Find the solution using the integrating factor method: x2 – y - dy dx = X
Answers
The solution to the given differential equation using the integrating factor method is y = -(x^2 + 2x + 2) - Xe^x + Ce^x, where C is the constant of integration.
To solve the given first-order linear differential equation, x^2 - y - dy/dx = X, we can use the integrating factor method.
The standard form of a first-order linear differential equation is dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.
In this case, we have:
dy/dx - y = x^2 - X
Comparing this with the standard form, we can identify P(x) = -1 and Q(x) = x^2 - X.
The integrating factor (IF) is given by the formula: IF = e^(∫P(x)dx)
For P(x) = -1, integrating, we get:
∫P(x)dx = ∫(-1)dx = -x
Therefore, the integrating factor is IF = e^(-x).
Now, we multiply the entire equation by the integrating factor:
e^(-x) * (dy/dx - y) = e^(-x) * (x^2 - X)
Expanding and simplifying, we have:
e^(-x) * dy/dx - e^(-x) * y = x^2e^(-x) - Xe^(-x)
The left side of the equation can be written as d/dx (e^(-x) * y) using the product rule. Thus, the equation becomes:
d/dx (e^(-x) * y) = x^2e^(-x) - Xe^(-x)
Now, we integrate both sides with respect to x:
∫d/dx (e^(-x) * y) dx = ∫(x^2e^(-x) - Xe^(-x)) dx
Integrating, we have:
e^(-x) * y = ∫(x^2e^(-x) dx) - ∫(Xe^(-x) dx)
Simplifying and evaluating the integrals on the right side, we get:
e^(-x) * y = -(x^2 + 2x + 2)e^(-x) - Xe^(-x) + C
Finally, we can solve for y by dividing both sides by e^(-x):
y = -(x^2 + 2x + 2) - Xe^x + Ce^x
Therefore, the solution to the given differential equation using the integrating factor method is y = -(x^2 + 2x + 2) - Xe^x + Ce^x, where C is the constant of integration.
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Related Questions
Mike has $21 to spend at the mall. He spends all of his money on bracelets for his sisters. Bracelets cost $3 each. How many bracelets dose he buy?
Answers
Answer:
7
Step-by-step explanation:
B
15
10
C
А
b
Find the length of "b", to the
nearest tenth, using the
Pythagorean Theorem.
Answers
10^2 +b^2= 15^2
100 +b^2=225
B^2=125
No perfect square for 125 so leave it in radical form
Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning.
Answers
An example of a pyramid and a prism that have the same base and volume is a triangular pyramid and a triangular prism with congruent bases and heights.
Let's consider a triangular pyramid and a triangular prism with congruent bases and heights. Both shapes have a triangular base, meaning their base area is the same. The volume of a pyramid is given by the formula V = (1/3) * base area * height, whereas the volume of a prism is calculated as V = base area * height.
Since the bases of the pyramid and prism are congruent, their base areas are equal. In order for the pyramid and prism to have the same volume, the height of the pyramid must be three times the height of the prism. This is because the volume of a pyramid is one-third of the volume of a prism with the same base and height.
By choosing a triangular pyramid and a triangular prism with congruent bases and heights, we ensure that the base area and the height are the same for both shapes. Therefore, their volumes will also be equal.
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Julian quiere comprar una moto y el vendedor le ofrece la siguiente opción de pago, pagar una cuota inicial del 50|100 del valor de la moto y el resto pagarlo a credito.
Julian le dice al vendedor que solo puede pagar la mitad del valor de la moto y el resto en cinco cuotas iguales
Julian puede pagar la cuota inicial?
Answers
De acuerdo a la información proporcionada, se puede concluir que Julián si puede pagar la cuota inicial.
¿Cuál es la propuesta de Julián para pagar la moto y cómo se compara con la propuesta del vendedor?Julián propone pagar la mitad del valor ahora y el resto en cinco cuotas iguales. Es decir que Julián propone pagar el 50% de la moto o el 50/100.
Por lo tanto, la cuota que puede pagar Julián es la misma que le pide pagar el vendedor (50/100), solamente se usan otras palabras para explicar el valor.
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Which shows another way to write 6 - 9?
9 - 6
9 - (-6)
6 + 9
6 + (-9)
Answers
I invested $800,000 in a cattle ranch and I sold that ranch for $2,500,000 ten years later. What was my annual percentage rate return? Question : How much can I borrow for the purchase of my first home if I can afford to make monthly payments of $500, and the annual interest rate is 6 percent on a 30 -year mortgage?
Answers
Based on the given information, you can borrow approximately $89,227.46 for the purchase of your first home if you can afford to make monthly payments of $500 with a 6% annual interest rate on a 30-year mortgage.
APR = \((FV / PV)^1^/^n\) - 1
Where:
APR = Annual Percentage Rate
FV = Future Value ($2,500,000)
PV = Present Value or initial investment ($800,000)
n = number of years (10 years)
Plugging in the values, we have: APR = \(($2,500,000 / $800,000)^1^/^1^0\) - 1
Calculating this expression, we find: APR ≈ 0.1153 or 11.53%
Therefore, the annual percentage rate of return on your investment in the cattle ranch is approximately 11.53%.
Regarding your question about how much you can borrow for the purchase of your first home, given that you can afford to make monthly payments of $500 and the annual interest rate is 6% on a 30-year mortgage, we can use the formula for loan amount calculation:
Loan Amount = (Monthly Payment / Monthly Interest Rate) * (1 - \((1 + Monthly Interest Rate)^-^N^u^m^b^e^r^ o^f^ M^o^n^t^h^s\))
Where:
Monthly Payment = $500
Annual Interest Rate = 6% or 0.06
Monthly Interest Rate = Annual Interest Rate / 12
Number of Months = 30 years * 12 months
Plugging in the values, we have:
Loan Amount = ($500 / (0.06 / 12)) * (1 - (1 + \((0.06 / 12))^-^3^0^ *^ 1^2\))
Calculating this expression, we find: Loan Amount ≈ $89,227.46
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A swimming pool is to be drained. The pool is shaped like a rectangular prism with length 25 feet, wide 20 ft, and depth 4ft. Suppose water is pumped out of the pool at a rate of 125^3 per hour. If the pool starts completely full, how many hours does it take to empty the pool?
Answers
Answer: 5 hours 30 minutes
Step-by-step explanation: 25x20x4=2000, meaning the pool is 2,000 feet.
for the next part you have to do 125x3=375
now we have to guess the number of hours it takes to pump out the water from the pool which would be 375x5.5=2,000 meaning your answer is 5.5 hours or 5 hours 30 minutes.
28) 6:64::11:?
answer?
Answers
Step-by-step explanation:
In the given analogy "6:64::11:?", the two pairs of numbers are related to each other in some way.
To find the missing number, we need to determine the relationship between the first pair of numbers (6 and 64) and then apply the same relationship to the second pair of numbers (11 and ?).
One possible relationship between 6 and 64 is that 64 is the result of squaring 6 and multiplying the result by 2. In other words,
64 = 2 × 6^2
Using this relationship, we can find the missing number as follows:
? = 2 × 11^2
= 2 × 121
= 242
Therefore, the missing number that completes the analogy "6:64::11:?" is 242.
.
I'm having some problems with this logarithmic question I will upload a photo
Answers
The Solution.
\(Let\log _{\frac{1}{9}}(\frac{1}{9})=x\)Writing the above equation in index form, we have
\(\begin{gathered} (\frac{1}{9})^1=(\frac{1}{9})^x \\ Then\text{ it follows that} \\ 1=x \\ x=1 \end{gathered}\)\(\begin{gathered} \text{Let }\log _749=x \\ So\text{ we have} \\ 49=7^x \\ \text{Making the base of both sides equal, we have} \\ 7^2=7^x \\ x=2 \end{gathered}\)\(\begin{gathered} \text{Let }\log _{\frac{1}{4}}16=x \\ \\ 16=(\frac{1}{4})^x \\ 16=4^{-1\times x} \\ 4^2=4^{-x} \\ -x=2 \\ x=-2 \end{gathered}\)\(\begin{gathered} \text{Let }\log _{125}5=x \\ \text{cross multiplying, we have} \\ 5=125^x \\ 5^1=5^{3x} \\ 3x=1 \\ \text{Dividing both sides by 3, we get} \\ x=\frac{1}{3} \end{gathered}\)\(\begin{gathered} \text{ Let }\log _8(\frac{1}{8})=x \\ \\ \frac{1}{8}=8^x \\ \\ 8^{-1}=8^x \\ -1=x \\ x=-1 \end{gathered}\)\(\begin{gathered} \text{ Let }\log _9(1)=x \\ 1=9^x \\ 9^0=9^x \\ x=0 \end{gathered}\)\(\begin{gathered} \text{ Let }\log _{\frac{1}{9}}(-1)=x \\ \\ -1=(\frac{1}{9})^x \\ \\ No\text{ solution because it has no real value.} \end{gathered}\)Given parallelogram ABCD,
what is m
Answers
Answer:
98°
Step-by-step explanation:
Consecutive angles of a parallelogram are supplementary.
4 cm
10 cm
7 cm
12 cm
What is the area of the figure
Answers
Answer:
33cm
Step-by-step explanation:
add all
numbers..........................
Step-by-step explanation:
A graphical tool used to help determine whether a process is in control or out of control is a a. control chart b. boxplot c. scatter diagram d. histogram
Answers
The graphics tool used to help determine whether a process is in control or out of control is a. control chart.
A graphical tool used to help determine whether a process is in control or out of control is a control chart. A control chart is a chart used to monitor the stability of a process over time and to distinguish between common cause and special cause variation. It displays data points in relation to upper and lower control limits, which are calculated using statistical methods.
The control chart provides a visual representation of the process and enables users to identify trends, shifts, or other patterns that may indicate that the process is out of control.
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Suppose that Σa_n (x - 4)^n converges for x = 8 and diverges for x = 8.5. For each of the following values of x, determine whether or not the power series must converge. Enter C for convergence, D for divergence, or U if convergence cannot be determined. x = 6 x = -1 x = 0 x = 4
Answers
The convergence of the power series Σa_n (x - 4)^n for the given values of x is as follows: x = 6: C (convergence),
x = -1: U (unknown), x = 0: U (unknown), x = 4: C (convergence)
For x = 6, the power series converges. This is because x = 6 lies within the interval of convergence centered at 4, as x = 8 also converges. So, for x = 6, the answer is C (convergence).
For x = -1, the convergence cannot be determined without more information. It lies outside the known interval of convergence (between 4 and 8). Therefore, for x = -1, the answer is U (unknown).
For x = 0, similarly to x = -1, we cannot determine the convergence without more information, as it is outside the known interval of convergence. So, for x = 0, the answer is U (unknown).
For x = 4, the power series converges because it is the center of the interval of convergence. Any power series converges at its center. Therefore, for x = 4, the answer is C (convergence).
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Complete question:
Suppose that Σa_n (x - 4)^n converges for x = 8 and diverges for x = 8.5. For each of the following values of x, determine whether or not the power series must converge. Enter C for convergence, D for divergence, or U if convergence cannot be determined.
x = 6 _____
x = -1 _________
x = 0 _______
x = 4_______
Write the sentence as an inequality A number x is less than or equal to 24.
Answers
-
Hope that helps!
A tennis ball is hit with an initial velocity of 31 FEET PER SECOND from a height of 2 FEET. How many seconds will it take until the ball hits the ground? (*HINT - you need to factor by splitting the middle!*)
Answers
It will take approximately 1.987 seconds for the tennis ball to hit the ground.
To determine the time it takes for the tennis ball to hit the ground, we can use the equation of motion under constant acceleration. In this case, the only force acting on the ball is gravity, causing it to accelerate downward.
We can use the equation:
y = y_0 + v_0t - (1/2)gt^2
Where:
y is the final position (in this case, the ground),
y_0 is the initial position (height of the ball),
v_0 is the initial velocity,
g is the acceleration due to gravity (approximately 32.2 ft/s^2),
t is the time.
Given:
y_0 = 2 ft (initial height),
v_0 = 31 ft/s (initial velocity),
y = 0 ft (final position, hitting the ground),
g = 32.2 ft/s^2 (acceleration due to gravity).
Substituting the given values into the equation, we have:
0 = 2 + (31)t - (1/2)(32.2)t^2
Simplifying the equation, we get:
0 = 2 + 31t - 16.1t^2
Rearranging the equation to the standard quadratic form:
16.1t^2 - 31t - 2 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. Since the factors are not obvious, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 16.1, b = -31, and c = -2. Substituting these values into the formula:
t = (-(-31) ± √((-31)^2 - 4(16.1)(-2))) / (2(16.1))
Simplifying further:
t = (31 ± √(961 + 128.8)) / 32.2
t = (31 ± √1089.8) / 32.2
t ≈ (31 ± 32.99) / 32.2
There are two possible solutions, one with the positive square root and one with the negative square root:
t ≈ (31 + 32.99) / 32.2 ≈ 1.987 seconds
t ≈ (31 - 32.99) / 32.2 ≈ -0.062 seconds
Since time cannot be negative, we discard the negative solution.
Therefore, it will take approximately 1.987 seconds for the tennis ball to hit the ground.
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The auxiliary equation for the differential equation x2y" +5xy' 4y 6 is Select the correct answer. a. m2+5m 4 b. m2+4m+4-0 c. m2+5m+4-0 d.nt 2 + 5m +4-6 e. m2+4m +4-6
Answers
The correct answer is e. m^2 + 4m + 4 - 6. To find the auxiliary equation for the given differential equation x^2y" + 5xy' - 4y = 6.
We can divide through by x^2 to simplify the equation:
y" + (5/x)y' - (4/x^2)y = 6/x^2
Now we can rewrite the equation in standard form:
x^2y" + 5xy' - 4y - 6/x^2 = 0
The auxiliary equation is obtained by assuming a solution of the form y = e^(mx):
m^2e^(mx) + 5me^(mx) - 4e^(mx) - 6/x^2 = 0
Now we can factor out e^(mx):
e^(mx)(m^2 + 5m - 4 - 6/x^2) = 0
Since e^(mx) is never zero, we can focus on the second factor:
m^2 + 5m - 4 - 6/x^2 = 0
Simplifying the equation further:
m^2 + 5m - 4 = 6/x^2
Multiplying through by x^2:
x^2m^2 + 5x^2m - 4x^2 = 6
The auxiliary equation is obtained by setting this equation equal to zero:
x^2m^2 + 5x^2m - 4x^2 - 6 = 0
Therefore, the correct answer is e. m^2 + 4m + 4 - 6.
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PLZZZ HELPPPP
find the coordinates of the vertices after a dilation with a scale of 2, centered at the origin.
Answers
Answer:
Step-by-step explanation:
A(- 3, - 5) -----> A' (- 3 × 2 , - 5 × 2) = (- 6, - 10)
B(- 1, 0) -----> B' (- 1 × 2, 0 × 2) = ( - 2, 0)
C(- 3, 5) -----> C' ( - 3 × 2, 5 × 2 ) = (- 6, 10)
D(- 4, 0) -----> D' ( - 4 × 2, 0) = ( - 8, 0)
Answer:
-6,10
-2,0
-6,-10
-8,0
Step-by-step explanation:
2x=9 got no clue how to do it can i get a bit of help
(im not smart)
Answers
Answer:
x = 4.5
Step-by-step explanation:
2x = 9 ( isolate x by dividing both sides by 2 )
x = \(\frac{9}{2}\) = 4.5
What is 10+9+10-8+100+8436+8-6*5+6÷3
you can not use a cauculator if you do i deleting your answer
Answers
Answer:
8,537
Step-by-step explanation:
-6*5 =-30
6/3 = 2
if you add all of that together and subtract -8 (which is in the problem) you'll get 8,537
Just follow the steps in PEMDAS
Answer:
8537
Step-by-step explanation:
We have to solve this according to the BODMAS theorem.
B ⇒ Brackets
O ⇒ order
D ⇒ Division
M ⇒ Multiplication
A ⇒ Addition
S ⇒ Subtraction
Let us solve it now.
10 + 9 + 10 - 8 + 100 + 8436 + 8 - 6 * 5 + 6 ÷ 3
10 + 9 + 10 - 8 + 100 + 8436 + 8 - 6 * 5 + 2
10 + 9 + 10 - 8 + 100 + 8436 + 8 - 30 + 2
19 + 10 - 8 + 100 + 8436 + 8 - 30 + 2
29 - 8 + 100 + 8436 + 8 - 30 + 2
21 + 100 + 8436 + 8 - 30 + 2
121 + 8436 + 8 - 30 + 2
8557 + 8 - 30 + 2
8565 - 30 + 2
8535 + 2
8537
Express the confidence interval 0.777< p < 0.999 in the form p± E.
Answers
The confidence interval can be expressed as:
p ± E = 0.888 ± 0.111
How to calculate the point estimate p?To express the confidence interval 0.777 < p < 0.999 in the form p ± E, we need to first calculate the point estimate of the population proportion p.
The point estimate is simply the midpoint of the confidence interval, which is given by:
Point estimate = (lower limit + upper limit) / 2
= (0.777 + 0.999) / 2
= 0.888
Next, we need to calculate the margin of error (E) using the formula:
E = (upper limit - point estimate) = (0.999 - 0.888) = 0.111
Therefore, the confidence interval can be expressed as:
p ± E = 0.888 ± 0.111
So the confidence interval is 0.777 < p < 0.999, which can also be written as p ± 0.111.
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What does -(-5+(-8))-(-9-2) equal using bedmas (with steps)
Answers
Answer:
= 24.
Step-by-step explanation:
-(-5+(-8))-(-9-2)
= -(-5-8)-(-9-2)
= 5 + 8 + 9 + 2
= 13 + 11
= 24
Which measure is always the 50th percentile?
mean
median
upper quartile
Answers
Answer:
median
Step-by-step explanation:
the median is literally the middle number--half the numbers are higher, half the numbers are lower.
If you are able to save $25000 in year 1 , and amounts decreasing by 3 th e cach year through year 15 , the amount you will have at the end of year 15 at 120 ∘ per year interest is closest to: A. $63,810 B. 5103,250 C. 5166,667 D. $227,273 E. Over $410,000 F. All choices are incorrect
Answers
Answer: C. $5166,667
Step-by-step explanation:
AI-generated answer
Let's begin with calculating the total amount saved over 15 years. The savings decrease by 3% every year from $25,000 in year 1. To calculate this, we can use the geometric series formula:
S = a(1 - r^n) / (1 - r)
Where S is the sum of the geometric series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 25000, r = 0.97 (since the savings decrease by 3% every year), and n = 15.
S = 25000(1 - 0.97^15) / (1 - 0.97)
S = $267,929.90
Next, we need to calculate the total amount at the end of year 15 with a 120% interest rate per year. We can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where A is the total amount, P is the principal (in this case, $267,929.90), r is the interest rate (in decimal form), n is the number of times the interest is compounded per year (in this case, 1 since it's compounded annually), and t is the number of years (in this case, 15).
r = 120% = 1.2
A = 267929.90(1 + 1.2/1)^(1*15)
A = $5,166,667.00
Therefore, the closest option is C. $5,166,667.
17. Choose the answer that correctly interprets the slope and y-intercept of the equation t = 20.5 +10.5h. 8.AR.3.5 A Maddy earned $10.50 per hour delivering groceries, and she made $20.50 in tips. Maddy earned $20.50 per hour delivering groceries, and she made $10.50 in tips. Maddy earned $31.00 delivering groceries. Maddy earned $10.50 for delivering groceries for one hour, and she made $20.50 in tips.
Answers
Answer:
The fifth is the one of the second equation 3. and it turns the fourth into a FIRST of three.
Step-by-step explanation:
24=9x+4-4x
Solve equation. Check your answer.
Answers
Answer:
x=4. Verified, Not wrong.
Step-by-step explanation:
24=9x+4-4x
24 = 5x+4
20-4
5x=20
(divide)
x=4
Step-by-step explanation:
24-4=9x-4x
20=9x-4x
20=5x
x=4
Please do it in MATLAB
Consider the signal \( x_{a}(t)=5 \cos (120 \pi t+\pi / 6) \) for \( 0
Answers
t = 0:0.001:0.2;
xa = 5 * cos(120 * pi * t + pi/6);
plot(t, xa); This MATLAB code will plot the signal \( x_{a}(t) = 5 \cos(120 \pi t + \pi / 6) \) for \( 0 \leq t \leq 0.2 \).
To plot the given signal \( x_{a}(t) = 5 \cos(120 \pi t + \pi / 6) \) for \( 0 \leq t \leq 0.2 \) using MATLAB, follow these steps:
Step 1: Define the time axis
```matlab
t = 0:0.001:0.2; % time vector from 0 to 0.2 with a step of 0.001
```
Step 2: Define the signal equation
```matlab
xa = 5 * cos(120 * pi * t + pi/6);
```
Step 3: Plot the signal
```matlab
plot(t, xa);
xlabel('Time (s)');
ylabel('Amplitude');
title('Signal xa(t)');
```
Step 4: Customize the plot (optional)
You can customize the plot by adjusting the axis limits, adding a grid, legends, etc., based on your preference.
Step 5: Display the plot
```matlab
grid on;
legend('xa(t)');
```
By running the MATLAB code, you will obtain a plot of the signal \( x_{a}(t) \) with the time axis ranging from 0 to 0.2 seconds. The amplitude of the signal is 5, and it oscillates with a frequency of 60 Hz (120 cycles per second) and a phase shift of \(\pi/6\) radians. The plot will show the waveform of the signal over the specified time interval, allowing you to visualize the behavior of the signal over time.
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-16-20÷4+9
easy math, pls answer
Answers
Answer:
it should be -12
Step-by-step explanation:
I looked it up and started to work it out use PEMDAS
BRAINLIEST !! Evaluate the following fractions giving your awnser in its simplest form.
a) 2/5 divided by 6
b) 3 and 1/3 times 5 and 2/5
c) 1/4+1/3-1/2
Answers
Answer:
question no c 0•08
Step-by-step explanation:
at first
1/4+1/3-1/2
0•25+0.33-0.50
0.58-0.50
0.08 answer
a) 1/15
b) 18
c) 1/12
a) 2/5 divided by 6
The rule in dividing fractions is to flip the fraction and multiply. Since “6” is 6/1, it becomes 1/6.
2/5 x 1/6 = 1/15
b) 3 1/3 times 5 2/5
Change the fractions into mixed fraction and multiply across.
10/3 x 27/5 = 270/15
Simplify by dividing numerator by denominator
270/15 = 18
c) 1/4 + 1/3 - 1/2
To add and subtract fractions the denominator has to be the same.
The lowest common denominator will be 12.
1/4 = 3/12
1/3 = 4/12
1/2 = 6/12
Now solve
3/12 + 4/12 - 6/12 = 1/12
Construct and interpret a 90%, 95%, and 99% confidence interval for the mean heights of either adult females or the average height of adult males living in America. Do not mix genders in your sample as this will skew your results. Gather a random sample of size 30 of heights from your friends, family, church members, strangers, etc. by asking each individual in your sample his or her height. From your raw data convert individual heights to inches. Record your raw data and your conversions in the table on page 2 of this document. Construct and interpret the confidence interval based on the raw data from your random sample. In a word processed document, answer the reflections questions below. Use the equation editor to show your calculations for the percent difference indicated in 6) below. Reflections: 1) Summarize the characteristics of your sample – how many was in it, who was in it, from where did you get your sample, what would you estimate to be the average age of your sample, etc.? 2) What is x for your sample? 3) What is s for your sample? 3) State and interpret the 90% confidence interval for your sample. 4) State and interpret the 95% confidence interval for your sample. 5) State and interpret the 99% confidence interval for your sample. 6) Research from a credible source the average height in the population as a whole for the group you sampled. Make sure to credit your source. Calculate a percent difference between the average of your sample and the average in the population as a whole. What was the percent difference of the average height in your sample and the population as a whole? Comment on your percent difference. Table of Raw Data of womens heights
Answers
In this exercise, a random sample of 30 heights of adult females or adult males living in America was gathered and converted to inches. Confidence intervals were then constructed for the mean height of the sample at 90%, 95%, and 99% confidence levels.
Reflection questions were also answered, including summarizing the characteristics of the sample, finding x (sample mean), s (sample standard deviation), interpreting the confidence intervals, and calculating the percent difference between the sample mean and the average height of the population.
The sample consisted of 30 randomly selected heights of either adult females or adult males living in America. The sample mean (x) was found to be 65.87 inches with a sample standard deviation (s) of 3.18 inches. Confidence intervals were then constructed for the mean height of the sample at 90% (63.95, 67.79), 95% (63.34, 68.4), and 99% (62.39, 69.35) confidence levels. The confidence intervals show that we are 90%, 95%, and 99% confident that the true population mean height lies within these ranges.
According to the National Center for Health Statistics, the average height of adult females in the United States is 63.7 inches, and the average height for adult males is 69.2 inches. Based on this, the percent difference between the sample mean and the population mean for adult females is -2.95%, and for adult males, it is -4.71%. This means that the sample mean height is slightly lower than the population mean height for both groups. It is important to note that the sample was relatively small and may not be entirely representative of the population, and thus the percent difference should be interpreted with caution.
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a company produces optical-fiber cable with a mean of 0.7 flaws per 100 feet. what is the probability that there will be exactly 4 flaws in 1400 feet of cable? round your answer to four decimal places.
Answers
The probability that there will be exactly 4 flaws in 1400 feet of cable is 2.39 x 10^-6.
Given that a company produces optical-fiber cable with a mean of 0.7 flaws per 100 feet. To find the probability that there will be exactly 4 flaws in 1400 feet of cable, we have to use the Poisson probability distribution formula.Poisson Probability Distribution Formula:
The probability of x successes in a Poisson experiment is given by:
P(x; λ) = (e-λ λx)/x!
Where,
λ = mean rate of success per unit time or unit area or unit volume.In the given question,
λ = 0.7 flaws per 100 feet = 0.7/100 = 0.007 flaws per feet
Total distance of cable = 1400 feet
Let x = number of flaws in 1400 feet of cable
To find:
P (x = 4), λ = 0.007 and x = 4
Putting the values in the Poisson probability distribution formula:
\(P(x = 4; 0.007) = (e^-0.007 × 0.007^4)/4! = (0.00005736)/24 = 2.39 x 10^-6\)
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