Total number of different chords of the circle passing through point P with integer lengths is equal to 7.
As given in the question,
Radius of the circle = 15units
Longest chord of the circle 'diameter' = 2 × 15
= 30units
Point 'P' is 9 units away from the circle.
Consider shortest distance( perpendicular) of point P for shortest chord.
Half length of the chord , shortest distance and radius form right triangle.
(1/2) length of the chord = √15² - 9²
= √225 - 81
= 12units
Length of the chord = 2(12)
= 24 units
Different Chords with integer length = 24, 25, 26, 27, 28 , 29 , 30
Total number of different chords = 7
total number of chords = 6 original + 6 mirror image + 1 longest chord
= 13
Therefore, total chords with different integer length passing through the point P of the circle is 7.
The complete question is :
A point P is at the 9 unit distance from the center of a circle of radius 15 unit. How many total number of different chord of the circle passing through the point P and have the integer lengths?
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determine the equation of the ellipse with foci (2,4)(2,4) and (2,-8)(2,−8), and co-vertices (10,-2)(10,−2) and (-6,-2)(−6,−2).
The equation of the ellipse with given foci (2,4) and (2,-8) and co-vertices (10, -2) and ( -6 , -2) is equal to (x - 2)²/40 + ( y + 2)²/36 = 1.
As given in the question,
Given foci of the ellipse :
( 2,4) and ( 2, -8)
Co-vertices of the ellipse :
( 10,-2) and (-6 , -2)
From the given foci and vertices ,
Centre 'c' = 2
b = 6
( h, k) = (2, -2)
a² = c² + b²
⇒a² = 4 + 36
⇒a = √40
⇒a = 2√10
Equation of the ellipse is given by:
( x - h)²/ a² + (y - k)²/b² =1
⇒( x - 2)²/ (2√10)² + (y + 2)²/6² = 1
⇒( x - 2)²/ 40 + (y + 2)²/36 = 1
Therefore, equation of the ellipse is equal to (x - 2)²/ 40 + (y +2)²/36 = 1.
The complete question is:
Determine the equation of the ellipse with foci (2,4) and (2,−8), and co-vertices (10,−2) and(−6,−2).
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plsss I need help
define a variable right and inequality and solve each problem. the sum of a number and -4 is at least 8
The inequality x - 4 ≥ 8 represents the sentence " the sum of a number and -4 is at least 8" and the solution of the inequality is the value of variable x is x ≥ 12.
What is inequality?Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
Let's assume that the number would be x
The sum of a number and -4 is at least 8
According to the given question, translate the phrase into an algebraic form :
x - 4 ≥ 8
Add 4 both sides of the above inequality,
x - 4 + 4 ≥ 8 + 4
x ≥ 12
Therefore, the inequality x - 4 ≥ 8 represents the sentence " the sum of a number and -4 is at least 8" and the solution of the inequality is the value of variable x is x ≥ 12.
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Train A has 75 cars. if train B has 15 cars, how many times more cars does train A have than train B?
Answer:
3
Step-by-step explanation:
75/15=
__3_________
15 / 75
- 75
---------------------------------
0
Have a great day
Triangle ABCis transformed with the center of dilation at the origin.
Pre-image: △ABC with vertices A(−3, 4), B(−1, 12), C(4, −2)
Image: △A′B′C′with vertices A′(−0.6, 0.8), B′(−0.2, 2.4), C′(0.8, −0.4)
What is the scale factor of the dilation that maps the pre-image to the image?
Enter your answer in the box.
Answer:
take any point on the triangle and any point on the image: for now let's take point A(-3,4) A`(-0.6
Step-by-step explanation:
sf =3/4
scale factor=5
Overall, 148 6th-grade scholars voted in the class president election. How many votes did each class president candidate receive?
The number of votes that each class president candidate receive is 59.2 votes and 88.8 votes respectively.
What is Percentage?To determine the quantity or percentage of something in terms of 100, use the percentage formula. Per cent simply means one in a hundred. Using the percentage formula, a number between 0 and 1 can be expressed. A number that is expressed as a fraction of 100 is what it is. It is primarily used to compare and determine ratios and is represented by the symbol %.
The number of votes gotten by each person will be:
A = Percentage × Number of votes
= 40% × 148
= 0.4 × 148.
= 59.2 votes.
B. = Percentage × Number of votes
= 60% × 148
= 0.6 × 148.
= 88.8 votes.
Therefore, A got 59.2 votes and B had 88.8 votes.
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⠀⠀
Walmart had 200 workers for Black Friday
but had to cut positions for after the
holidays to 152. What percent did the
amount of workers decrease?
Answer:
24%
Step-by-step explanation:
Walmart has 152/200 employees, simplify that to a number over 100, you get 76/100. 100-76=24
Answer:
Step-by-step explanation:
152/200 = .76
100-76 = 24
=24%
Therefore, the percentage the amount of workers decrease is by 24%.
Please do tell if this helps!
Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function C(t) = −5t2 + 10t, where the time, t, is hours after injection. Part A: What are the domain and range of the function C(t) based on the context of the problem? Show all necessary calculations. (5 points) Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body. (5 points)
The domain and range of the function C(t) based on the context of the problem is
domain 0 ≤ t ≤ 2range 0 ≤ C ≤ 2How to find the range and domainThe range is calculated from minima and maxima, this will be achieved using derivative
C(t) = -5t² + 10t
C'(t) = 0 = -10t + 10
t = 1
substituting t = 1
C(t) = -5(1)² + 10t = 5
so at point (1, 5) it is either minima or maxima,
The second derivative test will help to confirm if point (1, 5) is minima or maxima
C'(t) = -10t + 10
C''(t) = -10
a negative value means it is a maxima
Hence the highest value of the concentration C(t) is 5 mg/L and this is at t = 1
The concentration cannot be less than zero or negative hence the lowest concentration is 0. and the range is 0 to 5
0 ≤ C ≤ 2
The time cannot be below zero so the lowest time will be zero
The domain is controlled by the input values. considering the least concentration of the drug which will be at C = 0
C(t) = -5t² + 10t
0 = -5t(t - 2)
-5t = 0
t = 0
t - 2 = 0
t = 2
The roots of the quadratic equation gives the domain to be 0 to 2
0 ≤ t ≤ 2
The graph is attached
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Given the piecewise function defined below; answer the questions at the bottom of the page regarding the continuity and differentiability of f at x = 1. In(52 4) + 3 for I < 1 412 for T > 1 f(c) lim f(2) I 1 lim f(z) I f( lim f' (€) I_] lim f' (1) So the function f () is Submnlt Answcr att = [
The given function is neither continuous nor differentiable at x = 1.
Given the piecewise function defined below, the function is not continuous at x = 1 because the left-hand limit is not equal to the right-hand limit. Specifically, the left-hand limit of f(x) as x approaches 1 is 3, while the right-hand limit of f(x) as x approaches 1 is 4. Furthermore, the function is not differentiable at x = 1 because the derivative does not exist at this point. This is because the derivative of the function is discontinuous at x = 1, with a derivative of 0 for x < 1 and a derivative of 1 for x > 1.
Therefore, the function f(x) is not continuous or differentiable at x = 1.
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what is the largest area of a rectangle that can be inscribed in a semicircle of radius 6? round to the nearest whole numb
The area of the largest rectangle is 46.47 square root.
Consider a semi-circle with a rectangle ABCD inscribed in it.
Let, O = centre of the semi-circle
A and B lies on the base of the semi-circle
OA = OB = x
D and C lie on the semi-circle
BC = AD = y
AB = CD = 2x
By Pythagorean theorem,
CB² + OB² = OC²
⇒ y² + x² = (6)²
⇒ y² = 36 - x²
⇒ y = √(36 - x²)
Now, area of rectangle in terms of x,
Area, A = 2x × y
= 2x × √(36 - x²)
Differentiating,
A' = 2 × √(36 - x²) - 2x²/(36 - x²)
When x = 0, y = 6 and when x = 6, y = 0, area = 0.
It implies that area is maximum when the value of x lies between 0 and 6.
This will occur where A’ = 0.
⇒ 2 × √(36 - x²) - 2x²/(36 - x²) = 0
⇒ 2 × √(36 - x²) = 2x²/(36 - x²)
On simplification, we get,
⇒ 2 × (36 - x²) = 2x²
⇒ 36 - x² = x²
⇒ 2x² = 36
⇒ x² = 36/2
⇒ x = √36/2
Now, y = √(36 - x²) becomes
⇒ y = √(36 - (√36/2)²)
⇒ y = √(36 - 36/2)
⇒ y = √(96 - 36)/2
⇒ y = √60/2
Maximum area = 2xy
= 2(√36/2)(√60/2)
= 46.47
Therefore, the area of the largest rectangle = 46.47 square units.
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joe has dime and nickels in his piggy bank totaling $1.45 the number of nickels he has is 5 more than twice the number of dimes , d . which equation could be used to find the number of dimes he has ?
The expression 0.3d = 0.95 could be used to find the number of dimes he has.
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have joe who has dime and nickels in his piggy bank totaling $1.45 the number of nickels. He has is 5 more than twice the number of dimes [d].
We can write the the number of nickels as -
[n] = 2d + 5
0.1d + 0.1(2d + 5) = 1.45
0.1d + 0.2d + 0.5 = 1.45
0.3d = 0.95
Therefore, the expression 0.3d = 0.95 could be used to find the number of dimes he has.
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It is a function written as f(x)=b^x where bis greater than 0, b is not equal to 1, and x is any real number.
The function with the following formula is an exponential function with base b:
Given: f(x) = b^x, where b > 0, b ≠ 1 is a real number.
If r is a rational number, we can determine what b means.
An exponential function's domain is all real numbers since b^x may be defined for all real values r.
Now, all you need to know is that any positive number b^x and any real number r can be used to approximate the value of b using this approximation.
B can be any real number greater than 0 in the range.
An exponential function is always positive. And if in addition 0 < b < 1, f is a decreasing function.
Hence, f(x) decreases as x increases.
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evaluate the cross product (3iˆ 5jˆ)×(−3iˆ 8jˆ), which expands to −9 iˆ×iˆ 24 iˆ×jˆ−15 jˆ×iˆ 40 jˆ×jˆ. write your answer in vector form physics
The required cross product of given vectors is 39k
What is cross product?Cross product is a parallel procedure on two vectors in three-layered space. It brings about a vector that is opposite to the two vectors. The Vector result of two vectors, an and b, is meant by a × b. Its resultant vector is opposite to an and b. Vector items are additionally called cross items.
According to question:A = 3i + 5j , B = -3i + 8j
Then, cross product of A and B,
A×B = i(5×0 - 8×0) - j(3×0 - (-3)×0) + k(3×8 - (-3)×5)
A×B = i(0) - j(0) + k(24 + 15)
A×B = 39k
Thus, cross product of A and B is 39k
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dr. rhodes notices an interaction in his factorial study. in describing this, which statement might he use to explain the link between independent variable a and independent variable b in predicting the dependent variable?
For the factorial study, Dr Rhodes can use the statement b) effect of variable A depends on variable B
In a factorial study, two intervention comparisons are undertaken side by side. As a result, in addition to behavioral treatment or standard medical care, individuals may be randomly randomized to receive either aspirin or a placebo. In factorial designs, three different sorts of outcomes are pertinent: main effects, interaction effects, and simple effects.
The primary effect is the impact of one independent variable on the dependent variable, averaged over the values of the other independent variable. Dr Rhodes can forecast the relationship between independent variables A and B using the assumption that the effect of variable A depends on variable B.
Complete question
Dr. Rhodes notices an interaction in his factorial study. In describing this, which statement might he use to explain the link between Independent Variable A and Independent Variable B in predicting the dependent variable? TRUE OR FALSE
Variable A cancels out Variable B.
The effect of Variable A depends on Variable B.
The effect of Variable A is confused by Variable B.
Variable A mainly affects Variable B.
The effect of Variable A is mediated by Variable B.
The effect of Variable A depends on Variable B.
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What is the unit rate for 20 pounds for $4.00? Which answer is correct?
(1): 5 pounds/$1.00
(2): 4 pounds/$1.00
(3): 1 pound/$4.00
(4): 1 pound/$5.00
What is the unit rate for 20 pounds for $4.00? Which answer is correct?
(1): 5 pounds/$1.00
(2): 4 pounds/$1.00
(3): 1 pound/$4.00
(4): 1 pound/$5.00
the circumference of the base of a 5 -foot-high cylinder is 46 inches. what is the volume of the cylinder? (use the formula v
The volume of the cylinder is 10,099.09 inches³.
Here we have to find the volume of the cylinder.
Given data:
Circumference = 46 inches
height(h) = 5 foot = 12 × 5 = 60 inches
Circumference = 2πr
46 = 2 ×π × r
r = 23/π
The formula for the volume of the cylinder = πr²h
Volume = π( 23/π)² 60
= 529× 60 / π
= 10,099.09 inches³
Therefore the volume is 10,099.09 inches³.
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The unit rate for inches
per centimeter is 0.39.
Answer:
The inches unit number 0.39 in converts to 1 cm, one centimeter. It is the EQUAL length value of 1 centimeter but in the inches length unit alternative.
Step-by-step explanation:
What is the range of g(x)=-1/2 |x-6|+1? A. (-∞, 1] B. [1, ∞) C. [6, ∞) D. (-∞, ∞)
PLEASE HELP RIGHT NOW!
The required range of the given function g(x) = -1/2 |x-6| + 1 will be g(x) ≤ 1 or in internal notation form as (-∞, 1]. which is the correct answer would be an option (A).
The function is given in the question below as:
g(x) = -1/2 |x-6| + 1
We have to determine the range of the given function.
As per the given question, we have
⇒ g(x) = -1/2 |x-6| + 1
The range of an absolute function of the form:-
c|ax+b|+k is :f (x) ≤ k
In this case k = 1
⇒ g(x) ≤ 1
[tex]\mathrm{Range\:of\:}-\frac{1}{2}\left|x-6\right|+1:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\le \:1\:\\ \:\mathrm{Interval\:Notation:}&\:(-\infty \:,\:1]\end{bmatrix}[/tex]
Therefore, the required range of the given function will be g(x) ≤ 1 or in internal notation form as (-∞, 1].
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a triangle is bounded by a semi circle and the x axis, what length and width should the rectangle be to maximize the area
So for the maximum area, the values of the width and length of the rectangle will be 2.121.
Since we know that the formula for the area of a rectangle is:
A = XY, where x and y are the width and length
since we are given the are, y=√(9-x²), so
A = √(9-x²)
taking the derivative on both sides and considering it to be zero, we get
A' = (x/√(9-x²)(-2x) + √(9-x²) = 0, after multiplying by √(9-x²)
A' = -x² + 9 -x² = 0
=>-2x² =-9
=>x² = 9/2
=> x = 3/√2
so x=2.121 so the for y = √(9-(2.121)^2) = √(9-4.498) = 2.121
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A rectangle is bounded by a semicircle y = radical 9 − x^2 and the x-axis. What length and width should the rectangle be to maximize the area
if four residential fires are idpenently reported on a single day, what is the probability that two are in family homes, one is in an apartment, and one is in another type of dwelling
The probability that out of four residential fires who independently reported on a single day, two are in family homes, one is in an apartment, and one is in another type of dwelling is 0.0895 or 8.95%.
The multinomial distribution deals specifically with events that have multiple discrete outcomes. The multinomial distribution is not limited to events with only discrete outcomes.
We have given that
The National Fire Incident Reporting Service stated, among all residential fires.
Let consider the number of events ,
X₁---> fires residential who are in home
X₂--> fires residential who are in appartment
X₃--> fires residential who are in another type of dwelling
The probability of ae X₁ occured (p₁)
= 73% = 0.73
The probability of event X₂ occured (p₂) = 20% =0.20
The probability of event X₃ occurred (p₃) = 7% = 0.07
Four residential fires are independently reported on a single day.
we have to calculate probability that two are in family homes, one is in an apartment, and one is in another type of dwelling?
Now, Using the Multinomial distribution,
X₁= 2 , X₂= 1, X₃ = 1 and n = 4
P( X₁, X₂,-----,Xₙ ) =( n!/(X₁! X₂! ----Xₙ!) )( p₁ˣ₁ × p₂ˣ₂ × ----× pₙˣₙ)
P( 2,1,1) = 4!/2! 1! 1! ( 0.73² × 0.2× 0.07¹)
= 12 ( 0.014 × 0.5329)
= 0.0895272
Hence, the required probability is 0.089..
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Complete question:
The National Fire Incident Reporting Service stated that, among residential fires, 73% are in family homes, 20% are in apartments, and 7% are in other types of dwellings.
a) If four residential fires are independently reported on a single day, what is the probability that two are in family homes, one is in an apartment, and one is in another type of dwelling?
If x varies inversely as y and x=32 when y=3,find y when x=11
[tex]\qquad \qquad \textit{inverse proportional variation} \\\\ \textit{\underline{y} varies inversely with \underline{x}} ~\hspace{6em} \stackrel{\textit{constant of variation}}{y=\cfrac{\stackrel{\downarrow }{k}}{x}~\hfill } \\\\ \textit{\underline{x} varies inversely with }\underline{z^5} ~\hspace{5.5em} \stackrel{\textit{constant of variation}}{x=\cfrac{\stackrel{\downarrow }{k}}{z^5}~\hfill } \\\\[-0.35em] ~\dotfill[/tex]
[tex]\begin{array}{llll} \textit{"x" varies}\\ \textit{inversely with "y"} \end{array}\implies x=\cfrac{k}{y}\hspace{5em}\textit{we also know that} \begin{cases} x=32\\ y=3 \end{cases} \\\\\\ 32=\cfrac{k}{3}\implies 96=k\hspace{10em}\boxed{x=\cfrac{96}{y}} \\\\\\ \textit{when x = 11, what's "y"?}\qquad 11=\cfrac{96}{y}\implies y=\cfrac{96}{11}[/tex]
is the sum of a certain pair of integers odd? (1) the difference between the integers is odd. (2) one of the integers is even and the other is odd.
For first case, It is even integer. The difference of odd integers is always even integer. For second case, It is odd integer. The difference of an odd and even integers is always odd integer.
What is integer?Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers. The inverse additives of the corresponding positive numbers are the negative numbers. The boldface Z is a common mathematical symbol for the set of integers.
Here,
a. If there is an odd pair of integers,
for example,
=3 and 5
The difference,
5-3=2
It is even integer. The difference of odd integers is always even integer.
b. If one integer is even and other is odd,
for example,
=6 and 3
The difference,
6-3=3
It is odd integer. The difference of an odd and even integers is always odd integer.
It is an even integer in the first scenario. Odd integer differences always result in even integer differences. It is an odd integer in the second case. An odd integer is always the difference between an odd and an even integer.
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Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents?.
It will cost his parents $187.5 to invite 15 friends to his party because it cost $12.5 to invite just one.
What is unitary method?The unitary approach involves calculating the value of a single unit, from which we may calculate the values of the necessary number of units. A single or distinct unit is referred to by the word unitary. Therefore, the goal of this strategy is to establish values in reference to a single unit. The unitary technique, for instance, may be used to calculate how many kilometers a car would go on one litre of gas if it travels 44 km on two litres of fuel.
Here,
For inviting 20 friends, The cost of party was $250.
So for 1 friend,
=250/20
=$12.5
Now he wish to invite 15 friends,
=12.5*15
=$187.5
The cost of inviting 15 friends to his party will cost $187.5 to his parents as cost of inviting 1 friend was $12.5.
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Find the measure of each angle listed in the picture.
**Please Show All Work**
Angle AEB:
Angle CED:
Angle C:
Step-by-step explanation:
so, if I understand correctly :
angle A = 22°
angle B = 53°
angle D = 38°
remember : the sum of all 3 angles in every triangle is always 180°.
so,
180 = 22 + 53 + AEB
angle AEB = 180 - 22 - 53 = 105°
per the attributes of crossing lines the angles between them are the same on both sides of any of the lines (just left-right mirrored).
so, CB and AD are crossing lines, therefore
angle AEB = angle CED = 105°
and then
180 = 105 + 38 + C
angle C = 180 - 105 - 38 = 37°
Complete the table: Rashida bought 3 tickets to a concert for $75. Using this rate, determine how much 1 ticket will cost? and how much 5 tickets will cost? Please write the answer like it is in the block.
In this case we are multiplying the numbers and dividing them.
Fist we got to work with are numbers given the equation.
Given:
[tex]1, 5[/tex]
Therefore, the equation is:
[tex]75 / 3\\5x3[/tex]
75 / 3 = 255 x 25 = 125Therefore, the correct answers are 1 = 25, 5 = 125 and 3 = 75
A tack of heet of tiue paper i about 2. 5 inche high each heet i about 0. 01 inch thick how many heet are in the tack
Using simple mathematical operations, we can conclude that there are 250 sheets in a stack.
What are mathematical operations?A mathematical function known as an operation converts zero or more input values into a precisely defined output value.
The quantity of operands affects the operation's arity.
The order of operations refers to the rules that govern how to solve an expression with several operations.
Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction are all referred to as PEMDAS (from left to right).
So, the number of sheets in the stack is represented by n.
According to the information provided, a stack of tissue paper is roughly 2.5 inches high.
The thickness of each sheet is roughly 0.01 inches. Determine the number of sheets in the stack, please.
N sheets' thickness would be.
We'll now compare the thickness of n sheets to the thickness of the stack as follows:
0.01n = 2.5
Now, solve for n as follows:
0.01n/0.01 = 2.5/0.01
n = 250
Therefore, using simple mathematical operations, we can conclude that there are 250 sheets in a stack.
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is it okay if you can help me out here.
Answer:
Part B:16.5
Step-by-step explanation:
Answer
b
Step-by-step explanation:
Factor the polynomial function over the complex numbersf(x)=x^4+9x^3+15x^2+9x+14
The polynomial function over the complex numbers is,
(x + 1i) ( x - 1i) (x+ 7) ( x + 2).
What is factoring a polynomial?
Factorization of polynomials, also known as polynomial factorization, is a mathematical and computer algebraic technique that combines irreducible factors with coefficients in the same domain to produce a polynomial with coefficients in a given field or in integers.
Consider, the given polynomial
f(x) = x^4 + 9x^3 + 15x^2 + 9x + 14
We can use a factoring "trick" here....write this as
x^4 + 9x^3 + 14x^2 + x^2 + 9x + 14 = 0 factor by grouping
x^2 ( x^2 + 9x + 14) + 1 ( x^2 + 9x + 14) = 0
(x^2 + 1) (x^2 + 9x + 14) = 0
(x^2 + 1) (x^2 + 9x + 14) = 0
⇒ (x^2 + 1) = 0, (x^2 + 9x + 14) = 0
x^2 = -1, x^2 + 7x + 2x + 14 = 0
x = ±i , x(x + 7) + 2(x + 7) = 0
x = ±i , (x + 7)(x + 2) = 0
⇒ (x + i)(x - i)(x + 7)(x - 7) = 0
(x + 1i) ( x - 1i) (x+ 7) ( x + 2) = 0
Hence, the polynomial function over the complex numbers is,
(x + 1i) ( x - 1i) (x+ 7) ( x + 2).
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10. Find the slope of the line.
Answer: y = 1.75x - 1
Step-by-step explanation: The slope of the line is 1.75
select the curve generated by the parametric equations. indicate with an arrow the direction in which the curve is traced as t increases. x
The curve with parametric equations x = [tex]e^{- t}[/tex] + t, y = [tex]e^t[/tex] − t, − 2 ≤ t ≤ 2 can be plotted on a graph. The arrow shows the direction where t increases in the interval (-2,2).
The length of a curve of the provided parametric equations is indicated by the arrow pointing in the direction wherein the curve is traced given data.
Think about the equations for the parameterized curve in the x-y plane.
x = x( t )
y = y( t )
where the argument is t. The graph of the curve is created by charting the series of points as the variable t changes (x,y).
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The question is -
Select the curve generated by the parametric equations. Indicate with an arrow the direction in which the curve is traced as t increases. x = [tex]e^{- t}[/tex] + t , y = [tex]e^t[/tex] − t , − 2 ≤ t ≤ 2
a simple random sample of size n is drawn from a population that is normally distributed. the sample mean, , is found to be , and the sample standard deviation, s, is found to be . (a) construct % confidence interval about if the sample size, n, is . (b) construct % confidence interval about if the sample size, n, is . (c) construct % confidence interval about if the sample size, n, is . (d) could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed?
A 90% confidence interval about μ if the sample size, n, is 25 is (111.6, 118.4), a 90% confidence interval about μ if the sample size, n, is 20 is (111.1, 118.9), a 80% confidence interval about μ if the sample size, n, is 25 is (112.4, 117.6) and we need population to be normal. So we can not make intervals.
In the given question,
Sample mean, x, is found to be 115.
Sample standard deviation, s, is found to be 10.
(a) We have to construct a 90% confidence interval about μ if the sample size, n, is 25.
We need to build a 90% confidence interval. So p=0.90.
t-score for confidence interval is given by the excel code is
t=1.7109
Using these values, the margin of error is
ME = 1.7109× 10/√25
ME = 3.4
So confidence interval is
CI = (115-3.4, 115+3.4)
CI = (111.6, 118.4)
(b) We have to construct a 90% confidence interval about μ if the sample size, n, is 20.
We need to build a 90% confidence interval. So p=0.90.
t-score for confidence interval is given by the excel code is
t=1.7291
Using these values, the margin of error is
ME = 1.7291× 10/√20
ME = 3.9
So confidence interval is
CI = (115-3.9, 115+3.9)
CI = (111.1, 118.9)
(c) We have to construct a 80% confidence interval about μ if the sample size, n, is 25.
We need to build a 80% confidence interval. So p=0.80.
t-score for confidence interval is given by the excel code is
t=1.3178
Using these values, the margin of error is
ME = 1.3178× 10/√25
ME = 2.6
So confidence interval is
CI = (115-2.6, 115+2.6)
CI = (112.4, 117.6)
(d) Now we have to check, could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed.
To use t-distribution, the population has to be normal.Otherwise we need a sample size of at least 30. Here sample size is less than 30.
So we need population to be normal. So we can not make intervals.
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The right question is:
A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 115, and the sample standard deviation, s, is found to be 10.
(a) Construct a 90% confidence interval about μ if the sample size, n, is 25.
(b) Construct an 90% confidence interval about μ if the sample size, n, is 20.
(c) Construct an 80% confidence interval about μ if the sample size, n, is 25.
(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed.