Answer:
Total number of ways will be 209
Step-by-step explanation:
There are 6 boys and 4 girls in a group and 4 children are to be selected.
We have to find the number of ways that 4 children can be selected if at least one boy must be in the group of 4.
So the groups can be arranged as
(1 Boy + 3 girls), (2 Boy + 2 girls), (3 Boys + 1 girl), (4 boys)
Now we will find the combinations in which these arrangements can be done.
1 Boy and 3 girls = [tex]^{6}C_{1}\times^{4}C_{3}=6\times4[/tex]=24
2 Boy and 2 girls=[tex]^{6}C_{2}\times^{4}C_{2}=\frac{6!}{4!\times2!}\times\frac{4!}{2!\times2!}=15\times6=90[/tex]
3 Boys and 1 girl = [tex]^{6}C_{3}\times^{4}C_{1}=\frac{6!}{4!\times2!}\times\frac{4!}{3!}=\frac{6\times5\times4}{3 \times2} \times4=80[/tex]
4 Boys = [tex]^{6}C_{4}=\frac{6!}{4!\times2!} =\frac{6\times 5}{2\times1}=15[/tex]
Now total number of ways = 24 + 90 + 80 + 15 = 209
If the trapezoid below is reflected across the x-axis, what are the coordinates of B”?
Answer:
B'(3, -8)
Step-by-step explanation:
The image is the mirror image of the trapezoid below the x-axis.
Each x-coordinate remains the same. Each y-coordinate becomes the opposite.
B'(3, -8)
Answer:
(3 , -8)
Step-by-step explanation:
The current coordinates of B. are (3,8). The x-axis is the horizontal line that runs across. This means that if the trapezoid were to be reflected, it would end up upside down. When this happens, only the y value changes its sign.
In short, your Y value would become negative, making 8 change to -8
The function f(x)= x(squared) is similar to: g(x)= -3(x-5)(squared)+4. Describe the transformations. Show Graphs
Answer:
Parent function f(x) is inverted, stretched vertically by 1 : 3, shifted 5 units right and 4 units upwards to form new function g(x).
Step-by-step explanation:
The parent function graphed is f(x) = x²
This graph when inverted (parabola opening down)function becomes
g(x) = -x²
Further stretched vertically by a scale factor of 1:3 then new function becomes as g(x) = -3x²
Then we shift this function by 5 units to the right function will be
g(x) = -3(x - 5)²
At last we shift it 4 units vertically up then function becomes as
g(x) = -3(x - 5)² + 4
Professor Jones has to select 6 students out of his English class randomly to participate in a regional contest. There are 36 students in the class. Is this a PERMUTATION or a COMBINATION problem? How many ways can Prof. Jones choose his students?
Answer: This is a combination.
There are 1947792 ways to choose his students.
Step-by-step explanation:
Since we have given that
Number of students in a class = 36
Number of students selected for his English class = 6
We would use "Combination" .
As permutation is used when there is an arrangement.
whereas Combination is used when we have select r from group of n.
So, Number of ways that Prof. Jones can choose his students is given by
[tex]^{36}C_6=1947792[/tex]
Hence, there are 1947792 ways to choose his students.
1.(a) Find the matrix that represent the linear transformation f: p1\rightarrowp1 with respect to the bais {1,x};
f(a+bx)=(8a+2b)+(5a-b)x
(b) Does there exist a basis of P1 with respect to which f has a diagonal matrix that represents it? If so, what is the basis.
Answer:
[tex]a) \quad A=\left[\begin{array}{cc}8&5\\5&-1\end{array}\right] \\\\\\b) \quad \{-1+5x, 2+x\}[/tex]
Step-by-step explanation:
To compute the representation matrix A of f with respect the basis {1,x} we first compute
[tex]f(1)=f(1+0x)=(8\cdot 1 + 2 \cdot 0) + (5 \cdot 1 - 0)x=8+5x \\\\f(x)=f(0 + 1\cdot x)=(8 \cdot 0 + 2\cdot 1)+(5 \cdor 0 - 1)x = 2-1[/tex]
The coefficients of the polynomial f(1) gives us the entries of the first column of the matrix A, where the first entry is the coefficient that accompanies the basis element 1 and the second entry is the coefficient that accompanies the basis element x. In a similar way, the coefficients of the polynomial f(x) gives us the the entries of the second column of A. It holds that,
[tex]A=\left[\begin{array}{ccc}8&5\\2&-1\end{array}\right][/tex]
(b) First, note that we are using a one to one correspondence between the basis {1,x} and the basis {(1,0),(0,1)} of [tex]\mathbb{R}^2[/tex].
To compute a basis P1 with respect to which f has a diagonal matrix, we first have to compute the eigenvalues of A. The eigenvalues are the roots of the characteristic polynomial of A, we compute
[tex]0=\det\left[\begin{array}{ccc}8-\lambda & 2\\ 5 & -1-\lambda \end{array}\right]=(8 - \lambda)(-1-\lambda)-18=(\lambda - 9)(\lambda +2)[/tex]
and so the eigenvalues of the matrix A are [tex]\lambda_1=-2 \quad \text{and} \quad \lambda_2=9[/tex].
After we computed the eigenvalues we use the systems of equations
[tex]\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right] = \left[\begin{array}{c}-2x_1\\-2x_2\end{array}\right] \\\\\text{and} \\\\\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1 \\ x_2\end{array}\right]=\left[\begin{array}{cc}9x_1\\2x_1\end{array}\right][/tex]
to find the basis of the eigenvalues. We find that [tex]v_1=(-1,5 )[/tex] is an eigenvector for the eigenvalue -2 and that [tex]v_2=(2,1)[/tex] is an eigenvector for the eigenvalue 9. Finally, we use the one to one correspondence between the [tex]\mathbb{R}^2[/tex] and the space of liear polynomials to get the basis [tex]P1=\{-1+5x, 2+x\}[/tex] with respect to which f is represented by the diagonal matrix [tex]\left[\begin{array}{ccc}-2&0\\0&9\end{array}\right][/tex]
Verify that y = c_1 + c_2 e^2x is a solution of the ODE y" - 2y' = 0 for all values of c_1 and c_2.
Answer:
For any value of C1 and C2, [tex]y = C1 + C2*e^{2x}[/tex] is a solution.
Step-by-step explanation:
Let's verify the solution, but first, let's find the first and second derivatives of the given solution:
[tex]y = C1 + C2*e^{2x}[/tex]
For the first derivative we have:
[tex]y' = 0 + C2*(2x)'*e^{2x}[/tex]
[tex]y' = C2*(2)*e^{2x}[/tex]
For the second derivative we have:
[tex]y'' = C2*(2)*(2x)'*e^{2x}[/tex]
[tex]y'' = C2*(2)*(2)*e^{2x}[/tex]
[tex]y'' = C2*(4)*e^{2x}[/tex]
Let's solve the ODE by the above equations:
[tex]y'' - 2y' = 0[/tex]
[tex]C2*(4)*e^{2x} - 2*C2*(2)*e^{2x} = 0[/tex]
[tex]C2*(4)*e^{2x} - C2*(4)*e^{2x} = 0[/tex]
From the above equation we can observe that for any value of C2 the equation is solved, and because the ODE only involves first (y') and second (y'') derivatives, C1 can be any value as well, because it does not change the final result.
Determine whether T : R^2 -->R^2,T((x.y)) = (x,y^2) is a linear transformation
Answer: No, the given transformation T is NOT a linear transformation.
Step-by-step explanation: We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :
[tex]T(x,y)=(x,y^2).[/tex]
We know that
a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R
[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]
So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have
[tex]T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)[/tex]
and
[tex]aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).[/tex]
Therefore, we get
[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]
Thus, the given transformation T is NOT a linear transformation.
Forty percent of the homes constructed in the Quail Creek area include a security system. Three homes are selected at random: What is the probability all three of the selected homes have a security system
Answer: 0.064
Step-by-step explanation:
Binomial probability formula :-
[tex]P(X)=^nC_x \ p^x\ (1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.
Given : The proportion of the homes constructed in the Quail Creek area include a security system : [tex]p=0.40[/tex]
Now, if three homes are selected at random, then the probability all three of the selected homes have a security system is given by :-
[tex]P(3)=^3C_3 \ (0.40)^3\ (1-0.40)^{3-3}\\\\=(0.40)^3=0.064[/tex]
Hence, the probability all three of the selected homes have a security system = 0.064
The probability that all three homes selected at random in the Quail Creek area have a security system is 6.4%.
The probability that all three of the selected homes in the Quail Creek area have a security system, given that 40% of the homes have a security system, can be calculated by using the rule for independent events in probability.
Since each house is selected at random, we can multiply the probability of each house having a security system together:
P(all three homes have a security system) = P(home 1 has security system) × P(home 2 has security system) × P(home 3 has security system)
As every home has a 40% (or 0.40) chance of having a security system:
P(all three) = 0.40 × 0.40 × 0.40 = 0.064
Therefore, there is a 6.4% chance that all three homes selected will have a security system.
Tessa's class had a math exam where the grades were between 0 and 10. N(g) models the number of students whose grade on the exam was ggg. What does the statement N(8)>2⋅N(5) mean?
Interpreting the situation, we can conclude that the statement means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.
N(g) is the number of students who got a grade of g in the exam.Thus, N(8) is the number of students who got a grade of 8, while N(5) is the number of students who got a grade of 5.
[tex]N(8) > 2N(5)[/tex]
It means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.
A similar problem is given at https://brainly.com/question/11271837
The statement [tex]\( N(8) > 2 \cdot N(5) \)[/tex] means that the number of students who scored a grade of 8 on the exam is greater than twice the number of students who scored a grade of 5 on the exam.
To understand this, let's break down the notation:
- N(g) represents the number of students who scored (g) on the exam.
- N(8) is the number of students who scored an 8.
- N(5) is the number of students who scored a 5.
The inequality [tex]\( N(8) > 2 \cdot N(5) \)[/tex] compares these two quantities. It states that the count of students with a grade of 8 exceeds two times the count of students with a grade of 5. This indicates that a higher number of students performed better (scoring an 8) than those who scored a 5, with the difference being more than the number of students who scored a 5. In other words, if we were to take the number of students who scored a 5 and double it, there would still be more students who scored an 8. This could be an indicator of the overall performance of the class, suggesting that more students achieved a higher grade than those who scored in the middle range of the grading scale.
Find the general solution of the following nonhomogeneous second order differential equation: y" - 4y = e^2x
Answer:
Solution is [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]
Step-by-step explanation:
the given equation y''-4y[tex]=e^{2x}[/tex] can be written as
[tex]D^{2}y-4y=e^{2x}\\\\(D^{2}-4)y=e^{2x}\\\\[/tex]
The Complementary function thus becomes
y=c_{1}e^{m_{1}x}+c_{2}e^{m_{2}x}
where [tex]m_{1} , m_{2}[/tex] are the roots of the [tex]D^{2}-4[/tex]
The roots of [tex]D^{2}-4[/tex] are +2,-2 Thus the comlementary function becomes
[tex]y=c_{1}e^{2x}+c_{2}e^{-2x}[/tex]
here [tex]c_{1},c_{2}[/tex] are arbitary constants
Now the Particular Integral becomes using standard formula
[tex]y=\frac{e^{ax}}{f(D)}\\\\y=\frac{e^{ax}}{f(a)} (f(a)\neq 0)\\\\y=x\frac{e^{ax}}{f'(a)}(f(a)=0)[/tex]
[tex]y=\frac{e^{2x}}{D^{2}-4}\\\\y=\frac{e^{2x}}{(D+2)(D-2)}\\\\y=\frac{1}{D-2}\times \frac{e^{2x}}{2+2}\\\\y=\frac{1}{4}\times \frac{e^{2x}}{D-2}\\\\y=\frac{1}{4}xe^{2x}[/tex]
Hence the solution is = Complementary function + Particular Integral
Thus Solution becomes [tex]y=c_{1}e^{2x}+c_{2}e^{-2x}+\frac{1}{4}xe^{2x}[/tex]
The final general solution is [tex]y(x) = C1e^2x + C2e^-2x + 1/2xe^2x[/tex].
To find the general solution of the given differential equation: y'' - 4y = e2x, we will follow these steps:
1. Solve the Homogeneous Equation
First, solve the homogeneous part: y'' - 4y = 0
The characteristic equation is: r2 - 4 = 0
Solving for r, we get: r = ±2
Thus, the general solution to the homogeneous equation is: yh(x) = C1e2x + C2e-2x
2. Find a Particular Solution
Next, find a particular solution, yp(x), to the non homogeneous equation through the method of undetermined coefficients. Assume a particular solution of the form: yp(x) = Axe2x
Differentiating, we get: yp' = Ae2x + 2Axe2x and yp'' = 4Axe2x + 2Ae2x
Substitute these into the original equation:
4Axe2x + 2Ae2x - 4(Axe2x) = e2x
which simplifies to: 2Ae2x = e2x
Thus, A = 1/2
So, the particular solution is: yp(x) = (1/2)xe2x
3. Form the General Solution
The general solution to the nonhomogeneous differential equation is the sum of the homogeneous solution and the particular solution:
y(x) = yh(x) + yp(x)
Therefore, the general solution is: [tex]y(x) = C1e2x + C2e-2x + (1/2)xe2x[/tex].
Evaluate 6 - 2(-1) + | -5 | =
Answer:
13
Step-by-step explanation:
The product of two negative numbers is positive. The absolute value of a number is its magnitude written with a positive sign.
6 -2(-1) +|-5|
= 6 + 2 + 5
= 13
jose has $18 to spend for dinner what is the maximum amount he spend on meal and drinks so that he can leave a 15% tip? what percent of 60 is 18?
Answer:
$15.30
30% of 60 is 18
Step-by-step explanation:
To find the maximum amount he can spend on a meal, you have to find how much he is going to tip.
So to find the tip you multiply 15% by 18 and you get 2.7
Then you subtract 18 by 2.7 to find out how much he can spend on the meal.
18 - 2.7 = 15.30
So he can spend $15.30 on his meal and tip $2.70
To find what percent of 60 is 18, you have to use this equation:
is over of equals percent over 100
So is/of = x/100 We have the x as the percent because that's what you're trying to figure out.
You would put 18 as is because it has the word is before it and put 60 as of because it has of before it.
So 18/60 = x/100
Now you would do Cross Product Property
18*100 = 1800
60*x = 60x
60x = 1800
Now divide 60 by itself and by 1800
1800/60 = 30
x = 30%
A manufacturer produces bearings, but because of variability in the production process, not all of the bearings have the same diameter. The diameters have a normal distribution with a mean of 1.2 centimeters (cm) and a standard deviation of 0.03 cm. The manufacturer has determined that diameters in the range of 1.17 to 1.23 cm are acceptable. What proportion of all bearings falls in the acceptable range? (Round your answer to four decimal places.)
Answer:
68%
Step-by-step explanation:
It is given that the diameters of bearing have a normal distribution.
Mean = u = 1.2 cm
Standard deviation = [tex]\sigma[/tex] = 0.03 cm
We have to find the proportion of values which falls in between 1.17 to 1.23
In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:
[tex]z=\frac{x- \mu}{\sigma}[/tex]
For 1.17:
[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]
For 1.23:
[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]
So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question.
According to the empirical rule:
68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1.
Therefore, the answer to this question is 68%
Find the m∠p.
54
90°
27°
36°
Problem 5.58. Supposef XY and g : Y Z are functions If g of is one-to-one, prove that fmust be one-to-one 2. Find an example where g o f is one-to-one, but g is not one-to-one
f : X → Y and g: Y → Z
Now we have to show:
If gof is one-to-one then f must be one-to-one.
Given:
gof is one-to-one
To prove:
f is one-to-one.
Proof:
Let us assume that f(x) is not one-to-one .
This means that there exist x and y such that x≠y but f(x)=f(y)
On applying both side of the function by the function g we get:
g(f(x))=g(f(y))
i.e. gof(x)=gof(y)
This shows that gof is not one-to-one which is a contradiction to the given statement.
Hence, f(x) must be one-to-one.
Now, example to show that gof is one-to-one but g is not one-to-one.Let A={1,2,3,4} B={1,2,3,4,5} C={1,2,3,4,5,6}
Let f: A → B
be defined by f(x)=x
and g: B → C be defined by:
g(1)=1,g(2)=2,g(3)=3,g(4)=g(5)=4
is not a one-to-one function.
since 4≠5 but g(4)=g(5)
Also, gof : A → C
is a one-to-one function.
Let f(x) = (x − 3)−2. Find all values of c in (2, 5) such that f(5) − f(2) = f '(c)(5 − 2). (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
The answer does not exist.
Note - The statement has typing mistakes. Correct form is presented below:
Let [tex]f(x) = (x-3)^{-2}[/tex]. Find all values of [tex]c[/tex] in (2, 5) such that [tex]f(5) - f(2) = f'(c) \cdot (5-2)[/tex]. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
In this question we should use the Mean Value Theorem, which states that given a secant line between points A and B, there is at least a point C that belongs to the curve whose derivative exists.
We begin by calculating [tex]f(2)[/tex] and [tex]f(5)[/tex]:
[tex]f(2) = (2-3)^{-2}[/tex]
[tex]f(2) = 1[/tex]
[tex]f(5) = (5-3)^{-2}[/tex]
[tex]f(5) = 1[/tex]
And the slope of the derivative is:
[tex]f'(c) = \frac{f(5) - f(2)}{5-2}[/tex]
[tex]f'(c) = 0[/tex]
Now we find the derivative of the function:
[tex]f'(x) = -2\cdot (x-3)^{-3}[/tex]
[tex]-2\cdot (x-3)^{-3} = 0[/tex]
[tex]-2 = 0[/tex] (ABSURD)
Hence, we conclude that the answer does not exist.
We kindly invite to see this question on Mean Value Theorem: https://brainly.com/question/3957181
Eight measurements were made on the inside diameter of forged piston rings used in an automobile engine. The data (in millimeters) are 74.001, 74.003, 74.015, 74.000, 74.005, 74.002, 74.007, and 74.000. Calculate the sample mean and sample standard deviation. Round your answers to 3 decimal places. Sample mean
Answer: The sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.
Step-by-step explanation:
The given values : 74.001, 74.003, 74.015, 74.000, 74.005, 74.002, 74.007, and 74.000.
[tex]\text{Mean =}\dfrac{\text{Sum of all values}}{\text{Number of values}}\\\\\Rightarrow\overline{x}=\dfrac{ 592.033}{8}=74.004125\approx74.004[/tex]
The sample standard deviation is given by :-
[tex]\sigma=\sum\sqrt{\dfrac{(x-\overline{x})^2}{n}}\\\\\Rightarrow\ \sigma=\sqrt{\dfrac{0.000177}{8}}=0.00470372193056\approx0.005[/tex]
Hence, the sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.
Assume that the heights of men are normally distributed. A random sample of 16 men have a mean height of 67.5 inches and a standard deviation of 3.2 inches. Construct a 99% confidence interval for the population standard deviation, σ. (2.2, 5.4) (2.2, 6.0) (1.2, 3.2) (2.2, 5.8)
Answer: (2.2, 5.8)
Step-by-step explanation:
The confidence interval for standard deviation is given by :-
[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),\alpha/2}}} , \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),1-\alpha/2}}}\right )[/tex]
Given : Sample size : 16
Mean height : [tex]\mu=67.5[/tex] inches
Standard deviation : [tex]s=3.2[/tex] inches
Significance level : [tex]1-0.99=0.01[/tex]
Using Chi-square distribution table ,
[tex]\chi^2_{(15,0.005)}=32.80[/tex]
[tex]\chi^2_{(15,0.995)}=4.60[/tex]
Then , the 99% confidence interval for the population standard deviation is given by :-
[tex]\left ( \sqrt{\dfrac{(15)(3.2)^2}{32.80}} , \sqrt{\dfrac{(15)(3.2)^2}{4.6}}\right )\\\\=\left ( 2.1640071232,5.77852094812\right )\approx\left ( 2.2,5.8 \right )[/tex]
Neneh is a florist. Neneh can arrange 20 bouquets per day. She is considering hiring her husband Mustapha to work for her. Together Neneh and Mustapha can arrange 35 bouquets per day. What is Mustapha’s marginal product?
Answer:
Mustapha can arrange 15 bouquets per day.
Step-by-step explanation:
Neneh can arrange 20 bouquets per day.
Together Neneh and Mustapha can arrange 35 bouquets per day.
So, Mustapha can arrange [tex]35-20=15[/tex] bouquets per day.
Therefore, Mustapha’s marginal product is 15 bouquets.
A data set lists earthquake depths. The summary statistics are nequals300, x overbarequals5.89 km, sequals4.44 km. Use a 0.01 significance level to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 5.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
Answer:
Null hypothesis [tex]H_0:\mu=5.00km[/tex]
Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]
The p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.
Step-by-step explanation:
It is given that a data set lists earthquake depths. The summary statistics are
[tex]n=300[/tex]
[tex]\overline{x}=5.89km[/tex]
[tex]s=4.44km[/tex]
Level of significance = 0.01
We need to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 5.00.
Null hypothesis [tex]H_0:\mu=5.00km[/tex]
Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]
The formula for z-value is
[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]z=\frac{5.89-5.00}{\frac{4.44}{\sqrt{300}}}[/tex]
[tex]z=\frac{0.89}{0.25634351952}[/tex]
[tex]z=3.4719[/tex]
The p-value for z=3.4719 is 0.000517.
Since the p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.
The null and alternative hypotheses are H0: µ = 5.00 km and HA: µ ≠ 5.00 km. A t-test is used to calculate the test statistic, and the p-value is compared to the significance level of 0.01 to either reject or not reject the null hypothesis. The final conclusion is made in consideration of the original claim.
Explanation:In statistics, hypothesis testing is a tool for inferring whether a particular claim about a population is true. For this question about earthquake depths, we would start by setting our null hypothesis (H0) and our alternative hypothesis (HA).
The null hypothesis would be H0: µ = 5.00 km, and the alternative hypothesis would be HA: µ ≠ 5.00 km.
The test statistic can be calculated using a t test, since we are dealing with a sample mean and we know the sample standard deviation (sequals4.44 km).
The p-value associated with this test statistic would then be calculated, and compared to the significance level of 0.01. If the p-value is less than 0.01, we reject the null hypothesis. If, however, the p-value is greater than 0.01, we cannot reject the null hypothesis.
The final conclusion must be stated in terms of the original claim. If we reject the null hypothesis, we conclude that the evidence supports the claim that the mean earthquake depth is not equal to 5.00 km (supporting the alternative hypothesis). If we do not reject the null hypothesis, we conclude that the evidence does not support the claim that the mean is not 5.00 km. The data does not provide sufficient evidence to support a conclusion that the mean earthquake depth is different than 5.00 km.
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Jim borrows $14,000 for a period of 4 years at 6 % simple interest. Determine the interest due on the loan. [4 marks
Answer: $ 3,360
Step-by-step explanation:
Given : The principal amount borrowed for loan : [tex]P=\ \$14,000[/tex]
Time period : [tex]t=4[/tex]
Rate of interest : [tex]r=6\%=0.06[/tex]
The formula to calculate the simple interest is given by :-
[tex]S.I.=P\times r\times t\\\\\Rightatrrow\ S.I.=14000\times4\times0.06\\\\\Rightatrrow\ S.I.=3360[/tex]
Hence, the interest due on the loan = $ 3,360
Write an equation of the circle with center (-7, -4) and radius 6 .
Answer:
(x+7)^2 + (y+4)^2 = 6^2
or
(x+7)^2 + (y+4)^2 = 36
Step-by-step explanation:
We can write the equation of a circle in the form
(x-h)^2 + (y-k)^2 = r^2
Where (h,k) is the center and r is the radius
(x--7)^2 + (y--4)^2 = 6^2
(x+7)^2 + (y+4)^2 = 6^2
or
(x+7)^2 + (y+4)^2 = 36
Write an equation of the line through(2-1) and perpendicular to 2yx-4 Write the equation in the form x The one the Enter your answer in the box and then click Check Answer parts showing i Type here to search
Answer:
[tex]2x+y=3[/tex]
Step-by-step explanation:
Here we aer given a point (2,-1) and a line [tex]2y=x-4[/tex]. We are supposed to find the equation of the line passing through this point and perpendicular to this line.
Let us find the slope of the line perpendicular to [tex]2y=x-4[/tex]
Dividing above equation by 2 we get
[tex]y=\frac{1}{2}x-2[/tex]
Hence we have this equation in slope intercept form and comparing it with
[tex]y=mx+c[/tex] , we get Slope [tex]m = \frac{1}{2}[/tex]
We know that product of slopes of two perpendicular lines in -1
Hence if slope of line perpendicular to [tex]y=\frac{1}{2}x-2[/tex] is m' then
[tex]m\times m' =-1[/tex]
[tex]\frac{1}{2} \times m' =-1[/tex]
[tex]m'=-2[/tex]
Hence the slope of the line we have to find is -2
now we have slope and a point
Hence the equation of the line will be
[tex]\frac{y-(-1)}{x-2}=-2[/tex]
[tex]y+1=-2(x-2)[/tex]
[tex]y+1=-2x+4[/tex]
adding 2x and subtracting on both sides we get
[tex]2x+y=3[/tex]
Which is our equation asked
The _____ measures how accurate the point estimate is likely to be in estimating a parameter. standard deviation degree of unbiasedness interval estimate margin of error confidence level Why are confidence intervals preferred over significance tests by most researchers? they provide a range of plausible values for the parameter they allow use to accept the null hypothesis if the hypothesis value is contained within the interval since confidence intervals have a level of confidence associated with them, they give us more confidence in our decision regarding the null hypothesis they indicate whether or not the hypothesis parameter value is plausible all of these An interval estimate is typically preferred over a point estimate because i) it gives us a sense of accuracy of the point estimate ii) we know the probability that it contains the parameter (e.g., 95%) iii) it provides us with more possible parameter values I only II only both I and II all of these III only
Answer:
Standard deviation.
Step-by-step explanation:
The standard deviation measures how accurate the point estimate is likely to be in estimating a parameter.
The confidence interval measures how accurate the point estimate is likely to be in estimating a parameter.
A confidence interval communicates how accurate our estimate is likely to be.
The confidence interval is a range of of all plausible values of the random variable under test at a given confidence level which is expressed in percentage such as 98%, 95% and 90% of confidence level.
The standard deviation is the parameter to signify the dispersion of data around the mean value of the data.
Researchers prefer it because on the basis of the percentage of certainty in the test result of null hypothesis are accepted or rejected as it includes some chance for errors too. (example 95% sure means 5% not sure) also this gives a range of values and hence good chance to normalize errors.
An interval estimate is typically preferred over a point estimate because
i) it gives us a sense of accuracy of the point estimate
ii) we know the probability that it contains the parameter (e.g., 95%)
iii) it provides us with more possible parameter values
I only
II only
both I and II
all of these
III only
All three statements above are true hence all of these is the answer.
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A simple random sample of electronic components will be selected to test for the mean lifetime in hours. Assume that component lifetimes are normally distributed with population standard deviation of 27 hours. How many components must be sampled so that a 99% confidence interval will have margin of error of 3 hours?
Answer:
540
Step-by-step explanation:
we have given E=0.3
σ = 27 hours
100(1-α)%=99%
from here α=0.01
using standard table [tex]Z_\frac{\alpha }{2}=Z_\frac{0.01}{2}=2.58[/tex]
[tex]n=\left ( Z_\frac{\alpha }{2}\times \frac{\sigma }{E} \right )^{2}[/tex] =
[tex]\left ( 2.58\times \frac{27}{3} \right )^{2}[/tex]
n = [tex]23.22^{2}[/tex]
n=539.16
n can not be in fraction so n=540
To obtain a 99% confidence interval with a margin of error of 3 hours, at least 602 components must be sampled.
Explanation:In order to determine the number of components that must be sampled so that a 99% confidence interval will have a margin of error of 3 hours, we can use the formula:
n = (z * s / E)^2
Where:
n = sample size
z = z-value corresponding to the desired confidence level (in this case, 99% confidence level)
s = population standard deviation
E = margin of error
Plugging in the given values, we have:
n = (2.576 * 27 / 3)^2
n = 601.3696
Rounding up to the nearest whole number, we need to sample at least 602 components.
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A sum of money amounting to P5.15 consists of 10 cents and 25 cents, If there are 32 coins in all, how many 25 cents are there? A. 14 pcs B. 13 pcs C. 15 pcs D. 12 pcs
Answer: Option 'B' is correct.
Step-by-step explanation:
Let the number of 10 cents pcs be 'x'.
Let the number of 25 cents pcs be 'y'.
Since we have given that
Total number of coins = 32
Sum of money = $5.15
As we know that
$1 = 100 cents
$5.15 = 5.15×100 = 515 cents
According to question, we get that
[tex]x+y=32-----------(1)\\\\10x+25y=\$515------------(2)[/tex]
Using the graphing method, we get that
x = 19
y = 13
So, there are 13 pcs of 25 cents.
Hence, Option 'B' is correct.
By creating a system of equations based on the total number of coins (32) and their total value (P5.15), we calculate that there are 13 pieces of 25-cent coins.
Explanation:The student's question involves figuring out the number of 25-cent coins among a total of 32 coins which altogether amount to P5.15. This problem can be solved by setting up a system of equations to account for the total number of coins and the total value in pesos.
Let's denote the number of 10-cent coins as t and the number of 25-cent coins as q. We know from the problem that there are 32 coins in total, so:
(1) t + q = 32
We also know that the total value of the coins is P5.15, or 515 cents. Therefore:
(2) 10t + 25q = 515
By solving this system of equations, we can find the value of q, the number of 25-cent coins. First, we can multiply equation (1) by 10 to eliminate t when we subtract the equations:
10t + 10q = 320
Subtracting this from equation (2) gives us:
15q = 195
Dividing both sides by 15, we find that:
q = 13
So, there are 13 pieces of 25-cent coins, which corresponds to option B.
Six different integers are picked from the numbers 1 through 10. How many possible combinations are there, if the the second smallest integer in the group is 3?
Please solve ASAP
Answer:
1680 ways
Step-by-step explanation:
We have to select 6 different integers from 1 to 10. It is given that second smallest integer is 3. This means, for the smallest most integer we have only two options i.e. it can be either 1 or 2.
So, the selection of 6 numbers would be like:
{1 or 2, 3, a, b, c ,d}
There are 2 ways to select the smallest digit. Only 1 way to select the second smallest digit. For the rest four digits which are represented by a,b,c,d we have 7 options. This means we can chose 4 digits from 7. Number of ways to chose 4 digits from 7 is calculated as 7P4 i.e. by using permutations.
[tex]7P4 = \frac{7!}{(7-4)!}=840[/tex]
According to the fundamental rule of counting, the total number of ways would be the product of the individual number of ways we calculated above. So,
Total number of ways to pick 6 different integers according to the said criteria would be = 2 x 1 x 840 = 1680 ways
Seventeen candidates have filed for the upcoming county council election. 7 are women and 10 are men a) Is how many ways can 10 county council members be randomly elected out of the 17 candidates? (b) In how many ways can 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men? c) If 10 county council members are randomly elected from 17 candidates, what is the probability that 5 are women and 5 are men? Round answer to nearest ten-thousandth (4 places after decimal).
Answer: (a) 19448 ways
(b) 5292 ways
(c) 0.2721
Step-by-step explanation:
(a) 10 county council members be randomly elected out of the 17 candidates in the following ways:
= [tex]^{n}C_{r}[/tex]
= [tex]^{17}C_{10}[/tex]
= [tex]\frac{17!}{10!7!}[/tex]
= 19448 ways
(b) 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men in the following ways:
we know that there are 7 women and 10 men in total, so
= [tex]^{7}C_{5}[/tex] × [tex]^{10}C_{5}[/tex]
= [tex]\frac{7!}{5!2!}[/tex] × [tex]\frac{10!}{5!5!}[/tex]
= 21 × 252
= 5292 ways
(c) Now, the probability that 5 are women and 5 are men are selected:
= [tex]\frac{ ^{7}C_{5} * ^{10}C_{5}}{^{17}C_{10}}[/tex]
= [tex]\frac{5292}{19448}[/tex]
= 0.2721
If you draw a card with a value of three or less from a standard deck of cards, I will pay you $43. If not, you pay me $11. (Aces are considered the highest card in the deck.) Step 1 of 2 : Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
If [tex]W[/tex] is a random variable representing your winnings from playing the game, then it has support
[tex]W=\begin{cases}43&\text{if you draw something with value at most 3}\\-11&\text{otherwise}\end{cases}[/tex]
There are 52 cards in the deck. Only the 1s, 2s, and 3s fulfill the first condition, so there are 12 ways in which you can win $43. So [tex]W[/tex] has PMF
[tex]P(W=w)=\begin{cases}\frac{12}{52}=\frac3{13}&\text{for }w=43\\1-\frac{12}{52}=\frac{10}{13}&\text{for }w=-11\\0&\text{otherwise}\end{cases}[/tex]
You can expect to win
[tex]E[W]=\displaystyle\sum_ww\,P(W=w)=\frac{43\cdot3}{13}-\frac{11\cdot10}{13}=\boxed{\frac{19}{13}}[/tex]
or about $1.46 per game.
The expected value of the proposition is $7.31.
Explanation:Expected Value Calculation:To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and then sum them up.
The probability of drawing a card with a value of three or less is 12/52 since there are 12 cards with values of three or less in a standard deck of 52 cards. The probability of not drawing a card with a value of three or less is 40/52.
Using these probabilities and the given payoffs, we can calculate the expected value as follows:
Expected Value = (Probability of Winning * Payoff if Win) + (Probability of Losing * Payoff if Lose)
Expected Value = (12/52 * 43) + (40/52 * -11)
Calculating this expression gives us an expected value of $7.31 (rounded to two decimal places).
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The average assembly time for a Ford Taurus is μ = 38 hrs. An engineer suggests that using a new adhesive to attach moldings will speed up the assembly process. The new adhesive was used for one month. During that month, the average assembly time for 36 cars was = 37.5 hours with a standard deviation s = 1.2 hours. Use α = 0.01. Based on the calculated P-value will you reject or fail to reject the null hypothesis? Select one: a. reject the null hypothesis / data is significant b. fail to reject the null hypothesis c. cannot be determined
Answer:
a) reject null hypothesis since p < 0.01
Step-by-step explanation:
Given that the average assembly time for a Ford Taurus is
[tex]μ = 38 hrs[/tex]
Sample size [tex]n=36[/tex]
[tex]x bar = 37.5\\s=1.2\\SE = 1.2/6 = 0.2[/tex]
Test statistic t = mean diff/se = 0.5/0.2 = 2.5
(Here population std dev not known hence t test is used)
df = 35
p value = 0.008703
a) reject null hypothesis since p < 0.01
Write equations for the horizontal and vertical lines passing through the point (-1, -7)
Answer:
see below
Step-by-step explanation:
The horizontal line will have the same y and the y value will be constant
y = -7
The vertical line will have the same x and the x value will be constant
x = -1