Answer:
After solving the power:
[tex]\bold{2(cos60^\circ+isin60^\circ)}[/tex]
Rectangular form:
[tex]\bold{1+i\sqrt3}[/tex]
Step-by-step explanation:
Given the complex number:
[tex]2(cos20^\circ+isin20^\circ)^3[/tex]
To find:
The indicated power by using De Moivre's theorem.
The complex number in rectangular form.
Rectangular form of a complex number is given as [tex]a+ib[/tex] where a and b are real numbers.
Solution:
First of all, let us have a look at the De Moivre's theorem:
[tex](cos\theta+isin\theta )^n=cos(n\theta)+isin(n\theta )[/tex]
First of all, let us solve:
[tex](cos20^\circ+isin20^\circ)^3[/tex]
Let us apply the De Moivre's Theorem:
Here, n = 3
[tex](cos20^\circ+isin20^\circ)^3 = cos(3 \times 20)^\circ+isin(3 \times 20)^\circ\\\Rightarrow cos60^\circ+isin60^\circ[/tex]
Now, the given complex number becomes:
[tex]2(cos60^\circ+isin60^\circ)[/tex]
Let us put the values of [tex]cos60^\circ = \frac{1}{2}[/tex] and [tex]sin60^\circ = \frac{\sqrt3}{2}[/tex]
[tex]2(\dfrac{1}{2}+i\dfrac{\sqrt3}2)\\\Rightarrow (2 \times \dfrac{1}{2}+i\dfrac{\sqrt3}2\times 2)\\\Rightarrow \bold{1 +i\sqrt3 }[/tex]
So, the rectangular form of the given complex number is:
[tex]\bold{1+i\sqrt3}[/tex]
STUCK Basic geometry A for senior year school
Answer:
(C) A reflection across a horizontal line and a horizontal translation
Step-by-step explanation:
We can see that, near the x-axis, these shapes are 3 y values away from the x-axis, meaning that if we reflect one over the x-axis we will be at the same y values as the other shape.
Reflecting these points shows that we’ve got the same shape, just skewed the one side. We can then translate this shape horizontally to get it to where we want it.
Hope this helped!
Jaclyn is one-fourth of a foot taller than John. John is 31/6 feet tall. How many feet tall is Jaclyn
Answer:
5 5/12
Step-by-step explanation:
31/6 feet + 1/4 foot
= 31/6 + 1/4
= [(31 * 4) / 6 * 4] + [(1 * 6) / 4 * 6]
= [ 124/24 ] + [ 6/24 ]
= (124 + 6) / 24
= 130 / 24
= 5 10/24
= 5 5/12
Hope this helps! Tell me if I'm wrong!
Consider F and C below.
F(x, y) = x2 i + y2 j
C is the arc of the parabola y = 2x2 from (−1, 2) to (2, 8)
(a) Find a function f such that F = ∇f. f(x, y) =
(b) Use part (a) to evaluate C ∇f · dr along the given curve C.
(a)
[tex]\dfrac{\partial f}{\partial x}=x^2\implies f(x,y)=\dfrac{x^3}3+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=\dfrac{\mathrm dg}{\mathrm dy}=y^2\implies g(y)=\dfrac{y^3}3+C[/tex]
[tex]\implies f(x,y)=\dfrac{x^3+y^3}3+C[/tex]
(b)
[tex]\displaystyle\int_C\nabla f\cdot\mathrm d\mathbf r=f(2,8)-f(-1,2)=\boxed{171}[/tex]
List the sides of ΔRST in ascending order (shortest to longest). m∠R=2x+11°, m∠S=3x+23°, and m∠T=x+42°
Answer:
ST, RS, RT
Step-by-step explanation:
Angles of a triangle add up to 180°.
2x + 11° + 3x + 23° + x + 42° = 180°
6x + 76° = 180°
x = 17⅓
m∠R = 2x+11° = 45⅔°
m∠S = 3x+23° = 75°
m∠T = x+42° = 59⅓°
The shortest side is opposite the smallest angle, and the longest side is opposite the largest angle.
ST, RS, RT
I buy 6 CDs costing £6.99.
How much change do I get from
£50?
Answer:
subtract the 6.99 from 50 and you get your answer
Step-by-step explanation:
Answer:
8.06
Step-by-step explanation:
Take the cost of a cd and multiply by 6
6.99 *6
41.94
Subtract this from 50.00
50.00 - 41.94
8.06
You will get 8.06 back
AC is included between __________
Answer:
AC is included between <A and <C.
Step-by-step explanation:
i think, not sure tho
A 95% confidence interval for the mean number of television per American household is (1.15, 4.20). For each of the following statements about the above confidence interval, choose true or false.
a. The probability that u is between 1.15 and 4.20 is .95.
b. We are 95% confident that the true mean number of televisions per American household is between 1.15 and 4.20.
c. 95% of all samples should have x-bars between 1.15 and 4.20 televisions.
d. 95% of all American households have between 1.15 and 4.20 televisions
e. Of 100 intervals calculated the same way (95%), we expect 95 of them to capture the population mean.
f. Of 100 intervals calculated the same way (95%), we expect 100 of them to capture the sample mean.
Answer:
a. False
b. True
c. False
d. False
e.True
f. True
Step-by-step explanation:
The 95% is confidence interval its not a probability estimate. The probability will be different from the confidence interval. Confidence interval is about the population mean and is not calculated based on sample mean. Every confidence interval contains the sample mean. There is 95% confidence that the number of televisions per American household is between 1.15 to 4.20.
idk how to put the picture but can someone just tell me the points where the two dots go plz will give good rate nd say thx plz answer fast
Graph -8x-y=8
Answer:
(-1,0) and (0,-8)
Step-by-step explanation:
Hey there!
Well first we’ll graph -8x - y = 8,
Look at the image below
By lookig at the image below we can tell the 2 points are at,
(-1,0) and (0,-8)
Hope this helps :)
A jar contains 8 pennies, 5 nickels and 7 dimes. A child selects 2 coins at random without replacement from the jar. Let X represent the amount in cents of the selected coins. Be very precise with your answers.
a. Find the probability x = 2 cents.
b. Find the probability x = 6 cents.
c. Find the probability x = 10 cents.
d. Find the probability x = 11 cents.
e. Find the probability x = 15 cents.
f. Find the probability x = 20 cents.
g. Find the expected value of x.
Answer:
a. The probability x = 2 cents = 7/22
b. The probability x = 6 cents = 35/66
c. The probability x = 10 cents = 5/33
d. The probability x = 11 cents= 28/33
e. The probability x = 15 cents = 20/33
f. The probability x = 20 cents = 14/33
g. The expected value of x = 5.9
Step-by-step explanation:
This is a binomial probability distribution. The number of trials is known .
a. The probability x = 2 cents.
Probability ( X=2) P( selecting 2 dimes)= 7C2 / 12c2
= 21 / 66 = 7/22
b. The probability x = 6 cents.
Probability ( X=6) P( selecting a nickel and a dime)= 5C1 * 7C1/ 12c2
= 5*7 / 66 = 35/66
c. The probability x = 10 cents.
Probability ( X=10) P( selecting two nickels )= 5C2 / 12c2)
= 10/ 66 = 5/33
d. The probability x = 11 cents.
Probability ( X=11) P( selecting a penny and a dime)= 8C1 * 7C1/ 12c2)
= 8*7 / 66 = 56/66= 28/33
e. The probability x = 15 cents.
Probability ( X=15) P( selecting a penny and a nickel)= 8C1 * 5C1/ 12c2)
= 8*5 / 66 = 40/66= 20/33
f. The probability x = 20 cents.
Probability ( X=20) P( selecting 2 pennies )= 8C2 / 12c2)
= 28 / 66 = 14/33
g. The expected value of x.
E(X) = np
E(X) = 2 * (8C2+ 5C2+ 7C2)/(8+5+7) = 2( 28+10+21)/20
=2(59)/20= 5.9
A professor of women's studies is interested in determining if stress affects the menstrual cycle. Ten women are randomly sampled for an experiment and randomly divided into two groups. One of the groups is subjected to high stress for two months while the other lives in a relatively stress-free environment. The professor measures the menstrual cycle (in days) of each woman during the second month. The following data are obtained.
High stress 20 23 18 19 22
Relatively stress free 26 31 25 26 30
Required:
1. The obtained value of the appropriate statistic is _______
a. tobt = -4.73
b. tobt = -4.71
c. tobt = -3.05
d. tobt = -.047
2. The df for determining tcrit are ____.
a. 4
b. 9
c. 8
d. 3
3. Using a = 0.052 tail, tcrit = ____.
a. +- 2.162
b. +- 2.506
c. +- 2.462
d. +- 2.306
4. Using a = .052 tail, your conclusion is ____.
a. Accept H0; stress does not affect the menstrual cycle
b. Retain H0; we cannot conclude that stress affects the menstrual cycle
c. Retain H0; Stress affects the menstrual cycle
d. Reject H0; stress affects the menstrual cycle
Answer:
1. a. tobt = -4.73
2. b. 9
3. a. +- 2.162
4. d. Reject H0; stress affects the menstrual cycle.
Step-by-step explanation:
The degrees of freedom is number of independent variable factors that affect the range of parameters. The degrees of freedom is the calculation of number values that are free to vary. The degrees of freedom is calculated by N-1. Standard error is the estimated deviation of standard deviation from its sample mean distribution. The null hypothesis is rejected or accepted on the basis of level of significance. When the p-value is greater than level of significance we fail to reject the null hypothesis and null hypothesis is then accepted.
(a^8)3/2 in simplest form
Answer:
[tex]\large\boxed{\frac{3}{2}a^{8}}[/tex]
Step-by-step explanation:
([tex]a^{8}[/tex]) * [tex]\frac{3}{2}[/tex]
Remove the parenthesis by multiplying
[tex]\frac{3}{2}[/tex][tex]a^{8}[/tex]
This expression cannot be simplified further
[tex]\large\boxed{\frac{3}{2}a^{8}}[/tex]
Hope this helps :)
In this diagram, bac~edf. if the area of bac= 6 in.², what is the area of edf? PLZ HELP PLZ PLZ PLZ
Answer:
2.7 in²
Step-by-step explanation:
Given that ∆BAC ~ is similar to ∆EDF, the ratio of the area of ∆BAC to the area of ∆EDF = the square of the ratio of their corresponding sides.
Thus, let x be the area of ∆EDF
[tex] \frac{6}{x} = (\frac{3}{2})^2 [/tex]
[tex] \frac{6}{x} = \frac{9}{4} [/tex]
Cross multiply
[tex] x*9 = 4*6 [/tex]
[tex] 9x = 24 [/tex]
[tex] \frac{9x}{9} = \frac{24}{9} [/tex]
[tex] x = 2.67 [/tex]
Area of ∆EDF = 2.7 in²
This??? What is wrong with it?
Answer:
15.8 sq. in. of paper will be required.
Step-by-step explanation:
The problem is that a drinking cup does not have a cover, so only the lateral surface area counts.
I.e. We need only the first term.
A = pi r l = pi * 1.5 * sqrt(3^2+1.5^2)
= 15.81 sq. in.
Which set of numbers is arranged from least to greatest?
Answer:
The correct answer is option B
For the given data value, find the standard score and the percentile. A data value 0.6 standard deviations above the mean.
Answer:
The standard score [tex]z = 0.6[/tex]
The the percentile [tex]P(z < 0.6 ) = 72.57\%[/tex]
Step-by-step explanation:
From the question we are told that
A data value is 0.6 standard deviations above the mean.
The above statement can be mathematically represented as
[tex]x = 0.6 \sigma + \mu[/tex]
Where [tex]\sigma[/tex] is the standard deviation and [tex]\mu[/tex] is the mean
Generally the standard score is mathematically represented as
[tex]z = \frac{x - \mu}{\sigma}[/tex]
=> [tex]z = \frac{(0.6 \sigma + \mu) - \mu}{\sigma}[/tex]
=> [tex]z = 0.6[/tex]
Now the percentile is obtained from the z-table , the value is
[tex]P(z < 0.6 ) = 0.7257[/tex]
=> [tex]P(z < 0.6 ) = 72.57\%[/tex]
Evaluating function expressions
-1•f(-8)-4•g(4)=
Answer: -7
Step-by-step explanation:
To find f(-8) look at the f function. Find the y value when x = -8
To find g(4) look at the g function. Find the y value when x = 3
Plug these values into the equation
-1 f(-8) - 4 f(4)
-1 (-5) - 4 (3)
5 - 12 = -7
Given y(x) = f(x)g(x). Find the slope of the tangent line to y(x) at x = 7.
Answer:
Step-by-step explanation:
Interesting problem.
At 6<x<8,
f(x) = x-7
at 5<x<8
g(x) = (15-x)/2
=>
y(x)
= f(x)*g(x)
= (x-7)(15-x)/2
= (x^2+22x-105)/2
differentiate y(x) with respect to x,
y'(x) = -x+11
at x = 7,
y'(7) = -(7) + 11 = 4
-8 + (-15)
Evaluate this expression
Answer:
-23
Step-by-step explanation:
-8+(-15) means that you are subtracting 15 from -8. So you end up with -8-15=-23.
How much will $1000 deposited in an account earning 7% interest compounded annually be worth in 20 years? (which formula do I use? I am confused...txs)
Answer:
$3870
Step-by-step explanation:
Hello, the initial deposit is $1000.
After one year, we will get 1000 + 7%*1000= 1000 * ( 1+7%) = 1000 * (1+0.07)
= 1000 * 1.07
And we want to compound it so the second year we will get
[tex]1000 * 1.07 * 1.07 = 1000 * 1.07^2[/tex]
And after n years, we will get
[tex]1000 * 1.07^n[/tex]
In that example, we want to know how much we will get after 20 years, so this is:
[tex]1000 * 1.07^{20}=3869.684462...[/tex]
Thank you.
When interest is compounded, it means that both the interest and the amount deposited will earn interest.
We are to determine the future value of $1000 with annual compounding.
The formula for calculating future value:
FV = P (1 + r)^n
FV = Future value
P = amount deposited = $1000
R = interest rate = 7%
N = number of years = 20
$1000 x ( 1.07)^20 = $3,869.68
To learn more about compound interest, please check: https://brainly.com/question/14295570?referrer=searchResults
Simplify to create an equivalent expression. 7n-(4n-3) a=3n+3 b=3n−3 c=11n+3 d=11n−3 I will rate you brainliest. :)
Answer:
3n +3
Step-by-step explanation:
7n-(4n-3)
Distribute the minus sign
7n -4n +3
3n +3
This solid shape is made from 5 cubes. Which of the diagrams show the plan of the solid? Please help!
Answer:
A Maybe
Step-by-step explanation:
Cogntive identification
Unable to answer mathematically or analytically
The Plan of the solid shape is shown by : (A)
What is the Meaning of solid shape?A solid shape can be defined as a shape that possesses three dimensions. that is to say they are three dimensional shapes.
A solid shape has both length, width and height. They are more tangible and look physical than two dimensional shape.
solid shapes can take up space in the universe because they are more tangible and realistic.
In conclusion, the Plan of the solid shape is shown by : (A)
Learn more about solid shape: https://brainly.com/question/16717260
#SPJ2
Carl recorded the number of customers who visited his new store during the week:
Day Customers
Monday 17
Tuesday 13
Wednesday 14
Thursday 16
He expected to have 15 customers each day. To answer whether the number of customers follows a uniform distribution, a chi-square test for goodness of fit should be performed. (alpha = 0.10)
What is the chi-squared test statistic? Answers are rounded to the nearest hundredth.
Answer:
The chi - square test can be [tex]\approx[/tex] 0.667
Step-by-step explanation:
From the given data :
The null hypothesis and the alternative hypothesis can be computed as:
Null hypothesis: The number of customers does follow a uniform distribution
Alternative hypothesis: The number of customers does not follow a uniform distribution
We learnt that: Carl recorded the number of customers who visited his new store during the week:
Day Customers
Monday 17
Tuesday 13
Wednesday 14
Thursday 16
The above given data was the observed value.
However, the question progress by stating that : He expected to have 15 customers each day.
Now; we can have an expected value for each customer as:
Observed Value Expected Value
Day Customers
Monday 17 15
Tuesday 13 15
Wednesday 14 15
Thursday 16 15
The Chi square corresponding to each data can be determined by using the formula:
[tex]Chi -square = \dfrac{(observed \ value - expected \ value )^2}{expected \ value}[/tex]
For Monday:
[tex]Chi -square = \dfrac{(17 - 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(2)^2}{15}[/tex]
[tex]Chi - square = \dfrac{4}{15}[/tex]
chi - square = 0.2666666667
For Tuesday :
[tex]Chi -square = \dfrac{(13- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(-2)^2}{15}[/tex]
[tex]Chi - square = \dfrac{4}{15}[/tex]
chi - square = 0.2666666667
For Wednesday :
[tex]Chi -square = \dfrac{(14- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(-1 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )}{15}[/tex]
chi - square = 0.06666666667
For Thursday:
[tex]Chi -square = \dfrac{(16- 15 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )^2}{15}[/tex]
[tex]Chi -square = \dfrac{(1 )}{15}[/tex]
chi - square = 0.06666666667
Observed Value Expected Value chi - square
Day Customers
Monday 17 15 0.2666666667
Tuesday 13 15 0.2666666667
Wednesday 14 15 0.06666666667
Thursday 16 15 0.06666666667
Total : 0.6666666668
The chi - square test can be [tex]\approx[/tex] 0.667
At level of significance ∝ = 0.10
degree of freedom = n - 1
degree of freedom = 4 - 1
degree of freedom = 3
At ∝ = 0.10 and df = 3
The p - value for the chi - square test statistics is 0.880937
Decision rule: If the p - value is greater than the level of significance , we fail to reject the null hypothesis
Conclusion: Since the p - value is greater than the level of significance , we fail to reject the null hypothesis and conclude that there is insufficient evidence to show that the number of customers does not follows a uniform distribution.
Answer:.67
Step-by-step explanation:
HELP
PLSFind all the missing elements:
Answer:
a = 6.7 , c = 2.0
Step-by-step explanation:
For side aTo find the missing side a we use the sine rule
[tex] \frac{ |b| }{ \sin(B) } = \frac{ |a| }{ \sin(A) } [/tex]From the question
B = 58°
b = 6
A = 109°
Substituting the values into the above formula we have
[tex] \frac{6}{ \sin(58) } = \frac{ |a| }{ \sin(109) } [/tex][tex] |a| \sin(58) = 6\sin(109) [/tex]Divide both sides by sin 58°
[tex] |a| = \frac{6 \sin(108) }{ \sin(58) } [/tex]a = 6.728791
a = 6.7 to the nearest tenthFor side cTo find side c we use the sine rule
That's
[tex] \frac{ |b| }{ \sin(B) } = \frac{ |c| }{ \sin(C) } [/tex]C = 13°
[tex] \frac{6}{ \sin(58) } = \frac{ |c| }{ \sin(13) } [/tex][tex] |c| \sin(58) = 6 \sin(13) [/tex]Divide both sides by sin 58°
[tex] |c| = \frac{6 \sin(13) }{ \sin(58) } [/tex]c = 1.591544
c = 2.0 to the nearest tenthHope this helps you
Answer:
B=58 a=6.7 c=1.6
Step-by-step explanation:
It was right on Acellus
Sorry I cant give a better explanation but this unit is killing me.
Complete the recursive formula of the geometric sequence -0.3,0.9,-2.7,8.1
Answer:
[tex]a_{n}[/tex] = - 3[tex]a_{n-1}[/tex] with a₁ = - 0.3
Step-by-step explanation:
The recursive formula for a geometric sequence is of the form
[tex]a_{n}[/tex] = r[tex]a_{n-1}[/tex]
where r is the common ratio
r = [tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{0.9}{-0.3}[/tex] = - 3 , thus
[tex]a_{n}[/tex] = - 3[tex]a_{n-1}[/tex] with a₁ = - 0.3
I am performing a before and after evaluation on 30 students who have taken a keyboarding class. I want to see if the course improved their words per minute keyed.
Required:
a. State the Null and Alternate Hypothesis.
b. The statistic that I would use is:_________
c. What would my t critical be for this calculation at a 0.10 level of significance?
d. If my t calculated = 1.62, would I reject or fail to reject the null hypothesis?
Answer:
a)
H₀ : µd = 0
H₁ : µd < 0
b)
The test statistic is
tₙ₋₁ = α / s√n
c)
at 0.10 level of significance,
tₙ₋₁ , ₐ
t₃₀₋₁ , ₀.₁₀ = t₂₉, ₀.₁₀ = 1.311
d)
given that T(critical) = 1.62
∴ T(critical) = 1.62 > t₂₉, ₀.₁₀ = 1.311
at 10% level of significance,
REJECT H₀
Since 1.62 > 1.311, we can reject the null hypothesis.
1. Why is money better than a bartering system?
A People might not have items to trade.
B It helps people to agree on the value of something.
C People might lose track of their money.
D Both A and B
E Both B and C
The correct answer is D. Both A and B
Explanation:
Bartering is an economic system in which products are directly exchanged for other products. For example, a pound of oranges is exchanged for a pound of rice. Due to this, in bartering, there is no money or elements such as coins or bills that represent the value of products or services. This system has both advantages and disadvantages in comparison to the use of money.
In terms of disadvantages, bartering implies individuals need products or services they can use to exchange, which might not be possible for all individuals as not all individuals might produce a product or have a product other are interested in. Also, in bartering the value of products varies, for example, a pound of blueberries can be equal to a pound of rice, three pounds of rice, or even half pound of rice, as values change according to the situation of those participating in the exchange. This means, in bartering the value fluctuates and it is more difficult to agree on the value of something, which does not occur if money is used as each product has a defined price which might just vary slightly. According to this, options A and B are advantages of money over bartering.
The locksmith is 37.9 kilometers west of the bank and 77.9 kilometers west of the garbage dump. The garbage dump is 128.6 kilometers east of the pet store. The pet store is 35.0 kilometers north of the hardware store. Which is closer to the pet store, the hardware store or the locksmith?
The correct answer is Hardware store
Explanation:
In this case, you need to determine the distance between the pet store and the locksmith to know if the distance is longer than the one from the pet store and the hardware (35 kilometers). Now this distance can be calculated by subtracting the distance from the locksmith to the garbage dump from the total distance between the pet store and the garbage dump. This is because the locksmith is between the pet store and the garbage, which means the distance between the pet store and the locksmith is a portion of the total distance, which is the distance from the pet store and the garbage dump. The process is shown below:
128.6 kilometers (distance from the garbage dump to the pet store) - 77.9 kilometers (distance from the garbage dump to the locksmith) = 50.7 (distance from the locksmith to the pet store
50.7 kilometers is greater than 35 kilometers, which means the hardware store is closer to the pet store
If is an angle bisector of ∠QOR and ∠QOP = 71°, then find the angle measure of ∠QOR. Question 8 options: A) 71° B) 142° C) 35.5° D) 35°
Answer: B. 142°.
Step-by-step explanation:
Given: OP is an angle bisector of ∠QOR and ∠QOP = 71°.
We know that an angle bisector divides an angle into two equal parts.
So, if OP is an angle bisector of ∠QOR and ∠QOP = 71°.
Then, the angle measure of ∠QOR = Twice of ∠QOP
⇒ Angle measure of ∠QOR = 2 (71°)
⇒ Angle measure of ∠QOR = 142°
Hence, correct option is B. 142°.
An article contained the following observations on degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle range:
418 421 421 422 425 428 431 435 437
438 445 447 448 453 458 462 465
(c) Calculate a two-sided 95% confidence interval for true average degree of polymerization. (Round your answers to two decimal places.) Note that it is plausible that the given sample observations were selected from a normal distribution and there are no outliers.
(___ , ___)
Does the interval suggest that 441 is a plausible value for true average degree of polymerization?
Yes or No
Does the interval suggest that 451 is a plausible value?
Yes or No
Answer:
Step-by-step explanation:
Form a set of values we get
n = 17
And with the help of a calculator
μ₀ = 438,47
σ = 14,79
Normal Distribution is : N ( 438,47 ; 14,79 )
c)
CI = 95 % means α = 5 % α/2 = 2,5 % α/2 = 0,025
and as n < 30 we should use t-student distribution with n -1 degree of freedom df = 16. t score for 0,025 and 16 s from t-table 2,120
By definition:
CI = [ μ₀ ± t α/2 ; n-1 * σ/√n ]
CI = [ μ₀ ± 2,120* 14,79/√17 ]
CI = [ μ₀ ± 7,60 ]
CI = [ 438,47 ± 7,60 ]
CI = [ 430,87 ; 446,07 ]
95% confidence interval for true average degree of polymerization is [430.87 ; 446.07] and this interval suggest that 441 is a plausible value for true average degree of polymerization and also this interval does not suggest that 451 is a plausible value.
Given :
Sample = [ 418, 421, 421, 422, 425, 428, 431, 435, 437, 438, 445, 447, 448, 453, 458, 462, 465 ]95% confidence interval.The total number of values given is, n = 17
Mean, [tex]\mu_0=438.47[/tex]
Standard Deviation, [tex]\sigma = 14.79[/tex]
The normal distribution is given by: N (438.47 ; 14.79)
If Cl is 95% then [tex]\alpha[/tex] is 5% and [tex]\alpha /2[/tex] is 2.5%
[tex]\alpha /2 = 0.025[/tex]
Now, use t-statistics distribution with (n-1) degree of freedom df = 16
So, the t score for 0.025 and 16 s from t-table 2.120.
[tex]\rm Cl = [\mu_0 \pm t_{\alpha /2};(n-1)\times \dfrac{\sigma}{\sqrt{n} }][/tex]
[tex]\rm Cl = [\mu_0 \pm 2.120\times \dfrac{14.79}{\sqrt{17} }][/tex]
[tex]\rm Cl = [\mu_0 \pm 7.60][/tex]
Cl = [430.87 ; 446.07]
Yes, the interval suggests that 441 is a plausible value for true average degree of polymerization.
No, the interval does not suggest that 451 is a plausible value.
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https://brainly.com/question/2561151
Find the maximum and minimum values of the function f(x,y)=2x2+3y2−4x−5 on the domain x2+y2≤100. The maximum value of f(x,y) is:
First find the critical points of f :
[tex]f(x,y)=2x^2+3y^2-4x-5=2(x-1)^2+3y^2-7[/tex]
[tex]\dfrac{\partial f}{\partial x}=2(x-1)=0\implies x=1[/tex]
[tex]\dfrac{\partial f}{\partial y}=6y=0\implies y=0[/tex]
so the point (1, 0) is the only critical point, at which we have
[tex]f(1,0)=-7[/tex]
Next check for critical points along the boundary, which can be found by converting to polar coordinates:
[tex]f(x,y)=f(10\cos t,10\sin t)=g(t)=295-40\cos t-100\cos^2t[/tex]
Find the critical points of g :
[tex]\dfrac{\mathrm dg}{\mathrm dt}=40\sin t+200\sin t\cos t=40\sin t(1+5\cos t)=0[/tex]
[tex]\implies\sin t=0\text{ OR }1+5\cos t=0[/tex]
[tex]\implies t=n\pi\text{ OR }t=\cos^{-1}\left(-\dfrac15\right)+2n\pi\text{ OR }t=-\cos^{-1}\left(-\dfrac15\right)+2n\pi[/tex]
where n is any integer. We get 4 critical points in the interval [0, 2π) at
[tex]t=0\implies f(10,0)=155[/tex]
[tex]t=\cos^{-1}\left(-\dfrac15\right)\implies f(-2,4\sqrt6)=299[/tex]
[tex]t=\pi\implies f(-10,0)=235[/tex]
[tex]t=2\pi-\cos^{-1}\left(-\dfrac15\right)\implies f(-2,-4\sqrt6)=299[/tex]
So f has a minimum of -7 and a maximum of 299.
