The diameter of the rod required to limit the total elongation to 0.10 inches is approximately 1-1/2 inches (or 1.441 inches to be more precise). Hence, the correct answer is option (A) with an elongation of 0.2358 inches and using a 1-1/2" diameter rod.
(a) To compute the total elongation, we can use the formula:
Elongation = (P * L) / (A * E)
where P is the axial tensile load, L is the length of the rod, A is the cross-sectional area of the rod, and E is the modulus of elasticity for the material.
Given:
P = 30,000 lb
L = 20 ft = 240 in
Diameter of the rod = 1-1/2 in
First, we need to calculate the cross-sectional area:
Area = π * (diameter/2)^2
Area = π * (1.5/2)^2
Area ≈ 1.767 in^2
Next, we need to determine the modulus of elasticity for the material. Assuming it's a standard structural steel, we can use a typical value of 29,000,000 psi.
Now we can plug the values into the formula:
Elongation = (30,000 * 240) / (1.767 * 29,000,000)
Elongation ≈ 0.2358 in
Therefore, the total elongation is approximately 0.2358 inches.
(b) If the total elongation must not exceed 0.10 inches, we need to determine the diameter of the rod that satisfies this requirement.
We can rearrange the formula for elongation to solve for the cross-sectional area:
A = (P * L) / (E * Elongation)
Using the given values:
A = (30,000 * 240) / (29,000,000 * 0.10)
A ≈ 2.069 in^2
To find the corresponding diameter, we use the formula:
Diameter = √(4 * A / π)
Diameter = √(4 * 2.069 / π)
Diameter ≈ 1.441 in
Therefore, the diameter of the rod required to limit the total elongation to 0.10 inches is approximately 1-1/2 inches (or 1.441 inches to be more precise). Hence, the correct answer is option (A) with an elongation of 0.2358 inches and using a 1-1/2" diameter rod.
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we used the Optional Stopping Theorem to solve the Gambler's Ruin Problem. Specifically, we showed that if Sn So +?=1X; is a biased random walk starting at So = 1, where the steps X; are independent and equal to +1 with probability p1/2 and equal to - 1 with the remaining probability q=1 – p, then the probability of hitting N (jackpot") before 0 ("bust") is (g/p) - 1 PJ So = 1) = (g/p)N-1 Recall that the key to this was the martingale Mn = (g/p)Sn, which is only useful when pq. (a) For any pe [0, 1], argue that P(T<) = 1, where T = inf{n> 1: Sne {0,1}} is the first time that the walk visits 0 or N. Hint: One way is to consider each time that the walk visits 1 before time T, and then compare with a geometric random variable. Note: This is the one condition in the Optional Stopping Theorem that we did not verify during the lecture. (b) Find P(J|So = n) when instead So = n, for some 1
(a) To argue that P(T < ∞) = 1, where T is the first time the walk visits 0 or N, we can consider each time the walk visits 1 before time T.
Suppose the walk visits 1 for the first time at time k < T. At this point, the random walk is in a state where it can either hit 0 before N or hit N before 0.
Let's define a new random variable Y, which represents the number of steps needed for the walk to hit either 0 or N starting from state 1. Y follows a geometric distribution with parameter p since the steps are +1 with probability p and -1 with probability q = 1 - p.
Now, we can compare the random variable T and Y. If T < ∞, it means that the walk has hit either 0 or N before reaching time T. Since T is finite, it implies that the walk has hit 1 before time T. Therefore, we can say that T ≥ Y.
By the properties of the geometric distribution, we know that P(Y = ∞) = 0. This means that there is a non-zero probability of hitting either 0 or N starting from state 1. Therefore, P(T < ∞) = 1, as the walk is guaranteed to eventually hit either 0 or N.
(b) To find P(J|So = n), where So = n, we need to determine the probability of hitting N before hitting 0 starting from state n.
Recall that the probability of hitting N before 0 starting from state 1 is given by (g/p)^(N-1), as shown in the Optional Stopping Theorem formula. In our case, since the walk starts at state n, we need to adjust the formula accordingly.
The probability of hitting N before 0 starting from state n can be calculated as P(J|So = n) = (g/p)^(N-n).
This probability takes into account the number of steps required to reach N starting from state n. It represents the likelihood of hitting the jackpot (N) before going bust (0) when the walk starts at state n.
It's worth noting that this probability depends on the values of p, q, and N.
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HELP PLS ITS ALMOST DUE PLS PLS PLS
Answer:
19. B
20. C
Step-by-step explanation:
PLSS HELP IMMEDIATELY!!! i’ll give brainiest if u don’t leave a link!
Answer:
it is A
Step-by-step explanation:
i remember doing this in middle school.
The number of pizzas consumed per month by university students is normally distributed with a mean of 12 and a standard deviation of 3. A. What proportion of students consume more than 13 pizzas per month? Probability = = B. What is the probability that in a random sample of size 10, a total of more than 110 pizzas are consumed? Probability = Note: You can earn partial credit on this problem.
The probability to consume more than 13 pizzas per month is 0.3707 and more than 110 pizzas in a random sample of size 10 is 0.9646.
The number of pizzas consumed per month by university students is normally distributed with a mean of 12 and a standard deviation of 3.
A. Probability that more than 13 pizzas consumed by students:
For finding the probability, we need to find the Z-score first.
z = (x - μ) / σz = (13 - 12) / 3z = 0.3333
Now, we have to use the z-table to find the probability associated with the z-score 0.3333.
The area under the normal distribution curve to the right of 0.3333 is 0.3707 (rounded off to 4 decimal places).
Thus, the probability that a student consumes more than 13 pizzas per month is 0.3707.
B. Probability that more than 110 pizzas consumed in a random sample of size 10:
Let x be the number of pizzas consumed in the random sample of size 10.
Then, the distribution of x is a normal distribution with the mean = 10 × 12 = 120 and standard deviation = √(10 × 3²) = 5.4772
We have to find the probability that the total number of pizzas consumed is greater than 110. i.e. P(x > 110).
For finding the probability, we need to find the Z-score first.z = (110 - 120) / 5.4772z = -1.8257
The area under the normal distribution curve to the right of -1.8257 is 0.9646 (rounded off to 4 decimal places).
Thus, the probability that more than 110 pizzas are consumed in a random sample of size 10 is 0.9646.
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The cost of a banquet at Nick's Catering is $215 plus $27.50 per person. If
the total cost of a banquet was $2827.50, how many people were invited?
Answer:
x = 95
Step-by-step explanation:
Given that,
The cost of a banquet at Nick's Catering is $215 plus $27.50 per person
The total cost of a banquet was $2827.50
We need to find the number of people invited. Let there are x people. So,
215+27.5x = 2827.50
27.5x = 2827.50 -215
27.5x = 2612.5
x = 95
So, there are 95 people that were invited.
Each letter in the word THEORETICAL is placed on a separate piece of paper
and placed in a hat. A letter is chosen at random from the hat. What is the
probability that the letter chosen is an E?
(Give answer in format 'a/b, no spaces, use slash for fraction bar)
Answer:
The answer is 1/11
Step-by-step explanation:
Explanation is in the picture above
please mark as brainliest
What is the midpoint of DC with endpoints C (6,−1) and D (−7,8)?
Answer:
Fraction form: (-1/2, 7/2) Decimal form: -.5, 3.5)
Step-by-step explanation:
use the midpoint formula of: (x1 + x2/2 , y1 + y2/2) in order to get (x,y) coordinates.
Let's call endpoint C x1 and y1. So 6=x1 and -1=y1.
This makes endpoint D x2 and y2. So -7=x2 and 8=y2.
Now plug it in and simplify!
x-coordinates: (6+-7)/2 = (6-7)/2 = -1/2 or -.5
y-coordinates: (-1+8)/2 = (8-1)/2 = 7/2 or 3.5
the midpoint of endpoints C and D is (-1/2, 7/2)
**decimal form: (-.5, 3.5)
Florida Immigration 9 Points 910 randomly sampled registered voters in Tampa, FL were asked if they thought workers who have illegally entered the US should be allowed to keep their jobs and apply for US citizenship. (ii) allowed to keep their jobs as temporary guest workers but not allowed to apply for US citizenship, (iii) lose their jobs and have to leave the country, or (iv) not sure. These voters were also asked about their political ideology, to which they responded one of the following: conservative, liberal, or moderate. Q4.4 Type I Error 3 Points Describe what it would mean if we made a Type I Error on this test. (You must discuss what decision we made, and what the actual truth about the population is.)
Type I Error: A Type I error is the first kind of error that can occur when testing a hypothesis. A Type I error occurs when a null hypothesis is rejected even when it is accurate.
If we make a Type I Error on this test, it would mean that we reject a null hypothesis that is true. This mistake would be made if we made a decision to reject the null hypothesis when there is no significant evidence to support that decision. The null hypothesis is the hypothesis that claims no change or no difference between the groups being compared. Null hypothesis is the opposite of the alternative hypothesis which is the hypothesis that claims that there is a difference between groups being compared.
In this context, making a Type I Error would mean that we reject the null hypothesis which is that all groups of voters would agree that workers who have illegally entered the US should be allowed to keep their jobs and apply for US citizenship. Making this error would mean we have come to the conclusion that they do not agree, which would be incorrect.
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Which expression is equivalent to the expression shown below? --8x – 2(5 + 4x)
-8x-2(5+4x)
Step-by-step explanation:
-8x-10-8x -8x-8x-10 -16x-10
What is the surface area of a cylinder with height 8 ft and radius 4 ft
The Surface area of the cylinder with a height of 8 ft and a radius of 4 ft is approximately 301.44 square feet.
The surface area of a cylinder, we need to consider the lateral surface area and the area of the two circular bases.
The lateral surface area of a cylinder can be determined by multiplying the height of the cylinder by the circumference of its base. The formula for the lateral surface area (A) of a cylinder is given by A = 2πrh, where r is the radius and h is the height of the cylinder.
In this case, the height of the cylinder is 8 ft and the radius is 4 ft. Therefore, the lateral surface area can be calculated as follows:
A = 2π(4 ft)(8 ft)
A = 64π ft²
The area of each circular base can be calculated using the formula for the area of a circle, which is A = πr². In this case, the radius is 4 ft. Therefore, the area of each circular base is:
A_base = π(4 ft)²
A_base = 16π ft²
Since a cylinder has two circular bases, the total area of the two bases is:
A_bases = 2(16π ft²)
A_bases = 32π ft²
the total surface area, we sum the lateral surface area and the area of the two bases:
Total surface area = Lateral surface area + Area of bases
Total surface area = 64π ft² + 32π ft²
Total surface area = 96π ft²
Now, let's calculate the numerical value of the surface area:
Total surface area ≈ 96(3.14) ft²
Total surface area ≈ 301.44 ft²
Therefore, the surface area of the given cylinder, with a height of 8 ft and a radius of 4 ft, is approximately 301.44 square feet.
In conclusion, the surface area of the cylinder with a height of 8 ft and a radius of 4 ft is approximately 301.44 square feet.
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6. Markets with elastic supply and demand curves: a) Have demand and supply curves that never intersect. B) Are very sensitive to a change in price. C) Have greater movements in quantity than prices. D) Are very sensitive to a change in quantity. E) Are only theoretical and do not exist in the real world.
Answer:
The correct statement is B (are very sensitive to change in price)
Step-by-step explanation:
Option B is correct because of the following reason -:
The degree to which a rise in price affects the quantity demanded or supplied is known as elasticity. In the case of elastic demand and supply, as the price rises, the quantity demanded falls and the quantity supplied rises more than proportionally. Inelastic price elasticity of demand and supply, on the other hand, induces a less than proportional change in quantity as prices change.
Hence , the correct option is B .
Giving away 30 points, have a good day
Answer:
For real???
Step-by-step explanation:
Tysm!! <3 you deserve so much!
Answer:thanks
Step-by-step explanation:
A hiker is lost in the forest, but has his cell phone with a weak signal. Cell phones with GPS can give an approximate location through triangulation, which works by giving distances from two known points. Suppose the hiker is within distance of two cell phone towers that are 22.5 miles apart along a straight highway (running east to west, double-dashed line). Based on the signal delay, it can be determined that the signal from the hiker's phone is 14.2 miles from Tower A and 10.9 miles from Tower B. Assume the hiker is traveling a straight path south reach the highway quickly. How far must the hiker travel to reach the highway
Answer:
The distance the hiker must travel is approximately 5.5 miles
Step-by-step explanation:
The distance between the two cell phone towers = 22.5 miles
The distance between the hiker's phone and Tower A = 14.2 miles
The distance between the hiker's phone and Tower B = 10.9 miles
The direction of the highway along which the towers are located = East to west
The direction in which the hiker is travelling to reach the highway quickly = South
By cosine rule, we have;
a² = b² + c² - 2·b·c·cos(A)
Let 'a', 'b', and 'c', represent the sides of the triangle formed by the imaginary line between the two towers, the hiker's phone and Tower A, and the hiker's hone and tower B respectively, we have;
a = 22.5 miles
b = 14.2 miles
c = 10.9 miles
Therefore, we have;
22.5² = 14.2² + 10.9² - 2 × 14.2 × 10.9 × cos(A)
cos(A) = (22.5² - (14.2² + 10.9²))/( - 2 × 14.2 × 10.9) ≈ -0.6
∠A = arccos(-0.6) ≈ 126.9°
By sine rule, we have;
a/(sin(A)) = b/(sin(B)) = c/(sin(C))
∴ sin(B) = b × sin(A)/a
∴ sin(B) = 14.2×(sin(126.9°))/22.5
∠B = arcsine(14.2×(sin(126.9°))/22.5) ≈ 30.31°
∠C = 180° - (126.9° - 30.31°) = 22.79° See No Evil
The distance the hiker must travel, d = c × sin(B)
∴ d = 10.9 × sin(30.31°) ≈ 5.5
Therefore, the distance the hiker must travel, d ≈ 5.5 miles.
Suppose that A and B are mutually exclusive events Select the correct answer below, O A. Since A and B are mutually exclusive events, then the probability that A or Boccur is 1. That is, P(A)*P(B)-1 OB. Since A and B are mutually exclusive events, then the probability that both A and B occur is O. That is, PA}{B} = 0, OC. Since A and B are mutually exclusive events, then the probability that A or B occur is 1. That is, PIA & B)=1, OD. Since A and B are mutually exclusive events, then the probability that both A and B occur is O. That is, P(A&B)=0
The correct answer is option D. Since A and B are mutually exclusive events, the probability that both A and B occur is 0. In other words, P(A&B) = 0.
When two events, A and B, are mutually exclusive, it means that they cannot occur simultaneously. If one event happens, the other event cannot happen at the same time. In this scenario, the correct answer is option D, which states that the probability of both events A and B occurring together is 0, i.e., P(A&B) = 0.
To understand this concept, consider a simple example. Let's say event A represents flipping a coin and getting heads, while event B represents flipping a coin and getting tails. Since getting heads and getting tails are mutually exclusive outcomes, it is impossible for both events A and B to occur simultaneously. Therefore, the probability of both A and B occurring together is 0. In summary, when events A and B are mutually exclusive, the correct answer is option D, which states that the probability of both A and B occurring together is 0, i.e., P(A&B) = 0.
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Find three numbers whose sum is 21 and whose sum of squares is a minimum. The three numbers are________ (Use a comma to separate answers as needed.)
the three numbers whose sum is 21 and whose sum of squares is a minimum are 7, 7, and 7.
To find three numbers whose sum is 21 and whose sum of squares is a minimum, we can use a mathematical technique called optimization. Let's denote the three numbers as x, y, and z.
We need to minimize the sum of squares, which can be expressed as the function f(x, y, z) = x² + y² + z²
Given the constraint that the sum of the three numbers is 21, we have the equation x + y + z = 21.
To find the minimum value of f(x, y, z), we can use the method of Lagrange multipliers, which involves solving a system of equations.
First, let's define a Lagrange multiplier, λ, and set up the following equations:
1. ∂f/∂x = 2x + λ = 0
2. ∂f/∂y = 2y + λ = 0
3. ∂f/∂z = 2z + λ = 0
4. Constraint equation: x + y + z = 21
Solving equations 1, 2, and 3 for x, y, and z, respectively, we get:
x = -λ/2
y = -λ/2
z = -λ/2
Substituting these values into the constraint equation, we have:
-λ/2 - λ/2 - λ/2 = 21
-3λ/2 = 21
λ = -14
Substituting λ = -14 back into the expressions for x, y, and z, we get:
x = 7
y = 7
z = 7
Therefore, the three numbers whose sum is 21 and whose sum of squares is a minimum are 7, 7, and 7.
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Find the lateral area of this square
based pyramid.
10 in
5 in
[ ? ] in
Answer:
100in
Step-by-step explanation:
1/2 *10*5=25
4(25)=100
Which transformation carries the parallelogram onto itself?
Answer: D) a rotation of 180 degrees Clockwise about the center of the parallelogram
Step-by-step explanation:
For f, g € L’[a,b], prove the Cauchy-Schwarz inequality |(f,g)| = ||$||||$||. = Hint: Define a function Q(t) = (f + tg, f + tg) for any real number t. Use the rules of inner product to expand this expression and obtain a quadratic polynomial in t; because Q(t) > 0 (why?), the quadratic polynomial can have at most one real root. Examine the discriminant of the polynomial.
Given that f, g ∈ L’[a, b], we need to prove the Cauchy-Schwarz inequality, |(f, g)| = ||$|| . ||$||.
The Cauchy-Schwarz inequality for inner product in L’[a, b] states that for all f, g ∈ L’[a, b],|(f, g)| ≤ ||$|| . ||$||Proof: Consider a function Q(t) = (f + tg, f + tg) for any real number t. Then, by using the rules of inner product, we can expand this expression and obtain a quadratic polynomial in t.$$Q(t) = (f + tg, f + tg) = (f, f) + t(f, g) + t(g, f) + t^2(g, g)$$$$ = (f, f) + 2t(f, g) + t^2(g, g)$$. Now, Q(t) > 0 because Q(t) is a sum of squares. So, Q(t) is a quadratic polynomial that can have at most one real root since Q(t) > 0 for all t ∈ R.
To find the discriminant of Q(t), we need to solve the equation Q(t) = 0.$$(f, f) + 2t(f, g) + t^2(g, g) = 0$$.
The discriminant of Q(t) is:$$D = (f, g)^2 - (f, f)(g, g)$$
Since Q(t) > 0 for all t ∈ R, the discriminant D ≤ 0.$$D = (f, g)^2 - (f, f)(g, g) ≤ 0$$$$\Right arrow (f, g)^2 ≤ (f, f)(g, g)$$$$\Right arrow |(f, g)| ≤ ||$|| . ||$||$$
Thus, |(f, g)| = ||$|| . ||$||, which proves the Cauchy-Schwarz inequality. Therefore, the given statement is true.
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Find the unit rate for each, then compare. Which is faster?
8 laps in 70 seconds
12 laps in 98 seconds.
Answer:
8 laps in 70 seconds is faster.
Step-by-step explanation:
If we divide 70/8 and 98/12 we get the following:
70/8= 8.75
98/12=8.16
8.75>8.16
The unit rate is 1 lap in 8.75 seconds and 1 lap in 8.16 seconds
The perimeter of a square (perimeter = 4 times one side) is less than 16 inches. One side of the square measures x. what are the viable solutions for the value of x?
Answer:
C
Step-by-step explanation:
The perimeter of the square has to be only positive values, and so there has to be restrictions on the values. We can rule out answers A and B. Because the perimeter the values of x have to be less than 4. If they were greater than 4, then 4x>16. So we can rule out answer d. The correct answer is C.
If a random variable has binomial distribution with n = 150 and p = 0.6. Using normal approximation the probability; P(X≥ 95) =---
The required probability is 0.2023.
Given random variable X with binomial distribution with n=150 and p=0.6.
The binomial distribution with parameters n and p has probability mass function:
$$f(x)= \begin{cases} {n\choose x} p^x (1-p)^{n-x} & \text{for } x=0,1,2,\ldots,n, \\ 0 & \text{otherwise}. \end{cases}$$
Now the mean, μ = np = 150 × 0.6 = 90 and standard deviation, σ = √(npq) = √(150 × 0.6 × 0.4) = 6
Using the normal approximation,
we have:
$$\begin{aligned}P(X ≥ 95) &\approx P\left(Z \geq \frac{95 - \mu}{\sigma}\right)\\ &\approx P(Z \geq \frac{95 - 90}{6})\\ &\approx P(Z \geq 0.8333) \end{aligned}$$
Using the standard normal table, the area to the right of 0.83 is 0.2023.
Therefore, P(X ≥ 95) = 0.2023.
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According to the given information, the required probability is 0.2019.
The random variable has a binomial distribution with n = 150 and p = 0.6.
We can use the normal approximation to the binomial distribution to find the probability P(X ≥ 95).
Normal Approximation:
The conditions for the normal approximation to the binomial distribution are:
np ≥ 10 and n(1 - p) ≥ 10
The expected value of the binomial distribution is given by the formula E(X) = np
and the variance is given by the formula [tex]Var(X) = np(1 - p)[/tex].
Let X be the number of successes among n = 150 trials each with probability p = 0.6 of success.
The random variable X has a binomial distribution with parameters n and p, i.e., X ~ Bin(150, 0.6).
The expected value and variance of X are:
[tex]E(X) = np = 150(0.6) = 90[/tex],
[tex]Var(X) = np(1 - p) = 150(0.6)(0.4) = 36[/tex].
The probability that X takes a value greater than or equal to 95 is:
[tex]P(X ≥ 95) = P(Z > (95 - 90) / (6))[/tex]
where Z ~ N(0,1) is the standard normal distribution with mean 0 and variance 1.
[tex]P(X ≥ 95) = P(Z > 0.8333)[/tex]
We can use a standard normal distribution table or a calculator to find this probability.
Using a standard normal distribution table, we find:
[tex]P(Z > 0.8333) = 0.2019[/tex]
Thus, [tex]P(X ≥ 95) = 0.2019[/tex] (rounded to four decimal places).
Therefore, the required probability is 0.2019.
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.16 with the 6 repeating to a fraction
help me find the answer please
Answer:
A x<1125
Step-by-step explanation:
1/(x+6)+(×+1)/x=13/(x+6)
Answer:
x = 3, 2
Step-by-step explanation:
Answer: x = 3, 2
Step-by-step explanation:
Find the value of the variable.
20
12
A. 10
В. 13
C. 16
D.18
Answer:
option c.
by Pythagoras theorem.
hypotenuse²=height ²+base²
20²=x²+12²
400=x²+144
400-144=x²
256=x²
256½=x
16=x
The ratio of boys to girls at the play was 4 to 3. If there were 15 girls, how many boys were there?
Answer:
20 boys
Step-by-step explanation:
If there are 4 boys for every 3 girls, multiply both numbers by 5 (3*5 = 15) to find the number of boys.
Answer:
20
Step-by-step explanation:
4/3 = ?/15
multiply both sides by 15
15*4/3 = ?
? = 20
If f(x) = (x + 7)2 and g(x) = x2 +9,
which statement is true?
A fo)
B f(-4) > g(-3)
C f(1) = g(1)
D f(2) > g(2)
ANSWER : D
EXPLANATION : 81 > 13 is true
If the ratio of boys to girls is 1:4 and there are 20 girls in your class, how many boys are there?
Answer:
Step-by-step explanation:
5 boys
Answer:
me
Step-by-step explanation:
beceaus im the best Guy
Plot the x-intercepts, the y-intercept, and the vertex of the graph (Must use Desmos!)
Answer:
x-intercept: (-1,0)
y-intercept: (0,3)
Vertex: (-2,-1)
Step-by-step explanation:
Let X1 and X2 be independent random variables with mean μ and variance σ2. Suppose that we have two estimators of μ: Math and 1 = X1+X2/2 and math2=x1 + 3x2/4
(a) Are both estimators unbiased estimators of μ? (b) What is the variance of each estimator? Hint: Law of expected values
(a) Math2 is not an unbiased estimator of μ. (b)Math1 has a variance of
σ[tex]^{2}[/tex] and Math2 has a variance of 5σ[tex]^2[/tex]/8
(a) Neither of the estimators, Math1 or Math2, is an unbiased estimator of μ. An unbiased estimator should have an expected value equal to the parameter being estimated, in this case, μ.
For Math1,
the expected value is
E[Math1] = E[([tex]X_{1}[/tex] + [tex]X_{2}[/tex]) / 2]
= (E[[tex]X_{1}[/tex]] + E[[tex]X_{2}[/tex]]) / 2
= μ/2 + μ/2 = μ,
which means Math1 is an unbiased estimator of μ.
For Math2,
the expected value is
E[Math2] = E[([tex]X_{1}[/tex] + [tex]3X_{2}[/tex]) / 4]
= (E[[tex]X_{1}[/tex]] + 3E[[tex]X_{2}[/tex]]) / 4
= μ/4 + 3μ/4
= (μ + 3μ) / 4
= 4μ/4
= μ/2.
(b) To calculate the variances of the estimators, we'll use the property that the variance of a sum of independent random variables is the sum of their variances.
For Math1,
the variance is Var[Math1]
= Var[([tex]X_{1}[/tex] + [tex]X_{2}[/tex]) / 2]
= (Var[[tex]X_{1}[/tex]] + Var[[tex]X_{2}[/tex]]) / 4
= σ[tex]^2[/tex]/2 + σ[tex]^2[/tex]/2
= σ[tex]^2[/tex]
For Math2,
the variance is Var[Math2]
= Var[([tex]X_{1}[/tex] + [tex]3X_{2}[/tex]) / 4]
= (Var[[tex]X_{1}[/tex]] + 9Var[[tex]X_{1}[/tex]]) / 16
= σ[tex]^2[/tex]/4 + 9σ[tex]^2[/tex]/16
= 5σ[tex]^2[/tex]/8
Math1 has a variance of σ[tex]^2[/tex]
and Math2 has a variance of 5σ[tex]^2[/tex]/8
Learn more about parameter here:
https://brainly.com/question/31608396
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