Answers
The expression x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8 is x - 2
How to divide the polynomialFrom the question, we have the following parameters that can be used in our computation:
x^4 - 16 is divided by x^3 + 2x^2 + 4x + 8
Using the long division method of quotient, we have
x^3 + 2x^2 + 4x + 8 | x^4 - 16
The division steps are as follows
x - 2
x^3 + 2x^2 + 4x + 8 | x^4 - 16
x^4 + 2x^3 + 4x^2 + 8x
--------------------------------------------------------------
-2x^3 - 4x^2 - 8x - 16
-2x^3 - 4x^2 - 8x - 16
--------------------------------------------------------------
0
Hence, the quotient is x - 2
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Related Questions
Juro circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of the common chord
Answers
By using Pythagorean theorem, the length of the common chord is approximately 4.58 cm.
What is the Pythagorean theorem?
The Pythagorean theorem is a fundamental result in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle.
In equation form, this can be written as:
[tex]c^2 = a^2 + b^2[/tex]
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides (also known as the legs) of the right-angled triangle.
Let O1 and O2 be the centers of the circles with radii 5cm and 3cm, respectively, and let A and B be the points of intersection of the circles.
Let O1 and O2 be the centers of the circles with radii 5 cm and 3 cm, respectively, and let A and B be the points of intersection of the circles. Then, by the Pythagorean theorem, we have:
[tex]$$\begin{aligned} AB^2 &= AO_1^2 - BO_1^2 \ &= (5^2 - 2^2) , \text{cm}^2 \ &= 21 , \text{cm}^2 \end{aligned}$$[/tex]
Therefore, the length of the common chord AB is:
[tex]$$AB = \sqrt{21} \approx 4.58 , \text{cm}$$[/tex]
Thus, the length of the common chord is approximately 4.58 cm.
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shawls has 3 times as many stickers as abigail they have a total of 12 stickers how many stickers does shayla have
Answers
Syal =3x Abigail
Syal+Abigail =12
(3 x abigail)+1 Abigail =12 Abigail
Abigail =12:4 = 3 striker
Syal =3x3= 9 striker
5/12d+1/6d+1/3d+1/12d=6 solve for d
Answers
Answer:
take lcm ...i.e.= 12d then solve it ... value of d=1/6....I need to pass this math question pls help
Answers
The length x of the similar triangle is 17.3 units.
How to find the sides of similar triangle?Similar triangles are the triangles that have corresponding sides in ratio to each other and corresponding angles congruent to each other.
Therefore, the triangles are similar to each other. Let's use the similarity to find the side length x.
Therefore,
10 / x = x / 30
cross multiply
30 × 10 = x²
x² = 300
square root both sides of the equation
x = √300
x = 17.3205080757
x = 17.3 units
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Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot x = –1 Plus or minus StartRoot StartFraction 4 Over 8 EndFraction EndRoot 8(x2 + 2x + 1) = 3 + 1 8(x2 + 2x) = –3
Answers
The steps he can use are [tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{16}[/tex], (x + 1)² = 5/8 and 8(x² + 2x + 1) = -3 + 8
How to determine the steps he can useFrom the question, we have the following parameters that can be used in our computation:
8x² + 16x + 3 = 0
Assuming he uses the quadratic formula method, which is represented as
[tex]x = -\frac{-b \± \sqrt{b\² - 4ac}}{2a}[/tex]
So, we have
[tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{2 \cdot 8}[/tex]
[tex]x = -\frac{-16 \± \sqrt{16\² - 4 \cdot 8 \cdot 3}}{16}[/tex]
If he uses completing the square, then we have
8x² + 16x = -3
So, we have
x² + 2x = -3/8
x² + 2x + 1 = -3/8 + 1
So, we have
(x + 1)² = 5/8
If he factorizes, then the expression is
8(x² + 2x + 1) = -3 + 8
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What is the volume of the triangular prism below?
Answers
Answer:
210 [tex]cm^3[/tex]
Step-by-step explanation:
The formula of solving the volume of a triangular prism is:
volume = (height x base x length)^3 / 2
First, put in the numbers:
V = [tex]\frac{(2cm*7cm*30cm)^3}{2}[/tex]
V = [tex]\frac{(14cm*30cm)^3}{2}[/tex]
V = [tex]\frac{420^3}{2}[/tex]
V = [tex]210[/tex] [tex]cm^3[/tex]
for which of the following functions can we use the intermediate value theorem to prove the existence of roots in the indicated interval? i. f(x)
Answers
The intermediate value theorem can be used to prove the existence of roots in the indicated interval for a function that is continuous on the interval.
The intermediate value theorem states that if a function is continuous on an indicated interval [a,b], and if f(a) and f(b) have opposite signs, then there must be at least one root in the interval [a,b].
Therefore, for the given function f(x), we can use the intermediate value theorem to prove the existence of roots in the indicated interval if the function is continuous on the interval and if f(a) and f(b) have opposite signs.
In order to determine if the function is continuous on the indicated interval, we need to check if there are any discontinuities or breaks in the function on the interval. If there are no discontinuities or breaks, then the function is continuous on the indicated interval.
Next, we need to check if f(a) and f(b) have opposite signs. If f(a) and f(b) have opposite signs, then there must be at least one root in the indicated interval.
In conclusion, we can use the intermediate value theorem to prove the existence of roots in the indicated interval for a function that is continuous on the interval and if f(a) and f(b) have opposite signs.
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What is the largest even number of 5,2,4,3
Answers
Answer:
The largest even number of 5243 is 14.
step by step:- 5+2+4+3
=14
so that ,14 is the largest even number.
DUE TODAY PLEASE HELP NOW!!!!!!!!!!!!!
Here is another triangle similar to DEF found in the lesson section labeled “Shrinking Triangles”.
• Label the triangle D”E”F”.
• What is the scale factor from triangle DEF to triangle D”E”F”?
• What are the coordinates of F”? Explain how you know.
• What are cos(D”), sin(D”), and tan(D”)?
Answers
The scale factor of dilation is 0.075 and the coordinates of F" are (0.9, 0.375)
Define dilation?A thing must be scaled down or altered during the dilation process. It is a transformation that reduces or enlarges the objects using the supplied scale factor. Pre-image refers to the original figure, whereas image refers to the new figure obtained following dilatation.
Label the triangle D” E” F”.
The label of the triangle is added as an attachment.
From question, we have
DE = 12 units
Then, we have
D"E" = 0.9 units
Using the above, we have the following:
Scale factor = D"E"/DE
Scale factor = 0.9/12
Scale factor = 0.075
Hence, the scale factor of dilation is 0.075.
The coordinates of F"
This is calculated as
F = Scale factor * F
So, we have
F = 0.075 * (12, 5)
F = (0.9, 0.375)
The trigonometry ratios the sine, cosine and tangent are calculated as follow:
sin(D") = EF/DF
cos(D") = DE/DF
tan(D") = EF/DE
So, we have
sin(D") = 0.375/1 = 0.375
cos(D") = 0.9/1 = 0.9
tan(D") = 0.375/0.9 = 0.416
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what 1+1+= to what then 2+2
Answers
On solving the question we have that If you meant "1+1=2, then what is equation 2+2?" the answer is four.
What is equation?A math equation is a mechanism for connecting two statements and indicating equivalence with the equals sign (=). To explain the connection between the two sentences put on each side of a letter, a statistical method can be employed. The software and the logo are usually interchangeable. 2x - 4 equals 2, for example. An equation is a logical expression that asserts the equality of some mathematical expressions in algebra. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14.
"1+1+" is an incomplete expression since it requires another operand or operator to form a valid mathematical statement.
Assuming you meant "1+1=," the answer is two.
If you meant "1+1=2, then what is 2+2?" the answer is four.
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On solving the question we get 2 + 2 = 4 because it is given 1 + 1 = 2 it means it is basic addition.
What is Addition?In addition we perform mathematical operation in which we add two or more no with other numbers to get the desire or final output. Output can be positive or negative it only based on your input values.
Mathematical operation is defined as sentence in which two or more number perform an operation which finally gave us result.
Mathematical operations are addition, Subtraction , Multiplication , Division, Percentage etc.
"1+1+" is an incomplete expression since it requires another operand or operator to form a valid mathematical statement.
It is given that "1+1=," so the answer was two.
it meant that if "1+1=2, then what is 2+2 =4 .
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What is the measure of T ?
ABCDRSTU
60°
100°
110°
125°
Answers
The measure of T in the quadrilateral can be found to be B. 100 degrees.
How to find the angle ?The two quadrilaterals given are trapeziums and they are similar. This means that they have the same angles meausures.
As this is a quadrilateral, the total measure of the interior degrees would be the value of 360 degrees.
The value of T is the only missing angles and so can be found to be:
T = 360 - 125 - 60 - 75
T = 360 - 260
T = 100 degrees
In conclusion, the measure of T is 100 degrees.
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△ABC and △DEF are similar triangles. Find BC.
Answers
Answer:
5 units
Step-by-step explanation:
We know that Triangles ABC and DEF are similar, therefore their values will be Equal
So , x + 7 + x - 4 = 12 + 5
2x + 3 = 15
2x = 15-3
2x = 12
x = 6
So in Triangle ABC , AC = x + 7 = 6 + 7 = 13
Therefore, by Pythagorean Therom,
BC^2 + AB^2 = AC^2
BC^2 + 144 = 169
BC^2 = 169 -144 = 25
BC = [tex]\sqrt{25} = 5[/tex]
Hope it helps.
A concrete sidewalk is being put around a square park. If the diagonal line across the center of a square park is 24 feet, what is the exact length of each side?
Answers
By answering the presented question, we may conclude that As a result, Pythagorean theorem each side of the square park is exactly [tex]12\sqrt(2)[/tex]feet long.
what is Pythagorean theorem?The Pythagorean Theorem, generally known as Pythagorean Theorem, is the foundational Euclidean arithmetic that links the three points of a right triangle. This rule states that the area of a cube only with hypotenuse side is equal to the total of the areas of triangles that have both two sides. The Pythagorean Theorem states that the square that spans the hypotenuse of a right triangle opposite the right angle is the total of all the squares that span its vertices. It is sometimes represented in broad algebraic notation as a2 + b2 = c2.
Let us solve this problem using the Pythagorean theorem. The diagonal in a square is the hypotenuse of a right triangle, with each side of the square acting as one of its legs. The Pythagorean theorem asserts that the sum of the squares of the two shorter sides of a right triangle equals the square of the hypotenuse.
x² + x² = 24²
2x² = 576
x² = 288
[tex]x = \sqrt (288)\\x = \sqrt(144 * 2)\\x = 12\sqrt (2)\\[/tex]
As a result, each side of the square park is exactly [tex]12\sqrt(2)[/tex]feet long.
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Use the normal approximation to the binomial to find the probabilities for the specific values of X. N=50, p=. 8, X=44
Answers
The probability of X = 44 is approximately 0.087 using the normal approximation to the binomial.
The normal approximation to the binomial is a technique used to estimate probabilities for large binomial distributions when calculating by hand becomes impractical. In this case, we have N = 50, p = 0.8, and we want to find the probability of X = 44.
To use the normal approximation, we need to first check if the binomial distribution is approximately normal. This can be done by checking if the conditions np >= 10 and n(1-p) >= 10 are met. In our case, np = 50 x 0.8 = 40 and n(1-p) = 50 x 0.2 = 10, so the conditions are met.
Next, we calculate the mean and standard deviation of the normal distribution using the formulas μ = np and σ = sqrt(np(1-p)). In our case, μ = 40 and σ = sqrt(50 x 0.8 x 0.2) ≈ 2.83.
Finally, we use the normal distribution with mean μ and standard deviation σ to find the probability of X = 44. We need to standardize X using the formula Z = (X - μ) / σ, which gives Z = (44 - 40) / 2.83 ≈ 1.41.
Using a standard normal table or calculator, we find that the probability of Z being less than 1.41 is approximately 0.921. This means that the probability of X being less than or equal to 44 is approximately 0.921.
Therefore, the probability of X being exactly 44 is approximately the difference between the probability of X being less than or equal to 44 and the probability of X being less than or equal to 43. Using a continuity correction, we adjust 43.5 to 43, which gives us:
P(X = 44) ≈ P(43.5 < X < 44.5) ≈ P(Z < (44.5 - 40) / 2.83) - P(Z < (43.5 - 40) / 2.83) ≈ 0.087.
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The following are NOT examples of using the "Distributive Property"
Example 1:4(x-2)=4•x-4•2
Example 2:4•26
4•(20+6)=4•20+4•6
A:True
B:False
Answers
False, Example 1:4(x-2)=4•x-4•2 and Example 2:4•26 and Example 3: 4•(20+6)=4•20+4•6 are NOT examples of using the "Distributive Property".
The Distributive Property is a fundamental property of algebra that is used to simplify expressions involving multiplication and addition. It states that the product of a number and the sum or difference of two or more numbers is equal to the sum or difference of the products of that number and each of the numbers in the sum or difference.
Example 1 uses the Distributive Property correctly, showing how 4 multiplied by the difference of x and 2 is equal to the difference of 4 times x and 4 times 2.Example 2 is not an example of using the Distributive Property because it is just a multiplication of two numbers, which does not involve addition or subtraction.Example 3 uses the Distributive Property correctly, showing how 4 multiplied by the sum of 20 and 6 is equal to the sum of 4 times 20 and 4 times 6.Learn more about the Distributive Property at
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A softball coach has ordered softballs for two different leagues. The Junior League uses an 11-inch softball priced at $2.50 each. The Senior League uses a 12-inch softball priced at $3.50 each. The softball coach ordered a total of 120 softballs for $350.
How many of each size softball did the softball coach order?
11-inch softballs:
12-inch softballs:
Answers
The number of 11-inch softballs ordered is 70 and the number of 12-inch softballs ordered is 50
How many of each size of softball was ordered?a + b = 120 equation 1
2.50a + 3.50b = 350 equation 2
Where:
a = number of 11-inch softballs ordered b = number of 12-inch softballs orderedThe elimination method would be used to determine the number of each size of softball ordered.
Multiply equation 1 by 2.5
2.5a + 2.5b = 300 equation 3
Subtract equation 3 from equation 2
b = 50
Substitute for b in equation 1
a + 50 = 120
a = 120 - 50
a = 70
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OA and OB are opposite rays. If x = 42 degree, what is the value of y?
Answers
Answer:
69
Step-by-step explanation:
2y + x = 180 ( Linear Pair)
As x = 42
2y + 42 = 180
2y = 180- 42
2y = 138
y = 138/2
y = 69
A dairy farmer milks his two cows every day. He determined the chance that he gets anywhere between 12 and 14 gallons of milk in one day is around 32%. Identify the method of probability the farmer used to reach this conclusion. Select the correct answer below: theoretical relative frequency
Answers
The dairy farmer used the relative frequency method of probability to reach his conclusion.
Relative frequency is a method of calculating probability that is based on the observation of how often an event occurs in a sample. The farmer likely observed how often he gets between 12 and 14 gallons of milk in a day and used that data to calculate the probability of it happening.
In contrast, theoretical probability is based on the assumption that all possible outcomes are equally likely. It is calculated by dividing the number of desired outcomes by the total number of possible outcomes.
Therefore, the correct answer is relative frequency.
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Tyee has a points card for a movie theater.
• He receives 45 rewards points just for signing up.
• He earns 11.5 points for each visit to the movie theater.
• He needs at least 160 points for a free movie ticket.
Use the drop-down menu below to write an inequality representing v, the number of
visits he needs to make in order to get a free movie ticket.
Answers
Given:
45 rewards points for signing up
11.5 points for each visit
Total Number of Points are:
45 + 11.5 · number of visits he makes
How many visits must Tyler make to earn a free movie ticket?45 + 11.5 · x ≥ 160 points for a free movie ticket
Where, x is how many visits must Tyler make to earn a free movie ticket
11.5 · x ≥ 160 - 45
x ≥ 115 / 11.5
x ≥ 10
Since the number of visits to reach 180 points is 10 then Tyler has to visit the movie theater 10 times or more.
Check our answer:
45 + 11.5 · 10 = 160 will earn him a free movie ticket
A company claims their batteries will last an average of 100 days with a standard deviation of 15 days. If 20 of these batteries are selected what is the probability that the average life of the selected batteries is between 95 and 105?
Answers
The probability that the average life of the selected batteries is between 95 and 105 is 0.9292.
The company claims their batteries will last an average of 100 days with a standard deviation of 15 days. If 20 of these batteries are selected, the probability that the average life of the selected batteries is between 95 and 105 can be determined by the central limit theorem. In statistics, the central limit theorem (CLT) states that as the sample size increases, the distribution of the sample means approaches a normal distribution, irrespective of the population's underlying distribution. Therefore, the distribution of the sample means is a normal distribution with a mean equal to the population mean and standard deviation equal to population standard deviation divided by the square root of the sample size. Hence, the probability that the average life of the selected batteries is between 95 and 105 is the probability that the sample mean lies between 95 and 105.Using the z-score formula, z = (x - μ) / (σ / sqrt(n)), the z-scores can be calculated as follows:z1 = (95 - 100) / (15 / sqrt(20)) = -1.79z2 = (105 - 100) / (15 / sqrt(20)) = 1.79From the standard normal distribution table, the probability of z-score between -1.79 and 1.79 is 0.9292.Therefore, the probability that the average life of the selected batteries is between 95 and 105 is 0.9292.
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Sanskar invested 20% of money h le received every month's his total investment is ₹ 18000 per month find his total money gain per month
Answers
By technicality, he earns 18000*5 which equals 90000, however after his investments, it is 72000.
dA/dr = d(∩r 2 )/dr = 2∩r,
when r=8 dA/dr=?
dA/dt = 0.4 x dA/dt=?
Answers
When the circle's radius is 8 cm and it is shrinking at a rate of 0.4 cm per second, the rate that the area is shrinking is -6.4 cm²/s.
How big is the r² area?Circle area formula: r² = r. The radius squared is multiplied to find the circumference of the circle. When a circle's radius is specified, its area is equal to r². When the diameter "d" is known, the circle's surface area is equal to d²/4.
What is the name for pi's half?90 degrees is equal to a quarter turn. A complete turn is 360 degrees, while a half turn is 180. Moreover, the rotation can be expressed in radians or fraction of pi. Under this system, a complete circle turn is equal to 2 radians, a half circle turn is equal to 1, and so on.
Step 1: Variables involved in the question:
The variables involved in the question are:
The radius of the circle r = 8 cm (constant value)
The rate at which the radius is decreasing (dr/dt) = -0.4 cm/s (negative because it's decreasing)
The rate at which the area of the circle is changing dA/dt = ?
Step 2: Chain rule:
We can use the chain rule to link the three rates:
dA/dt = dA/dr x dr/dt
Step 3:
From the given information, we know that (dr/dt) = -0.4 cm/s.
Step 4: Equation for dA/dr
The formula for the area of a circle is A = πr², where r is the radius of the circle. We can differentiate both sides of the equation with respect to r to get the following formula for dA/dr:
dA/dr = 2πr
Step 5:
Substituting the given value of r = 8 cm in the equation for dA/dr, we get:
dA/dr = 2π(8) = 16π cm
Step 6: dA/dt:
Now we can use the chain rule to find dA/dt by substituting the values we have found for dA/dr and dr/dt:
dA/dt = dA/dr x dr/dt
dA/dt = 16π cm x -0.4 cm/s
dA/dt = -6.4π cm²/s
Therefore, the rate at which the area of the circle is decreasing when its radius is 8cm and decreasing at -0.4 cm/s is -6.4π cm²/s.
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3. In a translation, the
remains the same.
Answers
Destanye replaces the light bulb
in the hall closet every 6 months
and replaces the air filter every 4
months. She just replaced both
items this month. After how many
months will she replace both the
light bulb and the air filter?
Select all that apply.
Answers
After 12 months she replace both the light bulb and the air filter
We can also use the concept of the Least Common Multiple (LCM) to find the answer to the problem.
The LCM of two numbers is the smallest number that is divisible by both of them. To find the LCM of 6 and 4, we can list the multiples of each number and look for the smallest multiple that is common to both lists. Alternatively, we can use the prime factorization of each number and multiply the highest powers of the common prime factors.
Prime factorization of 6: 2 x 3
Prime factorization of 4: 2^2
To find the LCM, we multiply the highest powers of each prime factor:
LCM = 2^2 x 3 = 12
Therefore, the LCM of 6 and 4 is 12. This means that the next time Destanye will replace both the light bulb and the air filter is in 12 months.
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use the distributive property or simplify to match the equivalent expressions 4(100-3)
Answers
Step-by-step explanation:
4 ( 100 -3) = 4 x 100 - 4 x 3
= 400 - 12 = 388
The polynomial of degree 4, P ( x ) , has a root of multiplicity 2 at x = 1 and roots of multiplicity 1 at x = 0 and x = − 3 . It goes through the point ( 5, and 128 ).
Find a formula for P ( x ).
P ( x ) =
Answers
The polynomials P(x) can be formulized as P(x) = (1/5)(x-1)²(x)(x+3).
The given problem states that a degree 4 polynomial, P(x), has the following properties:
Multiplicity of 2 at x = 1Multiplicity of 1 at x = 0Multiplicity of 1 at x = -3The polynomial goes through the point (5,128)The degree of the polynomial is 4, so the polynomial can be written as;
P(x) = a(x-1)²(x-0)(x+3)
To find the value of 'a', substitute the given point (5,128) into the equation, P(x);
P(5) = a(5-1)²(5-0)(5+3) = 128
P(5) = a(4)²(5)(8) = 128
Simplifying, we get;
128 = 640a,
a = 1/5
Thus, the formula for P(x) is; P(x) = (1/5)(x-1)²(x)(x+3)
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Select the correct answer.
A sport statistician gathered data on the number of points scored in each game by high school basketball players across the country. She found
a population mean of 20. 15 and a standard deviation of 3. 7. Each sample size was 20 players. By the central limit theorem, which interval do
68% of the sample means fall within?
A. 19. 32 and 20. 98
B. 17. 67 and 22. 63
C. 19. 96 and 20. 34
D. 18. 50 and 21. 81
Answers
68% of the sample means fall within 19.32 and 20.98.
By the central limit theorem, we know that the distribution of sample means will approach a normal distribution with a mean equal to the population mean (20.15) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (3.7 / sqrt(20) ≈ 0.828).
To find the interval within which 68% of the sample means fall, we need to find the z-scores corresponding to the 16th and 84th percentiles of the normal distribution (which is the range that encompasses 68% of the data in a normal distribution).
Using a z-table or a calculator, we can find that the z-score for the 16th percentile is approximately -1.0 and the z-score for the 84th percentile is approximately 1.0.
Therefore, the interval within which 68% of the sample means fall is:
20.15 - 1.0(0.828) = 19.32
to
20.15 + 1.0(0.828) = 20.98
So the answer is option A: 19.32 and 20.98.
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Jina opened a savings account with $500 and was paid simple interest at an annual rate of 3%. When Jina closed the account, she was paid $30 in interest. How long was the account open for, in years?
Answers
We can conclude after answering the provided question that Therefore, interest the account was open for 2 years.
what is interest ?In mathematics, interest is the amount of money earned or owed on an initial investment or loan. You can use either simple or compound interest. Simple interest is calculated as a percentage of the original amount, whereas compound interest is calculated on the principal amount plus any previously earned interest. If you invest $100 at a 5% annual simple interest rate, you will earn $5 in interest every year for three years, for a total of $15.
We know that the amount of interest earned, I, is given by the formula:
[tex]I = P * r * t\\P = $500\\r = 0.03 \\I = $30\\t = I / (P * r)\\t = $30 / ($500 * 0.03)\\t = 2 years[/tex]
Therefore, the account was open for 2 years.
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Suppose that there are three factories that manufacture light-bulbs. For factory i, every manufactured light-bulb (independently) has a chance of being defective with a probability pi ,p1 =0.05,p2 =0.1,p3=0.3. Initially, I thought that when I order a box of light-bulbs it is equally likely to come from any of the three factories. Upon receiving the box I found 8 out of 100 to be defective. What is my posterior probability that the box came from factoryi,i∈{1,2,3}?
Answers
The posterior probability that the box of light-bulbs came from factory 1 is 0.2111, from factory 2 is 0.5219, and from factory 3 is 0.2669.
Bayes' theorem formula can be used to answer this question.The formula is as follows:P(A|B) = P(B|A) P(A) / P(B)Here, A is the event that the box of light bulbs came from a particular factory (i.e., A = {i | i ∈ {1,2,3}}), and B is the event that 8 out of 100 bulbs in the box are defective.
First, we need to find the probability of observing 8 defective light-bulbs out of 100 for each of the three factories. The probability of observing k defective light-bulbs out of n total light-bulbs is given by the binomial distribution: P(k) = n! / (k!(n-k)!) * pk * (1-p)n-k
Factory 1: p1 = 0.05n = 100P(8) = 100! / (8!(100-8)!) * (0.05)8 * (1-0.05)100-8 = 0.0993Factory 2:p2 = 0.1n = 100P(8) = 100! / (8!(100-8)!) * (0.1)8 * (1-0.1)100-8 = 0.2452Factory 3:p3 = 0.3n = 100P(8) = 100! / (8!(100-8)!) * (0.3)8 * (1-0.3)100-8 = 0.1041
The sum of these probabilities gives the marginal likelihood:P(B) = P(8|1)P(1) + P(8|2)P(2) + P(8|3)P(3) = 0.0993 * 1/3 + 0.2452 * 1/3 + 0.1041 * 1/3 = 0.1495 Using Bayes' theorem, we can now calculate the posterior probabilities:P(1|8) = P(8|1) P(1) / P(B) = 0.0993 * 1/3 / 0.1495 = 0.2111P(2|8) = P(8|2) P(2) / P(B) = 0.2452 * 1/3 / 0.1495 = 0.5219P(3|8) = P(8|3) P(3) / P(B) = 0.1041 * 1/3 / 0.1495 = 0.2669
Therefore, the posterior probability that the box of light-bulbs came from factory 1 is 0.2111, from factory 2 is 0.5219, and from factory 3 is 0.2669.
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Sophia who took the Graduate Record Examination (GRE) scored 160 on the Verbal Reasoning section and 157 on the Quantitative Reasoning section. The mean score for Verbal Reasoning section for all test takers was 151 with a standard deviation of 7, and the mean score for the Quantitative Reasoning was 153 with a standard deviation of 7.67. Suppose that both distributions are nearly normal.
(a) Write down the short-hand for these two normal distributions.
(b) What is Sophia’s Z-score on the Verbal Reasoning section? On the Quantitative Reasoning
section? Draw a standard normal distribution curve and mark these two Z-scores.
(c) What do these Z-scores tell you?
(d) Relative to others, which section did she do better on?
(e) Find her percentile scores for the two exams.
(f) What percent of the test takers did better than her on the Verbal Reasoning section? On the
Quantitative Reasoning section?
(g) Explain why simply comparing raw scores from the two sections could lead to an incorrect
conclusion as to which section a student did better on.
(h) If the distributions of the scores on these exams are not nearly normal, would your answers to
parts (b) - (f) change? Explain your reasoning.
Answers
Short-hand for these two normal distributions: N(151, 7) for Verbal Reasoning and N(153, 7.67) for Quantitative Reasoning, z-score is 1 and 0.52 and she performed better in the verbal reasoning section.
Sophia’s Z-score on the Verbal Reasoning section is (160-151)/7 = 1, and her Z-score on the Quantitative Reasoning section is (157-153)/7.67 = 0.52. The Z-scores are marked on the standard normal distribution curve as shown below.
The Z-scores tell us that Sophia scored higher than the mean score of other test takers in the Verbal Reasoning section (1 is to the right of the mean, 0) and lower than the mean score of other test takers in the Quantitative Reasoning section (0.52 is to the left of the mean, 0).
Relative to others, Sophia did better on the Verbal Reasoning section.
Sophia’s percentile score for the Verbal Reasoning section is 84% and for the Quantitative Reasoning section is 64%.
Approximately 84% of the test takers did better than Sophia on the Verbal Reasoning section, and 64% of the test takers did better than Sophia on the Quantitative Reasoning section.
Comparing raw scores from the two sections can lead to an incorrect conclusion as to which section a student did better on because raw scores do not take into account the number of students who scored higher or lower than the student in question. For example, a student may have scored higher than another student in one section but scored lower than many other students in that same section.
If the distributions of the scores on these exams are not nearly normal, then my answers would change. This is because a non-normal distribution would not follow the standard normal distribution curve, and thus the Z-scores and percentiles would be different.
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