Answer:
a) [tex] E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_1[/tex] is an unbiased estimator of [tex]\mu[/tex]
[tex] E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_2[/tex] is an unbiased estimator of [tex]\mu[/tex]
b) [tex] Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2} [/tex]
[tex] Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8} [/tex]
Step-by-step explanation:
For this case we know that we have two random variables:
[tex] X_1 , X_2[/tex] both with mean [tex]\mu = \mu[/tex] and variance [tex] \sigma^2[/tex]
And we define the following estimators:
[tex] \hat \theta_1 = \frac{X_1 + X_2}{2}[/tex]
[tex] \hat \theta_2 = \frac{X_1 + 3X_2}{4}[/tex]
Part a
In order to see if both estimators are unbiased we need to proof if the expected value of the estimators are equal to the real value of the parameter:
[tex] E(\hat \theta_i) = \mu , i = 1,2 [/tex]
So let's find the expected values for each estimator:
[tex] E(\hat \theta_1) = E(\frac{X_1 +X_2}{2})[/tex]
Using properties of expected value we have this:
[tex] E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_1[/tex] is an unbiased estimator of [tex]\mu[/tex]
For the second estimator we have:
[tex]E(\hat \theta_2) = E(\frac{X_1 + 3X_2}{4})[/tex]
Using properties of expected value we have this:
[tex] E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_2[/tex] is an unbiased estimator of [tex]\mu[/tex]
Part b
For the variance we need to remember this property: If a is a constant and X a random variable then:
[tex] Var(aX) = a^2 Var(X)[/tex]
For the first estimator we have:
[tex] Var(\hat \theta_1) = Var(\frac{X_1 +X_2}{2})[/tex]
[tex] Var(\hat \theta_1) =\frac{1}{4} Var(X_1 +X_2)=\frac{1}{4} [Var(X_1) + Var(X_2) + 2 Cov (X_1 , X_2)] [/tex]
Since both random variables are independent we know that [tex] Cov(X_1, X_2 ) = 0[/tex] so then we have:
[tex] Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2} [/tex]
For the second estimator we have:
[tex] Var(\hat \theta_2) = Var(\frac{X_1 +3X_2}{4})[/tex]
[tex] Var(\hat \theta_2) =\frac{1}{16} Var(X_1 +3X_2)=\frac{1}{4} [Var(X_1) + Var(3X_2) + 2 Cov (X_1 , 3X_2)] [/tex]
Since both random variables are independent we know that [tex] Cov(X_1, X_2 ) = 0[/tex] so then we have:
[tex] Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8} [/tex]
Both θ₁^ and θ₂^ are unbiased estimators of μ. The variance of θ₁^ is σ² / 2, while the variance of θ₂^ is 5σ² / 8.
Let's analyze the provided estimators of the mean (">").
a) Unbiased Estimators
An estimator heta is unbiased if E[θ] = μ. We have:
θ₁^ = (X₁ + X₂) / 2
The expected value of θ₁^ is:
E[θ₁^] = E[(X₁ + X₂) / 2] = (E[X₁] + E[X₂]) / 2 = (μ + μ) / 2 = μ
Thus, θ₁^ is an unbiased estimator of μ.
θ₂^ = (X₁ + 3X₂) / 4
The expected value of θ₂^ is:
E[θ₂^] = E[(X₁ + 3X₂) / 4] = (E[X₁] + 3E[X₂]) / 4 = (μ + 3μ) / 4 = 4μ / 4 = μ
Thus, θ₂^ is also an unbiased estimator of μ.
b) Variance of Each Estimator
The variance of an estimator θ is given by Var(θ). Considering the given variances of X₁ and X₂ (both σ²):
For θ₁^:Var(θ₁^) = Var[(X₁ + X₂) / 2] = (1/2)²Var(X₁) + (1/2)²Var(X₂) = (1/4)σ² + (1/4)σ² = σ² / 2
For θ₂^:Var(θ₂^) = Var[(X₁ + 3X₂) / 4] = (1/4)²Var(X₁) + (3/4)²Var(X₂) = (1/16)σ² + (9/16)σ² = (1/16 + 9/16)σ² = (10/16)σ² = 5σ² / 8
Thus, the variance of θ₁^ is σ² / 2 and the variance of θ₂^ is 5σ² / 8.
Find the angle between the vectors. Use a calculator if necessary. (Enter your answer in radians. Round your answer to three decimal places.) (√3,1) and <0, 5 > Find the angle between the vectors. Use a calculator if necessary.<0,4,4> and <3,-3,0>
Answer:
a. The angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(\sqrt{3}, 1\right)[/tex] and [tex]\mathbf{v}=\left(0, 5\right)[/tex] is [tex]\phi=\frac{\pi}{3}[/tex] or [tex]60\º[/tex].
b. The angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(0, 4, 4\right)[/tex] and [tex]\mathbf{v}=\left(3, -3, 0\right)[/tex] is [tex]\phi=\frac{2 \pi}{3}[/tex] or [tex]120\º[/tex].
Step-by-step explanation:
a. To calculate the angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(\sqrt{3}, 1\right)[/tex] and [tex]\mathbf{v}=\left(0, 5\right)[/tex] you must:
Step 1: Calculate the dot product.
The dot product is given as [tex]\displaystyle{\large{{\left({u}_{{x}},{u}_{{y}}\right)}\cdot{\left({v}_{{x}},{v}_{{y}}\right)}={u}_{{x}}\cdot{v}_{{x}}+{u}_{{y}}\cdot{v}_{{y}}}}[/tex].
So,
[tex]\left(\sqrt{3}, 1\right)\cdot\left(0, 5\right)=\left(\sqrt{3}\right)\cdot\left(0\right)+\left(1\right)\cdot\left(5\right)=5[/tex]
Step 2: Find the lengths of the vectors.
[tex]\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2}=\sqrt{\left(\sqrt{3}\right)^2+\left(1\right)^2}=2[/tex]
[tex]\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2}=\sqrt{\left(0\right)^2+\left(5\right)^2}=5[/tex]
Step 3: The angle is given by [tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}[/tex]
[tex]\frac{5}{2 \cdot 5}=\frac{1}{2}[/tex]
[tex]\phi=cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}=60^0[/tex]
b. To calculate the angle [tex]\phi[/tex] between the vectors [tex]\mathbf{u}=\left(0, 4, 4\right)[/tex] and [tex]\mathbf{v}=\left(3, -3, 0\right)[/tex] you must:
Step 1: Calculate the dot product.
[tex]\left(0, 4, 4\right)\cdot\left(3, -3, 0\right)=\left(0\right)\cdot\left(3\right)+\left(4\right)\cdot\left(-3\right)+\left(4\right)\cdot\left(0\right)=-12[/tex]
Step 2: Find the lengths of the vectors.
[tex]\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2+\left(u_z\right)^2}=\sqrt{\left(0\right)^2+\left(4\right)^2+\left(4\right)^2}=4 \sqrt{2}[/tex]
[tex]\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2+\left(v_z\right)^2}=\sqrt{\left(3\right)^2+\left(-3\right)^2+\left(0\right)^2}=3 \sqrt{2}[/tex]
Step 3: The angle is given by [tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}[/tex]
[tex]\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}=\frac{-12}{4 \sqrt{2} \cdot 3 \sqrt{2}}=- \frac{1}{2}[/tex]
[tex]\phi=cos^{-1}\left(- \frac{1}{2}\right)=\frac{2 \pi}{3}=120^0[/tex]
To find the angle between two vectors, use the dot product formula. For the vectors (√3,1) and <0, 5 >, the angle is π/3 radians. For the vectors <0,4,4> and <3,-3,0>, the angle is 2π/3 radians.
Explanation:To find the angle between two vectors, we can use the dot product formula. Let's start with the first set of vectors (√3,1) and <0, 5>. We can calculate the dot product by multiplying the corresponding components and adding them up: (√3)(0) + (1)(5) = 5. Next, we need to calculate the magnitudes of each vector. The magnitude of (√3,1) is √(√3)^2 + 1^2 = √4 = 2. The magnitude of <0, 5> is √0^2 + 5^2 = √25 = 5. Now, we can calculate the angle between the vectors using the formula: cosθ = dot product / (magnitude 1 * magnitude 2). Plugging in the values, cosθ = 5 / (2 * 5) = 1/2. Taking the inverse cosine of 1/2 gives us the angle in radians: θ = acos(1/2) = π/3 ≈ 1.047.
For the second set of vectors <0,4,4> and <3,-3,0>, we will follow the same steps. The dot product is (0)(3) + (4)(-3) + (4)(0) = -12. The magnitude of <0,4,4> is √0^2 + 4^2 + 4^2 = √32 = 4√2. The magnitude of <3,-3,0> is √3^2 + (-3)^2 + 0^2 = √18 = 3√2. Plugging in the values, cosθ = -12 / (4√2 * 3√2) = -1/2. Taking the inverse cosine of -1/2 gives us the angle in radians: θ = acos(-1/2) = 2π/3 ≈ 2.094.
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A recent survey estimated that 19 percent of all people living in a certain region regularly use sunscreen when going outdoors. The margin of error for the estimate was 1 percentage point. Based on the estimate and the margin of error, which is an appropriate conclusion?
Final answer:
The appropriate conclusion from the survey is that the true percentage of all people who regularly use sunscreen in the region is estimated to be between 18% and 20%, considering the margin of error of 1 percentage point.
Explanation:
Based on the estimate and the margin of error, an appropriate conclusion is that it can be said with a certain level of confidence, usually 95%, that the true percentage of all people living in that region who regularly use sunscreen when going outdoors is between 18% and 20%. The margin of error gives a range around the estimate to account for the sampling error, and it implies that the actual value in the population should fall within this range. This helps to understand the precision of the survey result and allows for the natural variability that comes from the fact that not every individual in the region was surveyed.
The variability also depends on the size of the sample, which relates to the confidence interval. For example, smaller samples tend to have larger margins of error, while larger samples tend to lead to smaller margins of error and thus, more precise estimates of the population parameter.
Subtracting fractions with unlike denominators 10/12 -1/4 show work
Answer:
7/12
Step-by-step explanation:
Step 1: Find common denominators
The common denominator is 12
10/12 is already good
1/4 * 3/3 = 3/12
Step 2: Subtract
10/12 - 3/12 = 7/12
Answer: 7/12
Final answer:
To subtract the fractions 10/12 and 1/4, find a common denominator, convert the fractions, and then subtract the numerators, resulting in the answer 7/12.
Explanation:
To subtract fractions with unlike denominators, you first need to find a common denominator. For the fractions 10/12 and 1/4, the common denominator is 12. You can convert 1/4 to 3/12, because 4 times 3 equals 12. Now you can subtract the fractions as follows:
10/12 - 3/12 = (10 - 3)/12 = 7/12
So, the answer to 10/12 - 1/4 is 7/12.
This process relies on the intuitive understanding that you can only subtract numerators if the denominators are the same. We never add or subtract the denominators themselves. When your fractions share the same denominator, the subtraction is straightforward as we are simply dealing with parts of the whole that are of the same size.
Suppose that we have a standard Solow model with a Cobb-Douglas production function. The central equation of the model is as follows: kt+1 = sAkα t + (1 − δ)kt . Consumption per worker is given by: ct = (1 − s)Akα t . (a) Solve for an expression for the steady state capital stock per worker. In doing so, assume that the level of productivity is fixed at some value A.
Answer:
The answer is shown in the attachment
Step-by-step explanation:
The detailed step by step and appropriate mathematical manipulation for the expression for the steady state capital stock per worker is as shown in the attachment.
A container initially containing10 L of water in which there is 20 g of salt dissolved. A solution containing 4 g/L of salt is pumped into the container at a rate of 2 L/min, and the well-stilled mixture runs out at a rate of 1 L/min. How much salt is in the tank after 40 min
Answer:
[tex] A(40)= \frac{-200}{10+40} +4 (10 +40)=-4+200 = 196 [/tex]
Step-by-step explanation:
For this case the solution flows at a rate of 2L/min and leaves at 1L/min. So then we can conclude the volume is given by [tex] V= 10 +t[/tex]
Since the initial volume is 10 L and the volume increase at a rate of 1L/min.
For this case we can define A as the concentration for the salt in the container. And for this case we can set up the following differential equation:
[tex] \frac{dA}{dt}= 4 \frac{gr}{L} *2 \frac{L}{min} - \frac{A}{10+t}[/tex]
Because at the begin we have a concentration of 8 gr/L and would be decreasing at a rate of [tex] \frac{A}{10+t}[/tex]
So then we can reorder the differential equation like this:
[tex] \frac{dA}{dt} +\frac{A}{10+t} =8[/tex]
We find the solution using the integration factor:
[tex] \mu = -\int \frac{1}{10+t} dt = -ln(10+t)[/tex]
And then the solution would be given by:
[tex] A = e^{-ln (10+t)} (\int e^{\int \frac{1}{10+t} dt})[/tex]
And if we simplify this we got:
[tex] A= \frac{1}{10+t} (c + \int (10 +t) 8 dt)[/tex]
And after do the integral we got:
[tex] A= \frac{c}{10+t} +4 (10 +t) [/tex]
And using the initial condition t=0 A= 20 we have this:
[tex] 20 = \frac{c}{10} +40[/tex]
[tex] c= -200[/tex]
So then we have this function for the solution of A:
[tex] A= \frac{-200}{10+t} +4 (10 +t) [/tex]
And now replacinf t= 40 we got:
[tex] A(40)= \frac{-200}{10+40} +4 (10 +40)=-4+200 = 196 [/tex]
The amount of salt in the tank after 40 minutes is determined by first calculating the total amount of salt added to the tank, then considering the amount of salt-water mixture that exited the tank during this period due to the outgoing flow.
Explanation:This question relates to a typical problem within the field of differential equations. You are asked to find the amount of salt in the tank after 40 minutes. The initial amount of salt in the tank is 20g, and additional salt-water solution is being pumped in at a rate of 2L/min with a salt concentration of 4g/L, hence adding 8g of salt per minute. At the same time, the well-stirred mixture is flowing out at a rate of 1L/min.
Therefore, after 40 minutes:
Additional salt added = 8g/min * 40 min = 320g.
Total salt content now = initial salt content + added salt = 20g + 320g = 340g.
However, within these 40 minutes,40 L of the mixture has flown out. Since the mixture is well-stirred, it retains the same concentration of salt as the mixture in the tank. Hence, if x is the amount of salt left in the tank, it will be on average the same concentration as that which flowed out and can be represented as follows: x/50 = (340-x)/40.
Solving for x we get the amount of salt left in the tank after 40 minutes.
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With a full tank of gas, if you can drive 485 miles, your car uses 18 miles per gallon. Write an equation to model this situation (use m for miles you can drive and g for gallons in tank).
Answer: m= 18g
Step-by-step explanation:
Let m= miles you can drive
g = g for gallons in tank
We know that there is directly proportional relation between the distance traveled by vehicle and the number of gallons of gas.
Then, rate of miles driven by you per gallon = [tex]\dfrac{m}{g}[/tex]
Since , rate of miles driven by you per gallon = 18 miles per gallon (given)
Then,
[tex]\dfrac{m}{g}=18\\\\ m= 18g[/tex]
Hence, the equation to model this situation : m= 18g
A newly proposed endoscopy method used to screen for colon cancer is tested among a random sample of the population. Given that 16 individuals from a test population of 100,000 individuals has colon cancer and 14 individuals test positive for colon cancer, the researchers should determine the sensitivity of the newly proposed screening method is 0.875.
Please explain why this is true
Answer:
The value of sensitivity obtained is equal to the one given in statement.
Step-by-step explanation:
Sensitivity is the ratio of number of positive test results to the number of individuals having disease. It is the ability of the test to detect all those with disease in the screened population.
Its formula is written as:
Sensitivity = [tex]\frac{number of true positives}{total with disease}[/tex]
So, in this case we have values
sensitivity = 14/16 = 0.875 So, the given statement is true.
The approximate heights of two of the world's tallest mountains are given. Determine which is the taller mountain and by how many meters. Round to the nearest meter. (1 in equals 2.54 cm; 1 ft equals 30.48 cm; 1 yd almost equals 0.9144 m; 1 mi almost equals 1.6 km1 in = 2.54 cm; 1 ft = 30.48 cm; 1 yd ≈ 0.9144 m; 1 mi ≈ 1.6 km) Lhotse: 8516 meters and Ngadi Chuli: 25,866 feet
Answer:
Height of Lhotse mountain is greater than Ngadi Chuli mountain.
Height of Lhotse mountain is 632.0432 meter more than height of Ngadi Chuli mountain.
Step-by-step explanation:
Height of Lhotse mountain = 8516 meters
Height of Ngadi Chuli mountain = 25,866 feet
1 ft = 30.48 cm
[tex]25,866 feet =25,866\times 30.48 m=788,395.68 cm[/tex]
1 cm = 0.01 m
[tex]788,395.68 cm=788,395.68\times 0.01 m=7883.9568 m[/tex]
8516 meters > 7883.9568 m
Height of Lhotse mountain is greater than Ngadi Chuli mountain.
Difference in their height :
8516 meters - 7883.9568 m = 632.0432 m
Height of Lhotse mountain is 632.0432 meter more than height of Ngadi Chuli mountain.
The height of Ngadi Chuli is approximately 7871 meters, while the height of Lhotse is 8516 meters. Therefore, Lhotse is the taller mountain and is about 645 meters taller than Ngadi Chuli.
Explanation:The height of Ngadi Chuli is given in feet, so we must first convert that to meters using the conversion factor 1 ft = 30.48 cm = 0.3048 m. Multiplying 25,866 feet by 0.3048 m/ft gives us an approximate height of 7871 meters for Ngadi Chuli.
Lhotse is given to be 8516 meters tall. Therefore, by comparing these two heights, we can determine that Lhotse is the taller mountain.
The difference in height is found by subtracting the height of Ngadi Chuli from that of Lhotse: 8516 m - 7871 m = 645 m. So, Lhotse is approximately 645 meters taller than Ngadi Chuli.
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At a used dealership, let X be an independent variable representing the age in years of a motorcycle and Y be the dependent variable representing the selling price of used motorcycle. The data is now given to you.
X = {5, 10; 12, 14, 15}; Y = {500, 400, 300, 200, 100}
(a) Write the regression model.
(b) Estimate the parameters of the model
(c) Write the prediction equation
(d) Calculate SSE.
Answer:
a) Selling price y= a + b (age x)
b)
a= 728.025
b= -38.217
c)
Selling price y = 728.025 - 38.217 age x
d)
SSE=8280.25
Step-by-step explanation:
a)
The regression model can be written as
y=a+bx
Here y=selling price and x is age.
So, the regression model will be
Selling price y= a + b (age x)
b)
We have to find the values of "a" and "b"
[tex]b=\frac{nsumxy-(sumx)(sumy)}{nsumx^{2} -(sumx)^2}[/tex]
sumx=5+10+12+14+15=56
sumy=500+400+300+200+100=1500
sumxy=5*500+10*400+12*300+14*200+15*100=14400
sumx²=5²+10²+12²+14²+15²=690
n=5
[tex]b=\frac{5(14400)-(56)(1500)}{5(690) -(56)^2}[/tex]
b=-12000/314
b=-38.217
ybar=a+bxbar
a=ybar-bxbar
ybar=sumy/n=1500/5=300
xbar=sumx/n=56/5=11.2
a=300-(-38.217)(11.2)
a=300+428.025
a=728.025
c)
Selling price y = a - b(age x)
Selling price y = 728.025 - 38.217 age x
d)
SSE known as sum of square of error can be calculated as
SSE=sum(y-yhat)²
y 500 400 300 200 100
x 5 10 12 14 15
yhat= 728.025 - 38.217 age x 536.940 345.855 269.421 192.987 154.770
y-yhat -36.940 54.145 30.579 7.013 -54.770
(y-yhat)² 1364.56 2931.68 935.08 49.18 2999.75
SSE=sum(y-yhat)²
SSE=1364.56 +2931.68 +935.08 +49.18 +2999.75
SSE =8280.25
Suppose x is a random variable best described by a uniform probability distribution with c=20 and d=40.
Find the probability P(20≤x≤35).
Answer:
P(20≤x≤35) = 0.75 .
Step-by-step explanation:
We know that the probability distribution function of Uniform Distribution is represented as :
If x follows Uniform(c,d) then,
f(x) = [tex]\frac{1}{d-c}[/tex] where c < x < d
To fond the given probability it is better to first calculate the Cumulative Distribution Function(CDF) of Uniform Distribution.
The CDF of Uniform Distribution is P(X<=x) = [tex]\frac{x-c}{d-c}[/tex] where d > c .
Therefore, P(20<=x<=35) = P(x<=35) - P(x<20)
P(x<=35) = [tex]\frac{x-20}{40-20}[/tex] because we are given c = 20 and d = 40.
= [tex]\frac{35-20}{40-20}[/tex] = 0.75
P(x<20) = [tex]\frac{x-20}{40-20}[/tex] = [tex]\frac{20-20}{40-20}[/tex] = 0
Hence, P(20<=x<=35) = 0.75 - 0 = 0.75.
The probability of x being between 20 and 35 is 0.75 or 75%.
Explanation:The uniform probability distribution is defined as:
P(x) = 1/(b-a) for a ≤ x ≤ b and 0 otherwise.
In this case, a=20 and b=40, so the probability P(20≤x≤35) can be calculated as:
P(20≤x≤35) = (35-20)/(40-20) = 15/20 = 0.75.
Therefore, the probability of x being between 20 and 35 is 0.75 or 75%.
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Angela took a general aptitude test and scored in the 91st percentile for aptitude in accounting. (a) What percentage of the scores were at or below her score?(b) What percentage were above?
Answer:
a) 91% of the scores were at or below her score.
b) 9% of the scores were above her score.
Step-by-step explanation:
When a value V is said to be in the xth percentile of a set, x% of the values in the set are lower than V and (100-x)% of the values in the set are higher than V.
In this problem, we have that:
Angela scored on the 91st percentile.
(a) What percentage of the scores were at or below her score?
Since Angela scored on the 91st percentile, 91% of the scores were at or below her score.
(b) What percentage were above?
Since Angela scored on the 91st percentile, 9% of the scores were above her score.
Angela scored higher than or equal to 91% of her peers, meaning 9% scored above her.
Explanation:If Angela scored in the 91st percentile on her general aptitude test, that means she scored higher than or equal to 91% of the other scores. Therefore, (a) 91% of scores were at or below Angela's score. As for (b), if you subtract her percentile from 100%, you can determine that 9% of scores were above hers.
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A consumer products company found that 44% of successful products also received favorable results from test market research, whereas 13% had unfavorable results but nevertheless were successful. That is, P(successful product and favorable test market) = 0.44 and P(successful product and unfavorable test market) = 0.13. They also found that 32% of unsuccessful products had unfavorable research results, whereas 11% of them had favorable research results, that is P(unsuccessful product and unfavorable test market) = 0.32 and P(unsuccessful product and favorable test market) = 0.11.
Find the probabilities of successful and unsuccessful products given known test market results, that is, P(successful product given favorable test market), P(successful product given unfavorable test market), P(unsuccessful product given favorable test market), and P(unsuccessful product given unfavorable test market).
Answer:
0.800.2890.200.711Step-by-step explanation:
Given:
[tex]P(S\cap F)=0.44\\P(S\cap F^{c})=0.13\\P(S^{c}\cap F^{c}) = 0.32\\P(S^{c}\cap F) = 0.11[/tex]
The rule of total probability states that:
[tex]P(A) = P(A\cap B) + P(A\cap B^{c})[/tex]
Compute the individual probabilities as follows:
[tex]P(S) = P(S\cap F) + P(S\cap F^{c})\\=0.44+0.13\\0.57[/tex]
[tex]P(S^{c}) = 1 - P(S)\\=1-0.57\\=0.43[/tex]
[tex]P(F) = P(S\cap F) + P(S^{c}\cap F)\\=0.44+0.11\\=0.55[/tex]
[tex]P(F^{c})=1-P(F)\\=1-0.55\\=0.45[/tex]
Conditional probability of an event A given B is:
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]
Compute the value of [tex]P(S|F)[/tex]:[tex]P(S|F)=\frac{P(S\cap F)}{P(F)}\\=\frac{0.44}{0.55}\\=0.80[/tex]
Compute the value of [tex]P(S|F^{c})[/tex][tex]P(S|F^{c})=\frac{P(S\cap F^{c})}{P(F^{c})}\\=\frac{0.13}{0.45}\\=0.289[/tex]
Compute the value of [tex]P(S^{c}|F)[/tex][tex]P(S^{c}|F)=\frac{P(S^{c}\cap F)}{P(F}\\=\frac{0.11}{0.55}\\=0.20[/tex]
Compute the value of[tex]P(S^{c}|F^{c})[/tex][tex]P(S^{c}|F^{c})=\frac{P(S^{c}\cap F^{c})}{P(F^{c})}\\=\frac{0.32}{0.45}\\=0.711[/tex]
The probability of successful product given favorable test market is 0.80, and the probability of successful product given unfavorable test market is 0.29. The probability of unsuccessful product given favorable test market is 0.20, and the probability of unsuccessful product given unfavorable test market is 0.71.
Explanation:To solve this, we first need to determine the total probability of each market test outcome. The probability that the market test is favorable (P(favorable test market)) can be found by summing the probabilities of a successful product with a favorable test market and an unsuccessful product with a favorable test market. Therefore, P(favorable test market) = 0.44 + 0.11 = 0.55. Similarly, P(unfavorable test market) = 0.13 + 0.32 = 0.45.
To find the conditional probabilities, we use the formula P(A|B) = P(A and B) / P(B):
P(successful product given favorable test market) = P(successful product and favorable test market) / P(favorable test market) = 0.44 / 0.55 = 0.80.P(successful product given unfavorable test market) = P(successful product and unfavorable test market) / P(unfavorable test market) = 0.13 / 0.45 = 0.29.P(unsuccessful product given favorable test market) = P(unsuccessful product and favorable test market) / P(favorable test market) = 0.11 / 0.55 = 0.20.P(unsuccessful product given unfavorable test market) = P(unsuccessful product and unfavorable test market) / P(unfavorable test market) = 0.32 / 0.45 = 0.71.Learn more about Conditional Probability here:https://brainly.com/question/32171649
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You select a sample of 50 scores from a population of 2,000 scores. You compute the range and standard deviation on the sample of 50 scores. You then select another sample of 50 scores from the same population. What measure of dispersion is likely to vary most between your first and second samples?
Answer:
The measure of dispersion which is likely to vary most between your first and second samples is the range.
Step-by-step explanation:
The range and standard deviation of a data are measures of dispersion, i.e. they measure the degree to which the data is dispersed.
The formula to compute the range is:
[tex]Range=X_{max}-X_{min}[/tex]
The formula to compute the sample standard deviation is:
[tex]s=\sqrt{\frac{1}{n-1}\sum (X-\bar X)^{2} }[/tex]
The sample size is: n = 50.
As the sample size is large (n = 50 > 30) the sample standard deviation (s) can be used to approximate the population standard deviation (σ). Thus, whatever the sample values be both the standard deviations can be used to approximate the population standard deviation. Hence, it can be said that both the sample standard deviations are approximately equal.Whereas the range of the two samples are very likely to vary since it is based on the minimum and maximum value of the data. For both the samples the minimum and maximum value may be differ. Thus providing different range values.Thus, the measure of dispersion which is likely to vary most between your first and second samples is the range.
Final answer:
The range is more likely to vary between two samples from a population due to its sensitivity to extreme values, whereas the standard deviation is a more stable measure of dispersion that accounts for all values in the dataset.
Explanation:
The measure of dispersion most likely to vary between the first and second samples from the population is the range. The range is sensitive to the particular values in the dataset because it only takes into account the smallest and largest values. Since each sample may have a different set of extreme values, the range can vary greatly from sample to sample. In contrast, the standard deviation is less likely to vary significantly between samples, as it measures the amount of variation or dispersion from the mean of a dataset and takes into account all the values in the set.
When drawing multiple samples from a population, the distribution of the sample means will tend to form a normal distribution around the population mean according to the Central Limit Theorem. In the case of a normally distributed population with a known mean and standard deviation, the sampling distribution of the sample mean will have a standard deviation equal to the population standard deviation divided by the square root of the sample size (known as the standard error).
What is the simplified form of this expression?
(-3x2 + 4x) + (2x2 − x − 11)
Answer:
option b
-x^2 + 3x - 11
Step-by-step explanation:
we have to add the X of the same order
x^2
-3x^2 + 2x^2 = -x^2
x^1
4x + ( -x ) = 3x
x^0
0 + (-11) = -11
finally we accommodate and get the final result
-x^2 + 3x - 11
Answer:-x^2 + 3x - 11
Step-by-step explanation:
I got it right on the test
The negative sign in 15t-2t belongs to the term____?
The expression 15t – 2t not equivalent to 2t – 15t because the negative sign in 15t – 2t belongs to the term 2t.
Solution:
Given expressions are 15t – 2t and 2t – 15t.
To determine 15t – 2t is equivalent to 2t – 15t or not.
Substitute t = 2 in above two expressions.
15t – 2t = 15(2) – 2(2)
= 30 – 4
= 26
2t – 15t = 2(2) – 15(2)
= 4 – 30
= –26
The values of the expressions are different when t = 2.
So, 15t – 2t is not equivalent to 2t – 15t.
Hence the expression 15t – 2t not equivalent to 2t – 15t because the negative sign in 15t – 2t belongs to the term 2t.
I need a better explanation how to find length and area of a triangle
Answer:
Height(h) = 24in
Area = 204in²
Step-by-step explanation:
we would use the heron formula because we have all the sides to calulate the area
S = (A+B+C)/2 = (25 + 17 + 26)/2 = 68/2 = 34
area = √s(s-a)(s-b)(s-c) = √34(34-25)(34-17)(34-26) = √34(9)(17)(8) = √34x9x17x8 = √41616 = 204
area= (1/2)bh
204 = (1/2)bh b= 17, h =?
204 = (1/2)17h
204 x 2 = 17h
408 = 17h
h = 408/17 = 24in
Height(h) = 24in
Area = 204in²
Following are measurements of soil concentrations (in mg/kg) of chromium (Cr) and nickel (Ni) at 20 sites in the area of Cleveland, Ohio. These data are taken from the article "Variation in North American Regulatory Guidance for Heavy Metal Surface Soil Contamination at Commercial and Industrial Sites" (A. Jennings and J. Ma, J Environment Eng, 2007:587–609).
Cr: 34 1 511 2 574 496 322 424 269 140 244 252 76 108 24 38 18 34 3O 191
Ni: 23 22 55 39 283 34 159 37 61 34 163 140 32 23 54 837 64 354 376 471
(a) Construct a histogram for each set of concentrations.
(b) Construct comparative boxplots for the two sets of concentrations.
(c) Using the boxplots, what differences can be seen between the two sets of concentrations?
Answer:
a) see attached
b) see attached
Step-by-step explanation:
There is existence of outlier in the Ni data and there is none is Cr data.
########################################
# You can try this out in R programming
cr = c(34, 1, 511, 2, 574, 496, 322, 424, 269, 140, 244, 252, 76, 108, 24,
38, 18, 34, 30, 191)
Ni = c(23, 22, 55, 39, 283, 34, 159, 37, 61, 34, 163, 140, 32,
23, 54, 837, 64, 354, 376, 471)
par(mfrow=c(1,2))
hist(cr, col='green')
hist(Ni, col='brown')
par(mfrow=c(1,2))
boxplot(cr, main = 'Boxplot of Cr')
boxplot(Ni, main = 'Boxplot of Ni')
boxplot(cr, Ni)
The question involves constructing histograms and boxplots for two sets of data on soil concentrations of certain metals. After construction, these should be analysed for differences in central tendency, spread, variability, skewness, or presence of outliers.
Explanation:This question seems to be related to Mathematics, specifically Statistics and Data Analysis. Firstly, to construct a histogram, you need to group the data into 'bins' or 'intervals', count how many data points fall into each bin, and then plot these counts as bars in your histogram. It's similar for both Chromium (Cr) and Nickel (Ni) data.
To construct comparative boxplots, it's essential to: calculate the quartiles (Q1, Q2 aka median, and Q3), identify the minimum and maximum points, and define possible outliers. With these statistical measures, you can draw the boxplots for both sets and analyse them. The dataset Cr seems to have several values clustered in the low range while Ni seems to have more spread, which can be verified by the boxplot comparison.
The interpretation of the boxplots should reflect on any differences in central tendency and spread between the two sets of concentrations. The variability, skewness, or outliers could also be significantly different. Any such inferred details from the boxplots may hint towards different contamination levels or heterogeneity in each of the datasets.
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Find the perimeter of the following shape, rounded to the nearest tenth:
Answer: the perimeter of the shape is 19.1
Step-by-step explanation:
To determine the length of each side of the quadrilateral, we would apply Pythagoras theorem which is expressed as
Hypotenuse² = opposite side² + adjacent side
For line AD,
AD² = 2² + 4² = 4 + 16 = 20
AD = √20 = 4.47
For line AB,
AB² = 1² + 5² = 1 + 25 = 26
AB = √26 = 5.1
For line BC,
BC² = 2² + 4² = 4 + 16 = 20
BC = √20 = 4.47
For line CD,
CD² = 1² + 5² = 1 + 25 = 26
CD = √26 = 5.1
The perimeter of a plane figure is the distance around the figure. Therefore
Perimeter = AB + AB + BC + CD
Perimeter = 4.47 + 5.1 + 4.47 + 5.1 =
19.1
Answer:
19.1
Step-by-step explanation:
Got it right on the test! <3
You have decided that you want to be a millionaire when you retire in 44 years. If you can earn an annual return of 11.06 percent, how much do you have to invest today? What if you can earn an annual return of 5.53 percent?
Answer:
1. Let amount deposited now be P.
So P*(1.1106^44) = 1,000,000, which means P=9896.06
2. Let amount deposited now be P.
So P*(1.0553^44) = 1,000,000, which means P=93639.42
Step-by-step explanation:
Answer:
Step-by-step explanation:
Let P represent the amount that you need to invest today. Thus, principal = $P
The return would be compounded annually. So
n = 1
The rate at which the principal would be compounded is 11.06%. So
r = 11.06/100 = 0.1106
The investment would be for 44 years. So
t = 44
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of t years.
A = $1000000
Therefore
1000000 = P(1+0.1106/1)^1×44
1000000 = P(1.1106)^44
P = 1000000/101.05
P = $9896.1
2) if you can earn an annual return of 5.53 percent, then
r = 5.53/100 = 0.053
1000000 = P(1+0.53/1)^1×44
1000000 = P(1.053)^44
P = 1000000/9.7
P = $103093
If a student determined that the volume of a sample of water is 11.4 mL, and its mass is 10.98 g, what should the student record as the experimental density of the water sample, with the correct number of significant figures
Answer:1.04 ml/g.
Step-by-step explanation:
The formula we use to find the density :
[tex]\text{Density}=\dfrac{\text{Volume}}{\text{Mass}}[/tex]
We are given that ,
Volume of a sample of water = 11.4 mL
Mass of sample of water = 10.98 g
Then , the experimental density of the water sample will be :-
[tex]\text{Density}=\dfrac{11.4}{10.98}=1.03825136\approx1.04[/tex]
Hence, the experimental density of the water sample is 1.04 ml/g.
Find the probability for the experiment of tossing a six-sided die twice.
1. The sum is 4.
2. The sum is 6.
3. The sum is at least 7.
4. The sum is at least 8.
5. The sum is less than 11.
6. The sum is 2, 3, or 12.
7. The sum is odd and no more than 7.
8. The sum is odd or prime.
Answer:
1. 1/12 or 0.083
2. 5/36 or 0.1389
3. 7/12 or 0.5833
4. 5/12 or 0.4167
5. 11/12 or 0.9167
6. 1/9 or 0.1111
7. 1/3 or 0.3333
8. 5/12 or 0.4167
Step-by-step explanation:
This problem can easily be understood by making sample space first.
Outcomes in sample space=S={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Total outcomes=36
The probability in the following scenario is calculated as
Probability=number of outcomes/Total outcomes
1.
The sum is 4
The sum is 4 ={(1,3),(2,2),(3,1)}
number of outcomes=3
P(The sum is 4)=3/36=1/12 or 0.083
2.
The sum is 6
The sum is 6 ={(1,5),(2,4),(3,3),(4,2),(5,1)}
number of outcomes=5
P(The sum is 6)=5/36 or 0.1389
3.
The sum is at least 7
The sum is at least 7=The sum is greater than or equal to 7
The sum is at least 7={(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,3),(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=21
P(The sum is at least 7)=21/36
P(The sum is at least 7)=7/12 or 0.5833
4.
The sum is at least 8
The sum is at least 8=The sum is greater than or equal to 8
The sum is at least 8={(2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=15
P(The sum is at least 8)=15/36
P(The sum is at least 8)=5/12 or 0.4167
5.
The sum is less than 11
The sum is less than 11={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(6,1),(6,2),(6,3),(6,4)}
number of outcomes=33
P(The sum is less than 11)=33/36
P(The sum is less than 11)=11/12 or 0.9167
6.
The sum is 2, 3, or 12
The sum is 2, 3, or 12={(1,1),(1,2),(2,1),(6,6)}
number of outcomes=4
P(The sum is 2, 3, or 12)=4/36
P(The sum is 2, 3, or 12)=1/9 or 0.1111
7.
The sum is odd and no more than 7
The sum is odd and no more than 7={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(6,1)}
number of outcomes=12
P(The sum is odd and no more than 7)=12/36
P(The sum is odd and no more than 7)=1/3 or 0.3333
8.
The sum is odd or prime
The sum is odd or prime={(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5)}
number of outcomes=15
P(The sum is odd or prime)=15/36
P(The sum is odd or prime)=5/12 or 0.4167
Probabilities are,
1. The sum is 4 = 1/12
2. The sum is 6 = 5/36
3. The sum is at least 7 = 7/12
4. The sum is at least 8 = 5/12
5. The sum is less than 11 = 11/12
6. The sum is 2, 3, or 12 = 1/9
7. The sum is odd and no more than 7 = 1/3
8. The sum is odd or prime = 19/36
When six sided die toss two times, Then total possible Outcomes shown below
[tex](1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),\\\\(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),\\\\(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}[/tex]
Total number of outcomes=36
Probability is calculated by dividing number of favourable outcomes by Total number of outcomes
1. Favourable outcomes for the sum is 4 [tex]={(1,3),(2,2),(3,1)}[/tex]
number of outcomes=3
The probability of The sum is 4. = [tex]3/36=1/12 = 0.083[/tex]
2.Favourable outcomes for the sum is 6[tex]={(1,5),(2,4),(3,3),(4,2),(5,1)}[/tex]
number of outcomes=5
The probability of The sum is 6, =5/36
3.The sum is at least 7,
={(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,3),(4,4),(4,5),(4,6),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=21
Probability of the sum is at least 7 =21/36=7/12 = 0.5833
4.The sum is at least 8
The sum is at least 8={(2,6),(3,5),(3,6),(4,4),(4,5),(4,6),(5,3),(5,4),(5,5),(5,6),(6,2),(6,3),(6,4),(6,5),(6,6)}
number of outcomes=15
The sum is at least 8=15/36 = 5/12 = 0.4167
5.The sum is less than 11
The sum is less than 11={(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(6,1),(6,2),(6,3),(6,4)}
number of outcomes=33
The sum is less than 11=33/36 = 11/12
6.The sum is 2, 3, or 12={(1,1),(1,2),(2,1),(6,6)}
number of outcomes=4
The sum is 2, 3, or 12=4/36 = 1/9
7.The sum is odd and no more than 7={(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(6,1)}
number of outcomes=12
The sum is odd and no more than 7=12/36 = 1/3
8.The sum is odd or prime={(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5), (3, 6), (6, 3), (4,5), (5, 4)
number of outcomes=19
The sum is odd or prime=19/36
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A study of 31,000 hospital admissions in New York State found that 4% of the admissions led to treatment-caused injuries. One-seventh of these treatment-caused injuries resulted in death, and one-fourth were caused by negligence. Malpractice claims were filed in one out of 7.5 cases involving negligence, and payments were made in one out of every two claims.a. What is the probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence? b. What is the probability a person admitted to the hospital will die from a treatment-caused injury?c. In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid?
Answer:
Step-by-step explanation:
a) probability a person admitted to the hospital will suffer a treatment-caused injury due to negligence
P(injury) = 4%
P(negligence) = 1/4 = 0.25
We need to find probability (injury)(negligence)
P(injury) * P(negligence) = 0.04*0.25 = 0.01
b) probability a person admitted to the hospital will die from a treatment-caused injury
P(injury) = 4%
P(death) = 1/7
P(Injury) *P(death) = 0.04/7 = 0.00571
c) In the case of a negligent treatment-caused injury, what is the probability a malpractice claim will be paid
P(claim) = 1/7.5
P(payment) = 1/2
P(claim)*P(payment) = 1/7.5 * 1/2 = 0.06
Answer:
a.) 0.01
b.) 0.006
c.)0.00067
Step-by-step explanation:
Researchers measure the body temperature of 52 randomly selected adults. They find a mean temperature of 98.2 degrees with a standard deviation of 0.682 degrees. Which of the following is the correct t-test statistic and p-value for a test of the following hypotheses?
H_o: mu = 98.6 degrees
H_a: mu notequalto 98.6 degrees
The test statistic is negative 1. 039.21.and the p-value is less than 0.000001.
The test statistic is negative 0.46.and the p-value is 2 times P(t_51 > -0.46).
The t-test statistic is negative 0.315.and the p-value is P(t_51 < -0.315).
The t-test statistic is negative 3.33.and the p-value is two times P(t_51 < -3.33).
Answer:
The t-test statistic is -3.33.and the p-value is two times P(t_51 < -3.33).
Step-by-step explanation:
Data given and notation
Assuming that the real sample mean is: [tex]\bar X=98.285[/tex] represent the sample mean
[tex]s=0.682[/tex] represent the sample standard deviation
[tex]n=52[/tex] sample size
[tex]\mu_o =98.6[/tex] represent the value that we want to test
[tex]\alpha[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to apply a two tailed test.
What are H0 and Ha for this study?
Null hypothesis: [tex]\mu = 98.6[/tex]
Alternative hypothesis :[tex]\mu \neq 98.6[/tex]
Compute the test statistic
The statistic for this case 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".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{98.2-98.6}{\frac{0.682}{\sqrt{52}}}=-3.298[/tex]
Now we can calculate the degrees of freedom and we got:
[tex] df = n-1 = 52-1 = 51[/tex]
P value
Since is a two tailed test the p value would be:
[tex]p_v =2*P(t_{51}<-3.3)=0.0018[/tex]
So the most appropiate conclusion for this case would be:
The t-test statistic is -3.33.and the p-value is two times P(t_51 < -3.33).
This problem involves two distinct sets of events, which we label A1and A2 and B1 and B2. The events A1and A2 are mutually exclusive and collectively exhaustive within their set. The events B1and B2 are mutually exclusive and collectively exhaustive within their set. Intersections can occur between all events from the two sets.Given P(A1)= 0.8, P(B1|A1) = 0.6, and P(B1|A2) = 0.2, what is P(A1|B1)?
Answer:
0.42
Step-by-step explanation:
Let the events be given as:
For twp mutually exclusive events, the probability of A1 is given as follows:
P (B1A1) = [tex]\frac{ P(B1) x P (BA1)}{P(A1)}[/tex]
= 0.6
Solving the equation above to get B1:
P (B1) = [tex]\frac{(0.8)x (0.6)}{(0.6)}[/tex]
= 0.8
Therefore, computing P (A1B1) gives P(A1) × P (B1)
= (0.8) × (0.6)
= 0.42 Ans
How do you convert $150,000 US dollars to $240,000 Australian dollars (1.31545 AU) using the exchange rate per US dollar?
Answer:
240,000 AUD to USD = 161,924.79 US Dollars or 150,000 USD to AUD = 222,316.63 Australian Dollars
Step-by-step explanation:
PLZ HELP A delivery truck is transporting boxes of two sizes: large and small. The large boxes weigh 55 pounds each, and the small boxes weigh 35 pounds each. There are 125 boxes in all. If the truck is carrying a total of 4950 pounds in boxes, how many of each type of box is it carrying?
Answer:
There were 28.75 large boxes and 96.25 small boxes.
Step-by-step explanation:
Create a system of equations to solve.
State your variables
let x be the number of small boxes
let y be the number of large boxes
35x + 55y = 4950 Equation for weight
x + y = 125 Equation for number of boxes
Rearrange equation for number of boxes to isolate "x".
x = 125 - y
Substitute the new equation into the equation for weight
35x + 55y = 4950
35(125 - y) + 55y = 4950
Expand the brackets
4375 - 35y + 55y = 4950 Combine like terms
4375 + 20y = 4950
Start isolating "y"
4375 + 20y = 4950
4375 - 4375 + 20y = 4950 - 4375 subtract 4375 from both sides
20y = 575
20y/20 = 575/20 divide both sides by 20
y = 28.75 number of large boxes
Calculate "x". Rearrange the equation for number of boxes to isolate "y".
y = 125 - x Substitute this expression
35x + 55y = 4950 equation for weight
35x + 55(125 - x) = 4950 expand the brackets
35x + 6875 - 55x = 4950 combine like terms
6875 - 20x = 4950 start isolating "x"
6875 - 6875 - 20x = 4950 - 6875 subtract 6875 on both sides
-20x = -1925
-20x/-20 = -1925/-20 divide both sides by -20
x = 96.25 number of small boxes
There is a mistake with this question because you should not be able to have partial boxes. However, the answer is:
There were 28.75 large boxes and 96.25 small boxes.
The brain volumes (cm cubed) of 20 brains have a mean of 1105.1 cm cubed and a standard deviation of 123.5 cm cubed. Use the given standard deviation and the range rule of thumb to identify the limits separating values that are significantly low or significantly high. For such data, would a brain volume of 1332.1 cm cubed be significantly high? Significantly low values are 858 cm cubed or lower. (Type an integer or a decimal. Do not round.)
Answer:
[tex]1332.1 \:cm^3[/tex] is neither significantly low nor significantly high.
Step-by-step explanation:
From the information given, we know that:
The mean is [tex]\bar{x}=1105.1 \:cm^{3}[/tex] and the standard deviation is [tex]\sigma=123.5 \:cm^{3}[/tex].
The range rule of thumb says that:
[tex]minimum \:usual \:value=\bar{x}-2\sigma\\\\maximum \:usual \:value=\bar{x}+2\sigma[/tex]
Applying these definitions, we get that
[tex]minimum \:usual \:value=1105.1-2(123.5)=858.1 \:cm^3\\\\maximum \:usual \:value=1105.1+2(123.5)=1352.1 \:cm^3[/tex]
We note that [tex]1332.1 \:cm^3[/tex] is between [tex]858.1 \:cm^3[/tex] and [tex]1352.1 \:cm^3[/tex], which indicates [tex]1332.1 \:cm^3[/tex] is neither significantly low nor significantly high.
A glass of orange juice contains 3.2 dL of juice. How many milliliters of orange juice are in the glass?
Answer:
There are 320 milliliters of orange juice in the glass.
Step-by-step explanation:
Unit conversion problems can be solved by rules of three.
We have that each dL is worth 100 mL(milliliters).
How many milliliters of orange juice are in the glass?
How many ml are 3.2dL of juice?
1 dL - 100 mL
3.2 dL - x mL
[tex]x = 100*3.2[/tex]
[tex]x = 320[/tex]
There are 320 milliliters of orange juice in the glass.
The measure of juice is 320 milliliters.
Given that
A glass of orange juice contains 3.2 dL of juice.
We have to determine
How many milliliters of orange juice is in the glass?
According to the question
To determine the measure of juice in milliliters following all the steps given below.
A glass of orange juice contains 3.2 dL of juice.
Here 1 dL = 100 milliliters
Then,
The measure of juice in milliliters is,
[tex]\rm = 3.2 \times 100\\\\= 320 \ milliliters[/tex]
Hence, The measure of juice is 320 milliliters.
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Suppose x is a normally distributed random variable with µ = 56 and σ = 3. Find a value x0 of the random variable x that satisfies the following equations or statements.a. 10% of the values of x are less than x0.b. 80% of the values of x are less than x0.c.1% of the values of x are greater than x0.
Answer:
a) 52.15
b) 58.53
c) 62.98
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 56
Standard Deviation, σ = 3
We are given that the distribution of x is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) We have to find [tex]x_0[/tex] such that
P(X < x) = 0.1
[tex]P( X < x) = P( z < \displaystyle\frac{x_0 - 56}{3})=0.1[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < -1.282) = 0.1[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = -1.282\\\\x_0 = 52.15[/tex]
b) We have to find [tex]x_0[/tex] such that
P(X < x) = 0.8
[tex]P( X < x) = P( z < \displaystyle\frac{x_0 - 56}{3})=0.8[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 0.842) = 0.8[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = 0.842\\\\x_0 = 58.53[/tex]
c) We have to find [tex]x_0[/tex] such that
P(X > x) = 0.01
[tex]P( X > x) = P( z > \displaystyle\frac{x - 56}{3})=0.01[/tex]
[tex]= 1 -P( z \leq \displaystyle\frac{x - 56}{3})=0.01 [/tex]
[tex]=P( z \leq \displaystyle\frac{x - 56}{3})=0.99 [/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 2.326) = 0.99[/tex]
[tex]\displaystyle\frac{x_0 - 56}{3} = 2.326\\\\x_0 = 62.98[/tex]
To find specific percentiles for a normally distributed variable x with mean 56 and standard deviation 3, you must use a z-score table or calculator to find the corresponding z-scores for the 10%, 80%, and top 1% percentages, then use the formula x0 = µ + z*σ to calculate x0.
Explanation:To find the value x0 of the normally distributed random variable x with mean µ = 56 and standard deviation σ = 3, that corresponds to specific percentiles, one would typically use a z-score table (or a calculator with a normal distribution function). However, in this scenario, the values provided for the x variable and the associated z-scores in the question are incorrect and would not provide a reasonable answer. Instead, I will explain the steps needed to find x0 without using the incorrect values provided.
Finding x0 for Each Given Percentile:For the 10% percentile: Find the z-score that corresponds to a cumulative probability of 0.10.After finding each z-score, use the formula x0 = µ + z*σ to calculate the corresponding x0 for each case.
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Let f be the function that determines the area of a circle (in square cm) given the radius of the circle in cm, r. That is, f(r) represents the area of a circle (in square cm) whose radius is r cm. Use function notation to complete the following tasks a. Represent the area (in square cm) of a circle whose radius is 4 cm. Preview syntax error b. Represent how much the area (in square cm) of a circle increases by when its radius increases from 10.9 to 10.91 cm. # Preview syntax error c. Represent the area of 5 circles that all have a radius of 12.7 cm *Preview syntax error d. A circle has a radius of 28 cm. Another larger circle has an area that is 59 square cm more than the first circle. Represent the area of the larger circle. # Preview) syntax error
Part(a):[tex]f(r)=f(4)[/tex]
Part(b):[tex]f(10.91)-f(10.9)[/tex]
Part(c):[tex]5 f(r)=5 f(12.7)[/tex]
Part(d):[tex]28+59 =f(28)+59[/tex]
Area of the circle:
The area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula,
[tex]A= \pi r^2[/tex]
where [tex]r[/tex] is the radius of the circle
The formula for the area of the circle is,
[tex]A=\pi r^2[/tex]
Part(a):
Given,
Radius([tex]r[/tex])=4 cm
So, the area is [tex]f(r)=f(4)[/tex]
Part(b):
Given,
[tex]r=10.91\\r=10.9[/tex]
The difference in area is,
[tex]f(10.91)-f(10.9)[/tex]
Part(c):
Area of 5 circles are,
[tex]5 f(r)=5 f(12.7)[/tex]
Part(d):
The area of the larger circle is,
Area of the circle of radius [tex]28+59 =f(28)+59[/tex]
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