Final answer:
To demonstrate that E[1/X] for a geometric random variable X with success probability p equals −p log(p)/(1−p), we use the pmf of a geometric distribution and turn the sum into an integral, ultimately showing the expected result through integration.
Explanation:
To show that for a geometric random variable X with parameter p, the expected value E[1/X] is −p log(p)/(1−p), we begin by recognizing that the probability mass function (pmf) of a geometric distribution with success probability p is given by P(X = x) = p(1-p)^(x-1) for x = 1, 2, 3, ... . To find E[1/X], we sum the values of 1/x multiplied by the probability of each x, which is ∑ (1/x)*p(1-p)^(x-1).
To evaluate this, we express each term α_i/i, where α_i = p(1-p)^(i-1), as an integral from 0 to 1-p of x^(i-1) dx. We then change the order of summation and integration, which is permissible under the conditions of the monotone convergence theorem:
∑ (α_i / i) = ∑ ∫_{0}^{1-p} x^(i-1) dx
= ∫_{0}^{1-p} (∑ x^(i-1)) dx
= ∫_{0}^{1-p} (p / (1 - x)) dx
= p ∫_{0}^{1-p} 1/(1 - x) dx
= -p log(1 - (1-p))
= -p log(p) / (1-p).
Given: dp/dt = k(M- P)
where P(t) measures the performance of someone learning a skill after a training time t, M is the maximum level of performance, and k is a positive constant. Solve this differential equation to find an expression for P(t). (Use P for P(t). Assume that P(0) = 0.)
Answer:
[tex]P=M(1-e^{-kt})[/tex]
Step-by-step explanation:
The relation between the variables is given by
[tex]\frac{dP}{dt} = k(M- P)[/tex]
This is a separable differential equation. Rearranging terms:
[tex]\frac{dP}{(M- P)} = kdt[/tex]
Multiplying by -1
[tex]\frac{dP}{(P- M)} = -kdt[/tex]
Integrating
[tex]ln(P-M)=-kt+D[/tex]
Where D is a constant. Applying expoentials
[tex]P-M=e^{-kt+D}=Ce^{-kt}[/tex]
Where [tex]C=e^{D}[/tex], another constant
Solving for P
[tex]P=M+Ce^{-kt}[/tex]
With the initial condition P=0 when t=0
[tex]0=M+Ce^{-k(0)}[/tex]
We get C=-M. The final expression for P is
[tex]P=M-Me^{-kt}[/tex]
[tex]P=M(1-e^{-kt})[/tex]
Keywords: performance , learning , skill , training , differential equation
The differential equation dp/dt = k(M - P) can be solved via separating variables, integrating and applying the initial condition. Result provides the equation for performance over time: P(t) = M(1 - e-kt).
Explanation:The subject of the question is around a differential equation. Firstly, you will rewrite the given equation dp/dt = k(M - P) in the form necessary for separation of variables: dp/(M - P) = k dt. Then, integrate both sides: ∫dp/(M - P) = ∫k dt. The left-hand side integral results in -ln|M - P|, and the right side is k*t + C, where C is the constant of integration. Finally, solve for P(t) by taking the exponential of both sides, and rearranging. The procedure results in the performance level equation.
P(t) = M - Ce-kt
Since we're given P(0) = 0, we can determine that C = M. Hence, we finally have the solution:
P(t) = M(1 - e-kt)
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A bakery decided to expand since business was going so well. Originally, the width was five times the length. Now they are adding 5 feet to the length. If the length was originally 15 feet, what will be the area of the new bakery?
Step-by-step explanation:
We need to find new area of bakery.
Old length = 15 feet
Old width = New width = 5 x Old length = 5 x 15 = 75 feet
New length = Old length + 5 feet = 15 + 5 = 20 feet
New area = New length x New width
New area = 75 x 20 = 1500 square feet.
New area of bakery = 1500 square feet.
Twenty-four blue socks and twenty-four red socks are lying in a drawer in a dark room. What is the minimum number of socks you need to pull out of the drawer so that you are sure to have at least two blue socks
Answer:
26 socks
Step-by-step explanation:
There are a total of 48 socks here. Let us assume you pulled out 24 socks at a go and all are red. Now, you would have exhausted the number of red socks here. You would be left with only blue socks which you can pull one after the other to give a total of 26 socks pulled out to have 2 blue socks at least.
A bag contains 222 red marbles, 222 green marbles, and 444 blue marbles.
If we choose a marble, then another marble without putting the first one back in the bag, what is the probability that the first marble will be red and the second will be green?
Answer:
1/14
Step-by-step explanation:
Assuming you mean that there are 2 red, 2 green, and 4 blue marbles, there are a total of 8 marbles.
On the first draw, the probability the marble is red is 2/8.
On the second draw, there's one less marble, so the probability of selecting a green marble is 2/7.
The total probability is:
2/8 × 2/7 = 1/14
The probability that the first marble is red and the second is green is approximately 0.0626 or 6.26%.
To find the probability we need to follow these steps:
Calculate the total number of marbles: 222 (red) + 222 (green) + 444 (blue) = 888 marbles.Determine the probability of drawing a red marble first.The probability of drawing a red marble first is:
P(Red) = Number of Red Marbles / Total Number of Marbles = 222 / 888 = 1/4 or 0.25.
After drawing a red marble, there are now 887 marbles left and still 222 green marbles in the bag.Calculate the probability of then drawing a green marble.The probability of drawing a green marble after a red one has been drawn is:
P(Green | Red) = Number of Green Marbles / Remaining Marbles = 222 / 887.
Finally, multiply these probabilities together to find the overall probability.The overall probability is:
P(Red then Green) = P(Red) * P(Green | Red) = (222 / 888) * (222 / 887) = (1/4) * (222 / 887).
Therefore, the probability that the first marble is red and the second is green is approximately 0.0626 or 6.26%.
A jogger runs around a circular track of radius 55 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (33, 44), her x-coordinate is changing at a rate of 15 ft/s. Find dy/dt.
Using related rates and the Pythagorean theorem, we can find that dy/dt, the rate of change of the y-coordinate, is 0 ft/s.
Explanation:To find dy/dt, we need to determine the rate of change of the y-coordinate of the jogger. Since the jogger is running on a circular track, we can use the concept of related rates to solve this problem.
Let's assume that the jogger completes a full lap around the track in a time interval of Δt. During this time interval, the x-coordinate of the jogger changes by Δx, the y-coordinate changes by Δy, and the distance traveled along the track is Δs.
Since the jogger is running at a constant speed, the distance Δs is equal to the distance traveled in a straight line, which is the hypotenuse of a right triangle with legs Δx and Δy. Using the Pythagorean theorem, we have:
Δs^2 = Δx^2 + Δy^2
Taking the derivative with respect to time, we have:
2Δs(dΔs/dt) = 2Δx(dΔx/dt) + 2Δy(dΔy/dt)
Substituting the given values, Δx is 15 ft/s, Δy is 0 (since the y-coordinate is not changing), and Δs is the distance around the circular track, which is equal to the circumference of the circle:
2π(55ft) = 2(15ft)(dΔx/dt) + 2(0)(dΔy/dt)
Simplifying, we have:
dΔx/dt = π(55ft)/15s = 11π/3 ft/s
Therefore, dy/dt = dΔy/dt = 0 ft/s, since the y-coordinate is not changing.
Please find the center of dialation given the information in the link. Real answers only.
Answer:
(-4, 0)
Step-by-step explanation:
The scale factor of 1/2 means each "dilated" point is 1/2 the distance from the center of dilation that the original point is. That is, the dilated point is the midpoint between the original and the dilation center.
If O is the origin of the dilation, then ...
(O + X)/2 = P . . . . . P is the dilation of point X
O +X = 2P
O = 2P -X = 2(0, 2) -(4, 4)
O = (-4, 0)
The center of dilation is (-4, 0).
_____
Another way to find the center of dilation is to realize that dilation moves points along a radial line from the center. Hence the place where those radial lines converge will be the center of dilation. See the attachment for a solution that way.
Rita purchased a prepaid phone card for $30. Long distance cost 16 cents a minute using the card. Rita used her card only once to make a long distance call. If the remaining credit on her card if $26.48, how many minutes did her call last?
Answer: her call lasted for 22 minutes
Step-by-step explanation:
Rita purchased a prepaid phone card for $30. This means that the total credit on her card is $30. Long distance cost 16 cents a minute using the card. Converting to dollars, it costs 16/100 = $0.16
Rita used her card only once to make a long distance call. If the number of minutes if long distance call that she made is x, total cost of x minutes long distance calls will be 0.16 × x = $0.16x
The remaining credit on her card would be 30 - 0.16x
If the remaining credit on her card if $26.48, it means that
30 - 0.16x = 26.48
0.16x = 30 - 26.48 = 3.52
x = 3.52/0.16 = 22 minutes.
A certain list consists of 21 different numbers. If n is in the list and n is 4 times the average(arithmetic mean) of the other 20 numbers in the list, then n is what fraction of the sum of the 21 numbers in the list?
(A) 1/20
(B) 1/6
(C) 1/5
(D) 4/21
(E) 5/21
Answer:
B. [tex]\frac{1}{6}[/tex]
Step-by-step explanation:
Let x be the sum of the 21 numbers,
In which n is one of the numbers,
Since,
[tex]\text{Average}=\frac{\text{Sum of the observations}}{\text{Number of observations}}[/tex]
So, the average of 20 numbers excluded n = [tex]\frac{x-n}{20}[/tex]
According to the question,
[tex]n = 4\times \frac{x-n}{20}[/tex]
[tex]n = \frac{x-n}{5}[/tex]
[tex]5n = x - n[/tex]
[tex]6n = x[/tex]
[tex]\imples n = \frac{1}{6}x = \frac{1}{6}\text{ of the sum of the 21 numbers}[/tex]
Hence, OPTION 'B' is correct.
Based on the Polynomial Remainder Theorem what is the value of the function below when x = 3.
Answer:
Remainder = 64
Step-by-step explanation:
Given equation,
[tex]x^4+3x^3-6x^2-12x-8[/tex]
Remainder theorem says a polynomial can be reset in terms of its divisor (a) by evaluating the polynomial at x=a
Plug x=3,
[tex]=3^4+3(3)^3-6(3)^2-12(3)-8\\=81+81-54-36-8\\=162-54-36-8\\=64[/tex]
Thus the remainder is 64 at x=3 ,using polynomial remainder theorem.
In a typical start for his baseball team, Rick throws 120 total pitches with a ratio of 3 fastballs thrown for every 2 curveballs. If Rick makes a relief appearance of 30 pitches using the same ratio, how many fastballs will he throw in the relief appearance?
Answer:
18 fastballs
Step-by-step explanation:
Let x represent each throw
fastball : curve ball = 3:2
For fastball we have 3x while for curve ball we have 2x
If Rick makes a relief appearance of 30 pitches with the same ratio,
3x + 2x = 30
5x = 30
x = 30/5
x = 6
fastball, 3x= 3*6
= 18
curveball = 2x = 2*6
= 12
Rick will throw 18 fast balls
Look at the graph of this system of equations: y = - x2 + 1 and y = x2. At which approximate points are the two equations equal? There more than one answer.
A.(-0.7, 0.5)
B.(0.5, 0.7)
C.(0.7, 0.5)
D.(-0.5, 0.7)
Answer:
option A and C, (-0.7, 0.5) and (0.7, 0.5)
Step-by-step explanation:
The two equations are equal means, the points at which the two graphs meet.
In that case the x and y coordinates satisfy both the graphs.
let the coordinates at the intersection point be (a,b).
Inserting in first equation,
[tex]b = -a^{2} + 1[/tex]
Inserting in second equation,
[tex]b = a^{2}[/tex]
Inserting value of b from second to first equation, we get
[tex]b = -b + 1[/tex]
[tex]b = \frac{1}{2} = 0.5[/tex]
Now inserting the value of b second equation, we get
[tex]\frac{1}{2} = x^{2}[/tex]
[tex]x = \sqrt{\frac{1}{2} } = +\frac{1}{1.414} or -\frac{1}{1.414} = +0.7 or -0.7[/tex]
Thus points are, (-0.7, 0.5) and (0.7, 0.5)
What is the value of h for the parallelogram?
Answer:
9.6 units
Step-by-step explanation:
The area of the parallelogram is the product of the base length and distance between parallel sides, either way you figure it.
16 × 6 = area = 10 × h
96 = 10h
h = 96/10 = 9.6 . . . . units
The exponential model Upper A equals 104.8 e Superscript 0.001 t describes the population, A, of a country in millions, t years after 2003. Use the model to determine when the population of the country will be 106 million.
Answer: The population of the country will be 106 millions in 2014.
Step-by-step explanation:
The exercise gives you the following exponential model, which describes the population "A" (in millions) of a country "t" years after 2003:
[tex]A=104.8 e^{0.001 t}[/tex]
In this case you must determine when the population of that country will be 106 millions, so you can identify that:
[tex]A=106[/tex]
Now you need to substitute this value into the exponential model given in the exercise:
[tex]106=104.8 e^{0.001 t}[/tex]
Finally, you must solve for "t", but first it is important to remember the following Properties of logarithms:
[tex]ln(a)^b=b*ln(a)\\\\ln(e)=1[/tex]
Then:
[tex]\frac{106}{104.8}=e^{0.001 t}\\\\ln(\frac{106}{104.8})=ln(e)^{0.001 t}\\\\ln(\frac{106}{104.8})=0.001 t(1)\\\\\frac{ln(\frac{106}{104.8})}{0.001}}=t\\\\t=11.38\\\\t\approx11[/tex]
Notice that in 11 years the population will be 106 millions, then the year will be:
[tex]2003+11=2014[/tex]
The population of the country will be 106 millions in 2014.
The average hourly wage of workers at a fast food restaurant is $6.75 with a standard deviation of $0.25. If the distribution is normal, what is the probability that a worker selected at random earns more than $7.00 an hour?
Answer:
25% would be the answer
Step-by-step explanation:
A boat leaves the entrance to a harbor and travels 150 miles on a bearing of Upper N 56 degrees Upper E. How many miles north and how many miles east from the harbor has the boat traveled?
Answer:
83.9 miles north124.4 miles eastStep-by-step explanation:
It can be helpful to draw a diagram. In the attached diagram, point H represents the harbor, point B represents the position of the boat, and point N represents a point directly north of the harbor and west of the boat.
The bearing N56E means the direction of travel is along a path that is 56° clockwise (toward the east) from north.
__
The mnemonic SOH CAH TOA reminds you of the relationships between the sides of a right triangle. Here, we are given the length of the hypotenuse, and we want to know the lengths of the sides opposite and adjacent to the angle. One of the useful relations is ...
Sin = Opposite/Hypotenuse
In our diagram, this would be ...
sin(56°) = BN/BH
We want to find length BN, so we can multiply by BH to get ...
BN = BH·sin(56°) = 150·0.829038 = 124.4 . . . . miles (east)
__
For the adjacent side, we use the relation ...
Cos = Adjacent/Hypotenuse
cos(56°) = HN/HB
HN = HB·cos(56°) = 150·0.559193 = 83.9 . . . . miles (north)
The boat has traveled 124.4 miles north and 83.9 miles east of the harbor entrance.
A 3 by 3 matrix Bis known to have eigenvalues 0, 1, 2. This information is enough to find three of these (give the answers where possible) : (a) the rank of B (b) thedeterminantofBTB (c) theeigenvaluesofBTB (d) the eigenvalues of (B2 + J)-1.
Answer with Step-by-step explanation:
We are given that a matrix B .
The eigenvalues of matrix are 0, 1 and 2.
a.We know that
Rank of matrix B=Number of different eigenvalues
We have three different eigenvalues
Therefore, rank of matrix B=3
b.
We know that
Determinant of matrix= Product of eigenvalues
Product of eigenvalues=[tex]0\times 1\times 2=0[/tex]
After transpose , the value of determinant remain same.
[tex]\mid B^TB\mid=\mid B^T\mid \mid B\mid =0\times 0=0[/tex]
c.Let
B=[tex]\left[\begin{array}{ccc}0&-&-\\-&1&-\\-&-&2\end{array}\right][/tex]
Transpose of matrix:Rows change into columns or columns change into rows.
After transpose of matrix B
[tex]B^T=\left[\begin{array}{ccc}0&-&-\\-&1&-\\-&-&2\end{array}\right][/tex]
[tex]B^TB=\left[\begin{array}{ccc}0^2&-&-\\-&1^2&-\\-&-&2^2\end{array}\right][/tex]
[tex]B^TB=\left[\begin{array}{ccc}0&-&-\\-&1&-\\-&-&4\end{array}\right][/tex]
Hence, the eigenvalues of [tex]B^TB[/tex] are 0, 1 and 4.
d.Eigenvalue of Identity matrix are 1, 1 and 1.
Eigenvalues of [tex]B^2+I=(0+1),(1+1),(2^2+1)=1,2,5[/tex]
We know that if eigenvalue of A is [tex]\lambda[/tex]
Then , the eigenvalue of [tex]A^{-1}[/tex] is [tex]\frac{1}{\lambda}[/tex]
Therefore, the eigenvalues of [tex](B^2+I)^{-1}[/tex] are
[tex]\frac{1}{1},\frac{1}{2},\frac{1}{5}[/tex]
The eigenvalues of [tex](B^2+I)^{-1}[/tex] are 1,[tex]\frac{1}{2}[/tex] and [tex]\frac{1}{5}[/tex]
Find the positive number such that the sum of 8 times this number and 7 times its reciprocal is as small as possible.
Answer:
√56/8
Step-by-step explanation:
Let the number be x
f(x) = 8x + 7(1/x)
f(x) = 8x + 7/x
Differentiate f(x) with respect to x
f'(x) = 8x - 7/x = 0
8 - 7/x^2 = 0
(8x^2 - 7)/2 = 0
8x^2 - 7 = 0
8x^2 = 7
x^2 = 7/8
x = √7/8
x = √7 /√8
x = (√7/√8)(√8/√8)
x = (√7*√8) / √8*√8)
x = √56/8
Find the values of x and y using trig
Answer:
Step-by-step explanation:
The triangle is a right angle triangle. This is because one of its angles is 90 degrees.
Let us determine x
Taking 47 degrees as the reference angle,
x = adjacent side
11 = hypotenuse
Applying trigonometric ratio,
Cos # = adjacent side / hypotenuse
# = 47 degrees
Cos 47 = x/11
x = 11cos47
x = 11 × 0.6820
x = 7.502
Let us determine y
Taking 47 degrees as the reference angle,
y = opposite side
11 = hypotenuse
Applying trigonometric ratio,
Sin # = opposite side / hypotenuse
# = 47 degrees
Sin 47 = y/11
x = 11Sin47
x = 11 × 0.7314
x = 8.0454
find one positive and one negative angle coterminal with an angle of 166 Question 4 options:
526°, –194°
516°, –14°
526°, –76
256°,-76
Answer: Choice A) 526 degrees, -194 degrees
==============================
Work Shown:
A coterminal angle points in the same exact direction as the original angle.
Because there are 360 degrees in a circle, this means we can add 360 to the original angle to get 166+360 = 526, which is one positive coterminal angle to 166 degrees.
Subtract 360 from the original angle and we'll get a negative coterminal angle
166 - 360 = -194
A government's congress has 376 members, of which 44 are women. An alien lands near the congress building and treats the members of congress as as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.085 and an upper bound of 0.149. What is wrong with the alien's approach to estimating the proportion of the human race that is female?
Answer:
Alien does not take the sample because alien choose the data of a Government's congress and congress contain less women.
On the other hand general population contain women greater than congress
So as compared to general population confidence interval is not representative.
Answer: The sample is not a simple random sample
Step-by-step explanation:
21. The parent function of the following graph is f(x) = 2^x. What is the equation of the following graph?
Answer:
5
Step-by-step explanation:
Plug x = 0 into f(x) to get
f(x) = 2^x
f(0) = 2^0
f(0) = 1
The y intercept (0,1) is on the graph of f(x).
The y intercept for the red curve shown is (0,3). It has been moved up two units compared to (0,1)
Therefore, g(x) = f(x)+2 where g(x) represents the red curve.
g(x) = f(x) + 2
[tex]g(x) = 2^x + 2[/tex] is the answer
The bill is $330. We want to split it but a friend gave us $50 for it. Therefore we each owe $140 after the $50. Since I have the $50 in my possession I wok give him $165 since were splitting the $50. Correct?
I don't think that's correct
Step-by-step explanation:
Why are you splitting the $50? you'd end up paying more than the bill and he'd be getting back more money than he put in. Sounds like a rip off. If he had given you each $50 than maybe you'd each owe $140. I assume there is 3 friends, the original bill price would have been $110 for each of you. But then one friend gave $50 to help pay the bill, if you had split the $50 you'd still not be paying back that much. Also why are YOU paying so much more? Everyone else is paying $140 and you're paying $165? You would not be giving him that much, all of you would not be paying an extra $30 either. you'd be splitting it to where it equals $50 all around, so instead it'd be around $93.00. Not $140 or $165. $16.7 multipled by 3 = $50.1
But at the end of the day, just tell him to take his money back. He really didn't help pay the bill that much with his $50, he still owes you $60 if he too had participated in whatever you guys were doing. So instead of going through the trouble, just tell him to take back his money.
Complete the sentence below The _____ _____, denoted Modifying Above p with caret, is given by the formula Modifying Above p with caret equals_____, where x is the number of individuals with a specified characteristic in a sample of n individuals. The _____ _____ mean proportion error variance deviation , denoted Modifying Above p with caret, ________is given by the formula Modifying Above p with caretequals Start Fraction n Over x End Fraction xn Start Fraction x Over n End Fraction , where x is the number of individuals with a specified characteristic in a sample of n individuals.
Answer:down t
Step-by-step explanation: j this is world. Theo t sit with this
Shawn bought fruit last week, consisting of 2.26 pounds of bananas, 1.5 pounds of grapes, and a watermelon that weighed 6.78 pounds. What is the total weight, in pounds, of the fruit that Shawn bought last week?
Answer:
10.54 pounds is right answer
Step-by-step explanation:
bananas 2.26 pounds
Grapes 1.5 pounds
water melon 6.78 pounds
total weight = 10.54 pounds
Answer:
10.54 pounds
Step-by-step explanation:
A submarine was stationed 700 feet below sea level. It ascends 259 feet every hour. If the submarine continues to ascend at the same rate when will the submarine be at the surface?
Answer: it will take 2.7 hours to get to the surface
Step-by-step explanation:
A submarine was stationed 700 feet below sea level. It means that the height of the submarine from the surface is 700 feet.
It ascends 259 feet every hour.
If the submarine continues to ascend at the same rate, the time it will take for it to get to the surface will be the distance from the surface divided by its constant speed.
Time taken to get to the surface
700/259 = 2.7 hours
In a game of poker a hand of five cards is dealt to each player from a deck of 52 cards. find the probablility of a hand containing a spade flush.
Answer:
0.00597
Step-by-step explanation:
Given,
Total number of cards = 52,
In which flush cards = 20,
Also, the number of spade flush cards = 5,
Since,
[tex]\text{Probability}=\frac{\text{Favourable outcomes}}{\text{Total outcomes}}[/tex]
Thus, the probability of a hand containing a spade flush, if each player has 5 cards
[tex]=\frac{\text{Ways of selecting a spade flush card}}{\text{Total ways of selecting five cards}}[/tex]
[tex]=\frac{^{20}C_5}{^{52}C_5}[/tex]
[tex]=\frac{\frac{20!}{5!15!}}{\frac{52!}{5!47!}}[/tex]
[tex]=\frac{15504}{2598960}[/tex]
= 0.00597
Mikayla is a waitress who makes a guaranteed $50 per day in addition to tips of 20% of all her weekly customer receipts, t. She works 6 days per week. Which of the following functions best represents the amount of money that Mikayla makes in one week?
A) f(t) = 50 + 20t
B) f(t) = 300 + 20t
C) f(t) = 50 + 0.2t
D) f(t) = 300 + 0.2t
Answer:
D. [tex]f(t)=300+0.20t[/tex]
Step-by-step explanation:
We have been given that Mikayla is a waitress who makes a guaranteed $50 per day.
Since Miklaya works 6 days per week, so the guaranteed income for one week would be [tex]\$50\times 6=\$300[/tex]
We are also told that she gets tips of 20% of all her weekly customer receipts, t. So amount earned from tips would be 20% of t, that is [tex]\frac{20}{100}t=0.20t[/tex].
Total amount earned in one week would be guaranteed income for 1 week plus 20% of t:
[tex]300+0.20t[/tex]
Therefore, our required function is [tex]f(t)=300+0.20t[/tex] and option D is the correct choice.
The workers at Sandbachian, Inc. took a random sample of 800 manhole covers and found that 40 of them were defective. What is the 95% CI for p, the true proportion of defective manhole covers, based on this sample?a) (37.26, 42.74)b) (.035, .065)c) (.047, .053)d) (.015, .085)
Answer: b) [tex](0.035,\ 0.065)[/tex]
Step-by-step explanation:
The confidence interval for proportion (p) is given by :-
[tex]\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
, where[tex]\hat{p}[/tex] = Sample proportion
n= sample size.
z* = Critical z-value.
Let p be the true proportion of defective manhole covers, based on this sample.
Given : The workers at Sandbachian, Inc. took a random sample of 800 manhole covers and found that 40 of them were defective.
Then , n= 800
[tex]\hat{p}=\dfrac{40}{800}=0.05[/tex]
Confidence interval = 95%
We know that the critical value for 95% Confidence interval : z*=1.96
Then, the 95% CI for p, the true proportion of defective manhole covers will be :-
[tex]0.05\pm (1.96)\sqrt{\dfrac{0.05(1-0.05)}{800}}\\\\=0.05\pm (1.96)(0.0077055)\\\\=0.05\pm0.01510278\\\\=(0.05-0.01510278,\ 0.05+0.01510278)\\\\=(0.03489722,\ 0.06510278)\approx(0.035,\ 0.065) [/tex]
Hence, the required confidence interval : b) [tex](0.035,\ 0.065)[/tex]
Asha invests rs 8000 at a certain rate for three yrs compounded annually . She finds at the end of the first year it amounts to rs 9200. Calculate(1) R%(2)interest acquired in second year (3)amount at the end of the third year
Answer:
= 12169 rs
Step-by-step explanation:
Asha invest total=rs 8000
Total years = 3
Amount after 1 year= 9200 rs
Interest on first year =9200-8000= 1200 rs
So for second and third years
A= P (1 + R/100)ⁿ
9200= 8000( 1+ R/100)¹
9200= 8000( 100+ R/100)
115= 100 +R
R = 15
So amount after second year
A= 8000 (1 +15/100)²
A= 10580
Interest on second is = 10580 –P-interest on ist
=10580-8000-1200
= 1380 rs
So total amount at the end
A = 8000 ( 1 +15/100) ³
=12169 rs
Please answer quick guys! 3. Given LJ ≅ MK, and LK ≅ MJ, prove ∠L ≅ ∠M (Picture is below)
Answer:
The Proof is given below.
Step-by-step explanation:
Given:
LJ ≅ MK
LK ≅ MJ
To Prove:
∠ L ≅ ∠ M
Proof:
In Δ LKJ and Δ MJK
LK ≅ MJ ……….{Given}
KJ ≅ KJ ………..{Reflexive Property}
LJ ≅ MJ ……….{Given}
Δ LKJ ≅ Δ MJK ....….{Side-Side-Side test}
∴∠ KLJ ≅ ∠ JMK .....{corresponding angles of congruent triangles (c.p.ct)}
i.e ∠ L ≅ ∠ M ............Proved.