The distribution of X is a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches. To find the probability that a person's height is between 64 and 69 inches, we need to determine the z-scores for each height value and use a z-table or calculator to find the probabilities.
Explanation:A) Give the distribution of X.
X ~ Normal(66, 2.5)
B) Find the probability that the person is between 64 and 69 inches.
P(64 ≤ X ≤ 69)
To find this probability, we need to calculate the z-score for each height value using the formula z = (X - μ) / σ. Then, we can use a z-table or a calculator to find the corresponding probabilities.
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The height X of Asian adult males follows a normal distribution with parameters X ~ N(66, 2.5). The probability that one randomly chosen male is between 64 and 69 inches tall can be found using Z-scores and the area under the normal curve.
The distribution of the height X for Asian adult males according to the given study is X ~ N(66, 2.5), where 66 is the mean height in inches and 2.5 is the standard deviation.
Probability that the height is between 64 and 69 inches
To find the probability that the height of a randomly chosen Asian adult male is between 64 and 69 inches, we can use the properties of the normal distribution. We first convert the raw scores to Z-scores using the formula Z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
The probability statement is P(64 < X < 69). To calculate this probability, we find the Z-scores for 64 and 69 inches and then find the area under the normal curve between these two Z-scores using a standard normal distribution table or a calculator with normal distribution functions.
The probability that a randomly selected teenager watched a rented video at least once during a week was 0.75. What is the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week? (Round your answer to four decimal places.)
Answer:
0.7564
Step-by-step explanation:
Let X be the event of watching a rented video at least once during a week,
Given,
The probability of watching a rented video at least once during a week was, p = 0.75,
So, the probability of not watching a rented video at least once during a week was, q = 1 - p = 0.25,
Binomial distributive formula,
[tex]P(x)=^nC_x p^x q^{n-x}[/tex]
Where,
[tex]^nC_x=\frac{n!}{x!(n-x)!}[/tex]
Hence, the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week,
P(X ≥ 5) = P(X=5) + P(X=6 )+ P(X=7)
[tex]=^7C_5 0.75^5 0.25^{7-5}+^7C_6 0.75^6 0.25^{7-6}+^7C_7 0.75^7 0.25^{7-7}[/tex]
[tex]=21 (0.75)^5 (0.25)^2 + 7 (0.75)^6 0.25 + 0.75^7[/tex]
[tex]=0.756408691406[/tex]
[tex]\approx 0.7564[/tex]
The probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week is 0.3015.
Explanation:The probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week can be calculated using the binomial probability distribution formula:
P(X ≥ k) = 1 - P(X < k)
where X is the number of teenagers who watched a rented movie at least once, k is the minimum number of teenagers (5 in this case), and P(X < k) is the probability that less than k teenagers watched a rented movie at least once.
In this case, the probability that a randomly selected teenager watched a rented video at least once during a week is 0.75. Therefore, the probability that a randomly selected teenager did not watch a rented video at least once is 1 - 0.75 = 0.25.
Using the binomial probability distribution formula, we can calculate the probability that less than 5 teenagers watched a rented movie at least once:
P(X < 5) = C(7, 0) * (0.25)^0 * (0.75)^7 + C(7, 1) * (0.25)^1 * (0.75)^6 + C(7, 2) * (0.25)^2 * (0.75)^5 + C(7, 3) * (0.25)^3 * (0.75)^4 + C(7, 4) * (0.25)^4 * (0.75)^3
where C(n, r) is the number of combinations of n items taken r at a time:
C(n, r) = n! / (r! * (n-r)!)
Substituting the values and evaluating the expression, we get:
P(X < 5) = 0.698486328125
Therefore, the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week is:
P(X ≥ 5) = 1 - P(X < 5) = 1 - 0.698486328125 = 0.301513671875
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F Find an equation for a circle san istying the gve a) Center (-1,4), passes through (3,7) Center (-1,4). passes through (3,7
Answer:
[tex](x+1)^{2} +(y-4)^{2} =25[/tex]
Step-by-step explanation:
In order to find the equation of the circle, first we need to know the circle's general equation, which is:
[tex](x-h)^{2} +(y-k)^{2} =r^{2}[/tex] where:
(h,k) is the center of the circle and r the radius of the circle.
Because the problem has given the center (-1,4) then h=-1 and k=4.
We need to find now the radius:
Using the distance equation: [tex]distance=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}[/tex] and because we have the center coordinates and an extra point (3,7) we can find the radius as:
[tex]distance=\sqrt{(3-(-1))^{2}+(7-4)^{2}}[/tex]
[tex]distance=\sqrt{4^{2}+3^{2}}[/tex]
[tex]distance=\sqrt{16+9}[/tex]
[tex]distance=\sqrt{25}[/tex]
[tex]distance=5[/tex] which means r=5
In conclusion, the equation for the given circle is [tex](x+1)^{2} +(y-4)^{2} =5^{2}[/tex] which also, can be written as [tex](x+1)^{2} +(y-4)^{2} =25[/tex]
2. According to a well-known legend, the game of chess was invented many centuries ago in northwestern India by a certain sage. When he took his invention to his king, the king liked the game so much that he offered the inventor any reward he wanted. The inventor asked for some grain to be obtained as follows: just a single grain of wheat was to be placed on the first square of the chessboard, two on the second, four on the third, eight on the fourth, and so on, until all 64 squares had been filled. If it took just 1 second to count each grain, how long would it take to count all the grain due to him?
Answer:
2^64. I know 2^20 is 1048576. Cube that and multiply by 16 or grab a calculator. I'm too lazy to solve this.
The time taken to count all the grain due to him is [tex]2^{64}-1[/tex] or 18,446,744,073,709,551,615 sec .
What is Geometric Progression?Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.
What is sum of Geometric Progression?. The sum of infinite, i.e. the sum of a GP with infinite terms is [tex]S_{∞} = \frac{a}{(1 - r) }[/tex]such that 0 < r < 1.
The formula used for calculating the sum of a geometric series with n terms is Sn = [tex]\frac{a( r^{n} -1 )}{(r - 1)} ,[/tex] where r ≠ 1.
According to the question
A chess board has 64 squares and all had been filled with grain
1st squares of chess board has grain = 1 = [tex]2^{0}[/tex]
2nd squares of chess board has grain = 2 = [tex]2^{1}[/tex]
3nd squares of chess board has grain = 4 = [tex]2^{2}[/tex]
4nd squares of chess board has grain = 8 = [tex]2^{3}[/tex]
so on ..
As this is an Geometric Progression
Where
First term (a) = 1
common ratio (r) = 2
Number of terms = n = 64
Now,
it took just 1 second to count each grain ,
Time taken to count all the grains
By using formula of sum of Geometric Progression
Sn = [tex]\frac{a( r^{n} -1 )}{(r - 1)} ,[/tex] where r ≠ 1.
substituting the values in formula
S₆₄ = [tex]\frac{1( 2^{64}-1 )}{(2-1)} ,[/tex]
S₆₄ = [tex]2^{64}-1[/tex]
S₆₄ = 18,446,744,073,709,551,615
Hence, The time taken to count all the grain due to him is [tex]2^{64}-1[/tex] or 18,446,744,073,709,551,615 sec .
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If a ball is drawn from a bag containing 13 red balls numbered 1-13 and 5 white balls numbered 14-18. What is the probability that a. the ball is not even numbered? b. the ball red and even numbered? c. the ball red or even numbered? d. the ball is neither red or even numbered?
Answer:
a. 50%
b. 33%
c. 17% (I'm assuming the exercise is wrong and it has to say "white" instead of "red", because if not is the same as b.)
d. 67%
Step-by-step explanation:
a. We have a total of 18 balls, 13 are red and 5 are white. They are numbered from 1 to 18. In this case, we don't care about the color of the ball, we just need it to be not even. We have to count how many not even numbers are between 1 and 18, that is 9. So, the chances of getting a ball not even numbered are 9 in 18, that's
[tex]\frac{9}{18}*100=50\%[/tex]
b. Now we do care about the color of the ball. The red balls are numbered from 1 to 13, so we have 6 balls even numbered. That makes the chances 6 in 18 (we still have 18 in total), that's
[tex]\frac{6}{18}*100=33\%[/tex]
c. (I'm assuming the exercise is wrong and it has to say "white" instead of "red", because if not is the same as b.)
The white balls are numbered from 14 to 18, so we have 3 balls even numbered. That makes the chances 3 in 18,
[tex]\frac{3}{18}*100=17\%[/tex]
d. Let's notice that "the ball is neither red or even numbered" is the complement (exactly the opposite) of "the ball is red and even numbered", that means
100% = Probability (ball red and even numbered) + Probability (ball neither red or even numbered)
So, Probability (ball neither red or even numbered) = 100% - Probability (ball red and even numbered) = 100% - 33% = 67%
An automobile emissions testing center has six inspectors and tests 50 vehicles per hour. Each inspector can inspect 12 vehicles per hour. How many inspectors would the center require to have a target utilization of 90 percent?
Answer:
The center require to have a 4.6 inspectors, but this on you decide if you have 4 or 5 inspectors, it may not have 4.6 people working.
So then if you decide to have 5 people their utilization percentage is 83.3%, but if you decide to have 4 people their utilization rate will be 104.17% at the risk of defaulting on demand.
Step-by-step explanation:
1. Define your variables
Demand rate= 50 vehicles per hour
Service rate = 12 vehicles per hour per inspector
Inspectors= a
2. Use the formule of the center´s utilization
U= [tex]\frac{Demand}{(Service) X a}[/tex]
0,9= [tex]\frac{50}{(12) X a }[/tex]
0,9a=[tex]\frac{50}{12}[/tex]
a=[tex]\frac{50}{12X0,9}[/tex]
a= 4.6
3. According to the center´s utilization formula the center require 4,6 inspectors, but approaches 5 because people cannot be divided. With this numbers of inspectors the utilization is:
U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]
U= [tex]\frac{50}{(12) X (5)}[/tex]
U= 83,3%
4. Other option that you will be used is that the center require 4 inspectors. With this numbers of inspectors the utilization is:
U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]
U= [tex]\frac{50}{(12) X (4)}[/tex]
U= 104,17%
The center would require 6 inspectors to have a target utilization of 90 percent.
To determine the number of inspectors required to have a target utilization of 90 percent, we can use the formula:
Number of inspectors = Total vehicles per hour / Vehicles inspected per inspector per hour / Target utilization
Given that the center tests 50 vehicles per hour, and each inspector can inspect 12 vehicles per hour, the formula becomes:
Number of inspectors = 50 / 12 / 0.9 = 5.56
Since we cannot have a fraction of an inspector, we round up to the nearest whole number. Therefore, the center would require 6 inspectors to have a target utilization of 90 percent.
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Calculate the amount of money you'll have at the end of the indicated time period You invest $2000 in an account that pays simple interest of 4 % for 20 years. The amount of money you'll have at the end of 20 years is S
Answer:
The amount would be $ 3600.
Step-by-step explanation:
Given,
The invested amount, P = $ 2000,
Annual rate of interest, r = 4 %,
Time, t = 20 years,
So, the simple interest would be,
[tex]I=\frac{P\times r\times t}{100}[/tex]
[tex]=\frac{2000\times 4\times 20}{100}[/tex]
[tex]=\frac{160000}{100}[/tex]
[tex]=\$1600[/tex]
Hence, the amount of money after 20 years,
[tex]A=P+I[/tex]
[tex]=2000+1600[/tex]
[tex]=\$ 3600[/tex]
M1Q6.) How many degrees should be used to represent convertables in the Pie Graph?
Answer:
80 degrees
Step-by-step explanation:
The entire circle represents 72 and the "slice of pie" is represented by a portion:
16/72 * 100
= 0.22 * 100
= 22.222%
.
22.222% of 360 degrees
= .22222 * 360
= 80 degrees
Each portion can be represented by 16/72
16/72 = 0.22
0.22... * 100% = 22.22...%
0.2222 * 360 = 80
Therefore, the answer is 80 degrees.
Best of Luck!
I need the answer to these math questions.
1) Multiply 8 minutes 31 seconds by 17.
Answer:
2 h 16 min 23 sec
Step-by-step explanation:
Hello
the time is expressed in the sexadecimal system, which uses the number 60 as an arithmetic base,hence
1 min=60 sec
1 hora =60 min
Now, we have
[tex](8 min + 31 sec)*17=136 min +527 sec\\\\\\we\ need\ to\ convert\ this\ in\ our\ base\ 60\, using\ a\ rule\ o\ three\\\\ 60 min=1\ hour\\136 min=x ?\\\\x=\frac{ 136 h}{60}\\ x=2.26 hours\\\\[/tex]
we take the whole part as an hour, and the decimal part is multiplied by 60 to get minutes
Step 1
[tex]8min*17=136 min =2.26 h\\\\2.26h = 2\ h + 0.26h(\frac{60 min}{1 h}) \\2.26h =2h+15.6 min\\\\[/tex]
we repeat the procedure to leave the minutes as a whole part
[tex]2.26\ h =2\ h+15\ min + 0.6\ min*(\frac{60 \sec}{1 m} )\\2.26\ h =2\ h\ 15\ min\ 36\ sec[/tex]
Step 2
[tex]\\527\s*(\frac{1 min}{ 60\ sec})=8.78\ min\\ \\8.78\ min= 8\min\0.78\ min\\8.78\ min=8\ min\ 0.78\min(\frac{60\ seg}{min})\\8.78\min=8\ min \ 47\ sec\\\\now, add\\\\8 min *17 =2\ h\ 15\ min\ 36\ sec\\31 sec *17 =8\ min \ 47\ sec\\(8\ min\ 31\ sec)*17=2\ h\ 15\ min\ 83\ sec\\83 s(\frac{1 min}{60 sec})=1.38 min\\1.38\ min\ =1\ min\ 0.38\ min*(\frac{60 sec}{1\ min})\\1.38\ min=1\min\ 23 s.\\( 8min 31 sec)*17=2 h 16 min 23 sec[/tex]
Have a great day
I need the answer to this math question.
1) Divide 251 days 21 hours by 13.
Then round to the nearest hundredth as necessary.
Answer: 465
Step-by-step explanation:
251 (days) x 24 (hours) = 6,024 hours
6,024+21 hours= 6,045
6045/13=465
Find the inverse of h(x) = [tex]\frac{2x+6}{5}[/tex]
show work please!
Answer:
The inverse of h(x) is [tex]\frac{5x-6}{2}[/tex]
Step-by-step explanation:
* Lets explain how to make the inverse of a function
- To find the inverse of a function we switch x and y and then solve
for new y
- You can make it with these steps
# write g(x) = y
# switch x and y
# solve for y
# write y as [tex]g^{-1}(x)[/tex]
* Lets solve the problem
∵ [tex]h(x)=\frac{2x+6}{5}[/tex]
# Step 1
∴ [tex]y=\frac{2x+6}{5}[/tex]
# Step 2
∴ [tex]x=\frac{2y+6}{5}[/tex]
# Step 3
∵ [tex]x=\frac{2y+6}{5}[/tex]
- Multiply each side by 5
∴ 5x = 2y + 6
- Subtract 6 from both sides
∴ 5x - 6 = 2y
- Divide both sides by 2
∴ [tex]y=\frac{5x-6}{2}[/tex]
# Step 4
∴ [tex]h^{-1}(x)=\frac{5x-6}{2}[/tex]
A cell of some bacteria divides into two cells every 10 minutes.The initial population is 3 bacteria. (a) Find the size of the population after t hours (function of t) (b) Find the size of the population after 7 hours. # Preview | Preview (c) When will the population reach 21? t42 Preview
Answer:
(a) [tex]P_{t}=3(2)^{6t}[/tex]
(b) [tex]3(2)^{42}[/tex]
(c) 28.07 minutes
Step-by-step explanation:
A cell of some bacteria divides itself into 2 cells in every 10 minutes and initial population of the bacteria was 3.
That means sequence formed will be 3, 6, 12, 24............
We can easily say that this sequence is a geometric sequence having common ratio (r) = [tex]\frac{T_{2}}{T_{1}}=\frac{6}{3}[/tex]
r = 2
Now we know the explicit formula of a geometric sequence is given by
[tex]P_{t}=P_{0}(r)^{\frac{60t}{10}}=P_{0}(r)^{6t}[/tex]
Where a = Initial population = 3 bacteria
r = common ratio = 2
and t = time in hours
So explicit formula will be [tex]P_{t}=3(2)^{6t}[/tex]
(a) Now we have to calculate the size of population after t hours
[tex]P_{t}=3(2)^{6t}[/tex]
(b) We have to find the size of population after 7 hours or 420 minutes
[tex]P_{t}=3(2)^{6\times7}[/tex]
= [tex]3(2)^{42}[/tex]
After 7 hours bacteria population will be [tex]3(2)^{42}[/tex]
(c) Time to reach population as 21
By the explicit formula
[tex]21=3(2)^{6t}[/tex]
[tex]2^{6t}=\frac{21}{3}=7[/tex]
Now we take log on both the sides of the equation
[tex]log(2^{6t})=log(7)[/tex]
6t log2 = log 7
6t(0.301) = 0.845
t(1.806) = 0.845
t = [tex]\frac{0.845}{1.806}=0.468[/tex] hours
Or t = 0.468×60 = 28.07 minutes
Therefore, after 28.07 minutes bacterial population will be 21
The bacteria population grows following an exponential pattern, therefore the population after t hours can be calculated using the exponential growth formula with the initial population as 3 and each cell dividing every 10 minutes. To calculate the time when the population reaches a certain size, solve the exponential growth equation for t.
Explanation:The growth of bacteria population can be described as exponential growth, with each cell dividing into two every 10 minutes. Given the initial population as 3 bacteria, we would need to calculate the number of divisions that occur within the specified time frame to calculate the population after t hours.
(a) To find the population after t hours, we convert the hours to minutes (since each division occurs every 10 minutes) and then calculate the number of divisions. Each bacterial division results in a doubling of the population, so we use the formula for exponential growth: N = N0 * 2^n, where N0 is the initial population (3), and n is the number of divisions (6t, because t hours is 60t minutes and each division occurs every 10 minutes, making a total of 6t divisions per hour). So the population after t hours is N = 3 * 2^(6t).
(b) To find the size of the population after 7 hours, we substitute t = 7 into the formula, to get N = 3 * 2^(6*7) = 3 * 2^42.
(c) To find out when the population reaches 21, we equate N to 21 in the formula and solve for t. So, 21 = 3 * 2^(6t). Solving this equation gives the time t in hours when the population will reach 21.
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FH←→ is tangent to circle E at point F.
What is the measure of ∠EFH?
80º
90º
160º
180º
Check the picture below.
Answer: SECOND OPTION.
Step-by-step explanation:
It is important to remember that a tangent to a circle is a line that touches it at one point. This point is called "Point of tangency". By definition, the angle between the tangent and the radius is 90 degrees.
In this case you can observe that EF is the radius of this circle, therefore, the angle between EF and FH measures 90 degrees.
Based on this, you can say the following:
[tex]\angle EFH=90\°[/tex]
This matches with the second option.
Math help ASAP!! Also both drop down boxes are the same.
Answer:
Domain: amount of fuel in the airplane's tank (in gallons)
The set of all real numbers from 0 to 200
Range: weight of airplane (In pounds)
The set of all real numbers from 3000 to 4400
Step-by-step explanation:
We have the following function
[tex]W=7F+3000[/tex]
Where W represents the weight of the plane in pounds and F represents the amount of fuel in gallons.
The domain of a function is the set of values "F" that can be entered in a function W(F) to obtain an output value of W.
In this case the range of the function W(F) is the whole set of values [tex]W_1, W_2, W_3, ..., W_n[/tex] that are obtained for [tex]F_1, F_2, F_3, ..., F_n[/tex]
Note that, in this case, equation W(F) is used to obtain the weight of the airplane from the amount of fuel F.
Then the domain of the function is the amount of fuel in the airplane tank (in gallons). Since the tank can only hold up to 200 gallons, and there are no negative volume units, then the domain is all real numbers between 0 and 200.
The range of the function is the weight of the plane (in pounds). Note that the minimum weight of the airplane with 0 gallons of fuel is 3000 pounds and the maximum weight with the full tank is 4400 pounds.
Then the range is all real numbers between 3000 and 4400
Find the general solution to 2y ′′ − y ′ − y = 0.
Answer: y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
Step-by-step explanation:
2y ′′ − y ′ − y = 0
The characteristic equation is:
[tex]2r^{2} - r - 1 = 0[/tex]
[tex]2r^{2} - 2r + r - 1 = 0[/tex]
2r(r-1) + 1(r-1) = 0
(r-1)(2r+1) = 0
[tex]r_{1} = 1 , r_{2} = \frac{-1}{2}[/tex]
∴ there are two distinct roots
so the general equation is as follows:
y(x) = [tex]C_{1} e^{r_{1}x } + C_{2} e^{r_{2}x }[/tex]
y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
Find the point on the plane 4x+3y+z=10 that is nearest to (2,0,1). What are the values of x, y, and z for the point? x= 28 / 13 y = 3 / 26 z= 27 / 26 (Type integers or simplified fractions.)
To find the point on the plane that is nearest to (2,0,1), we minimize the squared distance between the two points using partial derivatives and set them equal to 0. The values of x, y, and z for the point are x = 28/13, y = 3/26, and z = 27/26.
To find the point on the plane that is nearest to (2,0,1), we need to find the coordinates that satisfy the equation 4x+3y+z=10 and minimize the distance between the point and (2,0,1).
This can be done by minimizing the squared distance between the two points. Using the formula for distance, we get the squared distance as:
d^2 = (x-2)^2 + y^2 + (z-1)^2
To minimize the squared distance, we can find the partial derivatives with respect to x, y, and z and set them equal to 0.
Solving these equations, we find that x = 28/13, y = 3/26, and z = 27/26.
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Use the graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer.
5=f(x)
To find the x for a given function f(x) =5, use the graph of the function and search for places where the y-coordinate is 5. The x-coordinates of these points are the solutions.
Explanation:The problem is asking for the value of x when f(x) = 5. To solve this problem, you would examine the graph of the function and look for the point(s) where the y-coordinate (the function value) is 5; the corresponding x-coordinate(s) would be your answer. For example, if seen directly above the number five on the y-axis, the line crosses at x=3, then x=3 is your solution. If it crosses again at x=-2, then x=-2 is another solution. Always remember that some problems may indeed have more than one answer, especially with functions that are not linear.
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A sample of ????=25 diners at a local restaurant had a mean lunch bill of $16 with a standard deviation of ????=$4 . We obtain a 95% confidence interval as (14.43,17.57) . Which action will not reduce the margin of error?
Answer:
"Decreasing the sample size."
Step-by-step explanation:
The margin of error is :
[tex]ME=z\times \frac{\sigma }{\sqrt{n}}[/tex]
Therefore, as n decreases, margin of error increases.
In the question we are asked " Which action will not reduce the margin of error."
Therefore, the correct option for which action will not reduce the margin of error is "Decreasing the sample size."
Solve the system for the exact special solution y = y(x): (keep the fraction and the square root without decimals.) 1. ydx + x[ In(x) - In(y) - 1]dy = 0 and y(1) = e for In(e) = 1.
Assume a solution of the form [tex]\Psi(x,y)=C[/tex]. Differentiating both sides gives
[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]
with [tex]\Psi_x=y[/tex] and [tex]\Psi_y=x(\ln x-\ln y-1)[/tex].
Divide both sides by [tex]x[/tex] and we have
[tex]\dfrac yx\,\mathrm dx+(\ln x-\ln y-1)\,\mathrm dy=0[/tex]
Notice that
[tex]\left(\dfrac yx\right)_y=\dfrac1x[/tex]
[tex]\left(\ln x-\ln y-1\right)_x=\dfrac1x[/tex]
so the ODE is exact. Now we can look for a solution [tex]\Psi[/tex] with
[tex]\Psi_x=\dfrac yx[/tex]
[tex]\Psi_y=\ln x-\ln y-1[/tex]
Integrating the first PDE with respect to [tex]x[/tex] gives
[tex]\Psi(x,y)=y\ln x+f(y)[/tex]
and differentiating this with respect to [tex]y[/tex] gives
[tex]\Psi_y=\ln x+f'(y)=\ln x-\ln y-1\implies f'(y)=-\ln y-1\implies f(y)=-y\ln y+C[/tex]
So this ODE has general solution
[tex]y\ln x-y\ln y=C[/tex]
Given that [tex]y(1)=e[/tex], we have
[tex]e\ln1-e\ln e=C\implies C=-e[/tex]
so the particular solution is
[tex]y(\ln x-\ln y)=-e[/tex]
[tex]y\ln\dfrac xy=-e[/tex]
[tex]\boxed{y\ln\dfrac yx=e}[/tex]
The time needed to complete a final examination in a particular college course is normally distributed with a mean of 79 minutes and a standard deviation of 8 minutes. Answer the following questions.
What is the probability of completing the exam in one hour or less (to 4 decimals)?
What is the probability that a student will complete the exam in more than 60 minutes but less than 75 minutes (to 4 decimals)?
Assume that the class has 60 students and that the examination period is 90 minutes in length. How many students do you expect will be unable to complete the exam in the allotted time (to the nearest whole number)?
Answer: a) 0.0088
b) 0.2997
c) 5
Step-by-step explanation:
Given : Mean : [tex]\mu = 79[/tex] minutes
Standard deviation : [tex]\sigma = 8[/tex] minutes
The formula for z-score :
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
a) For x = 60 minutes
[tex]z=\dfrac{60-79}{8}=-2.375[/tex]
The p-value =[tex]P(z\leq-2.375)=0.0087745\approx0.0088[/tex]
b) For x = 75 minutes
[tex]z=\dfrac{75-79}{8}=-0.5[/tex]
The p-value =[tex]P(60<x<75)=P(-2.375<z<-0.5)[/tex]
[tex]=P(-0.5)-P(-2.375)=0.3085-0.0088=0.2997[/tex]
c) For x = 90 minutes
[tex]z=\dfrac{90-79}{8}=1.375[/tex]
The p-value =[tex]P(z>1.375)=1-P(z<1.375)[/tex]
[tex]=1-0.9154342=0.0845658[/tex]
If the number of students in the class = 60 .
Then , the number of students will be unable to complete the exam in the allotted time =[tex]0.0845658\times60=5.073948\approx5[/tex]
The probability of completing the exam in one hour or less is 0.0087. The probability that the exam is completed in more than 60 minutes but less than 75 minutes is 0.2998. We expect about 5 students to not finish the exam in the given 90 minutes.
Explanation:In statistics, when a data set is normally distributed, we use a z-score to describe the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The formula to calculate a z-score is Z = (X - μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation.
To answer the questions:
Probability of completing the exam in one hour or less: Here, we need to calculate the z-score for 60 minutes (which is one hour) using the given mean (79 minutes) and standard deviation (8 minutes). Using the Z score formula, Z = (60-79)/8 = -2.375. You would then look up this z-score in a Z-table (also known as standard normal table) to find the probability, which is around 0.0087 to four decimal places. So the probability of completing the exam in one hour or less is 0.0087.Probability that a student will complete the exam in more than 60 minutes but less than 75 minutes: We need to calculate the z-scores for 60 minutes and 75 minutes. We know the z-score for 60 minutes from before is -2.375. The z-score for 75 minutes is (75-79)/8 = -0.5. The probabilities in the Z-table for these z-scores are about 0.0087 and 0.3085 respectively. We need to subtract the two probabilities to get the answer: 0.3085 - 0.0087 = 0.2998. So the probability that the exam is completed in more than 60 minutes but less than 75 minutes is 0.2998.Expected number of students unable to complete the exam in the 90 minutes examination period: Here we need to find the probability that a student will take more than 90 minutes to finish the exam. The z-score for 90 minutes is (90-79)/8 = 1.375. The probability associated with this z-score in the Z-table is about 0.9157. This essentially means the probability of completing the exam in 90 minutes or less is 0.9157. So, the probability of not completing in time is 1 - 0.9157 = 0.0843. If there are 60 students in the class, we expect about 60*0.0843 = 5.058, which rounds to about 5 students, not to finish the exam in the given 90 minutes time.Learn more about Normal Distribution here:https://brainly.com/question/34741155
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In a certain study comma the chance of encountering a car crash on the roadstudy, the chance of encountering a car crash on the road is stated as 66%. Express the indicated degree of likelihood as a probability value between 0 and 1 inclusive.
Answer:
The probability is 0.66.
Step-by-step explanation:
Given,
The chance of encountering a car crash on the road is stated as 66%,
That is, out of 100% cases the percentage of the number of car crash cases is 66%,
⇒ Total outcomes = 100%, favourable outcome = 66 %
So, the probability of occurrence a car crash = [tex]\frac{66\%}{100\%}[/tex]
[tex]=\frac{66/100}{100/100}[/tex]
[tex]=\frac{66}{100}[/tex]
[tex]=0.66[/tex]
Where, 0 < 0.66 < 1.
Hence, the indicated degree of likelihood as a probability value between 0 and 1 inclusive is 0.66.
It is estimated that one third of the general population has blood type A A sample of six people is selected at random. What is the probability that exactly three of them have blood type A?
Answer: 0.2195
Step-by-step explanation:
Binomial distribution formula :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of x successes in the n independent trials of the experiment and p is the probability of success.
Given : The probability of that the general population has blood type A = [tex]\dfrac{1}{3}[/tex]
Sample size : n=6
Now, the probability that exactly three of them have blood type A is given by :-
[tex]P(3)=^6C_3(\dfrac{1}{3})^3(1-\dfrac{1}{3})^{6-3}\\\\=\dfrac{6!}{3!3!}(\dfrac{1}{3})^3(\dfrac{2}{3})^{3}\\\\=0.219478737997\approx0.2195[/tex]
Therefore, the probability that exactly three of them have blood type A = 0.2195
a. Given a = 13 and b = 195, find b div a and b mod a. b. Given a = 24 and b = 377, find b div a and b mod a.
Answer:
A. 15 and 0. B. 15 and 17
Step-by-step explanation:
A. a= 13 and b = 195.
b div a = 195 div 13. The result is the integer part of the division, so
195 div 13 = 15.
b mod a = 195 mod 13. The result is the residue of the division. In this case the division is exact, so
195 mod 13 = 0.
B. a = 24 and b=377.
b div a = 377 div 24. The result is the integer part of the division, so
377 div 24 = 15.
b mod a = 377 mod 24. The result is the residue of the division. In this case the division is not exact, so
377 = 24*15+17, then
377 mod 24 = 17.
A given binomial experiment has n=100 trials and p=1/3. Is it more likely to get x=20 successes or x=45 successes. Why?
Answer:
The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.
Step-by-step explanation:
Given information: n=100 and p=1/3.
According to the binomial distribution, the probability of getting r success in n trials is
[tex]P(x=r)=^nC_rp^rq^{n-r}[/tex]
where, n is total trials, p is probability of success and q is probability of failure.
Total trials, n = 100
Probability of success, p = [tex]\frac{1}{3}[/tex]
Probability of failure, q = [tex]1-\frac{1}{3}=\frac{2}{3}[/tex]
The probability of 20 successes is
[tex]P(x=20)=^{100}C_{20}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{100-20}[/tex]
[tex]P(x=20)=\frac{100!}{20!(100-20)!}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{80}\approx 0.001257[/tex]
The probability of 45 successes is
[tex]P(x=45)=^{100}C_{45}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{100-45}[/tex]
[tex]P(x=45)=\frac{100!}{45!(100-45)!}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{55}\approx 0.004296[/tex]
The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.
Two marbles are drawn without replacement from a box with 3 White, 2 green, 2 red, and 1 Blue Marble.
Find the probability. Both marbles are white
Answer:
[tex]\frac{3}{28}[/tex]
Step-by-step explanation:
Te meaning of without replacement is once the ball is picked from the stock it cannot be put back
Hre there are total 8 balls and 3 white balls
probability of picking 1st white is [tex]\frac{3}{8}[/tex]
now only 2 white and total is 7( one ball has already been picked)
therefore probability of picking 2nd white ball is [tex]\frac{2}{7}[/tex]
both the action are independent events
therefore, probability of picking 2 white balls is [tex]\frac{3}{8}[/tex] × [tex]\frac{2}{7}[/tex]
= [tex]\frac{3}{28}[/tex]
Answer:[tex]\frac{3}{28}[/tex]
Step-by-step explanation:
Given a box contains 3 White ,2 green,2 red & 1 Blue marble
We have to draw two marbles without replacement
therefore for first draw we have 3 white marble to choose among 8 marbles
i.e.
[tex]_{1}^{3}\textrm{C}[/tex] choices among the total of [tex]_{1}^{8}\textrm{C}[/tex] options
For second draw we 2 white marbles left therefore no of ways in which a white marble can be choosen is
[tex]_{1}^{2}\textrm{C}[/tex]
Therefore required probability is =[tex]\frac{favourable\ outcome}{Total\ outcome}[/tex]
P[tex]\left ( required\right )[/tex]=[tex]\frac{3\times2}{8\times7}[/tex]
P[tex]\left ( required\right )[/tex]=[tex]\frac{3}{28}[/tex]
A survey conducted by the U.S. department of Labor found the 48 out of 500 heads of households were unemployed. Compute a 99% confidence interval for the proportion of unemployed heads of households in the population. Round to three decimal places.
Answer:
(0.062, 0.130)
Step-by-step explanation:
Sample size = n = 500
Number of heads that were unemployed = x = 48
Proportion of heads that were unemployed = p = [tex]\frac{x}{n}=\frac{48}{500}=0.096[/tex]
Proportion of heads that were not unemployed = q = 1 - p = 1 - 0.096 = 0.904
Confidence Level = 99%
z-value for 99% confidence level = z = 2.58
The confidence interval about a population proportion is calculated as:
[tex](p-z\sqrt{\frac{pq}{n}} , p+z\sqrt{\frac{pq}{n}})[/tex]
Using the values, we get:
[tex](0.096-2.58\sqrt{\frac{0.096 \times 0.904}{500}},0.096+2.58\sqrt{\frac{0.096 \times 0.904}{500}})\\\\ = (0.062,0.130)[/tex]
Thus, 99% confidence interval for the proportion of unemployed heads of households in the population is (0.062, 0.130)
The 99% confidence interval for the proportion of unemployed heads of households in the population is approximately 0.096 ± 0.029.
To compute the 99% confidence interval for the proportion of unemployed heads of households, we can use the formula:
Confidence interval = sample proportion ± margin of error
1. Find the sample proportion:
Divide the number of unemployed heads of households (48) by the total number of heads of households surveyed (500).
Sample proportion = 48 / 500 = 0.096
2. Calculate the margin of error:
The margin of error depends on the level of confidence and the sample size. For a 99% confidence level, we need to find the critical value, which corresponds to 99% confidence and 500 as the sample size.
The critical value for a 99% confidence level and 500 as the sample size is approximately 2.576.
Margin of error = critical value * sqrt((sample proportion * (1 - sample proportion)) / sample size)
Margin of error = 2.576 * sqrt((0.096 * (1 - 0.096)) / 500) ≈ 0.029
3. Calculate the confidence interval:
Confidence interval = sample proportion ± margin of error
Confidence interval = 0.096 ± 0.029
For more such information on: confidence interval
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PLEASE HELP ME GET THESE FINISHED
Answer:
g(-0.5) = -1
g(0.2) = 0
g(0.5) = 1
Step-by-step explanation:
We are given the value of g(x) for which the x is defined.
Solving
g(-0.5) = -1
As given g(x) = -1 if -1.5 ≤ x ≤ 0.5
g(0.2) = 0
As given g(x) = 0 if -0.5 < x < 0.5
g(0.5) = 1
As given g(x) = 1 if 0.5 ≤ x < 1.5
Imagine the average time to complete a “4-year” bachelors degree is actually 4.3 years based on national data. You collect data on the 20 psychology students who started school during the same semester as you, finding an average time to complete at 4.5 years, with a sample standard deviation of 0.5 years. What is your 95% confidence interval? (for your FINAL answers, round to the nearest TWO decimal places)
Answer:
To answer you question, we need the confidence internals formula, μ±Zc*(σ/[tex]\sqrt{N}[/tex]); N is the 20 psychology students, 0.5 the desviation standar and 4.3 is the medium based on national data, every fact in years.
Step-by-step explanation:
You need to consult the Zc value, you can find this table attached in this question; for 95%, we have a Zc value of 1.96.
Next step replace values: 4.3±1.96*(0.5/[tex]\sqrt{20}[/tex]) = 4.3±0.22 (rounding the result)
So, the confidence interval are:
4.08cm≤X≤4.52cm
A true false test with 10 questions is given. Compute the probability of scoring exactly 80% by guessing
Answer: 0.04395
Explanation:
Given: 10 true-false questions.
So, we will have 50% chances (probability = 0.5) of being correct.
Prob( Exactly 80% score) = Prob (exactly 8 answers correct)
As we observe, if X= number of correct answers, then X~ Binomial (n=10, p=0.5)
So, Prob( Exactly 80% score) = Prob (exactly 8 answers correct)
=[tex]\binom{10}{8}\times(1/2)^{8}}\times(1/2)^{2}[/tex]
= 0.0439453125
= 0.04395
A study was conducted to measure the effectiveness of a diet program that claims to help manage weight. Subjects were randomly selected to participate. Before beginning the program, each participant was given a score based on his or her fitness level. After six months of following the diet, each participant received another score. The study wanted to test whether there was a difference between before and after scores. What is the correct alternative hypothesis for this analysis?
Answer: u (sub d) is inequal to zero
Step-by-step explanation: Because this is a paired t-test, our alternative hypothesis would be u(sub d) is inequal to zero.
Final answer:
The alternative hypothesis for a study on the effectiveness of a diet program would express the expectation of a statistically significant change in fitness level scores, likely a decrease if lower scores indicate better fitness, after participating in the program compared to before.
Explanation:
The correct alternative hypothesis for a study that aims to measure the effectiveness of a diet program in managing weight would look at whether there is a statistically significant difference in the fitness level scores of the participants before and after following the program. As the question suggests evaluating the effectiveness of a diet program, we are particularly interested in seeing an improvement, which would mean expecting a lower score after the program if the score represents a measure where lower is better.
Thus, the alternative hypothesis (H1) should reflect the expectation of improvement. If the fitness score is such that a lower score indicates better fitness, the alternative hypothesis would be:
H1: The mean fitness level score after the program is lower than the mean fitness level score before the program.
This implies that the diet program is effective in improving fitness levels. On the contrary, if a higher fitness score indicates better fitness, the alternative hypothesis would be framed to reflect an expected increase in the score after the program.
Find a simplified weighted voting system which is equivalent to
[8: 9, 3, 2, 1] and
[20: 8, 6, 3, 2, 1].
Answer: The explanation is as follows:
Step-by-step explanation:
(a) [8: 9, 3, 2, 1]
q = 8
Here, coalition is as follows:
[P1, P2, P3, P4] = [9, 3, 2, 1]
for the above coalition, the combined weight is
[P1, P2, P3, P4] = 9+3+2+1 = 15 ⇒ combined weight
For simplified weighted voting system;
q = combined weight ⇒ both the terms have to be equal for a simplified weighted voting system.
But, here 8 ≠ 15
∴ It is not a simplified weighted voting system.
(b) [20: 8, 6, 3, 2, 1]
q = 20
Here, coalition is as follows:
[P1, P2, P3, P4, P5] = [8, 6, 3, 2, 1]
for the above coalition, the combined weight is
[P1, P2, P3, P4, P5] = 8+6+3+2+1 = 20 ⇒ combined weight
For simplified weighted voting system;
q = combined weight
Since, 20 = 20
∴ It is a simplified weighted voting system.