The point of intersection will be the x, and y coordinates where the two lines cross.
To determine the point(s) of intersection algebraically, we need to find the solutions of the system of equations:
y = x² - x + 1
We can re-arrange the equation to solve for x:
x² - x + 1 - y = 0
We can then use the quadratic formula to find the solutions:
x = (1 ± √(1 - 4(1)(-y)))/2
x = (1 ± √(1 + 4y))/2
Now we can substitute values for y to find the x-coordinates of the points of intersection.
To verify the results, we can use a graphing utility to graph both equations and visually see where they intersect.
The point of intersection will be the x, and y coordinates where the two lines cross.
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Which represents the reflection of f(x) = StartRoot x EndRoot over the y-axis?
Answer:
I tink D?
Step-by-step explanation:
The function after the reflection over y-axis is f' ( x ) = √( -x )
What is Reflection?Reflection is a type of transformation that flips a shape along a line of reflection, also known as a mirror line, such that each point is at the same distance from the mirror line as its mirrored point. The line of reflection is the line that a figure is reflected over. If a point is on the line of reflection then the image is the same as the pre-image. Images are always congruent to pre-images.
The reflection of point (x, y) across the x-axis is (x, -y). When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is taken to be the additive inverse. The reflection of point (x, y) across the y-axis is (-x, y).
Given data ,
Let the function be represented as f ( x )
Now , the value of f ( x ) is
f ( x ) = √x
Now , when you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is taken to be the additive inverse. The reflection of point (x, y) across the y-axis is (-x, y)
So , the reflected function is f' ( x ) = √( -x ) and the graph is plotted
Hence , the reflected function is f' ( x ) = √( -x )
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A fashion store is planning its end–of–season sale. It sells 40 of its signature jeans every week at $175 per pair. Last year, the store's sales increased to 60 pairs per week when the price was dropped by $25. By what amount should the store discount the price this year to maximize its revenue?
A. $112.50
B. $103.50
C. $62.50
D. $71.50
Answer:
A
Step-by-step Explanation:
Every $1 decrease in price corresponds to a 0.8 increase in the number of pairs of jeans sold.
So, if [tex]x[/tex] pairs of jeans are sold, the revenue in dollars is [tex](175-x)(40+0.8x)[/tex].
The roots of this function are [tex]x=175[/tex] and [tex]x=-50[/tex]. The maximum is achieved halfway between these roots at [tex]x=62.5[/tex].
When 62.5 pairs of jeans are sold, the price is $112.50.
The store should discount the price to $112.50 to maximize its revenue
What is Discount?Discount is a deduction from the usual cost of something, typically given for prompt or advance payment or to a special category of buyers.
From the above question, every $1 decrease in price corresponds to a 0.8 increase in the number of pairs of jeans sold.
So, if pairs of jeans are sold, the revenue in dollars is (175 - x)(40 + 0.8x)
The roots of this function are x = 175 and x = -50. The maximum is achieved halfway between these roots at x = 62.50.
Therefore, When 62.5 pairs of jeans are sold, the price is $112.50.
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When adding two vectors, I should do the following: O Pick an angle and just go with it. O Write each vector in component form and add the components separately. O Combine all the vector components into one number and add them. O Calculate the magnitude of each vector and add them.
Write each vector in component form and add the components separately. This new vector will represent the sum of the two original vectors. This is the most straightforward way to add two vectors.
1. Write each vector in component form. This means breaking down each vector into its x- and y-components.
2. Add the x-components of each vector together and the y-components of each vector together.
3. The result will be a single vector with both x- and y-components.
To add two vectors, I should write each vector in component form, breaking them down into their x and y components. Then, I should add the components together, resulting in a new vector with components that are the sum of the two original vectors' components. This new vector will represent the sum of the two original vectors. This is the most way to add two vectors.
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how might the additional integration techniques learned in chapter 6 sections 6.1-6.2 be useful to you in the real world? briefly state.
Integration techniques from chapter 6, such as the power rule, substitution rule, and integration by parts, can be used to solve real-world problems involving calculus.
For example, the power rule can be used to calculate the area of an object by integrating its equation. The substitution rule can be used to calculate the volume of a solid by substituting the equation of its surface. Finally, integration by parts can be used to calculate the arc length of a function given its equation. All these techniques provide a way to calculate the area, volume, and arc length of various objects given their equations, allowing us to apply calculus to real-world situations. Integration techniques from chapter 6, such as the power rule, substitution rule, and integration by parts, can be used to solve real-world problems involving calculus.
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In the command window in MATLAB generate a random 6 x 6 matrix with integer entries using the command G = round(10* rand(6)) For each of the following questions below, execute the given commands and match the result with the best explanation of how MATLAB got the answer (some explanations will not match). G(:,:) G(:) G(7,6) ✓ [Choose ] maximum entry in each column of G reshapes all elements of G into a single column vector assigns the value 4 to all entries of G that are greater than 4 reshapes the matrix G into a single row vector returns the matrix G in its original form maximum entry in each row of G returns an indexing error entries of G that are greater than 4 sum of the entries in each column of G sum of the entries in each row max(G) sum(G) [ Choose ] G(G>4) [Choose ] G(G>4)=4 [Choose]
G(:,:) returns the matrix G in its original form. This command takes all elements of G and reshapes them into a single row vector. This means that all elements of G are included in the output, and the output is in the same shape as the original matrix.
G(:) reshapes all elements of G into a single column vector. This command takes all elements of G and places them into a single column. This means that all elements of G are included in the output, and the output is in a column vector.
G(7,6) returns an indexing error. This command attempts to access an element outside of the range of the matrix G. Since G is a 6 x 6 matrix, there is no element at the index (7,6).
max(G) returns the maximum entry in each column of G. This command takes the maximum value from each column and returns them as a single row vector. This means that the output contains the maximum value from each column and the output is in a single row vector.
sum(G) returns the sum of the entries in each column of G. This command takes the sum of all entries in each column and returns them as a single row.
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In which of the following expressions does the number 4 fill in the blank so that the equation is true? Select all that apply. A) ___(6 + 8) = 24 + 32 B) 9(3 + ___) = 27 + 36 C) ___(7 + 6) = 35 + 30 D) 2(2 + 9) = ___ + 18
: Pretty Sure u need the question to number 4-
Step-by-step explanation:
determine whether the quantitative variable is discrete or continuous amount of water in a dogs bowl
The quantitative variable is continuous for any given value of the amount of water in a dog's bowl.
In order to give an answer to this question, we need to recall the definition of continuous and discrete variables. Then according to the definition, we get the answer if the quantitative variable is discrete or continuous for the given amount of water in a dog's bowl.
As we understand that a discrete variable is a variable whose value is fetched by counting. A discrete random variable holds countable possible values. A random variable is one whose value is numerical. On the other hand, a continuous variable is a variable whose value is fetched by measuring.
The amount of water in a dog's bowl has any value. So the amount of water in a dog's bowl is a continuous variable.
Hence, the quantitative variable is continuous for any given amount of water in a dog's bowl.
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In a basketball game,Benito score 3 points less than twice the number of points that Carnell scored. Benito scored 13 points how many points did Carnell score
In a basketball game,Benito score 3 points less than twice the number of points that Carnell scored. Benito scored 13 points therefore the number of points Carnell scored was 16 points.
To get Carnell pointsBenito scored 13 points which means that the number of points Carnell scored was:
8 multiplied by 2 = 16
When we subtract 3 from 16 we get 13 which is Benito score mentioned in the question thereby making it the correct choice.
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Two angles are supplementary. If the mZA is five times the sum of the mZB plus 7.2°, what is mZB?
Due 01.
The value of angle mZB is 28. 8 degrees
How to determine the anglesIt is important to note that supplementary angles are described as pair of angles that sum up to angles on a straight line.
Also, angles on a straight is equal or equivalent to 180 degrees.
From the information given, we have that;
mZB+ mZA = 180mZA = 5mZB + 7. 2 degreesNow, susbtitute the values, we get;
mZB + mZA = 180
5mZB + 7. 2 + mZB = 180
collect like terms, we get;
5mZB + mZB = 180 - 7.2
Add or subtract the like terms
6mZB = 172.8
Now, divide both sides by the coefficient of mZB, we get;
mZB = 172.8/6
Divide the values
mZB = 28. 8 degrees
Hence, the value is 28. 8 degrees
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segment from ( − 8 , 10 )to ( − 8 , − 5 )that partitions the segment into a ratio of 2 to 1?
The coordinate of the point that partitions the given line in the ratio 2:1 is; (-8, 0)
How to partition line segments?If the coordinates are given as;
P (x₁ , y₁) and Q (x₂ , y₂) and the partition of the line segment is divided internally in the ratio m : n, then its coordinate ( x, y) given by :
( x ,y ) = [(mx₂ + nx₁)/ ( m + n), (my₂ + ny₁)/ ( m + n)]
We are given the coordinates as ( − 8 , 10 )to ( − 8 , − 5 ) and if it is partitioned in the ratio 2: 1, the coordinates of the points of partition are;
(x, y) = [(2(-8) + 1(-8))/ (2 + 1), (2(-5) + 1(10))/ (2 + 1)]
(x, y) = (-24/3), (0/3)
(x, y) = (-8, 0)
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4. Explain how to use division to find an
equivalent fraction for 20/16
Answer:
5/4
Step-by-step explanation:
divide 20 and 16 by 4
20/4=5
16/4=4
The sum of two numbers is twenty-five. Using x to represent the smaller of the two numbers, translate "the difference between four more than the larger number and twice the smaller number" into a variable expression. Then simplify.
Answer:
If the sum of the two numbers is 25, we can represent the larger number as x + (smaller number)
The difference between four more than the larger number and twice the smaller number is:
(x + 4) - 2(x) = x + 4 - 2x = -x + 4
Now we can simplify the expression:
-x + 4
That is the variable expression for the difference between four more than the larger number and twice the smaller number.
Step-by-step explanation:
Bicycle Speed
A bicycle is traveling at 19 miles per hour.
How many feet will it cover in 15 seconds?
Round your answer to the nearest tenth of a foot.
The distance covered by the bicycle at a speed of 19 miles per hour is 418 feet
What is speed?The rate of change in an object's position with regard to a scalar quantity of reference and time is what is meant by speed, according to its definition. Although it may appear difficult, speed is essentially speeding in a particular direction. Due to the fact that it is a scalar quantity, speed must be defined in terms of both magnitude (speed) and direction. It has a meter per second SI unit (ms-1). A body is considered to be accelerating if there is a change in the magnitude or the direction of its speed.
we know that 1 mile is 5280 feets
so, the speed of 19 miles per hour will be 100320 feet in 3600 seconds
we know
distance = speed × time
distance = 100320/ 3600 ×15
distance = 418 feet
hence the distance covered is 418 feet.
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Solve the equation for all real values of x.
3 tan²x =√3 tanx
The solution of the equation for all real values of x is πk, π/6.
How to solve the equation for all real values of x?Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, specifically right triangles.
3 tan²x = √3 tan x
Divide both sides by 3tan x:
tan x = (√3)/3
x = tan⁻¹((√3)/3)
x = 30°
x = π/6 radian
Note: multiply 30° by π/180 to convert to radian. 180° = π radian
Since π/6 (i.e. 30°) falls between 0 and 180°.
Thus, the solution of the equation for all real values of x is πk, π/6. We can choose πk, π/6.
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let a and be positive integers satisfying (ab+1)/(a+b) < 3/2. The maximum possible value of (a^3b^3+1)/(a^3+b^3) is p/q. where p and q are relatively prime positive integers. Find p+q.
p/q = (2x^3+1)/2x^3, which implies that p = 2x^3+1 and q = 2x^3. p + q = 2x^3+1 + 2x^3 = 3x^3.
Let a and b be positive integers satisfying (ab+1)/(a+b) < 3/2. The maximum possible value of (a^3b^3+1)/(a^3+b^3) is given by the equation p/q, where p and q are relatively prime positive integers. We can expand the numerator and denominator as follows: a^3b^3+1 = a^3b^3 + a^3 + b^3 and a^3+b^3 = a^3 + a^2b + ab^2 + b^3. Therefore, the expression is equal to (a^2b^2+a^2+ab+1)/(a^2b+ab^2+1). Since (ab+1)/(a+b) < 3/2, we have (ab+1)/(a+b) < 5/3 and (a^2b^2+a^2+ab+1)/(a^2b+ab^2+1) < 5/3.To find the sum of p and q, we can use the fact that the maximum value occurs when a and b are equal. Thus, setting a = b = x, we have (x^3x^3+1)/(x^3+x^3) = (x^6+1)/2x^3 = (2x^3+1)/2x^3. Thus, p/q = (2x^3+1)/2x^3, which implies that p = 2x^3+1 and q = 2x^3. Therefore, p + q = 2x^3+1 + 2x^3 = 3x^3.
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3x+12y=-8 show ur work
Here is the attached image for the answer. If you write it in in slope-intercept form, y=mx+b. You will get the answer below in the attached image.
Find the surface area of each pyramid. Round to the nearest tenth if necessary.
triangular pyramid: base side lengths, 6 yd; base height: 5.2 yd; slant height, 11.7 yd
a.
136.5 yd2
b.
90.7 yd2
c.
226.2 yd2
d.
120.9 yd2
The total surface area of the triangular pyramid will be 106.86 yd².
What is the surface area of triangular pyramid?The formula to calculate the total surface area of a triangular pyramid is -
T.S.A. = [tex]\frac{1}{2}[/tex](a × b) + [tex]\frac{3}{2}[/tex](b × s).
Given is a triangular pyramid with dimensions -
base side lengths : 6 yd base height : 5.2 ydslant height : 11.7 ydThe total surface area of the triangular prism will be -
T.S.A. = [tex]\frac{1}{2}[/tex](a × b) + [tex]\frac{3}{2}[/tex](b × s).
T.S.A. = 1/2(5.2 x 6) + 3/2(5.2 x 11.7)
T.S.A. = 31.2/2 + 182.52/2
T.S.A. = 15.6 + 91.26
T.S.A. = 106.86 yd²
Therefore, the total surface area of the triangular pyramid will be 106.86 yd².
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Qasim is deciding between two truck rental companies. Company A charges an initial fee of $15 for the rental plus $1 per mile driven. Company B charges an initial fee of $45 for the rental plus $0.50 per mile driven. Graph each function and determine which company would be cheaper if Qasim needs to drive 80 miles with the rented truck.
Equation for the cost of company A: f(x) = 15 + x
Equation for the cost of company B: g(x) = 45 + x/2
For 80 miles, company B is cheaper than company A.
What is linear equation?A linear equation is an algebraic equation where each term has an exponent of 1 and when this equation is graphed, it always results in a straight line.
Given,
Company A charges an initial fee of $15 for the rental plus $1 per mile driven
f(x) = 15 + x
Company B charges an initial fee of $45 for the rental plus $0.50 per mile driven
g(x) = 45 + 0.5x
g(x) = 45 + x/2
x is number of miles
For 80 miles company A charges
f(80) = 15 + 80
= 95
For 80 miles company B charges
g(80) = 45 + 80/2
= 85
By the above calculation, Company B is cheaper than Company A for 80 miles drive.
Hence,
f(x) = 15 + x is Equation for the cost of company A
g(x) = 45 + x/2 is Equation for the cost of company B
Company B is cheaper than company A for drive of 80 miles.
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TRUE OR FALSE a survey of high school students was conducted to see if students who take advanced classes are offered more scholarship money than students who do not take advanced classes.
True, a survey of high school students was conducted to see if students who take advanced classes are offered more scholarship money than students who do not take advanced classes.
Science is used in statistics to gather and arrange data into patterns or trends. A statistical research collects data about a population or group of people and attempts to generalize the findings for use by different people.
The observational study is one type of statistical study. These studies frequently occur early in a given area of health study since they may swiftly and affordably gather a lot of general information about a community.
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Consider the expression given below. Simplify this expression into the form A/B and place the appropriate expressions into the boxes shown below: V4219 Answer: 4x19 = AB, where A= 2x9 âand B = 4.19 Ð¥
the simplified expression is 2x9/4.19, which can be written as A/B, where A = 2x9 and B = 4.19. Therefore, the value of A is 18 and the value of B is 4.
To simplify the expression V4219, first divide it into two separate terms by breaking the number 4219 into 4 and 2119.
Now, we need to simplify the terms 4 and 2119 separately.
To simplify the term 4, we need to divide it by 4, which gives us 1.
To simplify the term 2119, we need to divide it by 19, which gives us 2x9.
Hence, the simplified expression is 2x9/4.19, which can be written as A/B, where A = 2x9 and B = 4.19. Therefore, the value of A is 18 and the value of B is 4.
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The Colossus Ferris wheel debuted at the 1984 New Orleans World's Fair. The ride is 180 ft tall, and passengers board the ride at an initial height of 15 ft above the ground. The height above ground, h, of a passenger on the ride is a periodic function of time, t. The graph displays the height above ground of the last passenger to board over the course of the 15 min.
Sine function model: h=-82.5 cos 3pi(t)+97.5 where h is the height of the passenger above the ground measured in feet and t is the time of operation of the ride in minutes
If the Colossus Ferris wheel debuted at the 1984 New Orleans World's Fair and the ride is 180 ft tall, and passengers board the ride at an initial height of 15 ft above the ground. The period of the sine function model is: 2/3 minutes.
How to find the period of the sine function model?Given equation:
h=-82.5 cos 3pi(t)+97.5
Where:
h= height of the passenger above the ground measured in feet
t = time of operation of the ride in minutes
So,
h=-82.5× cos [2 × π ×t /T) +97.5
2 ×π / T = 3 × π
T = 2/3 minutes
Therefore the period of the sine function model is: 2/3 minutes.
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The graph below is an example of which type of function?
O quadratic
O linear
O cubic
The graph below is an example of graph function is cubic.
What is graph function?We must identify the kind of the graph after receiving it.
A parabola represents the graph of a quadratic function. As a result, the presented graph is not a quadratic function graph.
A straight line is represented by a linear function. The presented graph is not a graph of a linear function as a result.
A square root function's graph resembles a half parabola. Therefore, it can't be the square root function graph.
A cubic function's final behaviour is "opposite at both ends." We can see that the provided graph satisfies this requirement.
As a result, the shown graph illustrates a cubic function.
The right graph is C.
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The table represents the number of pairs of shoes bought when customers bought a dress from Famous Formalwear,
Which value represents the correlation coefficient of this data set?
O 0.18
O 0.32
O 0.43
O 0.99
Number of dresses purchased
1
4
3
1
4
6
2
6
2
Number of Pairs of Shoes Purchased
2
3
3
1
2
2
1
2
1
The correlation coefficient for this data-set is given as follows:
r = 0.43.
How to obtain the correlation coefficient for the data-set?The coefficient is obtained inserting the points in a data-set in a correlation coefficient calculator.
The points for the data-set in this problem are given as follows:
(1,2), (4,3), (3,4), (1,1), (4,2), (6,2), (2,1), (6,2), (2,1).
Inserting these points into a correlation coefficient calculator, the correlation coefficient for this data-set is given as follows:
r = 0.43.
Meaning that the correct option is given by the third option.
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Write an expression for "5 decreased by 2.
Answer:
5 (less than sign) 2
Step-by-step explanation:
Answer:
5n-20
Step-by-step explanation:
Product means multiply and so the product of 5 and n is 5n. To decrease this value by 20 , means to subtract 20 from it hence: 5n - 20 is algebraic expression for the statement
Joe's Coffee Shop makes a blend that is a mixture of two types of coffee type a coffee cost Joe $4.65 per pound and type B coffee cost $5.95 per pound this month Joe made $150 of the blend for a total cost of $89.30 how many pounds of type B coffee did he use
Using simultaneous equations, the number of pounds of Type B coffee that Joe used for the blend was 94 pounds.
What are simultaneous equations?Simultaneous equations are two or more equations solved concurrently or simultaneously.
Simultaneous equations are also known as a system of equations.
Cost per pound of Type A coffee blend = $4.65
Cost per pound of Type B coffee blend = $5.95
The total pounds of the blend made for the month = 150 pounds
The total cost of 150 pounds = $819.70
Let the number of Type A coffee pounds = a and Type B coffee pounds = b.
Equations:4.65a + 5.95b = 819.7 ... Equation 1
a + b = 150 ... Equation 2
Multiply Equation 2 by 5.95:
5.95a + 5.95b = 892.5 ... Equation 3
Subtract Equation 1 from Equation 3:
5.95a + 5.95b = 892.5
-
4.65a + 5.95b = 819.7
1.3a = 72.8
a = 56 pounds
b = 150 - a
= 150 - 56
= 94 pounds
Thus, we can conclude that Joe's Coffee Shop used 94 pounds of Type B coffee this month.
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Question Completion:This month Joe made 150 pounds of the blend for a total cost of $819.70 how many pounds of type B coffee did he use?
Please answer this question quickly for me
The reversal of the statement would give the converse statement and this is done in option A.
What is a converse statement?We know that in logic a statement would need to have a premise and a conclusion. The premise is what would naturally lead to the conclusion. The statement is true when the premise does lead to the conclusion.
We know that the converse of a statement can be said to be obtained when we switch the positions of the premise and the conclusion in the statement. The conclusion becomes the premise and the premise becomes the conclusion.
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if you were in a spaceship accelerating at 1g how long would it take you to reach the speed of light
To reach the speed of light, the spaceship will take 3.1 × 10⁷ seconds. The result is obtained by using the formula for acceleration.
What is acceleration?Acceleration is the rate of change in velocity of an object with respect to time. It can be expressed as
[tex]a = \frac{\Delta v}{\Delta t}[/tex]
Where
a = accelerationΔv = change in velocityΔt = change in timeIf we were in a spaceship with an acceleration of g, determine the time we reach the speed of light!
We have
Acceleration, a = 1g = 9.8 m/s²Speed of light, c = 3 × 10⁸ m/sTo find the time so that we can reach the speed of light, we use the formula for acceleration.
[tex]a = \frac{\Delta v}{\Delta t}[/tex]
[tex]g = \frac{c - 0}{\Delta t}[/tex]
[tex]9.8 = \frac{3 \times 10^{8}}{\Delta t}[/tex]
[tex]\Delta t = \frac{3 \times 10^{8}}{9.8}[/tex]
Δt = 0.306 × 10⁸
Δt = 3.1 × 10⁷ s
Hence, the spaceship will take 3.1 × 10⁷ seconds to reach the speed of light.
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When rounded to the thousands place, which of the numbers below round to 812,000? Select all that apply... A) 811,895 B) 812,345 C) 811,398 D) 811,501 E) 812,499
The correct option D) 811,501, is the calculated number the numbers below, if rounded to the thousandths place, equal 812,000.
Define the term rounding off the number?Rounding Decimal Numbers:
Choose the final digit you want to keep.
If the following digit is less than 5, leave it alone (this is known as rounding down).But if the following digit is 5 or higher, increase it by 1. (this is known as rounding up).Decimal rounding:
Determine the number that will remain after we are done.
When rounding to tenths, one number is left just after decimal point.When rounding to hundredths, two integers are left after the decimal point, etc.For the stated question:
To get the number 812,000 when rounded to the thousands place,
number used is 811,501, as all other number are either larger than or smaller than 812,000.
Thus, option D) 811,501, is the calculated number the numbers below, if rounded to the thousandths place, equal 812,000.
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William and his children went into a restaurant that sells hamburgers for $8 each and drinks for $2.50 each. William has $95 to spend and must buy no less than 18 hamburgers and drinks altogether. If xx represents the number of hamburgers purchased and yy represents the number of drinks purchased, write and solve a system of inequalities graphically and determine one possible solution.
Answer: We can set up the following system of inequalities to represent the situation:
8x + 2.5y <= 95 (the total cost of the hamburgers and drinks must be less than or equal to $95)
x >= 18-y (William must buy no less than 18 hamburgers and drinks altogether)
x >= 0, y >= 0 (non-negative values for the number of hamburgers and drinks purchased)
To graph the system of inequalities, we can start by graphing the inequality 8x + 2.5y <= 95, which represents a boundary line in the xy-plane. Since it's a less or equal sign, we will shade the area that is below the line.
The second inequality represents a boundary line which is the line x = 18-y. Since x must be greater than or equal to 18-y, we will shade the area that is on or above the line.
The third inequality is a non-negative constraint for both x and y, so we will graph the x and y axis and shade the area that is above them.
After combining all the shaded areas, we'll get a feasible solution region that represents possible values of x and y that satisfies the system of inequalities.
One possible solution to the system of inequalities is x=24 and y=6, which is a point in the feasible region that satisfies all the inequalities.
It's worth noting that there could be other solutions as well, depending on the number of hamburgers and drinks William wants to buy.
Step-by-step explanation:
the manager of a large company that sells pet supplies online wants to increase sales by encouraging repeat purchases. the manager believes that if past customers are offered $10 off their next purchase, more than 40 percent of them will place an order. to investigate the belief, 90 customers who placed an order in the past year are selected at random. each of the selected customers is sent an e-mail with a coupon for $10 off the next purchase if the order is placed within 30 days. of those who receive the coupon, 38 place an order.
At the significance level of a = 0.05, there is no strong statistical support for the manager's assertion.
Accepting it would be a type II error because the null hypothesis might not be true.
The 'null' and 'alternative' hypotheses should be
p 0.4 for H0 and p > 0.4 for Ha
q= 1-p= 1-0.4= 0.6
The alpha level for significance is 0.05.
For one-tailed tests, the crucial area is Z> ± 1.645
The sample proportion is p^= x/n= 38/90=0.42222
Using the z statistic
z= p^- p/ √pq
z= 0.422-0.4/ √0.4*0.6
z= 0.04536
We are unable to reject the null hypothesis since the calculated value of z=0.04536 does not fall within the crucial zone Z> 1.645.
The manager's assertion is not sufficiently supported by the available data.
When we reject the actual null hypothesis, we commit a type I mistake.
When we accept the incorrect null hypothesis, we make type II errors.
Accepting it would be a type II error because the null hypothesis might not be true.
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