Answer: a) 2401 b) 840 c)343 d)343 e) 686
Step-by-step explanation:
a) Total are 7 letters
There are 4 possible places in 4-letter word for each of 7 letter.
So total amount of the words is
7*7*7*7= 2401
b) No letter can be repeated mens that in 1st place can stay any of 7 letters,
in 2-nd place- any of 6 letters
in 3-rd place- any of 5 letters
in 4-th place any of 4 letters
Total amount of words is
7*6*5*4=840
c) It is not written if the letters can be repeated. Assume that they can.
1st place occupied by letter A. So at 2-nd, 3-rd and 4th places can stay any of 7 letters
Total amount of words is
1*7*7*7=343
d) Similar case like c). At 1-st ,2-nd, 3-rd can stay any of 7 letters, and at 4th place the letter C (1 letter). Total amount of words is
7*7*7*1=343
e) There are 2 vowels A and E.
So at 1st place can stay any of 7 letters, at 2-nd place -any of 2 letters, at 3-rd place- any of 7 letters and at 4th place any of 7 letters. Total amount of words is
7*2*7*7=686
The total number of 4-letter words possible is 2401
The total number of 4-letter words possible with no letter repeated is 840
The total number of 4-letter words possible with the first letter being A is 343
The total number of 4-letter words possible with the letter C at the end is 294.
The total number of 4-letter words possible with the second letter being a vowel is 1029.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.com
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
(a) No condition is imposed:
There are 7 choices for the first letter, 7 choices for the second letter, 7 choices for the third letter, and 7 choices for the fourth letter.
The total number of 4-letter words possible.
7 × 7 × 7 × 7 = 2401
(b) No letter can be repeated in a word:
For the first letter, there are 7 choices.
For the second letter, there are only 6 choices left (since we cannot repeat the first letter). Similarly, for the third letter, there are only 5 choices left, and for the fourth letter, there are only 4 choices left.
The total number of 4-letter words possible with no letter repeated.
7 × 6 × 5 × 4 = 840
(c) Each word must begin with the letter A:
For the first letter, there is only 1 choice (the letter A). For the second letter, there are 7 choices (since any letter can be used).
For the third letter, there are 7 choices, and for the fourth letter, there are 7 choices.
The total number of 4-letter words is possible with the first letter being A.
1 × 7 × 7 × 7 = 343
(d) The letter C must be at the end:
For the fourth letter, there is only 1 choice (the letter C).
For the first letter, there are 7 choices (since any letter can be used).
For the second letter, there are 7 choices, and for the third letter, there are 6 choices (since the letter C cannot be used).
The total number of 4-letter words possible with the letter C at the end.
7 × 7 × 6 × 1 = 294
(e) The second letter must be a vowel:
For the first letter, there are 7 choices (since any letter can be used). For the second letter, there are 3 choices (the vowels A, E, and I).
For the third letter, there are 7 choices, and for the fourth letter, there are 7 choices.
The total number of 4-letter words possible with the second letter being a vowel.
7 × 3 × 7 × 7 = 1029
Thus,
The total number of 4-letter words possible is 2401
The total number of 4-letter words possible with no letter repeated is 840
The total number of 4-letter words possible with the first letter being A is 343
The total number of 4-letter words possible with the letter C at the end is 294.
The total number of 4-letter words possible with the second letter being a vowel is 1029.
Learn more about expressions here:
https://brainly.com/question/3118662
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a jacket originally sold for $45. this week it went on sale for 20% off. what is the discount and what is the sales price?
Answer:
discount = 9
new price = 36
Step-by-step explanation:
The discount is the price times the discount percent
45 * 20%
Change to decimal form
45*.20
9
The new price is the original price minus the discount
45-9 = 36
Translate to an equation and solve 133 is the product of -19 and n
Answer:
Step-by-step explanation:
133 is the product of -19 and n
Translate to an equation is:
133 = -19n
then:
n = 133/-19
n = -7
Check:
133 = -19*7
How many feet of chain fence are necessary to enclosed a dog pen that is square and has a area of 64 sq feet
Answer:
32 feet
Step-by-step explanation:
area of square is given by side^2
Perimeter of square is given by 4*side
_______________________________
Given
area of square = 64 sq feet
side^2 = 64
side^2 = 8^2
side = 8
thus side = 8 feet
_______________________________________
The dog pen is fenced with chain, hence chain will be fence at the edge of square and at the perimeter.
Thus, length of chain required will be same as the Perimeter of square.
Perimeter of given dog pen with side length 8 feet = 4*8 = 32 feet.
Thus, 32 feet of chain fence is required.
What is the value of $x$ if $-\frac23(x-5) = \frac32(x+1)$?
Answer:
x = (-29)/5
Step-by-step explanation:
Solve for x:
(2 (x - 5))/3 = (3 (x + 1))/2
Multiply both sides by 6:
(6×2 (x - 5))/3 = (6×3 (x + 1))/2
6/3 = (3×2)/3 = 2:
2×2 (x - 5) = (6×3 (x + 1))/2
6/2 = (2×3)/2 = 3:
2×2 (x - 5) = 3×3 (x + 1)
2×2 = 4:
4 (x - 5) = 3×3 (x + 1)
3×3 = 9:
4 (x - 5) = 9 (x + 1)
Expand out terms of the left hand side:
4 x - 20 = 9 (x + 1)
Expand out terms of the right hand side:
4 x - 20 = 9 x + 9
Subtract 9 x from both sides:
(4 x - 9 x) - 20 = (9 x - 9 x) + 9
4 x - 9 x = -5 x:
-5 x - 20 = (9 x - 9 x) + 9
9 x - 9 x = 0:
-5 x - 20 = 9
Add 20 to both sides:
(20 - 20) - 5 x = 20 + 9
20 - 20 = 0:
-5 x = 9 + 20
9 + 20 = 29:
-5 x = 29
Divide both sides of -5 x = 29 by -5:
(-5 x)/(-5) = 29/(-5)
(-5)/(-5) = 1:
x = 29/(-5)
Multiply numerator and denominator of 29/(-5) by -1:
Answer: x = (-29)/5
Answer:
Step-by-step explanation:
Multiplying both sides by $6$ to get rid of the fractions gives\[6\left(-\frac23\right)(k-6) = 6\left(\frac32\right)(k+6),\]so\[-4(k-6) = 9(k+6).\]Expanding both sides gives $-4k+24 = 9k + 54.$ Adding $4k$ to both sides gives $24 = 13k+54.$ Subtracting $54$ from both sides gives $-30=13k.$ Dividing both sides by $13$ gives $k =-30/13}.$
Help with finding the lettered angles please!
Answer:
i = 77.5°
k = 77.5°
j = 102.5°
h = 155°
Step-by-step explanation:
Since we have an isosceles triangle, we know that ∠i and ∠k are equal. So,
180 - 25 = 2x
155 = 2x
x = 77.5
So m∠i = m∠k = 77.5°
To find m∠j, we use Supplementary Angles:
180 - 77.5 = 102.5°
To find m∠h, we also use Supplementary Angles:
180 - 25 = 155°
You would like to have extra spending money, so you decided to work part-time at the local gym. The job pays $15.00 per hour and you work 20 hours per week. Your employer withholds 10% of your gross pay for federal taxes, 7.65% for FICA taxes, and 3% for state taxes.
Required:
a. What is your weekly gross pay?
b. How much is withheld per week for federal taxes?
c. How much is withheld per week for FICA taxes?
d. How much is withheld per week for state taxes?
e. What is your weekly net pay?
f. What percentage of your gross pay is withheld for taxes? Round to the nearest tenth of a percent.
Answer:
a. Weekly gross pay = $300
b. Federal taxes, F = $30
c. Fica taxes, K = 22.95
d. State taxes, S = $9
e. Weekly net pay = 238.05
Step-by-step explanation:
Gross pay, G = 15 $/h * 20 h = 300 / week
Fed taxes, F = 10%*G = $30
FICA, K = 7.65%*G = $22.95
State taxes, S = 3%*G = $9
a. Weekly gross pay = $300
b. Federal taxes, F = $30
c. Fica taxes, K = 22.95
d. State taxes, S = $9
e. Weekly net pay = 300 - (30+22.95+9) = 300 - 55.95 = 238.05
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A management consulting firm recommends that the ratio of middle-management salaries to management trainee salaries be 9:4. Using this recommendation, what is the annual middle
management salary if the annual management trainee salary is $24,000? (Round your answer to the nearest dollar.)
Enter a number
Answer:
67500
Step-by-step explanation:
you would set up the equation:x/y=5/4
The belief is that the mean number of hours per week of part-time work of high school seniors in a city is 10.5 hours. Data from a simple random sample of 23 high school seniors indicated that their mean number of part-time work was 11.5 with a standard deviation of 1.4. Test whether these data cast doubt on the current belief. (use α = 0.05)
Required:
a. State your null and alternative hypotheses.
b. Sketch the rejection region.
c. Calculate the test statistic. Plot this value in your sketch in part b.
d. Determine the P-value for your test.
e. State your conclusions clearly in complete sentences.
Answer:
a) See step by step explanation
b) See annex
c) t(s) = 3,4255
d) p- value = 0,00146 or 0,15 %
e) See step by step explanation
Step-by-step explanation:
As n < 30 we use a t-student distribution
Population mean μ₀ = 10,5
Sample size n = 23
Degree of freedom n - 1 = 22
Sample mean μ = 11,5
Sample standard deviation s = 1,4
Confidence Interval 95 %
a) Null Hypothesis H₀ μ = μ₀
Alternative Hypothesis Hₐ μ > μ₀
As CI = 95 % α = 5% or α = 0,05
We are solving a one tail-test
With df = 22 and α = 0,05 in t-table we find t(c) = 1,7171
c) t(s) = ( μ - μ₀ ) / s / √n
t(s) = ( 11,5 - 10,5 ) / 1,4 / √23
t(s) = 1 * 4,7958 / 1,4
t(s) = 3,4255
We compare t(s) and t(c)
t(s) > t(c) and t(s) is in the rejection region
Then we reject the null hypothesis.
d) P-value for t(s) = 3,4255 is from t-table equal to:
We find for df = 22 α = 0,001 and α = 0,005
values of t
t 3,505 2, 819 Δt = 0,686
α 0,001 0,005 Δα = 0,004
with these values we interpolate by rule of three
0,686 ⇒ 0,004
(3,505 - 3,4255) ⇒ x
x = 0,000463
and P-value = 0,00146 or 0,15 %
e) The p-value indicates we are far away to consider the accptance of H₀
what is the solution of the inequality shown below
Answer:
there is no inequality..
Step-by-step explanation:
Answer:
???
Step-by-step explanation:
to prove triangleABC is isosceles, which of the following statements can be used in the proof?
(idk the answer)
Answer:
Step-by-step explanation:
An isosceles triangle is a triangle in which two of its sides are equal. This also means that in the triangle, two angles are equal. The angles are usually the base angles. Looking at the given triangle ABC, the base angles are angle Angle A and Angle B, thus angle A = ang B
Therefore, the statement that can be used in the proof is
Angle CAB = angle CBA
9) brainliest & 10 + points!
Answer:
no supplement
Step-by-step explanation:
Supplementary angles add to 180 degrees,
This angle is larger than 180 degrees by itself, so it has no supplement
Yolonda wanted to see if there was a connection between red hair and green eyes. She observed people walking past her on the street
Answer:
I saw one person that way
Step-by-step explanation:
she had red hair and green eyes with pale skin
Given the polynomial function below, find F(-1)
F(x)= -x^3-x^2+1
A. -3
B. 3
C. 1
D. -1
two lines, 3y-2x=21 and 4y+5x=5, intersect at the point Q. find the coordinates of Q.
Answer:
Q = (- 3, 5 )
Step-by-step explanation:
Given the 2 equations
3y - 2x = 21 → (1)
4y + 5x = 5 → (2)
Multiplying (1) by 5 and (2) by 2 and adding will eliminate the x- term.
15y - 10x = 105 → (3)
8y + 10x = 10 → (4)
Add (3) and (4) term by term to eliminate x
23y = 115 ( divide both sides by 23 )
y = 5
Substitute y = 5 into either of the 2 equations and solve for x
Substituting into (2)
4(5) + 5x = 5
20 + 5x = 5 ( subtract 20 from both sides )
5x = - 15 ( divide both sides by 5 )
x = - 3
Solution is (- 3, 5 )
Given that is both the median and altitude of triangle ABC, congruence postulate SAS is used to prove that triangle ABC is what type of triangle?
Answer:
triangle ΔABC is an isosceles triangle.
Step-by-step explanation:
Given : Given that is both the median and altitude of triangle ABC.
To find : congruence postulate SAS is used to prove that triangle ABC is what type of triangle.
Solution : We have given that both the median and altitude of triangle ABC.
Let AD represent both the median and altitude of triangle ABC.
A median divides the side in two equal parts.
So , BD=BC.
An altitude is a perpendicular drawn .
A perpendicular makes an angle of 90°.
Hence <ADB = <ADC = 90°
AD is the side common to both the triangles ADB and ADC.
Hence, Δ ADB≅ΔADC (SAS congruence postulate).
So AB=AC by c.p .c .t.c(congruent parts of congruent triangles are congruent)
Hence by definition of Isosceles triangle ΔABC is an isosceles triangle.
Therefore, triangle ΔABC is an isosceles triangle.
Let ????(t)=⟨t2,1−t,4t⟩r(t)=⟨t2,1−t,4t⟩. Calculate the derivative of ????(t)⋅????(t)r(t)⋅a(t) at t=2t=2, assuming that ????(2)=⟨2,5,−3⟩a(2)=⟨2,5,−3⟩ and ????′(2)=⟨4,−3,9⟩
Answer:
The derivative is [tex]\frac{ d (r(t) \cdot a(t))}{dt} = 82[/tex]
Step-by-step explanation:
From the question we are told that
[tex]r(t) = (t^2 ,1 - t , 4t)[/tex]
[tex]a(2) = (2, 5, -3)[/tex] and [tex]a'(2) = (4,-3 , 9)[/tex]
At t = 2
[tex]r(t) = (2^2 ,1 - 2 , 4(2))[/tex]
[tex]r(t) = (4 ,-1 , 8 )[/tex]
Now the derivative of r(t) is
[tex]r'(t) = (2t, -1 ,4)[/tex]
At t = 2
[tex]r'(t) = (2(2), -1 ,4)[/tex]
[tex]r'(t) = (4, -1 ,4)[/tex]
Now the derivative of [tex]r(t) \cdot a(t)[/tex] At t = 2 is
[tex]= r'(2) a(2) + a'(2)r(2)[/tex]
[tex]= (4,-1,4)(2,5,-3) + (4,-3,9)(4,-1,8)[/tex]
[tex]= (8 - 5 -12) + (16+3+72)[/tex]
[tex]= -9 + 91[/tex]
[tex]\frac{ d (r(t) \cdot a(t))}{dt} = 82[/tex]
Let $x$ be the smallest multiple of $11$ that is greater than $1000$ and $y$ be the greatest multiple of $11$ less than $11^2$. Compute $x - y$.
Answer:
891
Step-by-step explanation:
x has to be 1001 and y has to be 11 * 10 = 110 so x - y = 1001 - 110 = 891.
Answer:
891
Step-by-step explanation:
[tex]$1001$ is the smallest integer greater than $1000$. It also happens to be a multiple of $11$, since $1001 = 11 \cdot 91$. So $1001$ is the smallest multiple of $11$ greater than $1000$ and thus $x = 1001$.The greatest multiple of $11$ that is less than $11^2 = 11 \cdot 11$ is$$11 \cdot (11 - 1) = 11 \cdot 10 = 110$$Thus $y = 110$, and we compute$$x - y = 1001 - 110 = \boxed{891}$$[/tex]
Hope this helped! :)
A local chess club claims that the length of time to play a game has a standard deviation of more than 13 minutes. Write sentences describing type I and type II errors for a hypothesis test of this claim.
Answer:
Step-by-step explanation:
A type I error occurs when the researcher rejects the null hypothesis when it is actually true.
A type II error occurs when the research fails to reject the null hypothesis when it is not true.
In this case study,
The null hypothesis is the standard deviation is less that or equal to 13min.
The alternative hypothesis would be that the standard deviation is greater than 13mins.
A type I error would occur when having done an experiment, the researcher rejects the null hypothesis when there is enough evidence that it is actually either less or equal to 13mins
A type II error would occur when the researcher fails to reject the null hypothesis when there is enough evidence that it is actually more than 13mins.
Find the slope of the line. m =
Answer: m=4
Step-by-step explanation:
To find the slope, we use the formula [tex]m=\frac{y_{2} -y_{1} }{x_{2}-x_{1} }[/tex]. We can use the two points to find the slope. The points on the graph are (-2,1) and (-3,-3).
[tex]m=\frac{-3-1}{-3-(-2)} =\frac{-4}{-1} =4[/tex]
To compare the production techniques used by foreign and local firms in Brazil, a random sample of 80 foreign firms and a random sample of 80 local firms are selected.This study uses_________ design
Answer:
An independent sample.
Step-by-step explanation:
In this scenario, to compare the production techniques used by foreign and local firms in Brazil, a random sample of 80 foreign firms and a random sample of 80 local firms are selected. We can safely conclude that this study uses an independent sample design.
An independent sample design can be defined as a research method that usually involves the use of multiple experimental groups (two or more). The samples or participants are only in one group and as such each group has no relationship with the other. This simply means that, the samples in a particular group is having no relationship with the other samples in another group.
Ultimately this implies, each samples are independent and satisfies only one condition of the independent sample design during the experiment to compare the production technique used by foreign and local firms in Brazil.
Hence, the researcher would use only two variables or conditions: a random sample of 80 foreign firms and a random sample of 80 local firms are selected.
Please answer this correctly
Answer:
5/12
Step-by-step explanation:
The probability of rolling a number greater than 1 is 5/6, because 5 numbers on a 6-sided dice are greater than 1.
The probability of rolling an even number is 3/6, because 3 numbers on a 6-sided dice are even numbers.
[tex]5/6 \times 3/6[/tex]
[tex]=15/36[/tex]
[tex]=5/12[/tex]
Does the point (3.28) lie on the line y = 19+ 3x
Answer:
yes
Step-by-step explanation:
y = 19+ 3x
Let x = 3 and y = 28
28 = 19 + 3*3
28 =19+9
28 = 28
This is true so the point is one the line
Find measure of arc or angle indicated
Please answer this correctly
Answer:
1/9
Step-by-step explanation:
The probability of picking a even number is 1/3
The probability of picking another even number is 1/3(if u put the first one back)
So u multiply 1/3 times 1/3 which gives u 1/9 which is ur answer hope this helps
Answer:
1/9
Step-by-step explanation:
3 cards total
1 even number
P(even) = even/total
1/3
Put the card back
3 cards total
1 even number
P(even) = even/total
1/3
P(even, replace, even) = P(even) * P(even) =1/3*1/3 = 1/9
A multiple of 6 is a number that has 6 as a factor. What is the sum of the two smallest multiples of 6 that are greater than 103?
Answer:
108 and 114
Step-by-step explanation:
Answer:
222
Step-by-step explanation:
We want the first two multiple of 6 that are greater than 103
103/6 =17 1/6
Rounding up
6*18 =108
The next multiple will be 6* 19
6*19 =114
The sum is 108+114 =222
an experiment consists of rolling two fair dice and adding the dots on the two sides facing u. Find the probability of the sum of the dots indicate. A sum less than or equal to 6
Answer:
41.67% probability of the sum of the dots indicate a sum less than or equal to 6
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes:
In this problem, we have these possible outcomes:
Format(Dice A, Dice B)
(1,1), (1,2), (1,3), (1,4), (1,5),(1,6)
(2,1), (2,2), (2,3), (2,4), (2,5),(2,6)
(3,1), (3,2), (3,3), (3,4), (3,5),(3,6)
(4,1), (4,2), (4,3), (4,4), (4,5),(4,6)
(5,1), (5,2), (5,3), (5,4), (5,5),(5,6)
(6,1), (6,2), (6,3), (6,4), (6,5),(6,6)
There are 36 possible outcomes.
Desired outcomes:
Sum of 6 or less. They are:
(1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (4,1), (4,2), (5,1)
15 desired outcomes
15/36 = 0.4167
41.67% probability of the sum of the dots indicate a sum less than or equal to 6
Consider a rat going through a maze, and each time the rat begins the maze he has 30% chance of finishing successfully. The rat goes through the maze over and over again until he is successful in finishing the maze. Whether or not the rat finishes the maze on one trial has no impact on his chance of finishing the maze on the next trial.
What is the probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt? Round your answer to two decimal places.
Answer:
3.53% probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt
Step-by-step explanation:
For each time that the rat goes through the maze, there are only two possible outcomes. Either he fails it, or he succeds. The trials are independent. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
What is the probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt?
Missing on the first six, each with 70% probability(P(X = 6) when p = 0.7, n = 6).
Succeeding on the 7th attempt, with p = 0.3. So
[tex]P = 0.3P(X = 6)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 6) = C_{6,6}.(0.7)^{6}.(0.3)^{0} = 0.117649[/tex]
[tex]P = 0.3P(X = 6) = 0.3*0.117649 = 0.0353[/tex]
3.53% probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt
Answer:
3.53% probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt
Step-by-step explanation:
For each time that the rat goes through the maze, there are only two possible outcomes. Either he fails it, or he succeds. The trials are independent. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
In which is the number of different combinations of x objects from a set of n elements, given by the following formula.
And p is the probability of X happening.
What is the probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt?
Missing on the first six, each with 70% probability(P(X = 6) when p = 0.7, n = 6).
Succeeding on the 7th attempt, with p = 0.3. So
In which
3.53% probability that the rat fails the maze 6 times in a row, and then succeeds on his 7th attempt
A set of prime number between 5 and 15.express it in listing and setbuilder methods
Answer:
A set of prime number between 5 and 15.
Listing method:{ 7,11,13}
Set builder method:{X:X is a set of prime number between 5 and 15}
In listing method,the elements are listed inside the brackets.Listing method are also called rooster method.
In set builder method,the elements are represented by a variable stating their common properties.Set builder method are also called rule method.
Hope this helps...
Good luck on your assignment..
If 1/6x+2/3y=8 what is the value of 2x+8y
Answer: 96
Step-by-step explanation:
Simply multiply the first question by 12 to get 2x+8y=96
Hope it helps <3
Poly(3-hydroxybutyrate) (PHB), a semicrystalline polymer that is fully biodegradable and biocompatible, is obtained from renewable resources. From a sustain-ability perspective, PHB offers many attractive proper-ties though it is more expensive to produce than standard plastics. The accompanying data on melting point (°C) for each of 12 specimens of the polymer using a differential scanning calorimeter appeared in the article "The Melting Behaviour of Poly(3-1-1ydroxybutyrate) by DSC. Reproducibility Study" (Polymer Testing, 2013: 215-220).
180.5 181.7 180.9 181.6 182.6 181.6
181.3 182.1 182.1 180.3 181.7 180.5
Compute the following:
a. The sample range
b. The sample variance S2 from the definition (Hint: First subtract 180 from each observation.]
c. The sample standard deviation
d. S2 using the shortcut method
Answer:
(a) 2.3
(b) 0.5245
(c) 0.7242
(d) 0.5245
Step-by-step explanation:
The data provided is:
S = {180.5, 181.7, 180.9, 181.6, 182.6, 181.6, 181.3, 182.1, 182.1, 180.3, 181.7, 180.5}
(a)
The formula to compute the sample range is:
[tex]\text{Sample Range}=\text{max.}_{x}-\text{min.}_{x}[/tex]
The data set arranged in ascending order is:
S' = {180.3 , 180.5 , 180.5 , 180.9 , 181.3 , 181.6 , 181.6 , 181.7 , 181.7 ,, 182.1 , 182.1 , 182.6}
The minimum value is, 180.3 and the maximum value is, 182.6.
Compute the sample range as follows:
[tex]\text{Sample Range}=\text{max.}_{x}-\text{min.}_{x}[/tex]
[tex]=182.6-180.3\\=2.3[/tex]
Thus, the sample range is 2.3.
(b)
Compute the sample variance as follows:
[tex]S^{2}=\frac{1}{n-1}\sum(x_{i}-\bar x)^{2}[/tex]
[tex]=\frac{1}{12-1}\times [(180.5-181.41)^{2}+(181.7-181.41)^{2}+...+(180.5-181.41)^{2}]\\\\=\frac{1}{11}\times 5.7692\\\\=0.524473\\\\\approx 0.5245[/tex]
Thus, the sample variance is 0.5245.
(c)
Compute the sample standard deviation as follows:
[tex]s=\sqrt{S^{2}}[/tex]
[tex]=\sqrt{0.5245}\\\\=0.7242[/tex]
Thus, the sample standard deviation is 0.7242.
(d)
Compute the sample variance using the shortcut method as follows:
[tex]S^{2}=\frac{1}{n-1}\cdot [\sum x_{i}^{2}-n(\bar x)^{2}][/tex]
[tex]=\frac{1}{12-1}\cdot [394913.57-(12\times (181.41)^{2}]\\\\=\frac{1}{11}\times [394913.57-394907.80]\\\\=\frac{5.77}{11}\\\\=0.5245[/tex]
Thus, the sample variance is 0.5245.