7
Alpha, α = 0.02
Number of people who agreed , x = 746
First, we design the null and the alternate hypothesis
[tex]H_{0}: p = 0.7\\H_A: p \neq 0.7[/tex]
This is a two-tailed test.
Formula:
[tex]\hat{p} = \dfrac{x}{n} = \dfrac{746}{1165} = 0.64[/tex]
[tex]z = \dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
Putting the values, we get,
[tex]z = \displaystyle\frac{0.64-0.7}{\sqrt{\frac{0.7(1-0.7)}{1165}}} = -4.46[/tex]
Now, [tex]z_{critical} \text{ at 0.02 level of significance } = \pm 2.33[/tex]
Since,
Since, the calculated z statistic does not lie in the acceptance region, we fail to accept the null hypothesis and reject it. We accept the alternate hypothesis.
Thus, there is enough evidence to support the claim that the true proportion differs from 70%.
Option D) There is sufficient evidence that the true population proportion is not equal to 70%.