The area bounded by the sin (100x² + 49y²) dA is calculated by using integration is 2π (1 - cos 1).
An area bounded by the curve: When the two curves intersect then they bound the region is known as the area bounded by the curve.
Evaluate the integral by making an appropriate change of variables.
The equation of the ellipse is 100x² + 49y² = 1
Let cos t = 10x and sin t = 7y. Then we have
or x = 1/10 cost , y = 1/7 sint.
Then
=> 100 (1/10cost)^2 + 49 (1/7 sint)^2 = 1
=> cos^2 t + sin^2t = 1 which suggests a change of variable will be:
[tex]\left \{ {{x(r,t) = r/10cost} \atop {y(r,t)=r/7sint}} \right.[/tex]
where 0≤r≤1 and 0≤t≤2[tex]\pi[/tex]. Then we also have,
100x² + 49y² = r²
So,
=> ∫∫ sin (100x^2 + 49y^2)dA
R
=> [tex]\\[/tex]2 ∫2[tex]\pi[/tex] ∫[tex]1[/tex] sinr^2 dr dt
0 0
=> 4[tex]\pi[/tex] ∫[tex]1[/tex] rsinr^2 dr
0
Now r² is replaced by r, then we get
=> 2[tex]\pi[/tex] ∫[tex]2[/tex] sinr^2 d(r^2)
0
=> - 2[tex]\pi[/tex] cos r^2 [tex]\left \{ {{r^2=1} \atop {r^2=0}} \right.[/tex]
=> -2[tex]\pi[/tex] (cos1-cos0)
=> -2[tex]\pi[/tex] (-1 + cos1)
=> 2[tex]\pi[/tex](1-cos1)
Evaluating the integral by making an appropriate change of variables 2 sin(100x^2 + 49y^2) dA.
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A company that manufactures memory chips for digital cameras uses the formula c = 3√n(40 6√n +9 4√n) to determine the cost, c, in dollars, for producing n chips. This formula can be written as c = 120 3√n +27 4√nb, where a and b are constants.
What are the values of a and b?
**answer choices in pic
The values of a and b are 3 and 2.
What is Expression?An expression is combination of variables, numbers and operators.
Given,
A company that manufactures memory chips for digital cameras uses the formula [tex]c=3\sqrt{n} (40\sqrt[6]{n}+9\sqrt[4]{n} )[/tex]
The cost, c, in dollars, for producing n chips.
We need to solve for values a and b.
[tex]c=3\sqrt{n} (40\sqrt[6]{n}+9\sqrt[4]{n} )[/tex]
[tex]c=3\sqrt{n} (40\sqrt[6]{n})+3\sqrt{n} (9\sqrt[4]{n} )[/tex]
[tex]c=120\sqrt{n} \sqrt[6]{n} +27\sqrt{n} \sqrt[4]{n}[/tex]
[tex]c=120n^{\frac{7}{2}} +27n^{\frac{5}{2}}[/tex]
c=[tex]120n^{\frac{3a}{2} } +27n^{\frac{4b}{2} }[/tex]
By solving this we get a=3 and b=2
Hence, the value of a and b is 3 and 2.
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Solve the following inequality for pp. Write your anwer in implet form. -5p -1 > -10p - 2
The value of the variable 'p' for the given inequality is found as p > -1/5.
Explain the term inequality?In mathematics, "inequality" refers to a relationship involving two variables or values that is not equal to one another and. Therefore, inequality emerges from a lack of balance.For the stated question-
-5p -1 > -10p - 2
Solve the inequality as
Add 1 both sides.
-5p -1 + 1 > -10p - 2 + 1
-5p > -10p - 1
Add 10p both sides.
-5p + 10p > -10p - 1 + 10p
5p > -1
Divide both sides by 5
p > -1/5
Thus, the value of the variable 'p' for the given inequality is found as p > -1/5.
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can y’all help me with these three questions
Determine if the expression for r(u, v) with domain D defines a smooth parametrized surface. Explain your answer. (Your instructors prefer angle bracket notation < > for vectors.) r(u, v) = (cos(u), sin(v), 1), D = {(u, v) | 0 Sus 41,0 S V S 41} r(u, v) with domain D ---Select--- v define a smooth parametrized surface. To be a smooth parametrized surface (", xru(u, v) # O for all (u, v) in the interior of the domain. In this case (?, *Tv , therefore there ---Select--- v (u, v) where (FoxPv)(u, v) = 7.
The Vector r(u, v) with the domain D does not define a smooth parametrized surface.
Vector: Vectors are quantities that can be identified by magnitude and direction. Examples are velocity and acceleration.
Given that:
Vector r(u, v) = (cos(u), sin(v), 1), D = {(u, v)
Vector [tex]r_{u}[/tex] = (-sinU, 0,0)
Vector [tex]r_{V}[/tex] = (0, cosv,0)
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] =
[tex]\left[\begin{array}{ccc}i&j&k\\-sinU&0&0\\0&cosv&0\end{array}\right][/tex]
Solving the vector multiplication. We get,
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] = 0 x i - j x 0 + (cosv x - sinU)
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] = 0 - sinUcosv
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] = - sinUcosv
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] = (0, 0,- sinUcosv)
The Vector r(u, v) with the domain D does not define a smooth parametrized surface.
Vector [tex]r_{u}[/tex] x Vector [tex]r_{V}[/tex] = (0, 0,- sinUcosv) therefore has some value.
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A line's y-intercept is 6, and its slope is 1. What is its equation in slope-intercept form?
Answer:y=x+6
Step-by-step explanation:
y= slopex+y-intercept
What is the equation of a line perpendicular to 3z + 4y - 12and passes thru the point (3.2)? Write the equation in slope-intercept form (y=mx+b)
Answer:
Step-by-step explanation:
After you know the equation of this line, you will be able to figure out the equation of the line that runs perpendicular to it. [11] To simplify. y − 6 = 3 2 ∗ ( x − 4) {\displaystyle {y}- {6}= {\frac {3} {2}}* ( {x}-4)} , first multiply all of the numbers in parentheses by the outer value to get.
The relationship between hours practice and coordinate video game is modeled by the equation Y = 10x graph this relationship on the graph below what is the slope and what does it mean for the situation
answer:
provided in the explanation.
step-by-step explanation:
the graph would contain the points (0,0), (1,10), (2,20), etc. the slope is 10, which represents the number of coordinate video games practiced in x number of hours. the representation for the slope depends on whether coordinate video game or hours practiced is on the x or y axis. hope that helps!
How much would $300 invested at 4% interest compounded monthly be worth after 8 years? round your answer to the nearest cent.
The amount will be $412.91 after 8 years if $300 is invested at 4% interest compounded monthly.
Given:
Principal (P) = $300
rate of interest (r) = 4%
time (t) = 8 years.
compounded monthly.
Amount (A) = ?
we know the formula as:
A(t) = P(1+r/n)^nt
we know t=8
A(8) = 300(1+0.04/12)^12(8)
A(8) = 300(1+0.04/12)^96
A(8) = 300(1.033)^96
A(8) = $412.91
Hence amount after 8 years will be $412.91.
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when three positive integers $a, b$, and $c$ are multiplied together, their product is $100$. suppose $a < b < c$. in how many ways can the numbers be chosen?
Together the numbers a , b and c can be chosen in 4 ways.
Given:
when three positive integers $a, b$, and $c$ are multiplied together, their product is $100$ and a < b < c .
The positive divisors of 100 are :
1 , 2 , 4 , 5 , 10 , 20 , 25 , 50 , 100.
It is clear that :
10 ≤ c ≤ 50.
so we apply from c value 10.
if c = 10 , then ( a , b , c ) = ( 2 , 5 , 10 )
if c = 20 , then ( a , b , c ) = ( 1 , 5 , 10 )
if c = 25 , then ( a , b , c ) = ( 1 , 4 , 25 )
if c = 50 , then ( a , b , c ) = ( 1 , 2 , 50 )
we can observe ways = 4.
Therefore the number of ways is 4.
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three points are chosen randomly and independently on a circle. what is the probability that all three pairwise distances between the points are less than the radius of the circle?
There is a 1/12 chance that the three pairwise distances between the locations are smaller than the radius of the circle.
Here,
The first point can be placed anywhere on the circle because it doesn't matter where it is chosen.
A total of 120 degrees and a probability of 1/3 must be provided by the next point, which must be within an arc of 60 degrees on either side. The second point must be 60 degrees or less from the first two.
The third point can be located within an arc with a minimum area of freedom of 60 degrees and a chance of 1/6 if the first two locations are separated by 60 degrees. The third point can be located with a maximum degree of freedom of a 120 degree arc and a 1/3 chance if the first two points are the same.
Up to a maximum of 60 degrees away from the starting point, the likelihood changes linearly.
By averaging the probabilities at either end, we may discover 1/4, the typical likelihood we can position the third point based on a fluctuating second point.
The overall probability is 1 1/3 4 = 1/12.
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when two regression models applied on the same data set have the same response variable but a different number of explanatory variables, the model that would evidently provide the better fit is the one with a .
The regression model that would offer the better fit is one with a smaller standard error of the estimation and a given elements coefficient of determination when two regression models used on the same data set have had the same dependent variables but a variable figure of explanatory factors.
The coefficient of determination is most frequently used to determine how well a regression model fits observed data. For instance, data fitting the regression model 60% of the time is indicated by a determination coefficient of 60%. In general, a greater coefficient denotes a better model fit.
The degree of deviation around the mean increases with the coefficient of variation. Typically, a percentage is used to indicate it. Without units, it enables comparison of value distributions whose measurement scales are incomparable.
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A student is taking a multiple-choice test in which each question has four possible answers. She knows the answers to 5 of the questions, can narrow the choices to 2 in 3 cases, and does not know anything about 2 of the questions. What is the probability that she will correctly answer A) 10, b) 9, c) 8 d) 7, e) 6, and f) 5 questions?
Do not need to answer every part if they are worked the EXACT same way.
the probability that she will correctly answer
A) 10 questions = 0
b) 9 questions = 0.00003
c) 8 questions = 0.00039
d) 7 questions =0.00309
e) 6 questions = 0.01622
f) 5 questions = 0.0584
Here the total number of question is 5+3+2= 10, and every question has four possible answers, for this problem, we will be using the binomial distribution, the formula is :
[tex]C_{n,k}[/tex][tex]p^{k}[/tex][tex]q^{n-k}[/tex] ,here C is the combination, p is the probability of success and q is the probability of having the success
for the given situation :
the p = 1/ 4 whereas, q=3/4
here n =10
now for different values of k which is the number of successes, we substitute the know values in the formula, and we get
P(k=10) = 0
P(k=9) = 0.00003
P(k=8) = 0.00039
(P=7)= 0.00309
(P=6)=0.01622
(P=5)= 0.0584
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a. a rectangular solar panel is made up of small squares. a particular solar panel is 5 squares by 27 squares. in what other rectangular arrays could the squares have been arranged? b. if there were 23 squares, how many rectangular arrays can there be?
(a) The rectangular arrays could the squares have been arranged as
1 by 135, 3 by 45, 5 by 27, 9 by 15, 15 by 9.
(b) The rectangular arrays is of 1 by 23.
What is solar panel?
An assembly of photovoltaic solar cells installed on a frame (often rectangular) is known as a solar cell panel, solar electric panel, photo-voltaic (PV) module, PV panel, or solar panel. A well-organized collection of PV panels is known as a photovoltaic system or solar array. Sunlight is used by solar panels to collect radiant energy, which is then transformed into direct current (DC) power.
(a) Let a particular solar panel is 5 squares by 27 squares.
5 by 27 = 135 which can be arranged as,
1 by 135
3 by 45
5 by 27
9 by 15
15 by 9
(b) There were 23 squares.
The factors of 23 are 1, 23 only so there can be only 1 rectangular array of 1 by 23.
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The function f(x)=lnx has a Taylor series at a=6. Find the first 4 nonzero terms in the series, that is write down the Taylor polynomial with 4 nonzero terms.
The Taylor series expansion of f of x is given by 11e to the fourth power + 22e to the fourth power times x minus two plus 22e to the fourth power multiplied by x minus two all squared, where f of x is equal to 11e to the power of two x in rising powers of x minus two.
What is Taylor series?An infinite sum of words that are expressed in terms of a function's derivatives at a single point is known as the Taylor series or Taylor expansion of a function in mathematics.
Near this point, the function and the sum of its Taylor series are equivalent for the majority of common functions.
For Brook Taylor, who introduced the Taylor series in 1715, they are named after him. When 0 is the point at which the derivatives are taken into account, a Taylor series is also known as a Maclaurin series in honor of Colin Maclaurin, who made great use of this unique situation of Taylor series in the middle of the 18th century.
The nth Taylor polynomial of a Taylor series is a polynomial of degree n that is created by the partial sum of the first n + 1 terms of a Taylor series.
Hence, The Taylor series expansion of f of x is given by 11e to the fourth power + 22e to the fourth power times x minus two plus 22e to the fourth power multiplied by x minus two all squared, where f of x is equal to 11e to the power of two x in rising powers of x minus two.
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Two students, Jessica and Steve, are solving the math problem below: A hiker is hiking at an elevation located 18 meters above sea level. Simultaneously a diver is diving at a location that is 12 meters below sea level. Find the distance between the two elevations. The students came up with different answers. Jessica said the answer is 6 meters. Steve said the answer is 30 meters. Who is correct?
The distance between the two elevation is 30 metres. Therefore, Steve is correct.
How to find the distance between the two elevations?A hiker is hiking at an elevation located 18 meters above sea level. Simultaneously a diver is diving at a location that is 12 meters below sea level.
The distance between the two elevation can be calculated as follows:
Jessica said the answer is 6 meters. Steve said the answer is 30 meters.
Let's find who is correct.
The hiker is 18 metres above sea level . The diver is 12 meters below sea level. For the diver to get to sea level he will travel 12 meters. For the diver to get to the hiker position he will travel extra 18 metres.
Therefore, the distance in elevation is 12 + 18 = 30 metres. Hence, Steve is correct.
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CNNBC recently reported that the mean annual cost of auto insurance is 1037 dollars. Assume the standard deviation is 243 dollars. You take a simple random sample of 69 auto insurance policies. (Do not use tables unless directed to do so.)
Find the probability that a single randomly selected value is more than 987 dollars.
P(XÂ > 987) =Â
Find the probability that a sample of size n=69 is randomly selected with a mean that is more than 987 dollars.
P(¯x > 987) =Â
Enter your answers as numbers accurate to 4 decimal places.
The probability that a single randomly selected value is more than 987 dollars is 0.5664
We know that probability is defined as the ratio of number of favorable outcomes to the total number of outcomes.
We know very well that z-value is calculate by the formula
z = (X-μ) / σ/√n
where z is the z-value,
X is randomly selected value,
μ is standard mean
σ is standard deviation and
n is the sample size.
For, X=987
=>z = (987-1037) / (243/√69)
=>z= -50 / (243 / 8.33)
=>z = -50 / (29.171)
=>z= -1.714
Now, we need to find the probability of A>987,so,we need to find what is the probability of A<=987.
P(X>987)= 1-P(X≤987)
=>P(X>987)=1-P(z<(-1.714))
Using the z-table ,we get that P(z<(-1.714))=0.43353
Therefore, P(X>987)=1-0.43353=0.5664
Hence, required probability is 0.5664
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(Complete question) is:
CNNBC recently reported that the mean annual cost of auto insurance is 1037 dollars. Assume the standard deviation is 243 dollars. You take a simple random sample of 69 auto insurance policies. (Do not use tables unless directed to do so.)
Find the probability that a single randomly selected value is more than 987 dollars. P(Â > 987).Enter your answers as numbers accurate to 4 decimal places.
Enter the data below into a calculator to find the correlation coefficient, the regression line equation, and the strength of the linear correlation:
x y
1 6
2 1
3 5
4 2
5 4
6 3
7 7
8 8
9 6
Correlation Coefficient (enter a decimal rounded to the nearest hundredth):
Equation of the regression line (enter as an equation in the form y=mx+b, no spaces, decimals rounded to the nearest hundredth):
Strength of the linear correlation (type one of the following exactly: "Perfect Negative", "Strong Negative", "Moderate Negative", "Weak to No", "Moderate Positive", "Strong Positive", or "Perfect Positive". Watch for spelling!):
Answer:
I guess it's 77 although I'm not in high school
A manufacturer of refrigerators must ship at least 100 refrigerators to its two warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25
Krefrigerators already, while warehouse B has 20 on hand. It costs $15 to ship a refrigerator to warehouse A and $10 to ship one to warehouse B. How many refrigerators
should be shipped to each warehouse to minimize cost? What is the minimum cost?
The minimum cost is 12.
Joey ha 8 pair of ock in hi drawer. Five of the pair are white. What percent of joey ock are colored?
Using percentages, we know that Joey has 37.5% of colored pairs of socks.
What is the percentage?%, which is a relative figure used to denote hundredths of any quantity.
Since one percent (symbolized as 1%) is equal to one-hundredth of something, 100 percent stands for everything, and 200 percent refers to twice the amount specified.
Percentage, which means "per 100," designates a portion of a total sum. 45 out of 100 is represented by 45%, for instance.
Finding the percentage of a whole in terms of 100 is what percentage calculation is.
Both manual calculation and the use of internet calculators are options.
So, the percentage of colored pair of socks are:
We know that Joey has a total of 8 pairs of socks.
We also know that 5 pairs of socks are white.
Hence, the colored pair of socks will be:
8 - 5 = 3
Calculate its percentage as follows:
3/8 * 100
0.375 * 100
37.5%
Therefore, using percentages, we know that Joey has 37.5% of colored pairs of socks.
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A ski rental service charges a $37.30 initial flat rate and then an additional $2.70 per hour. In this situation, what is the value of the y-intercept?
What is 8.123 as a decimal number
Answer: 8.123 is a decimal
Step-by-step explanation:
A decimal is a whole number and a fractional part
there is a decimal point
Joseph has a loyalty card good for a 9% discount at his local grocery store. What would his total in dollars and cents be, after the discount and before tax, if the total cost of all the items he wants to buy is $37? Round to the nearest cent.
Answer: $40.33
Step-by-step explanation:
9% of $37 = 3.33
37 + 3.33 = 40.33
Which inequality is shown in the graph?
A. y−5 < −5/3(x+6)
B. y−5 > −5/3(x+6)
C. y+5 < −5/3(x−6)
D. y+5 > −5/3(x−6)
which sentance describe points that miguel should consider in the goal setting process before he starts to invest
five-team tournament, each team plays one game with every other team. each team has a 50% chance of winning any game it plays. (there are no ties.) let m n be the probability that the tournament will produce neither an undefeated team nor a winless team, where m and n are relatively prime integers. find m n
The value of integers m and n is 17 and 32 respectively.
Here, we are given that in a five-team tournament, each team plays one game with every other team.
Chance of winning = 50% or 1/2
We need to find the probability that at least one team wins all games or at least one team loses all games. We can use complementary counting to do so.
Each team will play 4 matches.
Probability that there is one team which wins all games = [tex]5*1/2^{4}[/tex]
= 5/16
Similarly, the probability that there is one team which loses all games = 5/16
The probability that one team wins all games and another team loses all games = [tex][5(1/2)^{4} ][4(1/2^{3})][/tex]
= (5/16)(4/8)
= 5/32
Now, 5/16 + 5/16 - 5/32 = 15/32
Since this is the complement of the probability we want, we subtract it from 1 to get (1- 15/32) = 17/32
Thus, the value of integers m and n is 17 and 32 respectively.
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what is the equation in point-slope form of a line that passes through the points (7, −8) and (−4, 6) ? responses y+6=−23(x−4) y plus 6 equals negative fraction 2 over 3 end fraction open parenthesis x minus 4 close parenthesis y+6=−1411(x−4) y plus 6 equals negative fracion 14 over 11 end fraction open parenthesis x minus 4 close parenthesis y−6=−23(x+4) y minus 6 equals negative fraction 2 over 3 end fraction open parenthesis x plus 4 close parenthesis y−6=−1411(x+4)
The equation of the line that passes through the points (7, -8) and (-4, 6) in point slope form is y - 6 = -14/11(x + 4)
The coordinates of the first point = (7, -8)
The coordinates of the second point = (-4, 6)
The slope of the line = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Substitute the values in the equation
The slope of the line = (6-(8) / (-4-7)
= 6+8 / -4-7
= 14 / -11
= - 14/11
Point slope form is
[tex]y-y_1=m(x-x_1)[/tex]
Substitute the value in the equation
y - 6 = -14/11(x - (-4))
y - 6 = -14/11(x + 4)
Therefore, the equation of the line is y - 6 = -14/11(x + 4)
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Which of the following graphs show the most reliable results (the x-axis shows the original text scores, and the y-axis shows the re-test scores).
The following is the graph that displays the most trustworthy findings:
Choice C
A correlation coefficient is defined.
A measure of correlation between two variables, the correlation coefficient between two variables assumes values between -1 and 1.
The relationship is direct proportional if it is positive, and inverse proportional if it is negative.
Strong correlation exists between two variables if the correlation coefficient's absolute value is higher than 0.6.
This description leads us to conclude that the graph with the highest correlation coefficient is the most accurate, making Option C the proper choice.
Lack of Information
The two solutions to this problem are as follows:
Option B: Create a graph with a r = 0 correlation coefficient.
Option C: Create a graph with a r = 0.87 correlation coefficient.
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Use point-slope form to write the equation of a line that passes through the point (-13,-19) with the slope 7/6
Answer: [tex]y = \frac{7}{6}x - \frac{23}{6}[/tex]
Work Shown:
[tex]y - y_1 = m(x - x_1)\\\\y - (-19) = \frac{7}{6}(x - (-13))\\\\y + 19 = \frac{7}{6}(x + 13)\\\\y + 19 = \frac{7}{6}x + \frac{7}{6}*13\\\\y + 19 = \frac{7}{6}x + \frac{91}{6}\\\\y = \frac{7}{6}x + \frac{91}{6} - 19\\\\y = \frac{7}{6}x + \frac{91}{6} - 19*\frac{6}{6}\\\\y = \frac{7}{6}x + \frac{91}{6} - \frac{114}{6}\\\\y = \frac{7}{6}x + \frac{91-114}{6}\\\\y = \frac{7}{6}x - \frac{23}{6}\\\\[/tex]
Can the following side lengths form a triangle?
5 inches
2 inches
3 inches
A Yes
B No
c Maybe
Answer:
Step-by-step explanation:
the longest sideis greater than others.
so the answer is
Answer:
Yes it can
Step-by-step explanation:
Three sides of any length will always form a triangle.
a company claims that the mean weight per apple they ship is 120 grams with a standard deviation of 12 grams. data generated from a sample of 49 apples randomly selected from a shipment indicated a mean weight of 122.5 grams per apple. is there sufficient evidence to reject the company’s claim? (useα
We fail to reject H0. So there is enough evidence to state that the hypothesis H1 has the risk to make an error equal to α.
What do you mean by standard deviation?
The term "standard deviation" refers to a measurement of the data's dispersion from the mean. A lower number deviation implies that the data are grouped around the mean, whereas a large standard deviation shows that the data are more dispersed.
What does it mean if standard deviation is negative?The standard deviation is positive, or greater than zero, whenever there are two uneven terms in the observations. The standard deviation is exactly zero if all of the observations are equally distributed. Thus, the standard deviation can never be negative or less than zero.
We have two hypotheses:
H0: μ= 120
H1: μ≠ 120
where μ is the mean weight and per apple the value of α is 0.05. If conditions are met we will run sample a.
random: 49 apples are randomly selected
normal: 49>=30 so CLF holds. Now,
formula of z-test for μ
Test statistic: z-test=(¯x−μ)/( σ/√n)
We are given with:
n=49
¯x=122.5
σ=12
putting values in formula:
z-test=(122.5-120)/(12/√49)
=1.458
P-value: P(z>1.458)=normal df(1.458, ∞,1)
=0.0724
And it is a two tailed test so
=2(0.0724)
=0.1448
Since P-value> α ; (0.0724>0.05)
So, we fail to reject H0. We do not have convincing evidence that there is a difference in mean weight per apple. So here we would say that there is enough evidence to state that the hypothesis H1 has the risk to make an error equal to α.
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