Suppose an investigator has data on the amount of shelf space x devoted to display of a particular product and sales revenue y for that product. The investigator may wish to fit a model for which the true regression line passes through (0, 0). The appropriate model is Y ¼ b1x + e. Assume that (x1, y1), . . ., (xn, yn) are observed pairs generated from this model, and derive the least squares estimator of b1. [Hint: Write the sum of squared deviations as a function of b1, a trial value, and use calculus to find the minimizing value of b1.]
Hence, the least square estimator is obtained when you substitute [tex]y_i[/tex] with [tex]Y_i[/tex] is
[tex]\beta _1=\frac{\sum_{i=1}^{n}x_iY_i}{\sum_{i=1}^{n}x_i^2}[/tex]
What is the line of regression?
A regression line indicates a linear relationship between the dependent variables on the y-axis and the independent variables on the x-axis.
Here given that,
Suppose an investigator has data on the amount of shelf space [tex]x[/tex] devoted to display of a particular product and sales revenue [tex]y[/tex] for that product.
The investigator may wish to fit a model for which the true regression line passes through [tex](0, 0)[/tex]. The appropriate model is [tex]Y=\beta x+E[/tex]
Assume that[tex](x_1, y_1), . . ., (x_n, y_n)[/tex] are observed pairs generated from this model, and derive the least squares estimator of [tex]b_1[/tex].
Here, our motive is to minimize the function
[tex]f(b_1)=[/tex][tex]\sum_{i=1}^{n}(y_i-b_1x_i)^2[/tex]
Now,
[tex]f`(b_1)=0\\\\f`(b_1)=2\sum{i=1}^{n}(y_i-b_1x_i).(-x_i)\\\\[/tex]
So, the least square estimate [tex]b_1[/tex] is
[tex]\sum_{i=1}^{n}x_iy_i=b_1\sum_{i=1}^{n}x_i^2[/tex]
Hence, the least square estimator is obtained when you substitute [tex]y_i[/tex] with [tex]Y_i[/tex] is
[tex]\beta _1=\frac{\sum_{i=1}^{n}x_iY_i}{\sum_{i=1}^{n}x_i^2}[/tex].
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The correct solution set for -7x>_56
Answer:
Answer in image
Step-by-step explanation:
A sequence of positive numbers is defined by
given that a1=2
find a2 and a3 leaving your answer in a surd form
The values of a2 and a3 that satisfy the sequence [tex]a_{n+1}[/tex] = [tex]\sqrt{a_{n} ^{2} + 3 }[/tex] are [tex]\sqrt{7}[/tex] and [tex]\sqrt{10}[/tex]
What is a sequence?
A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4 etc. denote the terms of a sequence, then 1,2,3,4 denotes the position of the term.
the rule of the sequence is [tex]a_{n+1}[/tex] = [tex]\sqrt{a_{n} ^{2} + 3 }[/tex] when n = 1
[tex]a_{2}[/tex] = [tex]\sqrt{a_{1} ^{2} + 3 }[/tex] but [tex]a_{1}[/tex] = 2
[tex]a_{2}[/tex] = [tex]\sqrt{(2) ^{2} + 3 }[/tex]
[tex]a_{2}[/tex] = [tex]\sqrt{7}[/tex]
when n = 2
[tex]a_{3}[/tex] = [tex]\sqrt{a_{2} ^{2} + 3 }[/tex]
[tex]a_{3}[/tex] = [tex]\sqrt{a_{2} ^{2} + 3 }[/tex] but [tex]a_{2}[/tex] = [tex]\sqrt{7}[/tex]
[tex]a_{3}[/tex] = [tex]\sqrt{(\sqrt{7}) ^{2} + 3 }[/tex]
[tex]a_{3}[/tex] = [tex]\sqrt{7 + 3}[/tex]
[tex]a_{3}[/tex] = [tex]\sqrt{10}[/tex]
In conclusion [tex]a_{2}[/tex] and [tex]a_{3}[/tex] are [tex]\sqrt{7}[/tex] and [tex]\sqrt{10}[/tex] respectively.
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4. Using the statistics in the following report generated for Community Hospital, calculate (round to two decimal places) the percentage of occupancy for the month of December. Remember to calculate the Total for Patient Care Units (IPSD, Bed Count and Percent of Occupancy).
The percentage of occupancy for the month of December is 86.5%.
What is percentage?
Percentage is a fundamental concept in mathematics that describes a portion or proportion of a whole number. It is expressed as a fraction or ratio of two numbers and is typically represented with a percent symbol (%). This concept is frequently used in business, finance, and other fields to compare and analyze data. For example, a company might use a percentage to track the success of a marketing campaign or to compare the amount of sales between two different stores. Percentage can also be used to calculate discounts, taxes, and other types of fees.
Given: Community Hospital December, 20XX
PCU Inpatient Service Days Bed Count Percentage of Occupancy
⇒ Med/Surg 1876, 70 = (1,876 x 100)/ (70 x 31)
= 187,600/ 2,170 = 86.45 = 86.5%
⇒ ICU 300, 10 (300 x 100)/ (10 x 31)
= 30,000/ 310 = 96.77 = 96.8%
⇒ CCU 240, 8 (240 x 100)/ (8 x 31)
= 24,000/ 248 = 96.77 = 96.8%
⇒ Rehabilitation 300, 12 (300 x 100)/ (12 x 31)
= 30,000/372 = 80.6%
⇒ Burn 75, 5 (75 x 100)/ (5 x 31)
= 7,500/ 155 = 48.38 = 48.4%
⇒ Psychiatry 200, 15 (200 x 100)/(15 x 31)
= 20,000/465 = 43.0%
⇒ Pediatric 870, 30 (870 x 100)/ (30 x 31)
= 87,000/ 930 = 93.5%
⇒ Total for patient care 3861, 150 (3,861 x 100)/ (150 x 31)
= 386,100/ 4,650 = 83.0%
⇒ Nursery 280, 10 (280 x 100)/ (910 x 31)
= 28,000/ 310 = 90.3%
For example, Med/Surg has 1,876 inpatient service days and 70 beds in the month of December (31 days). The percentage of occupancy is therefore the inpatient service days multiplied by 100 and divided by the product of the number of beds (70) and the number of days (31) to get 86.5%
⇒ (1,876 x 100)/ (70 x 31)
= 187,600/ 2,170 = 86.45 = 86.5%
Hence, the percentage is 86.5%.
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on 3 of 10
Which of the numbers below are whole numbers? Check all that apply.
A. 99.6
B. 4538
C. 0.716
1
☐D. 6
E. 207.69
F. 773
A certain lottery requires players to select 6 different numbers, in any order, from 1 to 53 inclusive. How many different sets of 6 numbers can be chosen?
221643611 ways of 6 numbers can be chosen
What is Combination?An arrangement of objects where the order in which the objects are selected does not matter.
Given,
A certain lottery requires players to select 6 different numbers in any order, from 1 to 53 inclusive.
We need to find different sets of 6 numbers can be chosen
Total=53
Numbers=6
The first number can be any of the 53 numbers
The second number can be any of the 53 numbers
The third number can be any of the 53 numbers
The fourth number can be any of the 53 numbers
And so on
Hence, number of selections is:
53×53×53×53×53×53
53⁶
221643611 ways
Hence, 221643611 ways of 6 numbers can be chosen
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Write an equation of the line that passes through the given points.
(−7,0),(−7,5)
Answer:
x= -7
Step-by-step explanation:
the 2 x values are the same so it is an undefined slope so we set this equation x=-7
hopes this helps please mark brainliest
I need help , Polygon ABCDE is reflected across the line x = -2. Graph and label the image A’B’C’D’E’. Is m/_ A = m/_ A’ Explain.
point A' is the contracted point of A. The coordinate of point A is (-4,4) and coordinate of point A' is (-2,4) as per the given question and available figure.
what is dilation?The extension or contraction of image or figure based on a scale factor is called dilation. In this way, an image is transformed from one position to another.
what is scale factor?Scale factor is defined as the ratio of original image and transformed image.
As stated in the question, the polygon ABCDE is contracted and moved to a shape A'B'C'D'E'. The coordinate of point A is (-4,4) and point A' is (-2,4) as shown in figure. Angle A is equal to angle A' that means the shape of polygon was not changed.
hence, A'B'C'D'E' is the reflected image of ABCDE
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choose an equivalent expression for [(2/3)^4 (2/3)^3]^2
By using exponent properties, we will find the equivalent expression:
[(2/3)^4*(2/3)^3]^2 = (2/3)^14
How to find an equivalent expression?There are two exponent properties that we can use here, these are:
a^x*a^y = a^{x + y}
(a^x)^y = a^{x*y}
Now we have the expression:
[(2/3)^4 (2/3)^3]^2
If we use the first property, we can rewrite the expression as:
[(2/3)^4 (2/3)^3]^2 = [(2/3)^{4 + 3}]^2
[(2/3)^{4 + 3}]^2 = [(2/3)^7]^2
Now we can use the second property to get:
[(2/3)^7]^2 = (2/3)^{7*2} = (2/3)^14
That is the equivalent expression.
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Find the volume of a sphere with a radius of 10 ft. Use 3.14 for . You may round to the nearest hundredth when necessary.
A) 4186.67ft3
B) 5712.86ft3
C) 6578.24ft3
D) 2473.58ft3
Volume of sphere is 4186.67 ft³ with a radius 10 ft.
Define volume of sphere.The capacity of a sphere is its volume. It is the area that the sphere occupies. Cubic measurements of a sphere's volume include m3, cm3, in3, etc. The sphere has a round, three-dimensional shape. Its shape is determined by three axes: the x, y, and z axes. A sphere's volume is equal to 4/3 r³, where r is the sphere's radius. Using Archimedes' principle, one may determine volume, a fixed quantity. In accordance with Archimedes, the amount of water displaced if a solid sphere is dropped into a container of water will be equal to the volume of the sphere.
Given
Radius of sphere = 10 ft
Volume of sphere
4/3πr³
4/3 × 3.14 × 10³
4/3 × 3.14 × 1000
4/3 × 3140
4186.67 ft³
Volume of sphere is 4186.67 ft³ with a radius 10 ft.
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The temperature in Austin is 33.5 ◦F and rising at a rate of 2.5 ◦F per hour. The temperature in San Antonio is 37.5 ◦F and rising at a rate of 1.5 ◦F per hour. Write an inequality that shows how long the temperature in Austin will remain colder than that in San Antonio.
The inequality that shows how long the temperature in Austin will remain colder than that in San Antonio is 33.5 + 2.5t < 37.5 + 1.5t and the time is less than 4 hours
How to determine the inequality and the intervals ?From the question, we have the following parameters that can be used in our computation:
Austin
Initial temperature = 33.5 degrees Fahrenheit
Rate of increment = 2.5 degrees Fahrenheit
San Antonio
Initial temperature = 37.5 degrees Fahrenheit
Rate of increment = 1.5 degrees Fahrenheit
The above parameters mean that
Current temperature = Initial temperature + Rate of increment x Number of hours
Austin: A(t) = 33.5 + 2.5t
an Antonio: B(t) = 37.5 + 1.5t
When Austin is colder than San Antonio, we have
Substitute the known values in the above equation, so, we have the following representation
33.5 + 2.5t < 37.5 + 1.5t
Divide both sides by 1
t < 4
Hence, the interval is less than 4 hours
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for a standard normal distribution, find: p(z < 0.74) express the probability as a decimal rounded to 4 decimal places.
Expressing the probability as a decimal rounded to 4 decimal places the value of p ( z < 0.74 ) is 0.7704.
Given :
for a standard normal distribution, find: p(z < 0.74) express the probability as a decimal rounded to 4 decimal places.
Standard normal distribution :
The standard normal distribution, also called the z-distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Probability :
Probability is a branch of mathematics that deals with the occurrence of a random event.
From z - score table :
the test statistic follows the standard normal distribution N(0,1).
the probability p = 0.7704
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What is the mean of the temperatures?
19°C
22°C
30°C
45°C
answers
A.19°C
B.29°C
C.45°C
D.116°C
The mean of the temperatures is 29°C, which corresponds to answer B.
Step-by-step explanation:1. Add all the values together.For convenience reasons we are going to avoid the use of the Celsius symbol (°C) during the calculations.
[tex]19+22+30+45=116[/tex]
2. Count the amount of given values.There are 4 values of temperature.
3. Divide the total of the sum by the amount of given values.[tex]\frac{116}{4}=29[/tex]
4. Conclude.The mean of the temperatures is 29°C, which corresponds to answer B.
A surveyor is 980 feet from the base of the world's tallest fountain at Fountain Hills, Arizona. The angle of elevation to the top of the column of water is 29.7 degrees. His angle measuring device is at the same level as the base of the fountain. Find the height of the column of water to the nearest 10 feet.
To find the height of the column of water in the Fountain Hills fountain, we can use the tangent function and the information given in the problem. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, we can use the angle of elevation to the top of the fountain, the distance from the surveyor to the base of the fountain, and the height of the column of water to form a right triangle.
We can then use the tangent function to find the height of the column of water as follows:
tan(29.7°) = opposite/adjacent
opposite = tan(29.7°) * adjacent
opposite = tan(29.7°) * 980 feet
opposite = 0.56 * 980 feet
opposite = 548.8 feet
Therefore, the height of the column of water in the Fountain Hills fountain is approximately 549 feet. To the nearest 10 feet, this is 550 feet.
- Frank has a gas can with 5 gallons of gas.
If he uses 0.36 gallons each time he cuts the
grass, how many times can he cut the grass?
write an equivalent expression for 1/2(X+8)
x/2 + 4 is an equivalent expression .
What is an expression in maths?
A mathematical expression is a phrase that has a minimum of two numbers or variables and at least one mathematical operation.
This mathematical operation may be addition, subtraction, multiplication, or division. An expression's structure is as follows: Number/variable, Math Operator, Number/Variable is an expression.
An expression in mathematics is made up of a mixture of variables, numbers, and functions (such as addition, subtraction, multiplication or division etc.) It is possible to compare expressions and phrases.
= 1/2(X+8)
= x/2 + 8/2
= x/2 + 4
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Use long division to rewrite the following rational expression.
Answer:
—4—
_x+1_/4x+3
-(4x+4)
———–––
-1
#4x+3/x+1=4-1/x+1
A rectangular box of sand measures 4 inches by 12 inches by 16 inches. When resting on its smallest face, the level of the sand is 4 inches from the top of the box. How many inches from the bottom of the box will the sand level reach when the box is placed on its largest face?
The level of the sand in the box when the box is placed at the largest face will be 3 inches.
What is volume?Volume is the scalar quantity of any object that specified occupied space in 3D.
Volume has units of cube example meter³,cm³, etc.
The volume of the cube = side³.
The volume of sand when placed on a small face (4 x 12) will be as follows,
Volume = 4 x 12 x (16 - 4) = 4 x 12 x 12
When the box was kept at the largest face (16 x 12) the volume was still the same.
Let's say it goes up to h.
h x 16 x 12 = 4 x 12 x 12
h = 12/4 = 3 inches
Hence "The level of the sand in the box when the box is placed at the largest face will be 3 inches".
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A hand soap container proudly boasts 30% more for the same price as their regular
container. The larger bottle has 9.75 ounces of soap. What is the original amount of
soap before the 30% increase?
Answer:
6.825 ounces
Step-by-step explanation:
9.75 x 0.3 = 2.925
9.75 - 2.925 = 6.825
Chloe is older than her brother, Michael, and she never lets him forget it! When Michael is y
years old, Chloe is y + 2.5 years old. Right now Michael is 9.5 years old.
How old is Chloe?
Write your answer as a whole number or decimal.
Answer: Chloe is 12 years old
y + 2.5
y = 9.5
9.5 + 2.5 = 12
If f(x)=x+7 and g(x)=1/x-13}, what is the domain of (f g)(x) ?
{x l x ≠6}
{x l x ≠-6}
{x l x ≠ 13}
{x l x ≠ 13}
domain of (f.g)(x) is {x l x ≠ 13} i.e option (c) where f(x) = x+7 and g(x) = 1/(x-13)
What is domain of function?Domain of a function is defined as all those values of x that can be substituted in function for which function is defined.
Given: f(x) = x+7
g(x) = 1/(x-13)
First we need to find (f.g)(x) i.e multiplication of f(x) and g(x)
(f.g)(x) = (x+7) ×1/(x-13) = [tex]\frac{x+7}{x-13}[/tex]
Domain of function is values of x over which function is defined
For (f.g)(x) it is defined at all values of x except when x-13=0 as at this point denominator will be zero and function will become infinite that is not defined.
So, x - 13 ≠ 0
Adding 13 on both sides, we get
x ≠ 13
So domain of (f.g)(x) is x∈R - {13} i.e {x | x≠ 13}
Hence, option (c) is correct answer.
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Solving
The product of 5 and the difference
between a number and 7 is 75. What is
the number?
The number whose product of 5 and the difference between a number and 7 is 75 is, 22.
What is a numerical expression?A numerical expression is a mathematical statement written in the form of numbers and unknown variables. We can form numerical expressions from statements.
The given statement is The product of 5 and the difference
between a number and 7 is 75.
Let, The unknown number be 'n'.
∴ 5(n - 7) = 75.
Now, 5 will move from the numerator to the denominator.
n - 7 = 75/5.
n - 7 = 15.
Now, we'll move the constant to the other side such that the variable and the constants are on different sides for our further calculation.
n = 15 + 7.
n = 22.
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f(x)=3x square + x f(-5)
Answer:
If you need the answer as f by simplifying both sides of the equation and then isolating the variable, the answer would be:
[tex]xf=\frac{3x^2}{5}-\frac{f(x)}{5}[/tex]
Or, if you need to evaluate the function itself, the answer would be:
[tex]f(x)=3x^2-5xf[/tex]
An unbiased die is tossed. Find the probability that the number
obtained is
(c) Greater than 6
(a) An event number (b) Less than 4
Step-by-step explanation:
greater than 6=0 cause no number on a dice is greater than six
an even number=1/2
less than 4=1/2
Corey has $89 go to shopping T shirt cost $12.50 each corey wants to buy some T shirt and have at least $14 for lunch
Answer:
6 shirts you can buy and still have $14 dollars for lunch.
Step-by-step explanation:
So let's do 89-14=75. So since we just took out the cost of the lunch it should be a lot easier to solve now.
s=shirts
So 75=12.50s
75/12.50=6
So Corey can buy 6 shirts.
I hope this helps!!!
For the following data set, calculate the percentage of data points that fall within one standard deviation of the mean and compare the result to the expected percentage of a normal distribution.
{8, 12, 27, 32, 45, 57, 61, 73, 82, 94}
A minimum of 68% is the expected percentage in a normal distribution and 60% is lower than what is expected.
What is the standard deviation?
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while a high standard deviation indicates that the values are spread out over a wider range.
The given data is:
{8, 12, 27, 32, 45, 57, 61, 73, 82, 94}
Now find the mean of the given data would be
Mean = sum of all points/n
= 491/10
= 49.1
Standard deviation(s) = [tex]\sqrt{\frac{\sum(x - \bar x)^2}{n-1}}[/tex]
Standard deviation(s) = 29.31988
Considering one standard deviation from the mean, the lower and upper limits are given by the following
[49.1-29.31988, 49.1+29.31988}
=[19.78012, 78.41988]
We can see that only 6 {27,32,45,57,61,73} out of the 10 points, which is 60%, confine to the interval of one standard deviation from the mean.
Hence, a minimum of 68% is the expected percentage in a normal distribution and 60% is lower than what is expected.
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Simplify and state the restrictions on the domain:
Answer:
Step-by-step explanation:
First factor the left expression to get
Then cancel out .[tex]\frac{(x+3)(x+2)}{(x+4)(x+2)}*\frac{x+4}{2x}\\ \\\frac{(x+3)}{2x}[/tex]
The x+2 cancel's as well as the x+4's.
The box plot below represents some data set. What is the interquartile range (IQR) of the data? 0, 25, 50, 75, 100
The box plot indicates that the interquartile range (IQR) of the data 0, 25, 50, 75, and 100 is 100.
What is the interquartile range?Find the median (middle value) of the bottom half and upper half of the data before calculating the interquartile range (IQR). Quartile 1 (Q1) and Quartile 3 are these values (Q3). The IQR represents the variation between Q3 and Q1. We use the median to summarize a typical value rather than the mean when a data collection contains outliers or extreme values. The interquartile range, which is the difference between the first and third quartiles, is a statistic that is frequently used to summarize variability in data sets with outliers.
Here,
The first and third quartiles would need to be located initially. Then, to obtain the IQR, you would divide the third quartile by the first.
IQR, in my opinion, is 100.
The interquartile range (IQR) of the data 0, 25, 50, 75, 100 will be 100 by the given box plot.
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The Sugar Sweet company is going to transport its sugar to the market. It will cost $7500 to rent trucks, and it will cost an additional $175 for each ton of sugar transported.
Let C represent the total cost ( in dollars) and let s represent the amount of sugar (in tons) transported. Write an equation to find the total cost to transport 12 tons of sugar.
The total cost to transport 12 tons of sugar is $9600.
Whet is the equation to find the total cost to transport 12 tons of sugar?From the information, Sugar Sweet company is going to transport its sugar to the market. It will cost $7500 to rent trucks, and it will cost an additional $175 for each ton of sugar transported.
Let C represent the total cost ( in dollars)
Let s represent the amount of sugar.
The equation to find the total cost to transport 12 tons of sugar will be:
C = 7500 + (175 × s)
C = 7500 + 175s
C = 7500 + (175 × 12)
C = 7500 + 2100
C = 9600
The cost is $9600.
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Twenty five out of 40 student athletes received over 3.0 gpa or better this semester. What percent of student athletes received under a 3.0 gpa?
A. 37.5
B. 35.7
C. 50
D. 39.5
Answer: A. 37.5%
Step-by-step explanation:
Technically, you can't even solve this, because you don't know how many athletes got exactly a 3.0 GPA.
Ignoring this major oversight, let's just say that no one received a 3.0 GPA. In that case, there would be 40 - 25 = 15 athletes who received a sub-3.0 GPA.
Therefore, the percentage of student athletes who received a sub-3.0 GPA is 15/40 = 3/8 = 37.5%.