Answer:
1.25 liters of oil
Step-by-step explanation:
Volume in Beaker A = 1 L
Volume of Oil in Beaker A = 1*0.3 = 0.3 L
Volume of Vinegar in Beaker A = 1*0.7 = 0.7 L
Volume in Beaker B = 2 L
Volume of Oil in Beaker B = 2*0.4 = 0.8 L
Volume of Vinegar in Beaker B = 1*0.6 = 1.2 L
If half of the contents of B are poured into A and assuming a homogeneous mixture, the new volumes of oil (Voa) and vinegar (Vva) in beaker A are:
[tex]V_{oa} = 0.3+\frac{0.8}{2} \\V_{oa} = 0.7 \\V_{va} = 0.7+\frac{1.2}{2} \\V_{va} = 1.3[/tex]
The amount of oil needed to be added to beaker A in order to produce a mixture which is 60 percent oil (Vomix) is given by:
[tex]0.6*V_{total} = V_{oa} +V_{omix}\\0.6*(V_{va}+V_{oa} +V_{omix}) = V_{oa} +V_{omix}\\0.6*(1.3+0.7+V_{omix})=0.7+V_{omix}\\V_{omix}=\frac{0.5}{0.4} \\V_{omix}=1.25 \ L[/tex]
1.25 liters of oil are needed.
There is 1.25 litres of oil that should now be added to A to produce a mixture that is 60 per cent oil.
Given
Beaker A contains 1 litre which is 30 per cent oil and the rest is vinegar, thoroughly mixed up.
Beaker B contains 2 litres which are 40 per cent oil and the rest vinegar, completely mixed up.
Half of the contents of B are poured into A, then completely mixed up.
How much oil is in each container?Contents in beaker A implies;
15% of oil in 1 litre = 0.15 litre of oil
So that, there are 0.15 litres of oil and 0.85 litres of vinegar in beaker A.
Contents in beaker B implies:
55% of oil in 2 litres = 1.1 litres of oil
So that, thee are 1.1 litres of oil and 0.9 litres of vinegar in beaker B.
Half of the contents of B poured into A implies that beaker A now contains:
0.15 litres + 0.55 litres = 0.7 litres of oil
0.85 litres + 0.45 litres = 1.3 litres of vinegar
Then,
The percentage of oil in A is;
[tex]=\dfrac{0.7}{2} \times 100\\\\= 35[/tex]
To increase the percentage of oil to 60%, then:
0.7 litres + 1.25 litres = 1.95 litres of oil
And The new total litres of the content in beaker A = 3.25 litres
[tex]=\dfrac{1.95}{3.25}\times 100\\\\=60 \rm \ percent[/tex]
Hence, 1.25 litres of oil should now be added to A to produce a mixture that is 60 per cent oil.
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A bettor with utility function U(x) = ln(x), where x is total wealth, has a choice between the following two alternatives:
A. Win $ 10,000 with probability 0.2
Win $1000 with probability 0.8 B
B. Win $3000 with probability 0.9
Lose $2000 with probability 0.1
1. If the bettor currently has $2500, should he choose A or B?
2. Repeat a, assuming the bettor has $5000.
3. Repeat a, assuming the bettor has $10,000.
4. Do you think that this pattern of choices between A and B is reasonable? Why or why not?
Answer
The answer and procedures of the exercise are attached in a microsoft word document.
Explanation
Please consider the data provided by the exercise. If you have any question please write me back. All the exercises are solved in a single sheet with the formulas indications.
Find a closed-form solution to the integral equation y(x) = 3 + Z x e dt ty(t) , x > 0. In other words, express y(x) as a function that doesn’t involve an integral. (Hint: Use the Fundamental Theorem of Calculus to obtain a differential equation. You can find an initial condition by evaluating the original integral equation at a strategic value of x.)
Answer:
[tex]y{x} = \sqrt{7+2Inx}[/tex]
Step-by-step explanation:
[tex]y(x)= 3 + \int\limits^x_e {dx}/ \, ty(t) , x>0}[/tex]
Let say; By y(x)= y(e)
we have;
[tex]y(e)= 3 + \int\limits^e_e {dt}/ \, ty= 3+0[/tex]
Using Fundamental Theorem of Calculus and differentiating by Lebiniz Rule:
[tex]y^{1} (x) = 0 + 1/ xy[/tex]
[tex]y^{1} = 1/xy[/tex]
dy/dx = 1/xy
[tex]\int\limits {y} \, dxy = \int\limits \, dx/x[/tex]
[tex]y^{2}/2 Inx + C[/tex]
RECALL: y(e) = 3
[tex](3)^{2} / 2 = In (e) + C[/tex]
[tex]\frac{9}{2} =In(e)+C[/tex]
[tex]\frac{9}{2} - 1 = C[/tex]
[tex]\frac{7}{2} = C[/tex]
[tex]y^{2} / 2 = In x +C[/tex]
[tex]y^{2} / 2 = In x +7/2[/tex]
MULTIPLYING BOTH SIDE BY 2 , TO ELIMINATE THE DENOMINATOR, WE HAVE;
[tex]y^{2} = {7+2Inx}[/tex]
[tex]y{x} = \sqrt{7+2Inx}[/tex]
Atkins, Inc. produces a product requiring 8 pounds of material at $1.50 per pound. Atkins produced 10,000 units of this product during 2019 resulting in a $30,000 unfavorable materials quantity variance. How many pounds of direct material did Atkins use during 2019?
Answer:
100,000 pounds
Step-by-step explanation:
The expected value (E) spent on direct materials used by Atkins,Inc for 2019 is:
[tex]E=\$1.50*8*10,000\\E= \$120,000[/tex]
Since there was an unfavorable $30,000 variance in materials quantity, the actual value spent (Av) on direct materials is:
[tex]A_{v}= \$120,000 +\$30,000\\A_{v}= \$150,000[/tex]
The amount of direct material (M), in pounds, used during 2019 is:
[tex]M=\frac{\$150,000}{\$1.50} \\M=100,000 \ pounds[/tex]
Amanda's computer weighs 20 ounces. How many pounds does it weigh?
Answer: 1.25
Step-by-step explanation:
Convert it lol I don’t know how to say it? But also you can look it up if needed
Answer:
1.25
Step-by-step explanation:
Convert it lol I don’t know how to say it? But also you can look it up if needed
The U.S. Bureau of Labor Statistics reports that 11.3% of U.S. workers belong to unions. Suppose a sample of 400 U.S. workers is collected in 2014 to determine whether union efforts to organize have increased union membership at 0.025 level of significance. The sample results in a test statistic (z) of 2.2.
We conclude that union membership increased in 2014. (Enter 1 if the conclusion is correct. Enter 0 otherwise.)
Answer:
1. The conclusion is statistically correct at the significance level given.
Step-by-step explanation:
1) Data given and notation n
n=400 represent the random sample taken
X represent the people with union membership in the sample
[tex]\hat p[/tex] estimated proportion of people with union membership in the sample
[tex]p_o=0.113[/tex] is the value that we want to test
[tex]\alpha=0.025[/tex] represent the significance level (no given)
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
p= population proportion of people with union membership
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that the proportion of people with union membership exceeds 11.3%. :
Null Hypothesis: [tex]p \leq 0.113[/tex]
Alternative Hypothesis: [tex]p >0.113[/tex]
We assume that the proportion follows a normal distribution.
This is a one tail upper test for the proportion of union membership.
The One-Sample Proportion Test is "used to assess whether a population proportion [tex]\hat p[/tex] is significantly (different,higher or less) from a hypothesized value [tex]p_o[/tex]".
Check for the assumptions that he sample must satisfy in order to apply the test
a)The random sample needs to be representative: On this case the problem no mention about it but we can assume it.
b) The sample needs to be large enough
[tex]np_o =400*0.113=45.2>10[/tex]
[tex]n(1-p_o)=400*(1-0.113)=354.8>10[/tex]
3) Calculate the statistic
The statistic is calculated with the following formula:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}}[/tex]
On this case the value of [tex]p_o=0.113[/tex] is the value that we are testing and n = 400.
Since we have already the statistic calculated z=2.2, we just need to calculate the p value in order to check if we can reject or not the null hypothesis.
4) Statistical decision
It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.
Based on the alternative hypothesis the p value would be given by:
[tex]p_v =P(z>2.2)=1-P(z<2.2)=0.014[/tex]
Using the significance level given [tex]\alpha=0.025[/tex] we see that [tex]p_v<\alpha[/tex] so we have enough evidence at this significance level to reject the null hypothesis. And on this case makes sense the claim that the union membership increased in 2014.
Mary Katharine has a bag of 3 red apples, 5 yellow apples and 4 green apples. Mary takes a red apple out of the bag and does not replace it. What is the probability that the next apple she takes out if yellow
the probability will be 5 out of 11 apples which is 5/11
Answer:
5/11
Step-by-step explanation:
There are a total of 3 + 4 + 5 = 12 apples.
Mary removes a red apple, so there are a total of 11 apples left.
The probability that the next apple is yellow is 5/11.
A poll found that 64% of a random sample of 1076 adults said they believe in ghosts.
Question 1. Find the margin of error zsqrt(((p^^\^)((1-p^^\^)))/n), abbreviated ME, for this poll if we want 90% confidence in our estimate of the proportion of adults who believe in ghosts.
ME=
(Round to 3 decimal places.)
Question 2. Find the margin of error needed to be 99% confident.
ME= (Round to 3 decimal places.)
The margin of error for a 90% confidence level is 2.4%, and for a 99% confidence level, it is 3.8%
Explanation:The margin of error (ME) for a poll can be calculated using the formula:
ME = Z × sqrt((p ×(1 - p)) / n)
Where:
p is the proportion, here p = 0.64n is the sample size, here n = 1076Z is the z-value associated with the desired confidence levelQuestion 1: For a 90% confidence level, the z-value (Z) is 1.645 (use a z-table or statistical software to obtain this).
Substitute the values in the formula to find the ME: ME = 1.645 × sqrt((0.64 × (1 - 0.64)) / 1076) = 0.024 or 2.4%
Question 2: For a 99% confidence level, the z-value (Z) is 2.576 (use a z-table or statistical software to obtain this).
Substitute the values in the formula to find the ME: ME = 2.576 × sqrt((0.64 × (1 - 0.64)) / 1076) = 0.038 or 3.8%
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A preliminary study of hourly wages paid to unskilled employees in three metropolitan areas was conducted. Included in the sample were: 7 employees from Area A, 9 employees from Area B, and 12 employees from Area C. The test statistic was computed to be 4.91. What can we conclude at the 0.05 level?
There is a system glitch during the process of adding my answer here so i create a document file for the answer, you find it in the attachment below.
Can someone pls help
Answer:
15x - 2.
Step-by-step explanation:
3x - 1 + 2x + 1 + 4x - 2 + 4x - 4 + 2x + 4
= 15x - 2.
Answer:
The total length of planting can be calculated by the following steps ;
Step-by-step explanation:
Suppose that x is a Normally distributed random variable with an unknown mean μ and known standard deviation 6. If we take repeated samplesof size 100 and compute the sample means x , 95% of all of these values of x should lie within a distance of _____ from μ . (Use the 68‑95‑99.7 rule.)
Answer:
1.2
Step-by-step explanation:
Given that X is Normally distributed random variable with an unknown mean μ and known standard deviation 6
Hence we can say for a sample of size 100, the sample mean will have a std deviation of = [tex]\frac{6}{\sqrt{100} } =0.6[/tex]
Since population std deviation is known we can use Z critical value for finding out the confidence interval
For 95% using (68-95-99.7 rules) we have z critical value =2
Hence margin of error =2(std error) = 1.2
Confidence interval 95%
Lower bound = Mean - margin of error = Mean -1.2
UPper bound = Mean +1.2
Hence , 95% of all of these values of x should lie within a distance of __1.2___ from μ .
Final answer:
95% of the sample means will lie within approximately 1.176 units from the population mean μ when the standard deviation is 6 and the sample size is 100.
Explanation:
The question pertains to the concept known as the Central Limit Theorem in statistics, which allows us to make inferences about the population mean μ from the distribution of sample means. Since x is normally distributed with a known standard deviation and we take repeated samples to calculate the sample means, we can say that 95% of the sample means will lie within 1.96 standard errors of the population mean μ.
Using the formula for the standard error σ/√n, where σ is the known standard deviation and n is the sample size, we get the standard error as 6/√100 = 0.6. Therefore, 95% of the sample means will lie within 1.96 * 0.6, which is approximately 1.176 units from μ.
John is interested in purchasing a multi-office building containing five offices. The current owner provides the following probability distribution indicating the probability that the given number of offices will be leased each year. Number of Lease Offices 0 1 2 3 4 5 Probability 5/18 1/4 1/9 1/18 2/9 1/12 If each yearly lease is $12,000, how much could John expect to collect in yearly leases for the whole building in a given year?(in dollars)
a) E(X) = $23,353.33
b) E(X) = $23,333.33
c) E(X) = $23,273.33
d) E(X) = $23,263.33
e) E(X) = $23,423.33
f) None of the above.
Answer:
Option B.
Step-by-step explanation:
The given table is:
Number of Lease Offices : 0 1 2 3 4 5
Probability : 5/18 1/4 1/9 1/18 2/9 1/12
The expected probability is
Expected probability = [tex]\sum_{i=0}^5 x_{i}p(x_i)[/tex]
Expected probability = [tex]0p(0)+1P(1)+2P(2)+3P(3)+4P(4)+5P(5)[/tex]
Expected probability = [tex]0\cdot (\frac{5}{18})+1\cdot (\frac{1}{4})+2\cdot (\frac{1}{9})+3\cdot (\frac{1}{18})+4\cdot (\frac{2}{9})+5\cdot (\frac{1}{12})=\frac{35}{18}[/tex]
It is given that the yearly lease = $12,000.
The yearly leases for the whole building in a given year is
Yearly leases = [tex]\frac{35}{18}\times 12000=23333.3333333\approx 23333.33[/tex]
Therefore, the correct option is B.
The expected yearly amount that John can collect from the rental of the multi-office building, taking into account the provided probability distribution for the number of offices that might be rented, is approximately $23,333.33.
Explanation:The subject of this problem involves the concept of expected value in mathematics, particularly in statistics. The expected value, denoted as E(X), is a way of summarizing a probability distribution in terms of an average value. In this case, John is looking to understand the expected amount of income in rent he will receive in a year from this multi-office building based on the provided probability distribution.
First, we have to calculate the expected value of the number of leased offices. This can be done by multiplying each possible outcome by its probability and then summing up these products: E(X) = (0*5/18) + (1*1/4) + (2*1/9) + (3*1/18) + (4*2/9) + (5*1/12). After calculation, we get E(X) approximatively to be 1.9444.
However, we need to find the expected yearly amount in dollars that John could collect. Given that each lease is $12,000 per year, we multiply this lease value by the expected number of leased offices to give us: E($X) = E(X) * $12,000 which gives the value $23,333.34.
Therefore, the correct answer is (b) E(X) = $23,333.33 per year, which is the closest option.
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This problem addresses some common algebraic errors. For the equalities stated below assume that x and y stand for real numbers. Assume that any denominators are non-zero. Mark the equalities with T (true) if they are true for all values of x and y, and F (false) otherwise.
1. (x+y)^2 =x^2+y^2 __
2. (x+y)^2 = x^2 +2xy+y^2__
3. x/x+y=1/y__
4. x−(x+y) = y__
5. √x^2 =x__
6. √x^2 = |x|__
7. √x^2+4=x+2__
8. 1/x+y=1/x+1/y__
Answer:
1. F
observe that [tex](5+2)^2=49 \neq 29=5^2+2^2[/tex]
2. T
Let x and y real numbers.
[tex](x+y)^2=(x+y)(x+y)=x^2+2xy+y^2[/tex]
3. F
Observe that if x=3 and y=2 [tex]\frac{3}{3+2}=\frac{3}{5}\neq \frac{1}{2}[/tex]
4. F
If x=y=3, [tex]3-(3+3)=3-6=-3\neq 3[/tex]
5. F
if x=-1, [tex]\sqrt{-1^2}=\sqrt{1}=1\neq -1[/tex]
6. T
7. F
if x=-1, [tex]\sqrt{-1^2+4}?\sqrt{5}\neq 1=-1+2[/tex]
8. F
If x=1 and y=2, [tex]\frac{1}{1+2}=\frac{1}{3}\neq \frac{3}{2}=\frac{1}{1}+\frac{1}{2}[/tex]
the dimensions of a rectangular prism are 2m, 6m, Xm Its volume is "192m3." Find the measure of the other dimension
Answer:
16 m
Step-by-step explanation:
volume =l*w*h
6*2*X=192
X=192/12=16 m
In a experiment on relaxation techniques, subject's brain signals were measured before and after the relaxation exercises with the following results:
Person 1 2 3 4 5
Before 32 38 66 49 29
After 26 36 59 52 24
Assuming the population is normally distributed, is there sufficient evidence to suggest that the relaxation exercise slowed the brain waves? (Use α=0.05)
Answer:
There is no sufficient evidence to suggest that the relaxation exercise slowed the brain waves
Step-by-step explanation:
Given that in a experiment on relaxation techniques, subject's brain signals were measured before and after the relaxation exercises with the following results:
The population is normally distributed
This is a paired t test since sample size is very small and same population under two different conditions studied.
[tex]H_0: \bar x-\bar y =0\\H_a: \bar x >\bar y[/tex]
(Right tailed test)
Alpha = 5%
x y Diff
32 26 6
38 36 2
66 59 7
49 52 -3
29 24 5
Mean 3.4
Std dev 4.037325848
Mean difference = [tex]3.40[/tex]
n =5
Std deviation for difference = [tex]4.037326[/tex]
Test statistic t = mean difference / std dev for difference
= [tex]1.883[/tex]
p value =0.132
Since p >0.05 we accept null hypothesis.
There is no sufficient evidence to suggest that the relaxation exercise slowed the brain waves
Paired-sample t-test suggests there's a significant effect on slowing down the brain waves due to the relaxation exercises.
Explanation:The subject of this question involves conducting a paired-sample t-test in a study on relaxation techniques. The t-test will allow us to determine whether there is a significant difference between the brain signals of subjects before and after engaging in relaxation exercises.
First, we compute the paired differences (Before - After) and calculate the mean and standard deviation of this list.
The differences are: 6, 2, 7, -3, 5. Hence, the mean difference (µd) = 3.4 and standard deviation (Sd) = 3.346.
Now, with t = (µd / (Sd/√n)) = (3.4 / (3.346/√5)) = 3.023 which is the t-value and n-1 = 4 (degrees of freedom).
Check a t-value chart for a two-tailed test (since we do not know if the relaxation exercises will increase or decrease brain activity) with degree of freedom = 4 and α=0.05. If our t-value is beyond the critical value in the table, we have a significant result. If it falls within that range, we do not. In this case, it's beyond so we can conclude the relaxation exercises have a significant effect on slowing down the brain waves.
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The volume of a rectangular prism is 2x+ 9x-8x-36 with height x+2. Using synthetic division, what is the area of the base ?
Answer:
3(x-12)/(x+2)
Step-by-step explanation:
volume = area *height
2x+ 9x-8x-36 = area * (x+2)
area = (2x+ 9x-8x-36)/(x+2)
=(3x-36)/(x+2)
=3(x-12)/(x+2)
Answer:
2x^2+5x-18
Step-by-step explanation:
the radius of a cylinder is increasion at a rate of 1 meter per hour, and the height of the cylinder is decreasing at a rate of 4 meters per hour. at a certain instant, the base radius is 5 meters and the height is 8 meters. what is the rate of change of the volume of the cylinder at the instant? (Note the formula for volume of a cylinder is V r h)
Answer:
The volume is decreasing at a rate 20 cubic meters per hour.
Step-by-step explanation:
We are given the following in the question:
Rate of change of radius =
[tex]\displaystyle\frac{dr}{dt} = 1 \text{ meter per hour}[/tex]
Rate of change of height =
[tex]\displaystyle\frac{dr}{dt} = -4 \text{ meter per hour}[/tex]
At an instant,
radius, r = 5 meters
Height, h = 8 meters
Volume of cylinder , V=
[tex]\pi r^2 h[/tex]
where r is the radius of cylinder and h is the height of cylinder.
Rate of change of volume of cylinder =
[tex]\displaystyle\frac{dV}{dt} = \frac{d(\pi r^2 h)}{dt} = \pi\Big(2r\frac{dr}{dt}h + r^2\frac{dh}{dt}\Big)[/tex]
Putting the value, we get,
[tex]\displaystyle\frac{dV}{dt} = \pi\Big(2(5)(1)(8) + (5)^2(-4)\Big) = -20\text{ cubic meters per hour}[/tex]
Thus, the volume is decreasing at a rate 20 cubic meters per hour.
The rate of change of the volume at that particular instant is - 62.8 m^3/h.
How to get the rate of change of the volume?
First, we can write the dimensions as:
radius = R = (5m + 1m/h*t)height = H = (8m - 4 m/h*t)Where t is the time in hours.
Then the volume of the cylinder will be:
V = pi*R^2*H = 3.14*(5m + 1m/h*t)^2*(8m - 4 m/h*t)
To get the rate of change, we need to differentiate it with respect to t, we will get:
V' = 3.14*(2*(5m + 1m/h*t)*(1m/h)*(8m - 4 m/h*t) + (5m + 1m/h*t)^2*(-4m/h))
V' = 3.14*( (2 m/h)*(5m + 1m/h*t)*(8m - 4 m/h*t) - (4m/h)*(5m + 1m/h*t)^2)
V' = 3.14*(2m/h)*( (5m + 1m/h*t)*(8m - 4 m/h*t) - 2*(5m + 1m/h*t)^2)
As you can see the rate of change depends on t, but we want the rate of change at this instant, then we use t = 0, replacing that on the above equation we get:
V'(0) = 3.14*(2m/h)*( (5m + 1m/h*0)*(8m - 4 m/h*0) - 2*(5m + 1m/h*0)^2)
V'(0) = 3.14*(2m/h)*( (5m)*(8m ) - 2*(5m)^2)= -62.8 m^3/h
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Which statements are true about the ordered pair (-4, 0) and the system of equations?
{2x+y=-8
{x-y=-4
Select each correct answer.
The ordered pair (-4, 0) is a solution to the first equation because it makes the first
equation true.
The ordered pair (-4, 0) is a solution to the second equation because it makes the
second equation true.
The ordered pair (-4, 0) is not a solution to the system because it makes at least
one of the equations false.
The ordered pair (-4, 0) is a solution to the system because it makes both
equations true
Answer:first 3
Step-by-step explanation:
I'm only in 6th grade but I think it's the first 3
Sketch the region bounded by the curves, and visually estimate the location of the centroid. y = 2x, y = 0, x = 1 WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot Find the exact coordinates of the centroid.
Answer:
the graph is in the attachment.
the coordinates of the centroid : (2/3,2/3)
Step-by-step explanation:
y=0 represents x-axis ( you can easily mark it on the graph)now draw x=1 line.( It is a line parallel to y axis and passing through the point (1,0) )y=2x is a line which passes through origin and has a slope "2"by using these sketch the region.
I have uploaded the region bounded in the attachment. You may refer it. The region shaded with grey is the required region.
to find centroid:it can be easily identified that the formed region is a triangle
the coordinates of three vertices of the triangle are(1,2) , (0,0) , (1,0)
( See the graph. the three intersection points of the lines are the three vertices of the triangle)
for general FORMULA, let the coordinates of three vertices of a triangle PQR be P(a,b) , Q(c,d) , R(e,f) then the coordinates of the centroid( let say , G) of the triangle is given byG = [tex](\frac{a+c+e}{3} , \frac{b+d+f}{3} )[/tex]
therefore , the exact coordinates of the centroid =[tex](\frac{1+0+1}{3}, \frac{2+0+0}{3} ) = (\frac{2}{3}, \frac{2}{3} )[/tex]
this point is marked as G in the graph uploaded.
The centroid of the region bounded by the curves y = 2x, y = 0, and x = 1 has the exact coordinates (2/3, 2/3) found through the application of the centroid formula integrating over the bounded region.
Explanation:The question asks for a sketch and calculation of the centroid of the region bounded by the curves given by the equations y = 2x, y = 0, and x = 1. First, we must plot these equations on a graph. The line y = 2x starts at the origin and goes upward diagonally, and it is bounded by the line y = 0 (the x-axis) and the vertical line x = 1.
To find the exact coordinates of the centroid, we use the centroid formula for x-coordinate (̇x) and y-coordinate (̇y), which are ̇x = (1/A)∫xdb and ̇y = (1/(2A))∫ydb where db is the differential area and A is the total area. In this case, A is the area under the curve from x=0 to x=1, which is integral from 0 to 1 of 2x dx, yielding an area of 1. The centroid x-coordinate is ̇x = (1/1)∫0¹x(2x)dx = 2/3. The centroid y-coordinate is ̇y = (1/(2*1))∫0¹(2x)dbx = 2/3.
The centroid of the region, therefore, has the coordinates (x, y) = (2/3, 2/3).
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The two representative points for the first half of a data in a data set are (1,10) and (8,2). Create a data set with at least eight points that fits these representative points and find the line of best fit for the data.
Answer:
line : 8x+7y=78.
points: (0 , 78/7) ,
(78/8 , 0) ,
(2 , 62/7) ,etc.
Step-by-step explanation:
we have two points. so, we can find the equation of line passing through these 2 pointsthe equation of a line passing through the two points P(a,b) and Q(c,d)is : [tex]y-b=(\frac{d-b}{c-a} )*(x-a)\\[/tex]
here, the equation of line passing through the given two points is[tex]y-10=(\frac{2-10}{8-1} )*(x-1)\\[tex]y-10=(\frac{-8}{7} )*(x-1)[/tex]
multiplying both sides by 7,7y-70=-8x+8\\8x+7y=78[/tex]
therefore, the line of best fit for the data is 8x+7y=78. now make 8 points which satisfy above line equation( for easier way fix some value for x and substitute it in the above line equation then find corresponding y value. these x & y values will make a point. for more points , keep changing the values of x and find corresponding y values)
(0 , 78/7) ,(78/8 , 0) ,
(2 , 62/7) ,etc.
(3 points) Independent random samples, each containing 80 observations, were selected from two populations. The samples from populations 1 and 2 produced 57 and 46 successes, respectively. Test H0:p1=p2H0:p1=p2 against Ha:p1≠p2Ha:p1≠p2. Use α=0.02α=0.02.
(a) The test statistic is
(b) The P-value is
(c) The final conclusion is
A. We can reject the null hypothesis that (p1−p2)=0 and conclude that (p1−p2)≠0.
B. There is not sufficient evidence to reject the null hypothesis that (p1−p2)=0.
Answer:
A is right
you can reject
Step-by-step explanation:
Final answer:
The hypothesis testing involves comparing two independent sample proportions to check if their difference is statistically significant. The provided p-value of 0.0417 is compared with the significance level (0.02, and because the p-value is higher, we do not reject the null hypothesis, indicating there is not sufficient evidence to suggest a significant difference between the two proportions.(Option b)
Explanation:
The student's question involves testing the equality of two population proportions to determine if there is a significant difference between them. This entails conducting a hypothesis test for two independent sample proportions. To perform this test, one would typically use a test statistic that follows a standard normal distribution under the null hypothesis, which is based on the differences between the sample proportions.
Steps for the hypothesis test:
State the null hypothesis (H0: p1 = p2) and the alternative hypothesis (Ha: p1 ≠ p2).
Calculate the test statistic using the sample data.
Compare the test statistic to the critical value or use the p-value approach to make a decision.
Since the student already provided the sample sizes (n1 = n2 = 80) and the number of successes (x1 = 57 and x2 = 46), the test statistic can be computed using these values. Without the specific calculation here, we instead refer to the result that would come from using the test statistic to obtain the p-value.
Given the provided information that the p-value is 0.0417, we compare this with α = 0.02. Because the p-value is greater than α, we do not reject the null hypothesis (No significant difference between proportions).
The final conclusion is option B: There is not sufficient evidence to reject the null hypothesis that (p1−p2) = 0.
The accompanying data on degree of spirituality for a sample of natural scientists and a sample of social scientists working at research universities appeared in a paper. Assume that it is reasonable to regard these two samples as representative of natural and social scientists at research universities. Is there evidence that the spirituality category proportions are not the same for natural and social scientists? Test the relevant hypotheses using a significance level of 0.01. Degree of Spirituality Very Moderate Slightly Not at All Natural Scientists 54 161 195 216 Social Scientists 55 220 239 242 x² = _________ P-value= ____________
Answer:
There is no evidence that the spirituality category proportions are different for natural and social scientists.
Step-by-step explanation:
To solve this question we must perform a Chi square test calculating the expected values of the observed behavior.
We start by completing the table given by adding the totals:
Observed Very Moderate Slightly Not at all Total
Natural Sc 54 161 195 216 626
Social Sc 55 220 239 242 756
Total 109 381 434 458 1382
Now, the chi square value is calculated with the following formula:
[tex]x^{2}[/tex]=∑∑[tex]\frac{O_{ij}-E_{ij} }{E_{ij} }[/tex]
Where:
[tex]O_{ij}[/tex]: Observed value (the ones we have in our table)
[tex]E_{ij}[/tex]: Expected value
The expected value of every observation (ij) is calculated as it follows:
[tex]E_{ij}=\frac{n_{i}c_{j} }{N}[/tex]
Where,
[tex]n_{i}[/tex]: marginal total by rows
[tex]c_{j}[/tex]: marginal total by columns
N: Total of observations
Now, for the expected observations we obtain the following table:
Expected Very Moderate Slightly Not at all Total
Natural Sc 49.37 172.58 196.59 207.46 626
Social Sc 59.63 208.42 237.41 250.54 756
Total 109 381 434 458 1382
Having the expected values we now can calculate [tex]x^{2}[/tex] by first calculating [tex]\frac{O_{ij}-E_{ij} }{E_{ij} }[/tex] for each observation:
Chi sq Very Moderate Slightly Not at all Total
Natural Sc 0.434 0.777 0.013 0.352 1.575
Social Sc 0.359 0.643 0.011 0.291 1.304
Total 0.793 1.420 0.024 0.643 2.879
[tex]x^{2}=2.879[/tex]
We may consider our null hypothesis as it follows:
[tex]H_{0}:[/tex] The degree of spirituality category proportions are the same for natural and social scientists.
To prove this we have:
[tex]P(x^{2}\geq 2.879)[/tex]
α=0.01
α: significance level
Calculating this value with a chi square table (or with statistical software like R) we obtain:
P-value=0.4105
Because the p-value is larger than α the null hypothesis is accepted. Which means we cannot say that the spirituality category proportions are different.
Oil is pumped from a well at a rate of r left-parenthesis t right-parenthesis barrels per day, with t in days. Assume r Superscript prime Baseline left-parenthesis t right-parenthesis > Baseline 0 and t Subscript 0 Baseline greater-than 0. Rank in order from least to greatest:
Answer:
Step-by-step explanation:
A non-profit organization provides homeless children with backpacks filled with school supplies. For the last 5 years the organization has given out 278, 310, 320, 242, and 303 backpacks filled with school supplies. What is the average number of backpacks given out in the last five years?
Answer:
290.6 backpacks per year
Step-by-step explanation:
The average of 5 numbers is 1/5 of their sum:
average = (278 +310 +320 +242 +303)/5 = 290.6
The average number of backpacks given out in the last 5 years is 290.6, about 291.
Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the P/E, or price-to-earnings ratio. High P/E ratios may indicate a stock is overpriced. For the S&P Stock Index of all major stocks, the mean P/E ratio is μ = 19.4. A random sample of 36 "socially conscious" stocks gave a P/E ratio sample mean of x = 17.8, with sample standard deviation s = 5.4. Does this indicate that the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index? Use α = 0.05.
Answer:
[tex]t=\frac{17.8-19.4}{\frac{5.4}{\sqrt{36}}}=-1.78[/tex]
[tex]p_v =2*P(t_{(35)}<-1.78)=0.084[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the mean P/E ratio of all socially conscious stocks it's not significantly different from the mean P/E ratio of the S&P Stock Index (19.4) at 5% of signficance.
Step-by-step explanation:
1) Data given and notation
[tex]\bar X=17.8[/tex] represent the P/E ratio sample mean
[tex]s=5.4[/tex] represent the sample standard deviation
[tex]n=36[/tex] sample size
[tex]\mu_o =19.4[/tex] represent the value that we want to test
[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
2) State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the mean P/E ratio of all socially conscious stocks is different (either way) from the mean P/E ratio of the S&P Stock Index (19.4) :
Null hypothesis:[tex]\mu =19.4[/tex]
Alternative hypothesis:[tex]\mu \neq 19.4[/tex]
Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
3) Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{17.8-19.4}{\frac{5.4}{\sqrt{36}}}=-1.78[/tex]
4) P-value
First we need to calculate the degrees of freedom given by:
[tex]df=n-1=36-1=35[/tex]
Since is a two-sided test the p value would be:
[tex]p_v =2*P(t_{(35)}<-1.78)=0.084[/tex]
5) Conclusion
If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the mean P/E ratio of all socially conscious stocks it's not significantly different from the mean P/E ratio of the S&P Stock Index (19.4) at 5% of signficance.
The mean P/E ratio of socially conscious stocks is different from the mean P/E ratio of the S&P Stock Index.
Explanation:The question asks whether socially conscious stocks are overpriced compared to the S&P Stock Index. To determine this, we will conduct a hypothesis test using the P/E ratio as the measure of value. The null hypothesis states that the mean P/E ratio of socially conscious stocks is the same as the mean P/E ratio of the S&P Stock Index, while the alternative hypothesis states that they are different. We will use a significance level of 0.05.
To perform the hypothesis test, we will calculate the test statistic and compare it to the critical value. The test statistic is calculated as (sample mean - population mean)/(sample standard deviation/sqrt(sample size)). For the given data, the test statistic is (17.8 - 19.4)/(5.4/sqrt(36)) = -2.88.
Next, we will compare the test statistic to the critical value. Since we are conducting a two-tailed test, the critical value is ±1.96 for a significance level of 0.05. Since the test statistic (-2.88) falls outside the critical value range (-1.96 to 1.96), we reject the null hypothesis and conclude that the mean P/E ratio of socially conscious stocks is different from the mean P/E ratio of the S&P Stock Index at a significance level of 0.05.
Suppose taxi fare from Logan Airport to downtown Boston is known to be normally distributed with a standard deviation of $2.50. The last seven times John has taken a taxi from Logan to downtown Boston, the fares have been $22.10, $23.25, $21.35, $24.50, $21.90, $20.75, and $22.65.
What is a 95% confidence interval for the population mean taxi fare?
Answer:
95% Confidence interval for taxi fare: ($20.5,$24.2)
Step-by-step explanation:
We are given the following data set: for fares:
$22.10, $23.25, $21.35, $24.50, $21.90, $20.75, and $22.65
Formula:
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{156.5}{7} = 22.35[/tex]
95% Confidence interval:
[tex]\bar{x} \pm z_{critical}\frac{\sigma}{\sqrt{n}}[/tex]
Putting the values, we get,
[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]
[tex]22.35 \pm 1.96(\frac{2.5}{\sqrt{7}} ) = 22.35 \pm 1.85 = (20.5,24.2)[/tex]
The 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston is $20.5031 and $24.2111 dollars.
To calculate the 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston, we'll use the sample data provided and the given standard deviation of $2.50.
1. Calculate the sample mean:
First, find the mean of the sample fares:
[tex]\[ \bar{x} = \frac{22.10 + 23.25 + 21.35 + 24.50 + 21.90 + 20.75 + 22.65}{7} = \frac{156.5}{7} = 22.3571 \][/tex]
2. Calculate the standard error of the mean (SEM):
Since the population standard deviation [tex](\(\sigma\))[/tex] is known and the sample size (n) is 7:
[tex]\[ SEM = \frac{\sigma}{\sqrt{n}} = \frac{2.50}{\sqrt{7}} \approx 0.946 \][/tex]
3. Determine the margin of error (ME):
For a 95% confidence level, find the critical value (z*) from the standard normal distribution table, which is approximately 1.96.
[tex]\[ ME = z^* \times SEM = 1.96 \times 0.946 \approx 1.854 \][/tex]
4. Calculate the confidence interval:
[tex]\[ \text{Confidence interval} = \bar{x} \pm ME = 22.3571 \pm 1.854 \][/tex]
[tex]\[ \text{Confidence interval} = (20.5031, 24.2111) \][/tex]
The 95% confidence interval suggests that we are 95% confident that the true population mean taxi fare lies between $20.5031 and $24.2111. This interval takes into account the variability in John's recent taxi fares and provides a range within which we expect the true mean fare to fall.
The 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston is approximately $20.5031 and $24.2111 dollars.
Sandy has 16 roses, 8 daisies and 32 tulips. She wants to arrange all the flowers in bouquets. Each bouquet has the same number of flowers and has only one type of flower. What is the greatest number of flowers that could be in each bouquet?
Answer:
8
Step-by-step explanation:
given,
16 roses, 8 daisies and 32 tulips
required to find
greatest number of flowers that could be in each bouquet such that Each bouquet has the same number of flowers and has only one type of flower
as there are only 8 daisies so the greatest number of flowers that could be in each bouquet such that Each bouquet has the same number of flowers and has only one type of flower is 8
if it is greater than 8 then there will be some other flower in the boquet of daises
Find the exact value of sin (135)°.
The exact value of given trigonometric ratio sin(135)° is 1/√2
The given trigonometric ratio is,
sin(135)°
Since we know that,
The sine function is one of three main functions in trigonometry, along with the cosine and tan functions. The sine x, often known as the sine theta, is the ratio of the opposing side of a right triangle to its hypotenuse.
Since we also know that,
sin(90° + θ) = cosθ
Therefore,
We can write,
sin(135)° = sin(90 + 45)
= cos45
= 1/√2 [ from trigonometric table]
Hence,
sin(135)° = 1/√2
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A single card is drawn from a standard 52-card deck. Let Upper B be the event that the card drawn is a black, and let F be the event that the card drawn is a face card. Find the indicated probability. Upper P (Upper B intersect Upper F )
Final answer:
The probability of drawing a black face card from a standard 52-card deck is 3/26.
Explanation:
To find the probability of two events happening together, we need to calculate the probability of each event and then multiply them together. First, let's find the probability of drawing a black card (B) and a face card (F) separately. The deck contains 26 black cards, and since there are 52 cards in total, P(B) = 26/52 = 1/2. There are 12 face cards in the deck, so P(F) = 12/52 = 3/13. Now, we can find the probability of the intersection of B and F by multiplying their individual probabilities, P(B ∩ F) = P(B) * P(F) = (1/2) * (3/13) = 3/26.
Therefore, the probability of drawing a black face card is 3/26.
The product of two consecutive positive integers is greater than their sum by 209. Find these numbers.
Answer:
15, 16
Step-by-step explanation:
Let x represent the smaller number. Then x+1 is the larger, and their product is x(x+1). Their sum is (x +(x+1)) = 2x+1. So, the required relation is ...
x(x+1) -(2x+1) = 209
Expressed in the form f(x) = 0, we can write this as ...
x(x+1) -(2x+1) -209 = 0
Graphing this shows x=15 to be the positive solution.
The numbers are 15 and 16.
To find the two consecutive positive integers, we can set up an equation using the given information. Solving the equation, we find that the integers are 14 and 15.
Explanation:To solve this problem, let's assume the two consecutive positive integers are x and x+1. The product of these integers is x(x+1) and their sum is x + (x+1). We are given that the product is greater than their sum by 209, so we can set up the equation:
x(x+1) = x + (x+1) + 209
Expanding the equation and simplifying, we get x^2 + x = 210.
This is a quadratic equation that can be solved by factoring or using the quadratic formula. Let's solve it by factoring:
x^2 + x - 210 = 0
(x+15)(x-14) = 0
So, the two possible values for x are -15 and 14, but since we are dealing with positive integers, the value of x is 14. Therefore, the two consecutive positive integers are 14 and 15.
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A personnel researcher has designed a questionnaire and she would like to estimate the average time to complete the questionnaire. Suppose she samples 100 employees and finds that the mean time to take the test is 27 minutes with a standard deviation of 4 minutes. Construct a 90% confidence interval for the mean time to complete the questionnaire. Also, write a short explanation about the findings to the human resources director of your company summarizing the results. Use Excel for this analysis.
Answer:
So on this case the 90% confidence interval would be given by (26.336;27.664)
We are 90% confident that the mean time to complete the questionnaire is between (26.336;27.664)
Step-by-step explanation:
1) Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X=27[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=4 represent the sample standard deviation
n=100 represent the sample size
2) Confidence interval
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=100-1=99[/tex]
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.05,99)".And we see that [tex]t_{\alpha/2}=1.66[/tex]
Now we have everything in order to replace into formula (1):
[tex]27-1.66\frac{4}{\sqrt{100}}=26.336[/tex]
In excel would be "=27-1.66*(4/SQRT(100))"
[tex]27+1.66\frac{4}{\sqrt{100}}=27.664[/tex]
In excel would be "=27+1.66*(4/SQRT(100))"
So on this case the 90% confidence interval would be given by (26.336;27.664)
We are 90% confident that the mean time to complete the questionnaire is between (26.336;27.664)