For (a), the probability that John wins on the third round given that he only won two rounds of the three is 5/6.
For (b), the suitability of modelling X after the geometric distribution is appropriate because the geometric distribution is the probability distribution of the number of Bernoulli trials needed to get one success. The probability of John winning on the fifth round is $(\frac{2}{5})^4(\frac{5}{6})=\frac{50}{648}$
For (c), any necessary assumptions required in order to suitably model Y after the binomial distribution is that the trials (i.e. rounds of play) are independent, and that the probability of success is the same for each trial. The parameters of this binomial distribution are: n = 10, p = 1/5.
For (d), assuming the assumptions in Question 2(c) hold, the expected value of Y (E(Y)) is 8 and the variance of Y (Var(Y)) is 2.4.
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Instant Meals sent out free samples to introduce its new product, Sesame soup. Each sample weighs 64 ounces. The post office charges $0.36 for every 1.5 pounds of weight. How much would Instant Meals spend on postage to mail out 188 samples?
The amount that Instant Meals spends on postage to mail out 188 samples is: 19458 cents
How to solve Algebra Word Problems?The parameters are given as:
W = 64 oz
Total quantity = 188 samples
Charge = 36 cents per 1.5 pounds
Thus converting pounds tp oz, we have;
Charge = 36 cents per 24 oz
Required:
Total amount to pay the parcels
Solution:
Multiply W to 188 samples,
Total weight = 69 oz. (188)
Total weight = 12,972 oz.
Use the charge as an conversion factor,
Total price to pay:
P = 12,972 oz. (36 cents / 24 oz.)
P = 19458 cents
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Consider a set of data in which the sample mean is 33. 7 and the sample standard deviation is 7. 2. Calculate the z-score given that x=30. 2. Round your answer to two decimal places
Since, Consider a set of data in which the sample mean is 33. 7 and the sample standard deviation is 7. 2. Calculate the z-score given that
x= 30. 2. Therefore, the Z score is -0.486111.
Z score:
Standard scores are often referred to as z-scores; the two terms are used interchangeably. Other equivalent terms used include z-score, normal score, standardized variable, and high-energy physical attraction.
In statistics, a standard score is the number of standard deviations by which a raw score (i.e. an observation or data point) has a value greater than or less than the observed or measured mean. Raw scores above the mean have positive standard scores, while raw scores below the mean have negative standard scores.
Given that:
Sample mean (μ) = 33.7
Standard deviation (σ) = 7.2
X = 30.2
We know that:
Z score = X -μ/ σ
z-score using the formula z = (x - μ) / σ,
where,
x is your data point,
μ is the mean, and
σ is the standard deviation.
Putting the values in the equation:
⇒ Z score = X -μ/ σ
= 30.2 - 33.7/ 7.2
= -0.486111
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what is the volume of the cereal box?
Answer:
See below.
Step-by-step explanation:
We are asked to find the volume of the cereal box.
This cereal box is a rectangular prism, meaning that the cereal box should have 3 given dimensions; Length, Width, and Height.
[tex]Volume = Length \times Width \times Height.[/tex]
We have all 3 dimensions already, now we can solve for the volume.
Substitute:
[tex]Volume = 3 \times 12 \times 18.[/tex]
[tex]Volume = 648in^3.[/tex]
Which number sentence is true?
Please I need help. Using y=k/x, you get k=1.5. So applying this, the first two are correct and in inverse proportion, however the last one doesn’t seem to work. Please help
Answer:
To determine if s is inversely proportional to f, we need to check if their product is constant. Let's multiply s and f for each row:
s * f = 0.2 * 5 = 1
s * f = 0.5 * 2 = 1
s * f = 1.4 * 0.714 ≈ 1
s * f = 7.5 * 0.133 ≈ 1
s * f = 3 * 0.3 = 0.9
As we can see, the product of s and f is approximately constant for all rows except the last one. This means that s and f are not inversely proportional in general.
However, we can see that the product of s and f is close to 1 for the first four rows. This suggests that s and f may be inversely proportional for values of s less than 3.
To confirm this, we can calculate the constant of proportionality k using the first two rows:
s * f = k
0.2 * 5 = k
k = 1
Therefore, the equation relating s and f is:
s * f = 1
or
f = 1/s
This shows that s and f are indeed inversely proportional for values of s less than 3. However, for s = 3, the product of s and f is 0.9 instead of 1, which means that s and f are not inversely proportional for this value of s.
Factor the polynomial completely.
x3 + 10x2 + 24x
A.x(x + 6)(x +4)
B. x( x2 + 10x +24)
C. x(x - 6)(x - 4)
D. x(x + 3)(x + 8)
The answer choice A is x(x + 6)(x + 4), which is the polynomial [tex]x^3 + 10x^2 + 24x[/tex] fully factored form.
The quadratic expression enclosed in parenthesis can be factored:
x(x + 6)(x + 4)
what is a polynomial?Algebraic expressions called polynomials include coefficients and variables. Indeterminates are another name for variables. For polynomial expressions, we can do mathematical operations like addition, subtraction, multiplication, and positive integer exponents but not division by variables.[tex]x^2+x-12[/tex] is an illustration of a polynomial with a single variable.
from the question:
The given polynomial is[tex]x^3 + 10x^2 + 24x.[/tex]
From each term, we may factor out the common factor of x:
[tex]x(x^2 + 10x + 24)[/tex]
The quadratic expression enclosed in parenthesis can be factored:
x(x + 6)(x + 4)
As a result, solution option A is the fully factored version of the polynomial [tex]x^3 + 10x^2 + 24x[/tex] , which is x(x + 6)(x + 4).
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For each table make a scatter plot of the data. Draw a trend line and write its equation.
20.
X Y
2 3
4 6
5 5
7 7
8 9
8 8
21.
X Y
3 9
5 8
5 6
6 5
6 6
8 3
22.
X Y
1 1
2 3
3 5
3 6
5 8
6 9
To make a scatter plot of the data from each table, begin by assigning the x-axis to represent the values from one of the tables and the y-axis to represent the values from the other table. Then plot each pair of values onto the graph, creating a "scatter" of points. To draw a trend line, look for the general pattern in the data points and draw a line that best fits them. To find the equation of the trend line, use the formula y = mx + b, where m is the slope of the line and b is the y-intercept. To find the slope of the line, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line. To find the y-intercept, plug the slope and any point on the line into the equation y = mx + b and solve for b.
The given data is as follows: 5, 66, and 9. We can plot these values in a scatter plot by following the steps mentioned below:Step 1: Plot the given data points on the Cartesian plane. Step 2: Find the equation of the line of best fit/trend line. We can use the least-squares regression method to find the line of best fit/trend line. This method gives us the equation of a line that best represents the relationship between two variables. In this case, the two variables are X and Y. Step 3: Draw the trend line on the scatter plot. Step 4: Write the equation of the trend line. The equation of the line of best fit/trend line is in the form of y = mx + c, where m is the slope of the line, and c is the y-intercept. . However, we can still draw a line of best fit/trend line that passes through the data points. The equation of the line of best fit/trend line can be found using the least-squares regression method. Using this method, we get the equation of the line of best fit/trend line as: y = -10.33x + 71.33Therefore, the equation of the trend line is y = -10.33x + 71.33.
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Four minus three times a number is less than twenty five
Answer:
x>-7
Step-by-step explanation:
show working and answer
There is no answer since we cannot take the square root of a negative trigonometry value. As a result, the issue as presented cannot be resolved.
what is trigonometry?The area of mathematics called trigonometry examines how triangle side lengths and angles relate to one another. The subject first came to light in the Hellenistic era, about in the third century BC, as a result of the use of geometry in astronomical investigations. The area of mathematics known as exact techniques deals with several trigonometric functions and possible computations using them. There are six common trigonometric functions in trigonometry. These go by the designations sine, cosine, tangent, cotangent, secant, and cosecant, respectively (csc). Trigonometry is the study of triangle characteristics, particularly those of right triangles. Consequently, studying geometry entails learning about the characteristics of all geometric forms.
Trigonometry can be used to resolve this issue. We'll abbreviate the angle between the stairs and the slide as "".
We must first determine the slide's height. The Pythagorean Theorem allows us to perform the following:
Height 2 equals (slide length)/2 - (distance from step bottom to bottom of slide)/2 Height 2 equals 4.2/2 - 4.9/2 Height 2 equals -7.55
There is no answer since we cannot take the square root of a negative value. As a result, the issue as presented cannot be resolved.
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If the following system of equations was written as a matrix equation in the form AX=C , and matrix A was expressed in the form
A=[acbd]
find the value of a-b+c+d
4x + 2y=7
5x-6y=9
matrix A was expressed in the form, a=[a c] [b d] find the value of a - b + c + d. 5x+7y=7 3x-2y=9. That is
a =5, c=7; and b=3, d=-2
a - b + c + d = 5 - 3 + 7 + (- 2) = 7
The given equations are
5x + 7y = 7
3x - 2y = 9
As a matrix equation,
A = [5 7
3 - 2]
X = [x
y]
C = [7
9]
That is
a =5, c=7; and b=3, d=-2
a - b + c + d = 5 - 3 + 7 + (- 2) = 7
A matrix equation is an equation of the structure Hatchet = b, where An is an m × n matrix, b is a vector in R m, and x is a vector whose coefficients x 1, x 2,..., x n are obscure.
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the complete question is :
If the following system of equations was written as a matrix equation in the form AX = C,? and matrix A was expressed in the form, a=[a c] [b d] find the value of a - b + c + d. 5x+7y=7 3x-2y=9.
A new club sent out 266 coupons to boost sales for next year's memberships. They provided 6 times as many to potential members than to existing members. How many coupons did they send to existing members?
the number of coupons sent to potential members was 228.
What is an Equations?
Equations are mathematical statements with two algebraic expressions on either side of an equals (=) sign. It illustrates the equality between the expressions written on the left and right sides. To determine the value of a variable representing an unknown quantity, equations can be solved. A statement is not an equation if there is no "equal to" symbol in it. It will be regarded as an expression.
According to the problem, the total number of coupons sent out was 266. Therefore, we can write an equation based on the information given:
x + 6x = 266
Simplifying and solving for x, we have:
7x = 266
x = 38
Therefore, the number of coupons sent to existing members was 38. To find the number of coupons sent to potential members, we can multiply this number by 6:
6x = 6(38) = 228
Therefore, the number of coupons sent to potential members was 228.
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how long will it take to travel 432 kilometres at an average speed of 96 per hour
2. the ratio of the measure of angle WXZ to the measure of angle ZXY is 11:25 what is the measure of angle ZXY
3. the ratio of the width to the length of a rectangle is 4:5. if the area of the rectangle is 500 square centimeters, what is the length of the rectangle?
2) The measure οf angle ZXY is 125 degrees
3) The length οf the rectangle is 25 cm.
What is the area οf the rectangle?If the rectangle has a length οf 'l' and a width οf 'b', then the area οf the rectangle can be cοmputed by using the fοrmula:
Area = length * width
If the ratiο οf the measure οf angle WXZ tο the measure οf angle ZXY is 11:25, we can write this as:
x:y = 11:25
where x is the measure οf angle WXZ, and y is the measure οf angle ZXY.
We knοw that the sum οf the measures οf these twο angles is 180 degrees, since they are angles in a triangle. Therefοre, we can set up an equatiοn:
x + y = 180
We can sοlve this equatiοn fοr x in terms οf y:
x = 180 - y
Nοw we can substitute this expressiοn fοr x intο the ratiο:
(180 - y) : y = 11 : 25
We can crοss-multiply tο get:
25(180 - y) = 11y
Expanding and simplifying:
4500 - 25y = 11y
36y = 4500
y = 125
Therefοre, the measure οf angle ZXY is 125 degrees.
Let's call the width οf the rectangle w and the length οf the rectangle l. We knοw that the ratiο οf width tο length is 4:5, sο we can write:
w:l = 4:5
We alsο knοw that the area οf the rectangle is 500 square centimeters, sο we can write:
w * l = 500
Nοw we can use the ratiο tο sοlve fοr οne οf the variables in terms οf the οther. Let's sοlve fοr w in terms οf l:
w:l = 4:5
w = (4/5)l
Substitute this expressiοn fοr w intο the area equatiοn:
(4/5)l * l = 500
Simplify:
[tex]4l^2/5 = 500[/tex]
[tex]4l^2 = 2500[/tex]
[tex]l^2 = 625[/tex]
[tex]l = 25[/tex]
Therefοre, the length οf the rectangle is 25 cm. Tο find the width, we can use the ratiο:
w:l = 4:5
w = (4/5)l = (4/5) * 25 = 20
Therefοre, the width οf the rectangle is 20 cm.
Hence,
2) The measure οf angle ZXY is 125 degrees
3) The length οf the rectangle is 25 cm.
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Which equation accurately represents this statement? Select three options. Negative 3 less than 4.9 times a number, x, is the same as 12.8. Negative 3 minus 4.9 x = 12.8 4.9 x minus (negative 3) = 12.8 3 + 4.9 x = 12.8 (4.9 minus 3) x = 12.8 12.8 = 4.9 x + 3
The equation which accurately represents the statement as given are; 4.9x - (-3) = 12.8; 3 + 4.9x = 12.8; and 12.8 = 4.9x + 3.
Which equations represent the statement?As evident from the task content; the given statement is; Negative 3 less than 4.9 times a number, x, is the same as 12.8.
Therefore, we have ;
4.9x - (-3) = 12.8
Or by evaluation;
3 + 4.9x = 12.8
Or by rearrangement;
12.8 = 4.9x + 3.
On this note, the equations which represent the given word phrase are; 4.9x - (-3) = 12.8; 3 + 4.9x = 12.8; and 12.8 = 4.9x + 3.
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I need Help me please
Answer:
they are not similiar
Step-by-step explanation:
..........
what proportion of tickets sold are adult tickets? (image)
Find the rule for this function table Input (x) Output.
2 8, 3 11, 4 14
Answer:
Based on the given input-output pairs (2,8), (3,11), and (4,14), it appears that the rule for this function table is to multiply the input by 3 and then add 2 to get the output. In other words, for an input x, the output is given by the function f(x) = 3x + 2.
Find the volume of the cylinder.
Either enter an exact answer in terms of t or use 3.14 for T
Answer:
the volume is 5 to the next power which is 3 u end up with multiplication which mean u multiply 5 with 3 to get ur units
Select the correct answer from the drop-down menu Central angle BACof circle Ameasures Tradians. Arc BChas a length of 63.48 centimeters. What is the radius of the circle? The radius of circle Ais 110.4 55.2 993.6 207 20 centimeters. Reset Next
If Central angle BAC of circle A measures 23/20 π T radians. Arc BC has a length of 63.48 centimeters. The radius of the circle A is: 55.2cm.
How to find the radius of circle A?In this case, we know the length of arc BC is 63.48 cm and the central angle BAC measures 23/20 π radians.
Now let find the radius of circle A:
So:
63.48π = (23/20 π)/2π × 2πr
63.48π = 1.15 rπ
Divide both side by 1.15
r = 63.48 /1.15
r =55.2 cm
Therefore we can conclude that the radius of circle A is approximately 55.2 centimeters.
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The complete question is:
Central angle BAC of circle A measures 23/20 π T radians. Arc BC has a length of 63.48 centimeters. What is the radius of the circle? The radius of circle A is
You are rolling two dice. Find the probability of rolling two even numbers.
A. 1/8
B. 1/24
C. 1/4
D. 1/18
Answer:
c, 1/4
Step-by-step explanation:
There are six possibilities when rolling a single dice (1, 2, 3, 4, 5, 6). When rolling two dice, there are 36 combinations because 6x6=36. 1/2 of the numbers on each piece of dice are even, so if we multiply 1/2x1/2, we get an answer of 1/4.
The width of a rectangle is the length minus 6 units. The area of the rectangle is 7 square units. What is the length, in units, of the rectangle?
The length of the rectangle is 7 units.
What is length?Length is a measure of distance or size. It is typically measured in meters, centimeters, feet, or inches. Length is a fundamental concept in mathematics, physics, and engineering. It is used to measure the size of an object or the distance between two points. Length is an important factor in the design of many structures, including buildings, bridges, and roads. Length can also be used to describe the duration of time, such as the length of a movie or a song.
To solve this problem, we will use the formula for the area of a rectangle, which is A = lw. We know that the width is the length minus 6 units, so we can rewrite the area equation as: A = (l-6)l. We then solve for l by multiplying both sides of the equation by l and dividing by l-6, giving us: l = A/(l-6). Since the area is 7 square units, we can plug in 7 for A and solve for l: l = 7/(l-6). We then solve for l by solving for the value of l on both sides of the equation, giving us: l = 7/(7-6) = 7/1 = 7. Therefore, the length of the rectangle is 7 units.
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determine whether the following events are mutually exclusive. choosing a jack or a spade out of a standard deck of cards.
The events of choosing a jack or a spade out of a standard deck of cards are not mutually exclusive.
Mutually exclusive events are events that cannot happen at the same time. In other words, if one event occurs, the other cannot.
In a standard deck of cards, there are four jacks and thirteen spades. One of the jacks is also a spade (the jack of spades). Therefore, it is possible to choose a card that is both a jack and a spade at the same time.
Since the events are not mutually exclusive, the probability of choosing a jack or a spade is the sum of the probabilities of each event minus the probability of both events occurring:
P(jack or spade) = P(jack) + P(spade) - P(jack and spade) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13
Therefore, the events of choosing a jack or a spade out of a standard deck of cards are not mutually exclusive.
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Four spheres of radius 10 sit on a flat table. Each sphere touches two of the other spheres, and the centres of the four spheres form a square. A fifth sphere of radius r is sitting on the table; it touches the table at a point directly beneath the centre of the square formed by the four other spheres. If the sphere of radius r touches each of the four other spheres, determine the value of r.
The radius of the fifth sphere or the value of r is 14.14
In order to determine the value of the radius r of the fifth sphere, we must first calculate the side length of the square that the four spheres of radius 10 form.
Since all four spheres are touching, the side length of the square is equal to 20. Thus, the area of the square is equal to 400. Since the fifth sphere is touching each of the four other spheres and the centre of the square formed by them, the fifth sphere is sitting at the midpoint of each side of the square.
Thus, the distance from the centre of the square to the centre of the fifth sphere is 10, which is half the side length of the square. Now, we can calculate the radius of the fifth sphere using the Pythagorean theorem.
The triangle we will use is a right triangle with the side lengths of 10 and 10.
Thus, the hypotenuse (which is the radius of the fifth sphere r) is equal to the square root of 200. Therefore, the radius of the fifth sphere is approximately 14.14.
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The figure shows a rectangle with sides of length 4 and 9 units, respectively. What is the proportion of the shaded area in the rectangle?
The proportion of the shaded area in the rectangle is 0.111.
The shaded area of the rectangle can be calculated by subtracting the area of the unshaded area from the total area of the rectangle. The total area of the rectangle is 36 units2 (Area = length x width). The unshaded area is 32 units2 (Area = 4 x 9). Therefore, the shaded area is 4 units2 (36-32).
The proportion of the shaded area in the rectangle can be calculated by dividing the area of the shaded area by the total area of the rectangle. The proportion of the shaded area is 4/36, or 1/9. This can also be expressed as a decimal, 0.111.
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hw06-MoreProbability: Problem (1 point) (Note that an Ace is considered a face card for this problem) In drawing a single card from a regular deck of 52 cards we have: (a) P( face card or a number card )= (b) P( black and a Queen )= (c) P( black and a face card )= (d) P( Queen and 3 )= (e) P( black or 3 3 )= Note: You can earn partial credit on this problem. You have attempted this problem 0 times. You have unlimited attempts remaining.
(a) The probability of getting a face card or a number card is:P(face card or number card) = (12/52) + (40/52) = 52/52 = 1
(b) The probability of drawing a black and a Queen is:P(black and a Queen) = (26/52) × (2/26) = 2/52 = 1/26
(c) The probability of drawing a black and a face card is:P(black and a face card) = (26/52) × (6/26) = 6/52 = 3/26.
(d) The probability of drawing a Queen and a 3 is 0.
(e) The probability of drawing a black or a 3 is:P(black or 3) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13.
The probability of getting a face card or a number card is:P(face card or number card) = P(face card) + P(number card)There are 12 face cards in a deck of 52 cards. There are 4 Kings, 4 Queens, and 4 Jacks.There are 52 - 12 = 40 cards which are not face cards. There are four 2's, four 3's, four 4's, four 5's, four 6's, four 7's, four 8's, four 9's, and four 10's.Therefore, the probability of getting a face card or a number card is:P(face card or number card) = (12/52) + (40/52) = 52/52 = 1
The probability of drawing a black and a Queen is:P(black and a Queen) = P(black) × P(Queen given black)The probability of drawing a black card is 26/52 since there are 26 black cards in a deck of 52 cards. The probability of drawing a Queen, given that it is a black card is 2/26, since there are two Queens among the 26 black cards.Therefore, the probability of drawing a black and a Queen is:P(black and a Queen) = (26/52) × (2/26) = 2/52 = 1/26
The probability of drawing a black and a face card is:P(black and a face card) = P(black) × P(face card given black)The probability of drawing a black card is 26/52 since there are 26 black cards in a deck of 52 cards. The probability of drawing a face card given that it is a black card is 6/26 since there are 6 face cards among the 26 black cards.Therefore, the probability of drawing a black and a face card is:P(black and a face card) = (26/52) × (6/26) = 6/52 = 3/26
The probability of drawing a Queen and a 3 is:P(Queen and 3) = 0Since there are no 3's among the Queens, the probability of drawing a Queen and a 3 is 0.
The probability of drawing a black or a 3 is:P(black or 3) = P(black) + P(3) - P(black and 3)The probability of drawing a black card is 26/52. The probability of drawing a 3 is 4/52. There are two 3's which are black cards.Therefore, the probability of drawing a black or a 3 is:P(black or 3) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13.
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Help fast!!!
ABC Is reflected to form A’B’C’.
The coordinates of point A are (4, 1), and the coordinates of point B are (6, 3), and the coordinates of point C are (2, 4).
Which reflection results in the transformation of ABC to A’B’C’?
Answer:
The answer to your problem is, B. reflection across the y-axis
Step-by-step explanation:
It is given that △ABC is reflected to form △A'B'C' .It is given that the vertices of triangle ABC are A(4,1), B(6,3) and C(2,4).From the given figure it is clear that vertices of triangle A'B'C' are A'(-4,1), B'(-6,3) and C(-2,4).The relation between pre-image and image is defined by the ruleThe relation between preimage and image is defined by the rule
( x,y ) —> ( -x,y )
Reflection across y-axis represented by the above rule.
It means the △ABC is reflected across the y-axis to form △A'B'C' .
Thus the answer to your problem is, B. reflection across the y-axis
Let (3,7 ) be a point on the terminal side of θ . Find the exact values of cosθ , vac θ , and tanθ.
Answer:
cosθ = 3root58 / 58
cscθ = root58 / 7
tanθ = 7/3
Step-by-step explanation:
Draw a picture of the point. You can make a right triangle. One leg is 3 and one leg is 7. You can find the third side with Pythagorean Theorem or distance formula.
see image.
Then use right triangle trigonometry to find cosθ, cscθ, tanθ.
cosθ is the adjacent side over the hypotenuse.
cscθ is sinθ flipped over.
tanθ is opposite side over adjacent side.
see image
Can someone please help me, and explain?
Answer:
angle 1 = 54° because this angle is a vertical angle of 54°, and vertican angles are always the same.
angle 2 = 126° because straight line = 180°, and there we have 54°, so 180-54=126°
angle 3 = 126° because this angle is vertical angle to 126°
angle 4 = 121° because this angle is alternative interior angle of the sum of the angles 12(54) and 67°
angle 5 = 59° because angle 5 and angle 4 makes a straight line, staring line = 180°, and 180°-angle 4(121°)= 59°
angle 6 = 121° because this anle is vertican angle of 121°
angle 7 = 59 because this angle is vertical angle of 59°
angle 8 = 59° because this angle is corresponding to angle 5 (59°), and corresponding angles are always the same
angle 9 = 54° because this angle is corresponding angle to 54°
angle 10 = 67 because angle 10, 9 and 8 creates straight line, straight line = 180°, angle 9(54) + angle 8(59)=113, and 180°-113°=67°, that means that angle 10 = 67°
angle 11 = 59 because this angle is corresponding to angle 7 (59°)
angle 12 = 54° because this angle is corresponding angle to angle 1 (54°)
that is it :)
Find the image of the following points under the rotation through +90° about the centre origin. (a) A(4, 5) (b) B(-2, 3) (c) C(-3, -5) (d) D(4, -1)
The image of the given points under the rotation through +90° about the center origin include the following;
A' (-5, 4).
B' (-3, -2).
C' (5, -3).
D' (1, 4).
What is a rotation?In Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).
By applying a rotation of 90° counterclockwise to the vertices of each figure, the coordinates of the vertices of the image are as follows:
(x, y) → (-y, x)
A (4, 5) = A' (-5, 4).
B (-2, 3) = B' (-3, -2).
C (-3, -5) = C' (-(-5), 3) = (5, -3)
D (4, -1) = D' (-(-1), 4) = (1, 4)
Read more on rotation here: brainly.com/question/28854313
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Consider the hypothesis test H0:μ1=μ2 against H1:μ1<μ2. Suppose the sample sizes are n1=n2=15, that x1=6.2 and x2=7.8and that s21=4 and s22=6.25. Assume that σ21=σ22 and the data are drawn from normal distributions. Use α=0.05..
a) Test the hypothesis and find the p-value.
b) Explain how the test could be conducted with a confidence interval.
c)What is the power of the test in pair a) if μ1 is three unit less than μ2?
a) The p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that μ1 < μ2.
b) We can construct a (1 - α) confidence interval for the difference between the population means μ1 and μ2 to conduct the test
c) The power of the test is approximately 0.012 when μ1 is three units less than μ2.
a) Test the hypothesis and find the p-value.The test statistic for this hypothesis test is:
t = (x1 - x2) / √(s1²/n1 + s2²/n2)
Plugging in the given values, we get:
t = (6.2 - 7.8) / √(4/15 + 6.25/15) = -2.72
Using a t-distribution with degrees of freedom equal to (15 + 15 - 2) = 28 and a significance level of α = 0.05, we find the critical value to be -1.701. Since the test statistic is less than the critical value, we reject the null hypothesis.
The p-value can be calculated using a t-distribution with degrees of freedom equal to 28:
p-value = P(T < -2.72) = 0.0068
Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis that μ1 < μ2.
b) Explain how the test could be conducted with a confidence interval.To conduct the test using a confidence interval, we can construct a (1 - α) confidence interval for the difference between the population means μ1 and μ2:
(x1 - x2) ± t(1-α/2, n1+n2-2) * √(s1²/n1 + s2²/n2)
Plugging in the given values and using a t-distribution with degrees of freedom equal to 28 and a confidence level of 95% (α = 0.05), we get:
(6.2 - 7.8) ± 2.048 * √(4/15 + 6.25/15)
-2.168 < μ1 - μ2 < -0.232
Since the confidence interval does not include zero, we can conclude that there is evidence to support the alternative hypothesis that μ1 < μ2.
c) What is the power of the test in pair a) if μ1 is three units less than μ2?The power of a hypothesis test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. In this case, the alternative hypothesis is μ1 < μ2, so we want to calculate the probability of rejecting the null hypothesis when μ1 is actually three units less than μ2.
To do this, we need to calculate the test statistic and find the corresponding probability using a t-distribution with degrees of freedom equal to 28 and a significance level of 0.05.
The test statistic for this case is:
t = (x1 - x2 - Δ) / √(s1²/n1 + s2²/n2) = (6.2 - 7.8 - 3) / √(4/15 + 6.25/15) = -4.14
Using a t-distribution with degrees of freedom equal to 28, we find the probability of correctly rejecting the null hypothesis to be:
power = P(T < -t_crit) = P(T < -1.701 - (-4.14)) = P(T > 2.439) ≈ 0.012
Therefore, the power of the test is approximately 0.012 when μ1 is three units less than μ2. This means that the test is not very sensitive to detecting a difference of three units between the two population means.
Learn more about hypothesis test at https://brainly.com/question/14892273
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