Find all x ∈ Z satisfying each of the following equations.
(a) 3x ≡ 2 (mod 7)
(b) 5x + 1 ≡ 13 (mod 23)
(c) 5x + 1 ≡ 13 (mod 26)
(d) 9x ≡ 3 (mod 5)
(e) 5x ≡ 1 (mod 6)
(f) 3x ≡ 1 (mod 6)
Answers
Answer:
answers are shown in the file attached
Step-by-step explanation:
The detailed step are as shown in the attached file.
The solution of the all the equations are:
a) x = {10 + 7k}, where k is an integer.
b) x = {2 + 23k}, where k is an integer.
c) x = {18 + 26k}, where k is an integer.
d) x = {2 + 5k}, where k is an integer.
e) x = {5 + 6k}, where k is an integer.
f) x = {1 + 6k}, where k is an integer.
To find all integer solutions (x ∈ Z) for each equation, we need to solve them using modular arithmetic.
For each equation, we will use different approaches depending on the specific form of the equation.
Let's solve them one by one:
(a) 3x ≡ 2 (mod 7):
To find x, we'll multiply both sides of the equation by the modular inverse of 3 modulo 7, which is 5 (since 3 × 5 ≡ 1 (mod 7)):
3x × 5 ≡ 2 × 5 (mod 7)
15x ≡ 10 (mod 7)
Now, we find the smallest non-negative integer solution by dividing both sides by the greatest common divisor (GCD) of 15 and 7 (which is 1 since 15 and 7 are coprime):
x ≡ 10 (mod 7)
The solution set is x = {10, 17, 24, 31, ...}.
Since we are looking for integer solutions, we can simplify it to x = {10 + 7k}, where k is an integer.
(b) 5x + 1 ≡ 13 (mod 23):
Subtract 1 from both sides:
5x ≡ 12 (mod 23)
Next, we'll find the modular inverse of 5 modulo 23, which is 14 (since 5 * 14 ≡ 1 (mod 23)):
5x × 14 ≡ 12 × 14 (mod 23)
70x ≡ 168 (mod 23)
Now, reduce the coefficients to the smallest positive residue modulo 23:
x ≡ 2 (mod 23)
The solution set is x = {2, 25, 48, ...}, which can be simplified to x = {2 + 23k}, where k is an integer.
(c) 5x + 1 ≡ 13 (mod 26):
This equation is the same as the previous one. Follow the same steps:
5x ≡ 12 (mod 26)
Find the modular inverse of 5 modulo 26, which is 21 (since 5 × 21 ≡ 1 (mod 26)):
5x × 21 ≡ 12 × 21 (mod 26)
105x ≡ 252 (mod 26)
Reduce the coefficients to the smallest positive residue modulo 26:
x ≡ 18 (mod 26)
The solution set is x = {18, 44, 70, ...}, which can be simplified to x = {18 + 26k}, where k is an integer.
(d) 9x ≡ 3 (mod 5):
To find x, we'll multiply both sides by the modular inverse of 9 modulo 5, which is 4 (since 9 × 4 ≡ 1 (mod 5)):
9x × 4 ≡ 3 × 4 (mod 5)
36x ≡ 12 (mod 5)
Reduce the coefficients to the smallest positive residue modulo 5:
x ≡ 2 (mod 5)
The solution set is x = {2, 7, 12, ...}, which can be simplified to x = {2 + 5k}, where k is an integer.
(e) 5x ≡ 1 (mod 6):
To find x, we'll multiply both sides by the modular inverse of 5 modulo 6, which is 5 (since 5 × 5 ≡ 1 (mod 6)):
5x × 5 ≡ 1 × 5 (mod 6)
25x ≡ 5 (mod 6)
Reduce the coefficients to the smallest positive residue modulo 6:
x ≡ 5 (mod 6)
The solution set is x = {5, 11, 17, ...}, which can be simplified to x = {5 + 6k}, where k is an integer.
(f) 3x ≡ 1 (mod 6):
To find x, we'll multiply both sides by the modular inverse of 3 modulo 6, which is 3 (since 3 × 3 ≡ 1 (mod 6)):
3x × 3 ≡ 1 × 3 (mod 6)
9x ≡ 3 (mod 6)
Reduce the coefficients to the smallest positive residue modulo 6:
x ≡ 1 (mod 6)
The solution set is x = {1, 7, 13, ...}, which can be simplified to x = {1 + 6k}, where k is an integer.
Learn more about modular arithmetic click;
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Related Questions
It is known that a cable with a cross-sectional area of 0.300.30 sq in. has a capacity to hold 2500 lb. If the capacity of the cable is proportional to its cross-sectional area, what size cable is needed to hold 70007000 lb?
Answers
Answer:
0.84 square in
Step-by-step explanation:
Since the capacity of the cable is proportional to its cross-sectional area. If a cable that is 0.3 sq in can hold 2500 lb then per square inch it can hold
2500 / 0.3 = 8333.33 lb/in
To old 7000 lb it the cross-sectional area would need to be
7000 / 8333.33 = 0.84 square in
Answer:
0.84 square in.
Step-by-step explanation:
Since the capacity of the cable is proportional to its cross-sectional area.
Mathematically,
C = k * A
A1 = 0.3 sq
C1 = 2500 lb
C2 = 7000 lb
C1/A1 = C2/A2
2500/0.3 = 7000/C2
= 7000 / 8333.33
= 0.84 square in.
what would 20% of 1,500 be?
Answers
Answer:
300
Step-by-step explanation:
20% of 1,500 = 300
(Source: Google)
Answer:
300
Step-by-step explanation:
A researcher is studying psychological factors in academic achievement among teenage girls. One variable he is particularly interested in is competitiveness. What information does a measure of variability for the variable competitiveness convey?
a.Do all teenage girls have the same amount of competitiveness?
b.How spread out are the values for competitiveness amoung teenage girls?
c.What is the central tendency for the variable competitiveness among teenage girls?
d.What is the most common value for the variable competitiveness among teenage girls?
Answers
Answer:
The information a measure of variability for the variable competitiveness conveys is
The first option is correct
a.) Do all teenage girls have the same amount of competitiveness?
The second option is correct
b.) How spread out are the values for competitiveness among teenage girls?
Step-by-step explanation:
The first option is correct
a.) Do all teenage girls have the same amount of competitiveness?
The second option is correct
b.) How spread out are the values for competitiveness among teenage girls?
The third option is NOT correct. The given problem is a measure of dispersion not central tendency.
c.) What is the central tendency for the variable competitiveness among teenage girls?
The fourth option is also NOT correct. The problem given does not talk about a specific value of the competitiveness variable but rather it talks about the variability of the competitiveness variable
d.) What is the most common value for the variable competitiveness among teenage girls?
The starting salaries of individuals with an MBA degree are normally distributed with a mean of $40,000 and a standard deviation of $5,000.What is the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $30,000?1) 0.97722) 0.50003) 0.99874) 0.0228
Answers
Answer:
1) 0.9772
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 40000, \sigma = 5000[/tex]
What is the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $30,000?
This is 1 subtracted by the pvalue of Z when X = 30000. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{30000 - 40000}{5000}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228.
1 - 0.0228 = 0.9772
0.9772 that a randomly selected individual with an MBA degree will get a starting salary of at least $30,000
Final answer:
Using the normal distribution, the probability that a randomly selected individual with an MBA degree will get a starting salary of at least $30,000 is approximately 0.9772.
Explanation:
The probability of a randomly selected individual with an MBA degree getting a starting salary of at least $30,000 is determined using the normal distribution with a mean of $40,000 and a standard deviation of $5,000. You calculate the z-score for $30,000, which is (30,000 - 40,000) / 5,000 = -2.
Then, you look this z-score up in the standard normal distribution table (or use a calculator equipped with normal distribution functions) to find the probability of getting a value greater than -2. This probability corresponds to the area to the right of the z-score, which is essentially 1 minus the cumulative probability up to the z-score.
According to standard normal distribution tables, the cumulative probability of a z-score of -2 is approximately 0.0228. Therefore, the probability of getting at least $30,000 is
1 - 0.0228, which equals approximately 0.9772.
Comparing this to available choices, option 1) 0.9772 is the correct answer.
Suppose the probability of an unsuccessful missile launch is 0.3. If missiles continue to be launched until an unsuccessful launch occurs, what is the probability that exactly 4 total launches will be performed (round off to second decimal place)?
Answers
Answer:
0.1029 or 10.29%
Step-by-step explanation:
P(F) =0.3
P(S) = 1-0.3 = 0.7
If missiles continue to be launched until an unsuccessful launch occurs, the probability that exactly 4 total launches will be performed is the probability that the first three launches will be successful while the fourth will be unsucessful:
[tex]P(L=4) = P(S)*P(S)*P(S)*P(F)\\P(L=4) = 0.7*0.7*0.7*0.3\\P(L=4) = 0.1029 = 10.29\%[/tex]
The probability of 10 total launches is 0.1029 or 10.29%.
Calculate the probability of three successful launches [tex](0.7^3)[/tex] followed by one unsuccessful launch (0.3), which equals 10.29%.
The subject of the question is probability, specifically related to geometric distributions. The probability of an unsuccessful missile launch is given as 0.3. To find the probability that exactly 4 total launches will be performed before the first unsuccessful launch occurs, we need to calculate the probability of having three successful launches followed by one unsuccessful launch.
The probability of a successful launch is therefore 1 - 0.3 = 0.7. Since each launch is independent, the probability of exactly three successes followed by one failure is [tex](0.7)^3[/tex] times 0.3. This is calculated as (0.7 imes 0.7 imes 0.7) times 0.3 which equals 0.1029, or 10.29% when rounded to two decimal places.
"There are 15 questions on an exam. In how many ways can the exam be answered with exactly 8 answers correct?"
Answers
Answer:
The exam can be answered with exactly 8 answers correct in 6435 ways.
Step-by-step explanation:
The order is not important.
For example, answering correctly the questions 1,2,3,4,5,6,7,8 is the same outcome as answering 2,1,3,4,5,6,7,8. So we use the combinations formula to solve this problem.
Combinations formula:
[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]
"There are 15 questions on an exam. In how many ways can the exam be answered with exactly 8 answers correct?"
Combinations of 8 questions from a set of 15. So
[tex]C_{15,8} = \frac{15!}{8!(15-8)!} = 6435[/tex]
The exam can be answered with exactly 8 answers correct in 6435 ways.
To find the number of ways to answer 15 exam questions with exactly 8 correct answers, you use the binomial coefficient formula. The calculation yields 15C8 = 6,435. Thus, there are 6,435 ways to answer the exam with 8 correct answers.
The formula for finding the number of ways to choose k items from n items is given by:
nCk = n! / [k!(n-k)!]
In this case, we need to find the number of ways to get exactly 8 correct answers out of 15 questions:
n = 15 (total questions)k = 8 (correct answers)Plugging these values into the formula, we get:
15C8 = 15! / [8!(15-8)!]
Which simplifies to:
15C8 = 15! / (8! * 7!)
Calculating the factorial values:
15! = 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 18! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 17! = 7 × 6 × 5 × 4 × 3 × 2 × 1By cancelling out the common terms, we get:
15C8 = (15 × 14 × 13 × 12 × 11 × 10 × 9) / (7 × 6 × 5 × 4 × 3 × 2 × 1) = 6435
Conclusion
There are 6,435 ways to answer the exam with exactly 8 answers correct.
Suppose you deposit $1,250 at the end of each quarter in an account that will earn interest at an annual rate of 15 percent compounded quarterly. How much will you have at the end of four years
Answers
Answer:
The amount at the end of 4 years is $2,252.79.
Step-by-step explanation:
The amount formula for the compound interest compounded quarterly is:
[tex]A=P[1+\frac{r}{4}]^{4t}[/tex]
Here,
A = Amount after t years
P = Principal amount
t = number of years
r = interest rate
Given:
P = $1,250, r = 0.15, t = 4 years.
The amount at the end of 4 years is:
[tex]A=P[1+\frac{r}{4}]^{4t}\\=1250\times[1+\frac{0.15}{4}]^{4\times4}\\=1250\times1.80223\\=2252.7875\\\approx2252.79[/tex]
Thus, the amount at the end of 4 years is $2,252.79.
Final answer:
To calculate the amount of money you will have at the end of four years with quarterly deposits and compounded interest, use the formula for compound interest:
[tex]A = P(1 + r/n)^{nt}[/tex]. Substituting the given values, the result is approximately $1,776.40.
Explanation:
To calculate the amount of money you will have at the end of four years with quarterly deposits and compounded interest, you can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the total amount of money after the specified time period
P is the principal amount (initial deposit)
r is the annual interest rate (15% in this case)
n is the number of times interest is compounded per year (4 in this case for quarterly compounding)
t is the specified time period in years (4 in this case)
Let's substitute the given values into the formula:
[tex]A = 1250(1 + 0.15/4)^{4*4}[/tex][tex]A = 1250(1 + 0.0375)^{16}[/tex][tex]A = 1250(1.0375)^{16}[/tex]A ≈ 1250 * 1.82212
A ≈ $1,1776.4
Therefore, you will have approximately $1,776.40 at the end of four years.
A group of 22 7th grade girls is to be divided into a varsity team and a junior varsity team of 11 each. How many different divisions are possible?
Answers
Answer:
705,432 ways
Step-by-step explanation:
Since no girl will be left out once both teams are selected when selecting the varsity team, the junior varsity team is automatically composed by the players not selected, the number of ways to select both teams is:
[tex]n = \frac{22!}{(22-11)!11!} \\n=705,432[/tex]
There are 705,432 ways to divide the girls into a varsity team and a junior varsity team.
An airline charges the following baggage fees: $25 for the first bag and $35 for the second. Suppose 51% of passengers have no checked luggage, 32% have one piece of checked luggage and 17% have two pieces. We suppose a negligible portion of people check more than two bags. The average baggage-related revenue per passenger is: _____.
Answers
Answer:
The average baggage-related revenue per passenger is $18.20
Step-by-step explanation:
$25 for the first bag
$35 for the second bag
Total passenger = 100
51 passenger have no checked baggage
32 passenger have one checked baggage. Therefore the airlines charges for the 32 passenger will be: [tex]25 * 32 = 800[/tex].
17 passenger have two checked baggage. Therefore the airline charges for the 17 passenger will be: [tex](25 *17) + (35*17) = 425+595=1020[/tex]
Average baggage related revenue per passenger is: Total Revenue / passenger
Total Revenue = 1020 + 800 = $1820
Average = 1820/100 = $18.20
In the 2012 presidential election, exit polls from the critical state of Ohio provided the following results:
Total Obama Romney
Non college degree (60%) 52% 45%
College degree (40%) 47% 51%
What is the probability that a randomly selected respondent voted for Obama?
Answers
Answer:
50% probability that a randomly selected respondent voted for Obama.
Step-by-step explanation:
We have these following probabilities:
60% probability that an Ohio resident does not have a college degree.
If an Ohio resident does not have a college degree, a 52% probability that he voted for Obama.
40% probability that an Ohio resident has a college degree.
If an Ohio resident has a college degree, a 47% probability that he voted for Obama.
What is the probability that a randomly selected respondent voted for Obama?
This is the sum of 52% of 60%(non college degree) and 47% of 40%(college degree).
So
[tex]P = 0.52*0.6 + 0.47*0.4 = 0.5[/tex]
50% probability that a randomly selected respondent voted for Obama.
PLEASE HELP ME FAST!!! (I have a timer on my test!!!)
What is the quotient of - 3/8 and - 1/3 ?
Answers
Answer:
-3/8 = -0,375
-1/3 = -0,33
Step-by-step explanation:
N/A
Answer:when you divide two number the answer is called the quotient.
Step-by-step explanation:
-3/8
=-0.375
and
-1/3
=-0.333
26. Define a relation ∼ ∼ on R 2 R2 by stating that ( a , b ) ∼ ( c , d ) (a,b)∼(c,d) if and only if a 2 + b 2 ≤ c 2 + d 2 . a2+b2≤c2+d2. Show that ∼ ∼ is reflexive and transitive but not symmetric.
Answers
Answer:
~ is reflexive.
~ is asymmetric.
~ is transitive.
Step-by-step explanation:
~ is reflexive:
i.e., to prove [tex]$ \forall (a, b) \in \mathbb{R}^2 $[/tex], [tex]$ (a, b) R(a, b) $[/tex].
That is, every element in the domain is related to itself.
The given relation is [tex]$\sim: (a,b) \sim (c, d) \iff a^2 + b^2 \leq c^2 + d^2$[/tex]
Reflexive:
[tex]$ (a, b) \sim (a, b) $[/tex] since [tex]$ a^2 + b^2 = a^2 + b^2 $[/tex]
This is true for any pair of numbers in [tex]$ \mathbb{R}^2 $[/tex]. So, [tex]$ \sim $[/tex] is reflexive.
Symmetry:
[tex]$ \sim $[/tex] is symmetry iff whenever [tex]$ (a, b) \sim (c, d) $[/tex] then [tex]$ (c, d) \sim (a, b) $[/tex].
Consider the following counter - example.
Let (a, b) = (2, 3) and (c, d) = (6, 3)
[tex]$ a^2 + b^2 = 2^2 + 3^2 = 4 + 9 = 13 $[/tex]
[tex]$ c^2 + d^2 = 6^2 + 3^2 = 36 + 9 = 42 $[/tex]
Hence, [tex]$ (a, b) \sim (c, d) $[/tex] since [tex]$ a^2 + b^2 \leq c^2 + d^2 $[/tex]
Note that [tex]$ c^2 + d^2 \nleq a^2 + b^2 $[/tex]
Hence, the given relation is not symmetric.
Transitive:
[tex]$ \sim $[/tex] is transitive iff whenever [tex]$ (a, b) \sim (c, d) \hspace{2mm} \& \hspace{2mm} (c, d) \sim (e, f) $[/tex] then [tex]$ (a, b) \sim (e, f) $[/tex]
To prove transitivity let us assume [tex]$ (a, b) \sim (c, d) $[/tex] and [tex]$ (c, d) \sim (e, f) $[/tex].
We have to show [tex]$ (a, b) \sim (e, f) $[/tex]
Since [tex]$ (a, b) \sim (c, d) $[/tex] we have: [tex]$ a^2 + b^2 \leq c^2 + d^2 $[/tex]
Since [tex]$ (c, d) \sim (e, f) $[/tex] we have: [tex]$ c^2 + d^2 \leq e^2 + f^2 $[/tex]
Combining both the inequalities we get:
[tex]$ a^2 + b^2 \leq c^2 + d^2 \leq e^2 + f^2 $[/tex]
Therefore, we get: [tex]$ a^2 + b^2 \leq e^2 + f^2 $[/tex]
Therefore, [tex]$ \sim $[/tex] is transitive.
Hence, proved.
The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.06 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel. (a) What is the probability that there are no surface flaws in an auto's interior
Answers
Answer:
There is a 54.88% probability that there are no surface flaws in an auto's interior.
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
[tex]e = 2.71828[/tex] is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution with a mean of 0.06 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.
For one square foot, we have 0.06 flaws.
So for 10 square feet, the mean is [tex]\mu = 10*0.06 = 0.6[/tex]
(a) What is the probability that there are no surface flaws in an auto's interior
This is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.6}*(0.6)^{0}}{(0.6)!} = 0.5488[/tex]
There is a 54.88% probability that there are no surface flaws in an auto's interior.
Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedurex1+4x2 =112x1+7x2=18Find the solution to the system of equations.(Simplify your answer. Type an ordered pair.)
Answers
Answer:
-5; 4
Step-by-step explanation:
The given linear system is:
[tex]x_1+4x_2=11\\2x_1+7x_2=18[/tex]
Multiplying the first equation by -2 and adding to the second gives:
[tex]x_1+4x_2=11\\2x_1-2x_1_+7x_2-8x_2=18 -22\\\\x_1+4x_2=11\\-x_2=-4[/tex]
Multiply the second equation by 4 and add to the first to find x1:
[tex]x_1+4x_2-4x_2=11-16\\-x_2=-4\\\\x_1=-5\\x_2=4[/tex]
The order pair for the solution of the system is -5; 4.
Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit. Let 1- HIT and 0 OUT. The output from the simulation was as follows 10001001001110000111 1000011001111 000001111 Estimate the probability that he gets a hit. A) 0.301 B) 0.286 C) 0.452 D) 0.476
Answers
Answer:
The answer is D) 0.476
Step-by-step explanation:
If 1 represents HIT, and 0 represents OUT, the probability that Sammy will get a hit = the number of HITS (1s) ÷ the output from the simulation (i.e., total number of HITs and OUTs in the simulation)
Where:
the number of HITS (1s) = 20
the output from the simulation (i.e., total number of HITs and OUTs in the simulation) = 42
therefore, the probability that Sammy will get a hit = 20 ÷ 42 = 0.476
This implies that the correct answer is D) 0.476
Find the balance on 1,800 deposited at 5% annually for 4 years
Answers
Answer:
The balance after 5 years will be $2,187.91.
Step-by-step explanation:
This problem can be solved by the following formula:
[tex]A = P(1 + r)^{t}[/tex]
In which A is the final amount(balance), P is the principal(the deposit), r is the interest rate and t is the time, in years.
In this problem, we have that:
[tex]P = 1800, r = 0.05, t = 4[/tex]
We want to find A
So
[tex]A = P(1 + r)^{t}[/tex]
[tex]A = 1800(1+0.05)^{5}[/tex]
[tex]A = 2,187.91[/tex]
The balance after 5 years will be $2,187.91.
Answer: 2188
Step-by-step explanation:
so the person deposited 1800 and expects an annual raise in the amount of 5 percent
so the equation for 1 year is
1800(1+(5/100))=the answer
but for four years u will have to power the bract by 4
180(1+(5/100))^4= 2187.9
aka 2188
Jenna is planning a 5-day resort vacation with three of her friends. It will cost her $279 for airfare, $300 for food and entertainment, and $65 per day for her share of the hotel. She has $550 saved for her vacation and can earn $25 per hour as an assistant in her uncle’s photography studio. How many hours must she work in order to have enough money for her vacation? Round up to the nearest whole number.
Answers
Answer:
Jenna will have to work for 210 hours
Step-by-step explanation:
i) vacation is of six days
ii) $279 for airfare
iii) $300 for food and entertainment
iv) $65 per day for her share of the hotel, for 6 days = $65 [tex]\times[/tex] 6 = $390
v) Total expenses = $279 + $300 + $390 = $969
vi) Let x be the number of hours she will have to work to save for the vacation, where she earns $25 per hours
vii) She has $550 saved for the vacation
viii) Total savings + total earnings = Total expenses
therefore 550 + 25x = 969
therefore x = (969 -550) / 25 = 209.5 hours
therefore Jenna will have to work for 209.5 hours, or 210 hours (rounded up to the nearest whole number)
Answer:
She must work 15 more hours to pay for her vacation
Step-by-step explanation:
279+300+5(65)
579+325=904
She has already made 550 dollars so for the next part of the problem;
904-550=354
354/25=14.16
Normally we would round down but since she needs to make the minimum amount for her trip we round up to 15 hours.
Hope that helps, this is the correct answer.
An appliance dealer sells three different models of upright freezers having 13.5, 15.9, and 19.1 cubic feet of storage space, respectively. Let X = the amount of storage space purchased by the next customer to buy a freezer. Suppose that X has the following pmf.
x 13.5 15.9 19.1
p(x) 0.17 0.57 0.26
a. Compute E(X), E(X2), and V(X).
b. If the price of a freezer having capacity X cubic feet is 28X − 8.5, what is the expected price paid by the next customer to buy a freezer? (Round your answer to the nearest whole number.)
c. What is the variance of the price 28X − 8.5 paid by the next customer? (Round your answer to the nearest whole number.)
d. Suppose that although the rated capacity of a freezer is X, the actual capacity is h(X) = X − 0.02X2. What is the expected actual capacity of the freezer purchased by the next customer? (Round your answer to three decimal places.)
Answers
Answer:
a) [tex] E(X) =13.5*0.17 + 15.9*0.57 + 19.1*0.26 = 16.324[/tex]
[tex] E(X^2) =13.5^2*0.17 + 15.9^2*0.57 + 19.1^2*0.26 = 269.9348[/tex]
[tex] Var(X) = E(X^2) -[E(X)]^2 = 269.9348-(16.324)^2 = 3.462[/tex]
b) [tex] E(Y)= E(28X-8.5) = E(28X) - E(8.5) = 28 E(X) -8.5[/tex]
And replacing the result from part a we got:
[tex] E(Y) = 28*16.324 -8.5= 448.572[/tex]
c) [tex] Var(28X-8.5) = Var (28X)= 28^2 Var(X)= 784*3.462=2714.208[/tex]
d) [tex] E(H) = E(X -0.02 X^2) = E(X) -0.02 E(X^2) = 16.324-0.02(269.9348)= 10.925[/tex]
Step-by-step explanation:
For this case we have the following probability function given:
x 13.5 15.9 19.1
p(x) 0.17 0.57 0.26
The expected value of a random variable X is the n-th moment about zero of a probability density function f(x) if X is continuous, or the weighted average for a discrete probability distribution, if X is discrete.
The variance of a random variable X represent the spread of the possible values of the variable. The variance of X is written as Var(X).
Part a
We can calculate the expected value with the following formula:
[tex] E(X) = \sum_{i=1}^n X_i P(X_i)[/tex]
And if we replace we got:
[tex] E(X) =13.5*0.17 + 15.9*0.57 + 19.1*0.26 = 16.324[/tex]
For the second moment we can use this definition:
[tex] E(X^2) = \sum_{i=1}^n X^2_i P(X_i)[/tex]
And if we replace we got:
[tex] E(X^2) =13.5^2*0.17 + 15.9^2*0.57 + 19.1^2*0.26 = 269.9348[/tex]
The variance is defined:
[tex] Var(X) = E(X^2) -[E(X)]^2 = 269.9348-(16.324)^2 = 3.462[/tex]
Part b
For this case we define this new random variable Y = 28 X -8.5. And we want to find the expected value, so we have this:
[tex] E(Y)= E(28X-8.5) = E(28X) - E(8.5) = 28 E(X) -8.5[/tex]
And replacing the result from part a we got:
[tex] E(Y) = 28*16.324 -8.5= 448.572[/tex]
Part c
For the variance we can use the following property:
[tex]Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y)[/tex]
And using this formula we have:
[tex] Var(28X -8.5) = Var(28X)+ Var(8.5)+ 2 Cov(28X,-8.5)[/tex]
The variance for a constant is 0 so then Var(8.5)=0 and Cov(28X, -8.5) = 0 since by properties if X is a random variable and a represent a constant [tex] Cov(X,a)=0[/tex], so then we just have this:
[tex] Var(28X-8.5) = Var (28X)[/tex]
Using the following property [tex] Var(aX)= a^2 Var(X)[/tex] we have:
[tex] Var(28X-8.5) = Var (28X)= 28^2 Var(X)= 784*3.462=2714.208[/tex]
Part d
For this case we define [tex] H = X -0.02 X^2[/tex]
And if we find the expected value we have this:
[tex] E(H) = E(X -0.02 X^2) = E(X) -0.02 E(X^2) = 16.324-0.02(269.9348)= 10.925[/tex]
To compute E(X), E(X²), and V(X), multiply storage space values by their corresponding probabilities. The expected price paid is found by substituting X into the price equation and computing E(28X - 8.5). To find the expected actual capacity, substitute X into the equation h(X) = X - 0.02X² and compute E(h(X)) by multiplying each storage space value by its corresponding probability and summing up the results.
Explanation:To compute the expected value (E(X)), we multiply each storage space value by its corresponding probability and sum up the results. For E(X²), we square each storage space value, multiply it by its corresponding probability, and sum up the results. To find the variance (V(X)), we subtract the square of E(X) from E(X²). The expected price paid is found by substituting X into the price equation and computing E(28X - 8.5). The variance of the price is found by substituting X into the price equation, computing V(28X - 8.5), and rounding to the nearest whole number. To find the expected actual capacity, we substitute X into the equation h(X) = X - 0.02X² and compute E(h(X)) by multiplying each storage space value by its corresponding probability and summing up the results.
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Average Earnings of Workers The average earnings of year-round full-time workers 25–34 years old with a bachelor’s degree or higher were $58,500 in 2003. If the standard deviation is $11,200, what can you say about the percentage of these workers who earn?
a. Between $47.300 and $69,700?
b. More than $80.900?
c. How likely is it that someone earns more than $100,000?
Answers
Answer:
a. 68% of the workers will earn between $47300 and $69700.
b. 2.5% of workers will earn above $89000
c. Approximately 0
Step-by-step explanation:
The standard normal distribution curve in the attached graph is used to solve this question.
a. The value $47300 is a standard deviation below the mean i.e. 58500-11200=47300. While $69700 is a standard deviation above the mean. I.e. 58500+12000=69700.
Between the first deviation below and above the mean, you have 34%+34%=68% of the salary earners within this range. So we have 68%of staffs earning within this range
b. The second standard deviation above the mean is $80900. i.e. 58500+11200+11200=$80900
We have 50%+13.5%+2.5%= 97.5% earning below $80900. Therefore, 100-97.5= 2.5% of the workers earn above this amount.
c. From the Standard Deviation Rule, the probability is only about (1 -0 .997) / 2 = 0.0015 that a normal value would be more than 3 standard deviations away from its mean in one direction or the other. The probability is only 0.0002 that a normal variable would be more than 3.5 standard deviations above its mean. Any more standard deviations than that, and we generally say the probability is approximately zero.
To answer the question, we use z-scores and a z-table to find the percentages of workers who earn within certain ranges or above certain amounts.
Explanation:To answer this question, we can use the concept of the standard normal distribution. First, we convert the given earnings into z-scores by subtracting the mean and dividing by the standard deviation. With these z-scores, we can then use a z-table to find the percentage of workers who earn within a certain range or above a certain amount.
a. To find the percentage of workers who earn between $47,300 and $69,700, we need to convert these values into z-scores and find the area between these two z-scores on the z-table.
b. To find the percentage of workers who earn more than $80,900, we need to convert this value into a z-score and find the area to the right of this z-score on the z-table.
c. To determine how likely it is that someone earns more than $100,000, we need to convert this value into a z-score and find the area to the right of this z-score on the z-table.
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A manufacturer of personal computers tests competing brands and finds that the amount of energy they require is normally distributed with a mean of 285 kwh and a standard deviation of 9.1 kwh. If the lowest 25% and the highest 30% are not included in a second round of tests, what are the upper and lower limits for the energy amounts of the remaining computers?
Answers
Answer:
[tex]a=285 -0.674*9.1=278.87[/tex]
So the value of height that separates the bottom 25% of data from the top 75% is 278.87.
[tex]a=285 +0.524*9.1=289.77[/tex]
So the value of height that separates the bottom 70% of data from the top 30% is 289.77.
The answer would be 278.87 and 289.77
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the amount of energy of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(285,9.1)[/tex]
Where [tex]\mu=285[/tex] and [tex]\sigma=9.1[/tex]
Lowest 25%
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.75[/tex] (a)
[tex]P(X<a)=0.25[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.25 of the area on the left and 0.75 of the area on the right it's z=-0.674. On this case P(Z<-0.674)=0.25 and P(z>-0.674)=0.75
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.25[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.25[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-0.674<\frac{a-285}{9.1}[/tex]
And if we solve for a we got
[tex]a=285 -0.674*9.1=278.87[/tex]
So the value of height that separates the bottom 25% of data from the top 75% is 278.87.
Highest 30%
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.30[/tex] (a)
[tex]P(X<a)=0.70[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.7 of the area on the left and 0.3 of the area on the right it's z=0.524. On this case P(Z<0.524)=0.7 and P(z>0.524)=0.3
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.7[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.7[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=0.524<\frac{a-285}{9.1}[/tex]
And if we solve for a we got
[tex]a=285 +0.524*9.1=289.77[/tex]
So the value of height that separates the bottom 70% of data from the top 30% is 289.77.
The lower limit for the energy amounts of the remaining computers is 279.03 kwh, and the upper limit is 290.74 kwh.
Explanation:To find the upper and lower limits for the energy amounts of the remaining computers, we need to find the z-scores corresponding to the lowest 25% and highest 30% of the distribution.
Using the invNorm function in a calculator, we find that the z-score for the lowest 25% is approximately -0.674 and the z-score for the highest 30% is approximately 0.524.
To find the corresponding energy amounts, we use the formula:
Lowest Energy Amount = Mean + (Z-score * Standard Deviation)
Highest Energy Amount = Mean + (Z-score * Standard Deviation)
Substituting the z-scores and the given values, we get:
Lowest Energy Amount = 285 + (-0.674 * 9.1) = 279.03 kwh
Highest Energy Amount = 285 + (0.524 * 9.1) = 290.74 kwh
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Industrial engineers periodically conduct "work measurement" analyses to determine the time required to produce a single unit of output. At a large processing plant, the number of total worker-hours required per day to perform a certain task was recorded for 50 days. a. Compute the mean, median, and mode of the data set. b. Find the range, variance, and standard deviation of the data set. c. Construct the intervals using the Empirical rule. d. Find the 70th percentile for the data on total daily worker-hours.
Answers
Answer:
a) [tex] \bar X =117.8[/tex]
[tex] Median= \frac{117+118}{2}=117.5[/tex]
The mode on this case is the most repeated value 128 with a frequency of 3
b) [tex] Range = Max -Min = 150-88=62[/tex]
[tex] s^2 = 225.334[/tex]
[tex] s= \sqrt{225.334}= 15.011[/tex]
c) [tex] y \pm s[/tex]
[tex] Lower = 117.8 -15.011=102.809[/tex]
[tex] Upper = 117.8 +15.011=132.831[/tex]
[tex] y \pm 2s[/tex]
[tex] Lower = 117.8 -2*15.011=87.797[/tex]
[tex] Upper = 117.8 +2*15.011=147.842[/tex]
[tex] y \pm 3s[/tex]
[tex] Lower = 117.8 -3*15.011=72.787[/tex]
[tex] Upper = 117.8 +3*15.011=162.85[/tex]
d) For this case we can calculate the position where we have accumulated 70% of the data below.
50*0.7 = 35
So on the position 35th from the dataset ordered we see that the value is 128 and this value would represent the 70th percentile on this case.
Step-by-step explanation:
For this case we consider the following data:
128,119,95,97,124,128,142,98,108,120,113,109,124,132,97,138,133,136,120,112,146,128,103,135,114,109,100,111,131,113,124,131,133,131,88,118,116,98,112,138,100,112,111,150,117,122,97,116,92,122
Part a
For this case we can calculate the mean with the following formula:
[tex] \bar X = \frac{\sum_{i=1}^{50} X_i}{50}[/tex]
And after replace we got [tex] \bar X =117.8[/tex]
In order to calculate the median first we order the dataset and we got:
88 92 95 97 97 97 98 98 100 100 103 108 109 109 111 111 112 112 112 113 113 114 116 116 117 118 119 120 120 122 122 124 124 124 128 128 128 131 131 131 132 133 133 135 136 138 138 142 146 150
The median would be the average between the position 25 and 26 from the data ordered and we got:
[tex] Median= \frac{117+118}{2}=117.5[/tex]
The mode on this case is the most repeated value 128 with a frequency of 3
Part b
the range is defined as the difference between the maximun and minimum so we got:
[tex] Range = Max -Min = 150-88=62[/tex]
The sample variance can be calculated with this formula:
[tex] s^2 = \frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}[/tex]
And after calculate we got: [tex] s^2 = 225.334[/tex]
And the deviation is just the square root of the variance and we got:
[tex] s= \sqrt{225.334}= 15.011[/tex]
Part c
For this case we can construct the interval with 1 , 2 and 3 deviation from the mean like this:
[tex] y \pm s[/tex]
[tex] Lower = 117.8 -15.011=102.809[/tex]
[tex] Upper = 117.8 +15.011=132.831[/tex]
[tex] y \pm 2s[/tex]
[tex] Lower = 117.8 -2*15.011=87.797[/tex]
[tex] Upper = 117.8 +2*15.011=147.842[/tex]
[tex] y \pm 3s[/tex]
[tex] Lower = 117.8 -3*15.011=72.787[/tex]
[tex] Upper = 117.8 +3*15.011=162.85[/tex]
Part d
For this case we can calculate the position where we have accumulated 70% of the data below.
50*0.7 = 35
So on the position 35th from the dataset ordered we see that the value is 128 and this value would represent the 70th percentile on this case.
Sneha is inspecting the average cleanliness of the rooms in a hotel. There are 10 floors in the hotel, each with 20 rooms. From each floor, she randomly selects 2 rooms and inspects them and then takes the average of these scores. What type of sampling did Sneha use
Answers
Answer: Stratified sampling
Step-by-step explanation:
Stratified sampling is a particular kind of random sampling technique.
Here , the researcher splits the entire population into distinct groups known as strata.
Then he draw a sample by taking participants from each strata and continue his work on sample.
As per given ,
Researcher = Sneha
Strata = Each floor
Since from each floor she randomly selects 2 rooms and inspects them and then takes the average of these scores.
Therefore , Sneha used the Stratified sampling technique.
Please answer need to turn in ASAP
Answers
Answer: it will take Sebastian's mom 45 minutes to catch up with the bus.
Step-by-step explanation:
By the time Sebastian's mom catches up with the bus, she would have covered the same distance with the bus.
Let t represent the time it will take for Sebastian's mom to catch up with the bus.
Distance = speed × time
Sebastian's school bus averages 28 miles per hour.
Distance covered by Sebastian's school bus in t hours is
28 × t = 28t
Sebastian's mom travelled at an average speed of 42mph. Since she left home 0.25 hour after Sebastian, the time it would take her to cover the same distance is
(t - 0.25) hour. Distance covered in
(t - 0.25) hour is
42(t - 0.25)
Since the distance covered is the same, then
28t = 42(t - 0.25)
28t = 42t - 10.5
42t - 28t = 10.5
14t = 10.5
t = 10.5/14
t = 0.75
Converting to minutes, it becomes
0.75 × 60 = 45 minutes
Solve by graphing. (If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 2x + y = −11 6x + 3y = 15
Answers
Answer:
The system is INCONSISTENT.
Equation of two parallel lines differ only by a constant.
Step-by-step explanation:
The given equations are:
2x + y = -11
6x + 3y = 15
From the graph we see that these lines are parallel. Any two parallel lines never meet. Hence, they do not have a solution.
Hence, the system is called an Inconsistent system.
Also, note that it can be seen easily from the equations are parallel without the help of a graph.
The equation 6x + 3y = 3(2x + y)
The terms (except for the constant term) are proportional. That means they represent parallel lines.
Hence, the answer.
In order to conduct an experiment, 55 subjects are randomly selected from a group of 4343 subjects. How many different groups of 55 subjects are possible?
Answers
Answer:
962,598 different groups of 5 subjects are possible.
Step-by-step explanation:
The order is not important.
For example, Math, English, Business, Geography and History is the same group as English, Math, Business, Geography and History.
So we use the combinations formula to solve this problem.
Combinations formula:
[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]
Combinations of 5 subjects from a set of 43
[tex]C_{43, 5} = \frac{43!}{5!(43-5)!} = 962598[/tex]
962,598 different groups of 5 subjects are possible.
An electronics company has two production facilities, denoted A and B. During an average week, facility A produces 2000 computer monitors and 10,000 flat panel televisions, and facility B produces 3000 computer monitors and 13,000 flat panel televisions.
Determine the number of weeks of production from A and B required to produce 28,000 monitors and 132,000 televisions.
Answers
Answer:
8 weeks of production from A and 4 weeks of production of B required to produce 28,000 monitors and 132,000 televisions.
Step-by-step explanation:
Let the number of weeks of production for team A be x and for team B be y.
In a week, facility A produces 2,000 computer monitors.
In a week ,facility A produces 3,000 computer monitors.
Total monitors to be produced = 28,000
[tex]2000x+3000y=28,000[/tex]
[tex]2x+3y=28[/tex]..[1]
In a week, facility A produces 10,000 flat panel televisions.
In a week ,facility A produces 13,000 flat panel televisions.
Total flat panel televisions to be produced = 132,000
[tex]10,000x+13,000y=132,000[/tex]
[tex]10x+13y=132[/tex]..[2]
Solving equation [1] and [2] by eliminationg method.
5 × [1] + (-1) × [2]
[tex]10x+15y=140[/tex]
[tex]-10x-13y=-132[/tex]
y = 4
[tex]x = \frac{28-3\times 4}{2}=\frac{16}{2}=8[/tex]
8 weeks of production from A and 4 weeks of production of B required to produce 28,000 monitors and 132,000 televisions.
To produce 28,000 monitors and 132,000 televisions, it would take approximately 3.5 weeks for monitors and 3.67 weeks for televisions.
Explanation:To determine the number of weeks of production required, we need to divide the total number of monitors and televisions needed by the production rate of each facility.
For monitors: Facility A produces a total of 2000 + 3000 = 5000 monitors per week, and Facility B produces 3000 monitors per week. Therefore, the total production rate for monitors is 5000 + 3000 = 8000 monitors per week.
For televisions: Facility A produces a total of 10,000 + 13,000 = 23,000 televisions per week, and Facility B produces 13,000 televisions per week. Therefore, the total production rate for televisions is 23,000 + 13,000 = 36,000 televisions per week.
To produce 28,000 monitors and 132,000 televisions, it would take:
Monitors: 28,000 / 8000 = 3.5 weeks
Televisions: 132,000 / 36,000 = 3.67 weeks
Therefore, the number of weeks required to produce the desired quantities is approximately 3.5 weeks for monitors and 3.67 weeks for televisions.
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How many 4-permutations of [10] have maximum element equal to 6? How many have maximum element at most 6?
Answers
I'm guessing that [10] refers to the set of the first 10 positive integers.
If the largest element of a given 4-permutation is 6, then the other three elements are pulled from the set {1, 2, 3, 4, 5}. This can be done in 5!/(5 - 3)! = 60 ways. Then there are four possible positions to place the 6, giving a total of 4 * 60 = 240 permutations.
If the largest element of a permutation is *at most* 6, then the maximal element is 4, 5, or 6.
If it's 4, then there are three other elements available; this can be done in 3!/(3 - 3)! = 6 ways; multiply by 4 to get a total of 24;If it's 5, then there are four other elements available, hence 4!/(4 - 3)! = 24 ways; multiply by 4 to get a total of 96;If it's 6, then the total is 240.Putting everything together, the total number of permutations in which the maximal element is at most 6 is 24 + 96 + 240 = 360.
The number of 4-permutations of [10] with a maximum element of 6 is 24. The number of 4-permutations of [10] with a maximum element at most 6 is 120.
Explanation:To find the number of 4-permutations of [10] with a maximum element of 6, we can consider the possibilities for the position of the maximum element in the permutation. There are 4 possible positions for the maximum element: first, second, third, or fourth. If the maximum element is in the first position, the remaining 3 elements can be any combination of the remaining 3 numbers (10, 9, and 8) which gives us 3! = 6 permutations. Similarly, if the maximum element is in the second, third, or fourth position, we will have 6 permutations for each position.
Therefore, the total number of 4-permutations of [10] with a maximum element of 6 is 4 * 6 = 24.
To find the number of 4-permutations of [10] with a maximum element at most 6, we need to consider all possible values for the maximum element. We already found that there are 24 permutations with a maximum element of 6. Now, we need to consider the possibilities where the maximum element is 5, 4, 3, 2, or 1.
If the maximum element is 5, the remaining 3 elements can be any combination of the remaining 4 numbers (10, 9, 8, and 6) which gives us 4! = 24 permutations. Similarly, if the maximum element is 4, 3, 2, or 1, we will have 24 permutations for each maximum element.
Therefore, the total number of 4-permutations of [10] with a maximum element at most 6 is 24 + 24 + 24 + 24 + 24 = 120.
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Two numbers total 63 and have a difference of 11. Find the two numbers.
Answers
Answer:
The two numbers are 37 and 26.
Step-by-step explanation:
This question can be solved by a simple system of equations.
Building the system:
I am going to say that our numbers are x and y.
Two numbers total 63
This means that [tex]x + y = 63[/tex]
difference of 11
This means that [tex]x - y = 11[/tex].
Solving the system
[tex]x + y = 63[/tex]
[tex]x - y = 11[/tex]
Using the addition method
x + x + y - y = 63 + 11
2x = 74
x = 37
[tex]x - y = 11[/tex]
[tex]y = x - 11[/tex]
[tex]y = 37 - 11 = 26[/tex]
The two numbers are 37 and 26.
Answer:
The total 2 numbers are 37 and 26
hope i helped
A single tree produces about 2.6x102 pounds of oxygen each year. the amazon rainforest has about 3.9 X 10 [ ] pounds per day enter your answer by filling in the box
Answers
Answer:
The answer to your question is 2.8 x 10¹¹
Step-by-step explanation:
Data
1 tree produces 2.6 x 10² pounds of oxygen/year
number of trees = 3.9 x 10¹¹
pounds of oxygen per day = ?
Process
1.- Divide the pounds of oxygen by 365, to get the pounds of oxygen per day.
2.6 x 10² / 365 = 0.712
2.- Multiply the number of trees by the pounds of oxygen per day
3.9 x 10¹¹ x 0.712 = 2.8 x 10¹¹ pounds of oxygen
A box contains three coins which have the same look. However, two of them are fair and theother one is biased withP(H) = 0.3. A coin is randomly selected from the box and tossed 10 times.(a)[15 points] What is the probability of observing exactly 3 heads?
Answers
Answer:
0.1673 (16.73%)
Step-by-step explanation:
The probability that a fair coin is chosen is 2/3 . if the fair coin is chosen the probability of getting 3 heads is determined by binomial distribution:
P(3 heads in 10 flips )=B(n=10,p=0.5,X=3) =0.1171
But if a coin that is not fair is chosen , then the probability of getting 3 heads also follows a binomial distribution but with p=0.3
P(3 heads)=B(n=10,p=0.3,X=3)=0.2668
Finally the probability is
P= 2/3*0.1171 + 1/3* 0.2678 = 0.1673 (16.73%)