Given that a circle is inscribed in an isosceles trapezoid with bases of 8 cm and 2 cm. We need to find the probability that a randomly selected point inside the trapezoid lies on the circle.
The isosceles trapezoid is shown below: [asy] draw((0,0)--(8,0)--(3,5)--(1,5)--cycle); draw((1,5)--(1,0)); draw((3,5)--(3,0)); draw((0,0)--(1,5)); draw((8,0)--(3,5)); draw(circle((2.88,2.38),2.38)); label("$8$",(4,0),S); label("$2$",(1.5,5),N); [/asy]Let ABCD be the isosceles trapezoid,
where AB = 8 cm, DC = 2 cm, and AD = BC.
Since the circle is inscribed in the trapezoid, we can use the following formula:2s = AB + DC = 8 + 2 = 10 cm
Where s is the semi-perimeter of the trapezoid. Also, let O be the center of the circle. We can draw lines OA, OB, OC, and OD as shown below: [asy] draw((0,0)--(8,0)--(3,5)--(1,5)--cycle); draw((1,5)--(1,0)); draw((3,5)--(3,0)); draw((0,0)--(1,5)); draw((8,0)--(3,5)); draw(circle((2.88,2.38),2.38)); label("$A$",(0,0),SW); label("$B$",(8,0),SE); label("$C$",(3,5),N); label("$D$",(1,5),N); label("$O$",(2.88,2.38),N); label("$8$",(4,0),S); label("$2$",(1.5,5),N); draw((0,0)--(2.88,2.38)--(8,0)--cycle); label("$s$",(3,0),S); label("$s$",(1.44,2.38),E); [/asy]Since O is the center of the circle, we have:OA = OB = OC = OD = rwhere r is the radius of the circle.
Also, we have:s = OA + OB + AB/2 + DC/2s = 2r + 2s/2s = r + 5 cmWe can solve for r:r + 5 cm = 10 cmr = 5 cmNow that we know the radius of the circle, we can find the area of the trapezoid and the area of the circle.
Then, we can find the probability that a randomly selected point inside the trapezoid lies on the circle as follows:Area of trapezoid = (AB + DC)/2 × height= (8 + 2)/2 × 5= 25 cm²Area of circle = πr²= π(5)²= 25π cm²Therefore, the probability that a randomly selected point inside the trapezoid lies on the circle is:
Area of circle/Area of trapezoid= 25π/25= π/1= π
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The probability that a randomly selected point inside the trapezoid lies on the circle is 0.399 or 39.9%. Therefore, option (A) is the correct answer.
The circle is inscribed in an isosceles trapezoid with bases of 8 cm and 2 cm.
Inscribed Circle of an Isosceles Trapezoid
Therefore, the length of the parallel sides (AB and CD) is equal.
Let the length of the parallel sides be ‘a’. Then, OB = OD = r (let)
It is also given that the lengths of the parallel sides of the trapezoid are 8 cm and 2 cm.
Then, its height is given by:
h = AB - CD / 2 = (8 - 2) / 2 = 3 cm
Therefore, the length of the base BC of the right-angled triangle is equal to ‘3’.
Then, the length of the other side (AC) can be given as:
AC = sqrt((AB - BC)² + h²) = sqrt((8 - 3)² + 3²) = sqrt(34) cm
The area of the trapezoid can be calculated as follows:
Area of the trapezoid = 1/2 (sum of the parallel sides) x (height)A = 1/2 (8 + 2) x 3A = 15 sq. cm.
The area of the circle can be given by:
Area of the circle = πr²πr² = A / 2πr² = 15 / (2 x π)
Therefore, r² = 2.39
r = sqrt(2.39) sq. cm.
Now, the probability that a randomly selected point inside the trapezoid lies on the circle can be calculated by dividing the area of the circle by the area of the trapezoid:
P (point inside the trapezoid lies on the circle) = Area of the circle / Area of the trapezoid
P = πr² / 15
P = π (2.39) / 15
P = 0.399 or 39.9%
The probability that a randomly selected point inside the trapezoid lies on the circle is 0.399 or 39.9%.
Therefore, option (A) is the correct answer.
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For the quadratic function show below the coordinates of its vertex are
Answer: 0,2
Step-by-step explanation:
Verify that Rolle's Theorem can be applied to the function f(x) = 3 - 102 +31-30 on the interval [2, 5]. Then find all values of c in the interval such that f' (c) = 0.
Enter the exact answers in increasing order.
To enter √a, type sqrt(a).
c =
C=
Show your work and explain, in your own words, how you arrived at your answers.
Equation Editor A- A T I
BIUS X₂ x²
Font Size Styles
Words: 0
The exact values of c in increasing order are: c = (10 - sqrt(7)) / 3, (10 + sqrt(7)) / 3
To verify that Rolle's Theorem can be applied to the function f(x) = x³ - 10x² + 31x - 30 on the interval [2, 5], we need to check if the following conditions are satisfied:
f(x) is continuous on [2, 5].f(x) is differentiable on (2, 5).f(2) = f(5).Let's check each condition:
f(x) = x³ - 10x² + 31x - 30 is a polynomial function and is continuous for all real values of x. So, it is continuous on [2, 5].
To check the differentiability, we need to find f'(x):
f'(x) = 3x² - 20x + 31.
The derivative f'(x) exists and is continuous for all real values of x. So, f(x) is differentiable on (2, 5).
Now, let's evaluate f(2) and f(5):
f(2) = (2)³ - 10(2)² + 31(2) - 30 = -10
f(5) = (5)³ - 10(5)² + 31(5) - 30 = 95
Since f(2) = -10 is not equal to f(5) = 95, we can conclude that Rolle's Theorem can be applied to the function f(x) = x³ - 10x² + 31x - 30 on the interval [2, 5] after differentiable.
To find the values of c in the interval (2, 5) such that f'(c) = 0, we need to solve the equation f'(c) = 3c² - 20c + 31 = 0.
Using quadratic formula:
c = (-(-20) ± sqrt((-20)² - 4(3)(31))) / (2(3))
c = (20 ± sqrt(400 - 372)) / 6
c = (20 ± sqrt(28)) / 6
c = (20 ± 2sqrt(7)) / 6
c = (10 ± sqrt(7)) / 3
The values of c in the interval (2, 5) such that f'(c) = 0 are:
c = (10 + sqrt(7)) / 3
c = (10 - sqrt(7)) / 3
Therefore, the exact values of c in increasing order are: c = (10 - sqrt(7)) / 3, (10 + sqrt(7)) / 3.
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Incomplete question:
Verify that Rolle's Theorem can be applied to the function f(x) = x³ - 10x² +31x-30 on the interval [2, 5]. Then find all values of c in the interval such that f' (c) = 0.
Enter the exact answers in increasing order.
To enter √a, type sqrt(a)
c = ?
Need help as soon as possible pls help
1. The value of x is:
180° - 40° - 30° = 110°
2. The value of x is:
90°- 38° = 52°
Answer:
Step-by-step explanation:
180 = 40 + x + 30
110 = x
90 = 38 + x
52 = x
Given the coordinates for the function below, which of the following are coordinates for its inverse?
The inverse of the given function is represented by Data Table B.
What is a Function?A function is a law that relates a dependent and an independent variable.
The Inverse of the function is determined by interchanging the values of a and y in an f(x,y) function and then express the equation of y in terms of x.
Th table of the function is
Miles to go Miles Travelled
0 0
100 310
200 450
340 550
650 650
The inverse of this data table will be
Miles Travelled Miles to go
650 650
340 550
200 450
100 310
0 0
This is represented by Data Table B.
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9. (10%) Consider a linear block code whose generator matrix G is. 1 0 0 1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1 1 (a) (2%) Find the parity check matrix H. (b) (3%) What is the minimum distance of the code?
(a) The parity check matrix H is:
1 1 1 0 1
0 1 0 1 1
0 1 1 0 0
To find the parity check matrix H, we can use the fact that H is the transpose of the generator matrix G with an identity matrix on the right side.
Given the generator matrix G:
1 0 0 1
1 1 1 0
1 0 1 1
0 1 0 0
1 1 0 1
We can rewrite G as:
1 0 0 1 1 1 0 0
1 1 1 0 0 1 1 0
1 0 1 1 1 0 1 1
0 1 0 0 1 1 0 0
1 1 0 1 0 1 1 1
Now, we can obtain the parity check matrix H by taking the transpose of G and removing the rightmost identity matrix:
H = Transpose(G without the rightmost identity matrix)
H =
1 1 1 0 1
0 1 0 1 1
0 1 1 0 0
Therefore, the matrix H is:
1 1 1 0 1
0 1 0 1 1
0 1 1 0 0
(b) The minimum distance of a linear block code is the smallest number of bit positions in which any two codewords differ. It determines the error detection and correction capability of the code.
To find the minimum distance of the code, we can examine the columns of the parity check matrix H. The number of non-zero entries in the smallest column gives us the minimum distance.
Looking at the parity check matrix H, we see that the smallest column has two non-zero entries in positions 1 and 2. Therefore, the minimum distance of the code is 2.
In conclusion, the minimum distance of the code is 2.
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whata 10 divided by 100
Answer:
it 10000
Step-by-step explanation:
79ib dlndokveik dinfl. kwbfb
Solve -72 = 8 (y - 3) pls
Step-by-step explanation:
Given
- 72 = 8 ( y - 3 )
or - 72 / 8 = y - 3
or, - 9 = y - 3
y = - 9 + 3
Therefore X = - 6
Hope it will help :)❤
Answer:
Step-by-step explanation:
-72=8y-24
-72+24=8y
-48=8y
-48/8=y
-6=y
divide 54 books between A and B in the ratio 13:14
Answer:
hi
Step-by-step explanation:
i think,
13:14=54
13k+14k=54
27k=54
k=54/27
k=2
13×2=26 and 14×2=28
to verify the correctness of the answer-26+28=56
have a nice day
hope it helps
The ratio of the divide is mathematically given as
26:28
What is the ratio of the divide?Question Parameters:
divide 54 books between A and B in the ratio 13:14
Generally, ratios are mathematically given as
X=13/27*54
X=26
Y=14/27*54
Y=28
In conclusion, the ratio of the divide would be
26:28
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x is 40% of 60
x=?
please help!!
Answer:
x = 24
Step-by-step explanation:
x = .4(60)
x = 24
40% of 60 can be found by converting 40% to a decimal and multiplying it by 60.
40% = 0.4
60x0.4=2.4
Now move the decimal point right one.
So x is 24.
---
hope it helps
2. A wooded area is in the shape of a a trapezoid whose bases measure 128 m and $2 m and its height is 40 m. A 4 m wide walkway is constructed which runs perpendicular from the two bases. Calculate the area of the wooded area after the addition of the watkway
Correction in the Question:
A wooded area is in the shape of a a trapezoid whose bases measure 128 m and 92 m and its height is 40 m. A 4 m wide walkway is constructed which runs perpendicular from the two bases. Calculate the area of the wooded area after the addition of the walkway.
Answer:
The wooded area after the addition of the walkway is 4240 [tex]m^2[/tex].
Step-by-step explanation:
we are given
length of the two bases = 128m and 92m
height of the trapezoid = 40m
the approximate figure of the given trapezoid is given as:
__ __ __ 92 __ __ _
/ | | | \
/ | 40 |4| \
/__ _| __ __ | |__ __ __ __ \
128
Area of a trapezoid = [(a + b)/2] * height, where a and b are representing the bases of the given trapezoid.
Area = [(92 + 128)/2] * 40
= [220/2] * 40
= 110 * 40
= 4400 [tex]m^2[/tex]
Now there is a 4m wide walkway is to be constructed in that trapezoid. The pathway will be a rectangle as it has 4m width and 40m height as it is perpendicular to both the bases.
Area of a rectangle = length * width
Area = 40 * 4
= 160
Since the walkway will reduce the area of the trapezoid as it is constructed upon it therefore the wooded area after the addition of the walkway is
4400 + (-160) = 4240 [tex]m^2[/tex].
I asked some friends how old they think they will be when they get married. Here are their answers:
{42, 38, 27, 53, 39, 29, 52}
Put this data in order from least to greatest.
Answer:
27,29,38,39,42,52,53
Step-by-step explanation:
Please help me out you can just give me the answer! PLEASE AND THANK YOU!
Answer:
(Disclaimer: all digits in the answers are in the measuring unit degrees)
1) 15
2) 16
3) 46
4) 59.
Step-by-step explanation:
The first one is said to add up to 89, so you have to do 89 subtract 44 and 30 as they are told. That = 15
The second one is a right angle, meaning it adds up to 90. You do 90 - 74 as that is the value told. That = 16
Angles on a straight line = 180 so you have to do 180 - 134 as that is the value told. This = 46
This is 54 as opposing angles on two lines are =. That means this = 59 too.
(3)(0.2)+9 help please
Answer:
9.6
Step-by-step explanation:
Answer:
[tex]9.6[/tex]
Step-by-step explanation:
1) Simplify 3 × 0.2 to 0.6.
[tex]0.6 + 9[/tex]
2) Simplify.
[tex]9.6[/tex]
Hence, the answer is 9.6
Find the radius of the button.
28 mm
radius:
mm
Answer:
14 mm
Step-by-step explanation:
Just write down the answer.
Answer:
28mm
Step-by-step explanation:
is the right answer
Use completing the square to find the equation of the following circle in standard form.
x2 + y2 - 4x + 12y - 16 = 0
Answer:
Here's a calculator that should help
Step-by-step explanation:
https://www.calculatorsoup.com/calculators/algebra/completing-the-square-calculator.php
how do I solve sin(4x)=sin(2x)?
Answer:
sin(4x) = sin(2x) is unsolvable.
Step-by-step explanation:
The two sides are not equal.
5. How many mililiters of a 3.5 M iron (II) nitrite (Fe(NO2)2) solution are needed to provide a tot
of 0.13 kg of Fe(NO),?
need help plss
Answer:
try this link!
Step-by-step explanation:
https://www.wylieisd.net/cms/lib09/TX01918453/Centricity/Domain/783/Math%20Connections%20Key.pdf
a triangle with an area of 23 cm² is dilated by a factor of 6. what is the area of the dilated triangle?
When a triangle is dilated by a scale factor, the area of the dilated triangle is equal to the scale factor squared times the area of the original triangle. The area of the dilated triangle is 828 cm².
In this case, the original triangle has an area of 23 cm². The triangle is dilated by a factor of 6, so the scale factor is 6.
To find the area of the dilated triangle, we use the formula:
Area of Dilated Triangle = (Scale Factor)^2 * Area of Original Triangle
Plugging in the values:
Area of Dilated Triangle = 6^2 * 23 cm²
= 36 * 23 cm²
= 828 cm²
Therefore, the area of the dilated triangle is 828 cm².
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Tbh you can guess for the fractions
Answer:
4:1 ratio
Sugar: 1 cup
Butter: 3/4 cups
Eggs: 2
Baking powder: 3/8 tsp
Flour: 5/8 tsp (you need way more flour for cookies)
Salt: 1/8 tsp (original # was hard to make out but I think it was 1/2 tsp)
Kai has 8 pints of buttermilk. He uses 4 ounces of buttermilk in
his receipe for a loaf of bread. How many loaves of bread can he make with the buttermilk that he has?
A. 2 loaves
B. 16 loaves
C. 24 loaves
D. 32 loaves
find the range of f(x)=2/5x + 5 for the domain (-4, -2, 0 3)
my answer is 3.4, 4.2, 5, 6.2 but im not fully sure
Answer:
Your answers are correct
Step-by-step explanation:
PLS HELP ME QUICK I NEED THIS GOOD GRADE PLS HELPPPP! Tell whether each statement is true or false. If false, provide a counterexample. The set of whole numbers contains the set of rational numbers. Every terminating decimal is a rational number. Every square root is a rational number. The integers are closed under addition.
Answer:
1. The set of whole numbers contains the set of rational numbers. FALSE.
The set of integers contains the set of rational numbers
2. Every terminating decimal is a rational number. TRUE
3. Every square root is a rational number. FALSE
Many square roots are irrational numbers, meaning there is no rational number equivalent.
4. The integers are closed under addition. TRUE
OMG HELP PLS IM PANICKING OMG OMG I GOT A F IN MATH AND I ONLY HAVE 1 DAY TO CHANGE MY GRADE BECAUSE TOMORROW IS THE FINAL REPORT CARD RESULTS AND I DONT WANNA FAIL PLS HELP-
AND CAN ANYONE PLS DRAW ME THE ANSWERS I CANT UNDERSTAND ANYTHING PLS I BEG U ILL GIVE U BRAINLEST
Answer:
here, i can help you out!
Step-by-step explanation:
Answer:
For 1. 2/5 is less full then 1/2
Find the mean, median, and mode of the data set.
{0,9, 3, 6, 10, 10, 7,1
MEAN:
MEDIAN:
MODE:
Prove that for a normal matrix A, eigenvectors corresponding to different eigenvalues are necessarily orthogonal.
Eigenvalues and eigenvectors play a crucial role in the study of linear transformations and matrices. For a normal matrix A, it can be proven that eigenvectors corresponding to different eigenvalues are necessarily orthogonal.
To understand why eigenvectors corresponding to different eigenvalues are orthogonal for a normal matrix, we need to consider the properties of normal matrices. A matrix A is normal if it commutes with its conjugate transpose A* (i.e., A * A* = A* A).
Now, let's consider two eigenvectors v₁ and v₂ corresponding to different eigenvalues λ₁ and λ₂, respectively. We want to show that v₁ and v₂ are orthogonal, meaning their dot product is zero (v₁ · v₂ = 0).
Let's denote the conjugate transpose of A as A*, and the eigenvalues and eigenvectors as follows:
A * A = A * A* (1)
Multiplying both sides of equation (1) by v₂* (the conjugate transpose of v₂) from the left gives:
v₂* A * A = v₂* A * A* (2)
Since v₂ is an eigenvector of A, we can express it as:
A * v₂ = λ₂ v₂ (3)
Substituting equation (3) into equation (2) gives:
v₂* λ₂ A = v₂* A * A* (4)
Now, let's multiply equation (4) by v₁ from the right:
v₂* λ₂ A v₁ = v₂* A * A* v₁ (5)
Since v₁ is an eigenvector of A, we can express it as:
A * v₁ = λ₁ v₁ (6)
Substituting equation (6) into equation (5) gives:
v₂* λ₂ λ₁ v₁ = v₂* λ₁ A* v₁ (7)
Notice that λ₁ and λ₂ are scalars, so we can move them around. Taking the conjugate transpose of equation (7), we get:
(λ₂ λ₁) v₁* v₂ = (λ₁ v₁)* A v₂ (8)
Now, we have v₁* v₂ on the left-hand side and (λ₁ v₁)* A v₂ on the right-hand side. If v₁ and v₂ are not orthogonal (v₁ · v₂ ≠ 0), then v₁* v₂ ≠ 0. However, the right-hand side of equation (8) is proportional to (λ₁ v₁)* A v₂, which is proportional to A v₂. This implies that A v₂ is a scalar multiple of v₁, which contradicts the assumption that v₁ and v₂ correspond to different eigenvalues.
Therefore, we conclude that eigenvectors corresponding to different eigenvalues for a normal matrix are necessarily orthogonal.
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a local ice cream shop has a special deal on thursdays: buy a waffle cone for $3 and get each scoop of ice cream for $1.50. what would be the rate of change in this word problem?
In the given word problem, the rate of change is the change in the cost of the ice cream concerning the change in the number of scoops.
That is, the rate of change is the ratio of the change in the cost of ice cream and the change in the number of scoops. Let's first calculate the initial rate of change or slope of the given deal: When we buy a waffle cone, the cost is $3, and we can buy one scoop of ice cream for $1.50.So, for one scoop of ice cream, the total cost would be 3 + 1.50 = $4.50.
We can represent the cost of one scoop of ice cream with the help of a linear equation: y = mx + b. Here, the slope or the rate of change, m = Change in cost of ice cream/ Change in the number of scoops= 1.5/1= 1.5Therefore, the rate of change of the ice cream with respect to the number of scoops is $1.50/scoop.
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The distribution of white blood cell count per cubic millimeter of whole blood is approximately Normal with mean 7500 and standard deviation 1750 for healthy patients Include an appropriately labeled and shaded Normal curve for each part. There should be three separate curves. 4. What is the probability that a randomly selected person will have a white blood cell count of between 2000 and 10,000?
The probability that a randomly selected person will have a white blood cell count of between 2000 and 10,000 is 0.9223 approximately.
The given mean, standard deviation, and the range of values are as follows:
Mean = 7500
Standard deviation = 1750
Range of values = Between 2000 and 10000
We are required to calculate the probability of a random person having a white blood cell count between 2000 and 10000.
Let's find the Z values for 2000 and 10000.Z1 = (2000 - 7500) / 1750 = -3Z2 = (10000 - 7500) / 1750 = 1.43
The required probability is the sum of the probability of the given range of values.
The probability of the first value is:P(X < 2000) = P(Z < -3) = 0.00135
The probability of the second value is:P(X > 10000) = P(Z > 1.43) = 0.0764
To find the probability for the given range, we will subtract the probability of the second value from the probability of the first value.
P(2000 < X < 10000) = 1 - P(X < 2000) - P(X > 10000)P(2000 < X < 10000)
= 1 - 0.00135 - 0.0764P(2000 < X < 10000) = 0.9223
The probability that a randomly selected person will have a white blood cell count between 2000 and 10,000 is 0.9223, approximately.
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If A, B and M are three collinear points, such that M divides AB internally in the ratio of 7:5 and P is any point not on the line AB, show that PM = PA + PB (4 marks] = 12 12
Given collinear points A, B, and M, with M dividing AB internally in the ratio of 7:5, and a point P not on the line AB, it can be shown that PM is equal to the sum of PA and PB.
Let's consider the line segment AB, where M is a point that divides it internally in the ratio of 7:5. This means that the ratio of AM to MB is 7:5.
Now, let's consider the triangle PAB, where P is a point not on the line AB. We want to show that PM is equal to the sum of PA and PB.
Since M divides AB internally in the ratio of 7:5, we can express AM and MB in terms of their lengths. Let's assume the length of AM is 7x and the length of MB is 5x.
Using this information, we can express the lengths of PA and PB in terms of x as well. Let's denote the length of PA as y and the length of PB as z.
Since M divides AB internally in the ratio of 7:5, we can write:
AM/MB = 7x/5x = 7/5
Similarly, we can express the ratios of PM to PA and PM to PB:
PM/PA = 7x/y
PM/PB = 5x/z
We need to show that PM is equal to PA + PB:
PM = PA + PB
Substituting the ratios we derived earlier:
(7x/y) = (5x/z) + 1
To simplify the equation, we can multiply both sides by yz:
7xz = 5xy + yz
Next, we can factor out the common factor of y:
7xz = y(5x + z)
Now, we can divide both sides by (5x + z):
PM = y
Therefore, we have shown that PM is equal to PA + PB.
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consider the polynomial function q(x)=-2x^8+5x^6-3x^5+50
What is the end behavior of the graph of q?
Choose 1 answer:
(Choice A) As x→∞, q(x)→∞, and as x→−∞, q(x)→∞
(Choice B) As x→∞, q(x)→-∞, and as x→−∞, q(x)→∞
(Choice C) As x→∞, q(x)→-∞, and as x→−∞, q(x)→-∞
(Choice D) As x→∞, q(x)→∞, and as x→−∞, q(x)→-∞
Answer:
C.
Step-by-step explanation:
Answer A and D are definitely incorrect. Hope this helps Zoey. #Zoeyiscute:)
Assume that women's heights are normally distributed with a mean given by = 62.5 in, and a standard deviation given by a = 2.1 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 63 in. (b) If 33 women are randomly selected, find the probability that they have a mean height less than 63 in.
(a) The probability is approximately __________ (Round to four decimal places as needed.)
(b) The probability is approximately_______________
(a) The probability is approximately 0.6915. (Round to four decimal places as needed.)
(b) The probability is approximately 0.9999.
(a) The probability that a randomly selected woman's height is less than 63 inches is approximately 0.6915.
To find this probability, we can use the standard normal distribution and z-scores. The z-score is calculated using the formula z = (x - μ) / σ, where x is the value we are interested in, μ is the mean, and σ is the standard deviation. In this case, x = 63 inches, μ = 62.5 inches, and σ = 2.1 inches.
Substituting these values into the formula, we get z = (63 - 62.5) / 2.1 = 0.2381. To find the probability corresponding to this z-score, we can look it up in the standard normal distribution table or use a statistical calculator. The probability associated with a z-score of 0.2381 is approximately 0.5915.
However, since we want to find the probability that the height is less than 63 inches, we need to find the area to the left of the z-score. Since the standard normal distribution is symmetrical, the area to the left of a positive z-score is equal to 1 minus the area to the right. Therefore, the probability that a randomly selected woman's height is less than 63 inches is approximately 1 - 0.5915 = 0.6915.
(b) The probability that a sample of 33 randomly selected women has a mean height less than 63 inches is approximately 0.9999.
When we have a sample of multiple individuals, the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution. This phenomenon is known as the Central Limit Theorem.
For this problem, we can assume that the mean height of the sample of 33 women follows a normal distribution with the same mean as the population (62.5 inches) but with a standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, the sample standard deviation would be 2.1 inches divided by the square root of 33, which is approximately 0.3669 inches.
To find the probability that the sample mean height is less than 63 inches, we can again use z-scores. The z-score is calculated using the formula z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Substituting the values into the formula, we get z = (63 - 62.5) / (0.3669) = 1.3662. The probability corresponding to this z-score can be found using a standard normal distribution table or a statistical calculator. The probability associated with a z-score of 1.3662 is approximately 0.9082.
However, since we want to find the probability that the sample mean height is less than 63 inches, we need to find the area to the left of the z-score. Thus, the probability that a sample of 33 randomly selected women has a mean height less than 63 inches is approximately 0.9999.
To know more about the Central Limit Theorem refer here:
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