Answer: 0.1880
Step-by-step explanation:
The binomial distribution formula is given by :-
[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
where P(x) is the probability of x successes out of n trials, p is the probability of success on a particular trial.
Given : The probability of new vehicles require warranty service within the first year : p =0.13.
Number of trials : n= 12
Now, the required probability will be :
[tex]P(x=0)=^{12}C_0(0.13)^{0}(1-0.13)^{12-0}\\\\=(1)(0.13)^{0}(0.87)^{12}=0.188031682201\approx0.1880[/tex]
Thus, the probability that none of these vehicles requires warranty service = 0.1880
A tank holds 300 gallons of water and 100 pounds of salt. A saline solution with concentration 1 lb salt/gal is added at a rate of 4 gal/min. Simultaneously, the tank is emptying at a rate of 1 gal/min. Find the specific solution Q(t) for the quantity of salt in the tank at a given time t.
The amount of salt in the tank changes with rate according to
[tex]Q'(t)=\left(1\dfrac{\rm lb}{\rm gal}\right)\left(4\dfrac{\rm gal}{\rm min}\right)-\left(\dfrac{Q(t)}{300+(4-1)t}\dfrac{\rm lb}{\rm gal}\right)\left(1\dfrac{\rm gal}{\rm min}\right)[/tex]
[tex]\implies Q'+\dfrac Q{300+3t}=4[/tex]
which is a linear ODE in [tex]Q(t)[/tex]. Multiplying both sides by [tex](300+3t)^{1/3}[/tex] gives
[tex](300+3t)^{1/3}Q'+(300+3t)^{-2/3}Q=4(300+3t)^{1/3}[/tex]
so that the left side condenses into the derivative of a product,
[tex]\big((300+3t)^{1/3}Q\big)'=4(300+3t)^{1/3}[/tex]
Integrate both sides and solve for [tex]Q(t)[/tex] to get
[tex](300+3t)^{1/3}Q=(300+3t)^{4/3}+C[/tex]
[tex]\implies Q(t)=300+3t+C(300+3t)^{-1/3}[/tex]
Given that [tex]Q(0)=100[/tex], we find
[tex]100=300+C\cdot300^{-1/3}\implies C=-200\cdot300^{1/3}[/tex]
and we get the particular solution
[tex]Q(t)=300+3t-200\cdot300^{1/3}(300+3t)^{-1/3}[/tex]
[tex]\boxed{Q(t)=300+3t-2\cdot100^{4/3}(100+t)^{-1/3}}[/tex]
Prove Corollary 6.2. If L : V ? W is a linear transformation of a vector space V into a vector space W and dim V=dim W, then the following statements are true: (a) If L is one-to-one, then it is onto. (b) If L is onto, then it is one-to-one.
Answer with explanation:
Given L: V\rightarrow W is a linear transformation of a vector space V into a vector space W.
Let Dim V= DimW=n
a.If L is one-one
Then nullity=0 .It means dimension of null space is zero.
By rank- nullity theorem we have
Rank+nullity= Dim V=n
Rank+0=n
Rank=n
Hence, the linear transformation is onto. Because dimension of range is equal to dimension of codomain.
b.If linear transformation is onto.
It means dimension of range space is equal to dimension of codomain
Rank=n
By rank nullity theorem we have
Rank + nullity=dimV
n+nullity=n
Nullity=n-n=0
Dimension of null space is zero.Hence, the linear transformation is one-one.
To prove Corollary 6.2, we show that a one-to-one linear transformation between vector spaces of equal dimensions is onto, and conversely, an onto transformation is one-to-one. This conclusion is based on the properties of linear transformations and the significance of preserving vector space dimensions.
Explanation:To prove Corollary 6.2 regarding a linear transformation L: V → W, where both vector spaces V and W have the same dimension, we must show two parts:
If L is one-to-one, then it is onto. Assuming L is one-to-one, for every vector v in V, there is a unique image in W, guaranteeing that all vectors in W can be reached since dim V = dim W. Thus, L covers all of W, making it onto.
If L is onto, then it is one-to-one. When assuming L is onto, every vector in W is the image of some vector in V. Because dim V = dim W, there can't be more vectors in V than in W, preventing multiple vectors in V from mapping to the same vector in W, thus L is one-to-one.
These parts rely heavily on the concept that linear transformations maintain vector operations, such as the addition and scalar multiplication, and the relationship between dimensions of the domain and codomain for linear transformations.
Learn more about Linear Transformation here:https://brainly.com/question/35061121
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Raul received a score of 77 on a history test for which the class mean was 70 with a standard deviation of 6. He received a score of 79 on a biology test for which the class mean was 70 with standard deviation 4. On which test did he do better relative to the rest of the class?
Answer:
He did better on biology test.
Step-by-step explanation:
For comparing the scores we need to find z-scores for both subjects,
We know that,
z-score or standard score for x is,
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where, [tex]\mu[/tex] is mean,
[tex]\sigma[/tex] is standard deviation,
In history test,
[tex]x=77[/tex]
[tex]\mu=70[/tex]
[tex]\sigma = 6[/tex]
Thus, the z-score would be,
[tex]z_1=\frac{77-70}{6}[/tex]
[tex]\approx 1.167[/tex]
In biology test,
[tex]x=79[/tex]
[tex]\mu = 70[/tex]
[tex]\sigma = 4[/tex],
Thus, the z-score would be,
[tex]z_2=\frac{79-70}{4}[/tex]
[tex]= 2.25[/tex]
∵ [tex]z_2>z_1[/tex]
Hence, he did better on biology test.
Jayanta is raising money for the homeless, and discovers each church group requires 2 hr of letter writing and 1 hr of follow-up calls, while each labor union needs 2 hr of letter writing and 3 hr of follow-up. She can raise $125 from each church group and $175 from each union. She has a maximum of 20 hours of letter writing and 14 hours of follow-up available each month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month.
Answer:
8 churches, 2 unions; $1350 per monthStep-by-step explanation:
Let x and y represent the numbers of churches and unions contacted in the month, respectively. Then Jayanta's limit on letter writing hours is ...
2x +2y ≤ 20
and her limit on follow-up call hours is ...
x + 3y ≤ 14
Graphing these inequalities (see below) results in a feasible region with vertices at (x, y) = (0, 4 2/3), (8, 2), and (10, 0). Of these, the mixture of groups producing the most money is ...
8 churches and 2 unions.
The money she can raise from that mixture is ...
8×$125 +2×$175 = $1350 in a month
Answer:
Sqdancefan's answer is correct.
Step-by-step explanation:
I misread the question.
A brand name has a 40% recognition rate. If the owner of the brand wants to verify that rate by beginning with a small sample of 5 randomly selected consumers, find the probability that exactly 2 of the 5 consumers recognize the brand name. Also find the probability that the number who recognize the brand name is not 2.
Answer:
1) 0.3456
2) 0.6544.
Step-by-step explanation:
Let X represents the event of recognizing the brand,
Given,
The probability of recognizing the brand, p = 40% = 0.40,
Thus, the probability of not recognizing the brand, q = 1 - 0.40 = 0.60,
Since, the binomial distribution formula,
[tex]P(x) = ^nC_r (p)^r(q)^{n-r}[/tex]
Where,
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
1) Thus, the probability that exactly 2 of the 5 consumers recognize the brand name is,
[tex]P(X=2)=^5C_2 (0.40)^2 (0.60)^{5-2}[/tex]
[tex]=10 (0.40)^2 (0.60)^3[/tex]
[tex]=0.3456[/tex]
2) Also, the probability that the number who recognize the brand name is not 2 = 1 - P(X=2) = 1 - 0.3456 = 0.6544.
You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 700 eggs and 1500 cups of cream. How many quarts of each flavor should you make in order to use up all the eggs and cream? HINT [See Example 5.]
Answer:
200 Creamy Vanilla
300 Continental Mocha
Step-by-step explanation:
3 cups cream for each:
500 Quarts total
Creamy Vanilla requires 400 eggs
Continental Mocha requres 300 eggs for a total of 700 eggs
The answer is:
[tex]\[ \boxed{V = 200, M = 300} \][/tex]
To solve this problem, we need to set up a system of equations based on the given information and then solve for the number of quarts of each flavour of ice cream that should be made.
Let [tex]\( V \)[/tex] represent the number of quarts of Creamy Vanilla ice cream and [tex]\( M \)[/tex] represent the number of quarts of Continental Mocha ice cream.
From the information given, we can derive the following equations:
For the eggs:
Each quart of Creamy Vanilla requires 2 eggs, so [tex]\( 2V \)[/tex] eggs are used for Creamy Vanilla.
Each quart of Continental Mocha requires 1 egg, so [tex]\( M \)[/tex] eggs are used for Continental Mocha.
Since there are 700 eggs in total, we have the equation:
[tex]\[ 2V + M = 700 \][/tex]
For the cream:
Each quart of Creamy Vanilla requires 3 cups of cream, so [tex]\( 3V \)[/tex] cups of cream are used for Creamy Vanilla.
Each quart of Continental Mocha also requires 3 cups of cream, so [tex]\( 3M \)[/tex] cups of cream are used for Continental Mocha.
Since there are 1500 cups of cream in total, we have the equation:
[tex]\[ 3V + 3M = 1500 \][/tex]
Now we have a system of two equations with two variables:
[tex]\[ \begin{cases} 2V + M = 700 \\ 3V + 3M = 1500 \end{cases} \][/tex]
To solve this system, we can simplify the second equation by dividing every term by 3:
[tex]\[ V + M = 500 \][/tex]
Now we have a simpler system:
[tex]\[ \begin{cases} 2V + M = 700 \\ V + M = 500 \end{cases} \][/tex]
We can subtract the second equation from the first to eliminate [tex]\( M \)[/tex] and solve for [tex]\( V \)[/tex] :
[tex]\[ (2V + M) - (V + M) = 700 - 500 \][/tex]
[tex]\[ V = 200 \][/tex]
Now that we have the value of [tex]\( V \)[/tex], we can substitute it back into the simplified second equation to find [tex]\( M \)[/tex]:
[tex]\[ 200 + M = 500 \][/tex]
[tex]\[ M = 500 - 200 \][/tex]
[tex]\[ M = 300 \][/tex]
Therefore, the factory should make 200 quarts of Creamy Vanilla and 300 quarts of Continental Mocha to use up all the eggs and cream.
The final answer is:
[tex]\[ \boxed{V = 200, M = 300} \][/tex]
Doing research for insurance rates, it is found that those aged 30 to 49 drive an average of 38.7 miles per day with a standard deviation of 6.7 miles. These distances are normally distributed. If a group of 60 drivers in that age group are randomly selected, what is the probability that the mean distance traveled each day is between 32.5 miles and 40.5 miles?
Answer: 0.4302
Step-by-step explanation:
Given : Mean : [tex]\mu=\text{38.7 miles }[/tex]
Standard deviation : [tex]\sigma=\text{6.7 miles }[/tex]
Sample size : [tex]n=60[/tex]
Also, these distances are normally distributed.
Then , the formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=32.5
[tex]\\\\ z=\dfrac{32.5-38.7}{6.7}=-0.925373134\approx-0.93[/tex]
For x=40.5
[tex]\\\\ z=\dfrac{40.5-38.7}{6.7}=0.268656\approx0.27[/tex]
The p-value = [tex]P(-0.93<z<0.27)[/tex]
[tex]=P(0.27)-P(-0.93)=0.6064198- 0.1761855=0.4302343\approx0.4302[/tex]
Hence, the required probability :-0.4302
For f(x) = 2|x+3| – 5, name the type of function and describe each of the three
transformations from the parent function f(x) = |x|
Answer:
There are three transformation i.e. vertically stretch, downward and towards left.
Step-by-step explanation:
Given : Parent function [tex]f(x)=|x|[/tex] and transformed function [tex]f(x)=2|x+3|-5[/tex]
To find : Name the type of function and describe each of the three transformations?
Solution :
Parent function [tex]f(x)=|x|[/tex] get transformed into [tex]f(x)=2|x+3|-5[/tex]
The transformations are as follow :
1) The function is multiplied by 2 which means there is a transformation vertically by a stretch factor '2'.
2) The function is subtracted by 5 which means there is a transformation downward by a factor '5'.
3) The function inside x value get added by 3 which means there is a transformation left by a factor '3'.
Therefore, There are three transformation i.e. vertically stretch, downward and towards left.
Round each of the following, using front-end rounding. [1.1] 4. 50,987 5. 851,004
Answer:
4) 50,000
5) 900,000
Step-by-step explanation:
In Front end rounding what we do is we focus on first two digit of the number if the second digit number is greater than or equal to 5 we add the first digit by 1.
4) 50,987
here we can clearly see that the first two digit is 50 and second digit is not equal to 5 then rounding will be equal to 50,000.
5) 851,004
here we can clearly see that the first two digit is 85 and second digit is equal to 5 then we will round it 8+1=9
hence rounded number will be 900,000
Final answer:
Front-end rounding of 50,987 is 50,000 and for 851,004 it's 900,000, based on the most significant digits, using zeros as placeholders after rounding.
Explanation:
Front-end rounding involves rounding numbers based on the most significant digits. Let's apply this method to the numbers provided.
50,987 rounded to the nearest ten thousand would be 50,000.For 851,004, rounded to the nearest hundred thousand would be 900,000.When rounding, if the digit immediately after the place you are rounding to is 5 or greater, you round up. Otherwise, you round down. Placeholder zeros are used to maintain the value's place in the number system.
Two particles travel along the space curves r1(t) = t, t2, t3 r2(t) = 1 + 2t, 1 + 6t, 1 + 14t . Find the points at which their paths intersect. (If an answer does not exist, enter DNE.)
Answer:
DNE
Step-by-step explanation:
Given that two particles travel along the space curves
[tex]r_1(t) = (t, t^2, t^3)\\ r_2(t) = (1 + 2t, 1 + 6t, 1 + 14t )[/tex]
To find the points of intersection:
At points of intersection both coordinates should be equal.
i.e. r1 =r2
Equate corresponding coordinates
[tex]t=1+2t\\t^2=1+6t\\t^3=1+14t[/tex]
I equation gives t =-1
Substitute in II equation to get [tex]t^2 = -5[/tex]
i.e. t cannot be real
Hence no point of intersection
DNE
Find the angle between the given vectors to the nearest tenth of a degree.
u = <6, -1>, v = <7, -4>
Answer:
A
Step-by-step explanation:
Given
u = <6, -1>
u = 6i-j
and
v=<7,-4>
v=7i-4j
The formula for angle is:
Let x be the angle
[tex]cos\ x = \frac{u.v}{||u||.||v||}[/tex]
where ||u|| is the length and u.v is the dot product or scalar product of both vectors
So,
[tex]||u|| = \sqrt{(6)^2+(-1)^2}\\ = \sqrt{36+1}\\ = \sqrt{37}\\ ||v||=\sqrt{(7)^2+(-4)^2}\\ = \sqrt{49+16}\\ = \sqrt{65}\\[/tex]
[tex]u.v = u_1u_2+v_1v_2\\= (6)(7)+(-1)(-4)\\=42+4\\=46[/tex]
[tex]cos\ x=\frac{46}{\sqrt{37}\sqrt{65}} \\= \frac{46}{\sqrt{2405} }\\Can\ also\ be\ written\ as:\\= \frac{46}{\sqrt{2405} } * \frac{\sqrt{2405} }{\sqrt{2405}} \\=\frac{46\sqrt{2405} }{2405}[/tex]
The calculated angle will be in radians. To find the angle in degrees:
[tex]x = \frac{180}{\pi} cos^{-1} (\frac{46\sqrt{2405} }{2405})\\x = 20.282\\x= 20.3\\[/tex]
Hence Option A is correct ..
Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years. Your answer should be a decimal rounded to the fourth decimal place.
Answer:
The probability is 0.0643
Step-by-step explanation:
* Lets revise some definition to solve the problem
- The standard deviation of the distribution of sample means is called σM
- σM = σ/√n , where σ is the standard deviation and n is the sample size
- z-score = (M - μ)/σM, where M is the mean of the sample , μ is the mean
of the population
* Lets solve the problem
- The mean of the washing machine is 9.3 years
∴ μ = 9.3
- The standard deviation is 1.1 years
∴ σ = 1.1
- There are 70 washing machines randomly selected
∴ n = 70
- The mean replacement time less than 9.1 years
∴ M = 9.1
- Lets calculate z-score
∵ σM = σ/√n
∴ σM = 1.1/√70 = 0.1315
∵ z-score = (M - μ)/σM
∴ z-score = (9.1 - 9.3)/0.1315 = - 1.5209
- Use the normal distribution table of z to find P(z < -1.5209)
∴ P(z < -1.5209) = 0.06426
∵ P(M < 9.1) = P(z < -1.5209)
∴ P(M < 9.1) = 0.0643
* The probability is 0.0643
A container contains 12 diesel engines. The company chooses 5 engines at random, and will not ship the container if any of the engines chosen are defective. Find the probability that a container will be shipped even though it contains 2 defectives if the sample size is 5.
Answer:[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]
Step-by-step explanation:
Given a total of 12 diesel engines
out of which 2 are defective
Company have to choose 5 good engines to ship.
Therefore no of ways in good engines are selected is [tex]^{10}C_5[/tex]
no of ways in which any 5 engines are selected from a total of 12 engines is
[tex]^{12}C_5[/tex]
Therefore the required probability =[tex]\frac{^{10}C_5}{^{12}C_5}[/tex]
=[tex]\frac{7}{22}[/tex]=0.318
Assuming that in a box there are 10 black socks and 12 blue socks, calculate the maximum number of socks needed to be drawn from the box before a pair of the same color can be made. Using the pigeonhole principle
Answer:
Step-by-step explanation:
The pigeonhole principal states that if n items are put into m containers, with n > m then at least one container must contain more than one item.
In other words, to find the case where maximum attempts are required, we eliminate all the cases where our criterion is not met and we will be left with the desired result.
For this case, the box has 10 black socks and 12 blue socks.
10 black socks = 5 left foot ones and 5 right foot ones
12 blue socks = 6 left foot ones and 6 right foot ones
If we draw all left foot or right foot ones then we will not have a pair till 11 draws have been made.
The next socks drawn will be a right foot one of blue or black color and a pair will be made.
Therefore, the maximum number of socks needed to be drawn from the box are 12.
Solve the following initial-value problem, showing all work, including a clear general solution as well as the particular solution requested (Label the Gen. Sol. and Particular Sol.). Write both solutions in EXPLICIT FORM (solved for y). x dy/dx = x^3 + 2y subject to: y(2) = 6
Answer:
General Solution is [tex]y=x^{3}+cx^{2}[/tex] and the particular solution is [tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]
Step-by-step explanation:
[tex]x\frac{\mathrm{dy} }{\mathrm{d} x}=x^{3}+3y\\\\Rearranging \\\\x\frac{\mathrm{dy} }{\mathrm{d} x}-3y=x^{3}\\\\\frac{\mathrm{d} y}{\mathrm{d} x}-\frac{3y}{x}=x^{2}[/tex]
This is a linear diffrential equation of type
[tex]\frac{\mathrm{d} y}{\mathrm{d} x}+p(x)y=q(x)[/tex]..................(i)
here [tex]p(x)=\frac{-2}{x}[/tex]
[tex]q(x)=x^{2}[/tex]
The solution of equation i is given by
[tex]y\times e^{\int p(x)dx}=\int e^{\int p(x)dx}\times q(x)dx[/tex]
we have [tex]e^{\int p(x)dx}=e^{\int \frac{-2}{x}dx}\\\\e^{\int \frac{-2}{x}dx}=e^{-2ln(x)}\\\\=e^{ln(x^{-2})}\\\\=\frac{1}{x^{2} } \\\\\because e^{ln(f(x))}=f(x)]\\\\Thus\\\\e^{\int p(x)dx}=\frac{1}{x^{2}}[/tex]
Thus the solution becomes
[tex]\tfrac{y}{x^{2}}=\int \frac{1}{x^{2}}\times x^{2}dx\\\\\tfrac{y}{x^{2}}=\int 1dx\\\\\tfrac{y}{x^{2}}=x+c[/tex][tex]y=x^{3}+cx^{2[/tex]
This is the general solution now to find the particular solution we put value of x=2 for which y=6
we have [tex]6=8+4c[/tex]
Thus solving for c we get c = -1/2
Thus particular solution becomes
[tex]y=x^{3}-\frac{1}{2}x^{2}[/tex]
A company that makes cola drinks states that the mean caffeine content per 12-ounce bottle of cola is 40 milligrams. You want to test this claim. During your tests, you find that a random sample of thirty 12-ounce bottles of cola has a mean caffeine content of 42.3 milligrams. Assume the population is normally distributed and the population standard deviation is 7.1 milligrams. At alphaequals0.04, can you reject the company's claim
Answer:
Step-by-step explanation:
Put hypotheses as:
[tex]H_0: mu =40 mg\\H_a: mu \neq 40 mg.[/tex]
(Two tailed test at 4%)
Since population std deviation is given we can do Z test.
[tex]Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296[/tex]
Test statistic t = mean diff/se = [tex]\frac{2.3}{1.296} =1.775[/tex]
p value=0.075
Since p >0.04 our alpha, we accept null hypothesis.
At alphaequals0.04, we cannot reject the company's claim
Step-by-step explanation:
Put hypotheses as:
\begin{gathered}H_0: mu =40 mg\\H_a: mu \neq 40 mg.\end{gathered}
H
0
:mu=40mg
H
a
:mu
=40mg.
(Two tailed test at 4%)
Since population std deviation is given we can do Z test.
\begin{gathered}Sample size n =30\\Sample mean = 42.3 mg\\Mean diff = 2.3 mg\\Std error of sample = \frac{7.1}{\sqrt{30} } =1.296\end{gathered}
Samplesizen=30
Samplemean=42.3mg
Meandiff=2.3mg
Stderrorofsample=
30
7.1
=1.296
Test statistic t = mean diff/se = \frac{2.3}{1.296} =1.775
1.296
2.3
=1.775
p value=0.075
Since p >0.04 our alpha, we accept null hypothesis.
At alphaequals0.04, we cannot reject the company's claim
The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites.
The correct option is to reject the null hypothesis. The mean life spans for whites and non-whites born in 1900 in the certain county are not the same.
To conduct the hypothesis test, we will use a two-sample z-test for the difference in two means. The null hypothesis (H0) states that there is no difference in the mean life spans between whites and non-whites, while the alternative hypothesis (Ha) states that there is a difference.
The formula for the z-test statistic is:
[tex]\[ z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \][/tex]
where:
- [tex]\(\bar{x}_1\)[/tex] and [tex]\(\bar{x}_2\)[/tex] are the sample means for whites and non-whites, respectively.
- [tex]\(\mu_1\)[/tex] and [tex]\(\mu_2\)[/tex] are the population means for whites and non-whites, respectively.
- [tex]\(\sigma_1\)[/tex] and [tex]\(\sigma_2\)[/tex] are the population standard deviations for whites and non-whites, respectively.
- [tex]\(n_1\)[/tex] and [tex]\(n_2\)[/tex] are the sample sizes for whites and non-whites, respectively.
Given:
- [tex]\(\bar{x}_1 = 45.3\) years, \(s_1 = 12.7\) years, \(n_1 = 124\)[/tex] (for whites)
- [tex]\(\bar{x}_2 = 34.1\)[/tex] years, [tex]\(s_2 = 15.6\) years, \(n_2 = 82\)[/tex] (for non-whites)
- [tex]\(\mu_1 = 47.6\)[/tex] years (for whites)
- [tex]\(\mu_2 = 33.0\)[/tex] years (for non-whites)
Since we do not have the population standard deviations, we will use the sample standard deviations as an estimate. This is appropriate given the sample sizes are large enough (generally [tex]\(n > 30\)[/tex] is considered sufficient).
First, we calculate the standard error (SE) of the difference in means:
[tex]\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{12.7^2}{124} + \frac{15.6^2}{82}} \][/tex]
[tex]\[ SE = \sqrt{\frac{161.29}{124} + \frac{243.36}{82}} \][/tex]
[tex]\[ SE = \sqrt{1.299 + 2.968} \][/tex]
[tex]\[ SE = \sqrt{4.267} \][/tex]
[tex]\[ SE \approx 2.066 \][/tex]
Now, we calculate the z-statistic:
[tex]\[ z = \frac{(45.3 - 34.1) - (47.6 - 33.0)}{2.066} \][/tex]
[tex]\[ z = \frac{11.2 - 14.6}{2.066} \][/tex]
[tex]\[ z = \frac{-3.4}{2.066} \][/tex]
[tex]\[ z \approx -1.646 \][/tex]
Next, we find the p-value for this z-statistic. Since we are conducting a two-tailed test, we will look at the probability of a z-score being less than -1.646 or greater than 1.646. Using a standard normal distribution table or a calculator, we find that the p-value is approximately 0.100.
Finally, we compare the p-value to our significance level (commonly denoted as [tex]\(\alpha\))[/tex]. If the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis. If we choose [tex]\(\alpha = 0.05\)[/tex], then since [tex]\(0.100 > 0.05\)[/tex], we fail to reject the null hypothesis.
However, if we choose a significance level of [tex]\(\alpha = 0.10\)[/tex], then since [tex]\(0.100 \leq 0.10\)[/tex], we would reject the null hypothesis, indicating that there is a statistically significant difference in the mean life spans between whites and non-whites in the county.
Given the p-value is on the boundary of common significance levels, the conclusion may vary depending on the chosen level of significance. However, the correct option based on the provided conversation is to reject the null hypothesis, suggesting that the mean life spans are not the same for whites and non-whites in the county.
We are doing a study on shampoo buyers in Walmart stores across the US and we send them online surveys. Based on past studies, we know that of the people who start the survey, 80% qualify for the study, and only 40% of these people who do qualify actually complete the survey. Of the people who complete the survey, 10% of respondents are removed due to invalid or low quality answers. If we want 1000 valid responses for this particular study, how many people do we need to have start the survey?
Answer:
2500
Step-by-step explanation:
They tell you that only 40% of the total complete the quality test, and you want to know how many represent 100%.
If this 40% is 1000 valid responses use cross multiplication to know how many people are the total.
40%-----1000 valid responses
100%----x=
[tex]\frac{100*1000}{40} =X\\2500=X[/tex]
So, you can tell they need at least 2500 persons to have 1000 valid responses.
Answer:
Number of people = 3473
Step-by-step explanation:
Probability is typically the rate at which an event or occurrence is likely to happen. Whenever we're not so sure about an event outcome, we can then deliberate about the probabilities or possibility of certain outcomes that is how likely they are to happen.
Probability of success, =0.8×0.4×0.9 =0.288
Expected, E=1000
Expected = np
1000=n×0.288
n=0.2881000≈34
Type the correct answer in each box. Use numerals instead of words.
Consider the systems of equations below.
Answer:
System A has 2 real solutions.
System B has 0 real solutions.
System C has 1 real solutions.
Step-by-step explanation:
If the graph of system of equation intersect each other at n points then the system of equation has n real solutions.
System A:
[tex]x^2+y^2=17[/tex] .... (1)
[tex]y=-\frac{1}{2}x[/tex] .... (2)
Plot the graph of these equations.
From graph (1) it is clear that the graph of equation (1) and (2), intersect each other at two points, (-3.688,1.844) and (3.688,-1.844).
Therefore, System A has two real solutions.
System B:
[tex]y=x^2-7x+10[/tex] .... (3)
[tex]y=-6x+5[/tex] .... (4)
Plot the graph of these equations.
From graph (2) it is clear that the graph of equation (3) and (4), never intersect each other.
Therefore, System B has 0 real solutions.
System C:
[tex]y=-2x^2+9[/tex] .... (5)
[tex]8x-y=-17[/tex] .... (6)
Plot the graph of these equations.
From graph (3) it is clear that the graph of equation (5) and (6), intersect each other at one points, (-2,1).
Therefore, System C has 1 real solutions.
Bill Mason is considering two job offers. Job 1 pays a salary of $41,300 with $5,525 of nontaxable employee benefits. Job 2 pays a salary of $40,400 and $7,125 of nontaxable benefits. Use a 25 percent tax rate.
Calculate the monetary value for both the jobs.
Final answer:
To calculate the monetary value of Bill Mason's job offers, you subtract the taxes from the salary and add the nontaxable benefits. Job 1 results in a total monetary value of $36,500, while Job 2 is higher, with a total monetary value of $37,425. Therefore, Job 2 offers a higher total monetary value.
Explanation:
Bill Mason is considering two job offers, each with a different combination of salary and nontaxable benefits. To determine the monetary value for both jobs, considering a 25 percent tax rate, we must calculate the Net Annual Income for each job separately. Net Annual Income is the salary after taxes have been subtracted. Nonetheless, nontaxable benefits do not affect the Net Annual Income as they do not get taxed.
Calculations for Job 1:
Gross Salary: $41,300
Taxable Income (Tax Rate 25%): 0.25 x $41,300 = $10,325
Net Salary after Tax: $41,300 - $10,325 = $30,975
Nontaxable Benefits: $5,525
Total Monetary Value: Net Salary + Benefits = $30,975 + $5,525 = $36,500
Calculations for Job 2:
Gross Salary: $40,400
Taxable Income (Tax Rate 25%): 0.25 x $40,400 = $10,100
Net Salary after Tax: $40,400 - $10,100 = $30,300
Nontaxable Benefits: $7,125
Total Monetary Value: Net Salary + Benefits = $30,300 + $7,125 = $37,425
The final answer for the total monetary value of Job 1 is $36,500 and for Job 2 is $37,425. As per the given calculations, Job 2 offers a higher total monetary value compared to Job 1 when taking into account the salary after tax plus nontaxable benefits. This explanation should provide clarity on how to assess job offers based on their financial merits.
You deposit $100 in an account earning 4% interest compounded annually. How much will you have in the account in 10 years? Round your answer to the nearest penny
Answer:
$148.02
Step-by-step explanation:
In the question
Principal = $100, rate(R) = 4% compounded annually, time(T)= 10 years
we know that the formula for compound interest
A=[tex]P\times_(1+\frac{R}{100} )^{T}[/tex] where A is amount
now putting values in the above formula we get
A=[tex]100\times_(1+\frac{4}{100} )^{10}[/tex]
therefore A= $148.024428
rounding off to the nearest penny we get amount as $148.02 and compound interest will be $48.02
if 4 oz of solution is required for 2 Gallons of water. How much of the solution should I use if the amount of water required is 22 ounces?
1 Gallon = 128 Ounces
Answer:
You will requiere 0.34375 oz of the solution for 22 ounces of water.
Step-by-step explanation:
We should start by matching the units: in the example we have 2 gallons of water, while the question refers to 22 ounces of water.
So, by the equivalence we are given, we now know that 2 gallons of water are equal to 256 ounces.
From this point the question can be solved using an arithmetic method known as cross-multiply (sometimes refered as the Rule of Three). This consist of wrinting an equation of the form:
[tex]\frac{a}{b} = \frac{c}{x}\\[/tex]
Where b is the amount of solution required for a ounces of water, and x (unknown variable) is the amount of solution for c ounces of water.
So we have the next equation:
[tex]\frac{256}{4} =\frac{22}{x}[/tex]
We apply the correspondent arithmetic rules so we can calculate the x as follows:
[tex]x=\frac{(4)(22)}{256}[/tex]
[tex]x=\frac{88}{256}[/tex]
[tex]x=0.34375[/tex]
This way we can reply that 0.34375 oz of solution are required for 22 oz of water.
1. Solve the equation x = (2x+ 3)1/2
2. Consider f1(x) = ln(x + 1) + ln (x-1) and f2(x) = ln(x^2-1).
a) State domains and ranges of f1 and f2.
b) Sketch the curves y = f1(x) and y = f2(x).
Answer:
1. [tex]x=3[/tex]; 2. Domain: [tex]x>1[/tex] Range: all real numbers
Step-by-step explanation:
Let's find the solutions.
1. Solve the equation [tex]x=\sqrt{2x+3}[/tex] so:
[tex](x)^2=(\sqrt{2x+3})^2[/tex]
[tex]x^2=2x+3[/tex]
[tex]x^2-2x-3=0[/tex]
[tex]x1=\frac{-b+\sqrt{b^{2}-4ac}}{2a}[/tex]
[tex]x1=\frac{2+\sqrt{(-2)^{2}-(4*1*(-3))}}{2*1}[/tex]
[tex]x1=3[/tex]
[tex]x2=\frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]
[tex]x2=\frac{2-\sqrt{(-2)^{2}-(4*1*(-3))}}{2*1}[/tex]
[tex]x2=-1[/tex]
Although we have two answers, remember that from the original equation the result of [tex]\sqrt{2x+3} > 0[/tex] is never negative. So -1 do not solve the equation.
In conlcusion, the equation is solved by x=3.
2A. Domains and ranges of f1(x) and f2(x)
[tex]f1(x)=ln(x+1)+ln(x-1)[/tex]
Using logarithmic property [tex]ln(a)+ln(b)=ln(a*b)[/tex] we have:
[tex]f1(x)=ln(x^2-1)[/tex] because:
[tex]ln(x)[/tex] is defined by [tex]x>0[/tex] then:
[tex]x^2-1>0[/tex]
[tex]x>\sqrt{1}[/tex] so the domain of f1(x) is [tex]x>1[/tex]
Now for the range:
[tex]f1(x)=ln(x+1)+ln(x-1)[/tex]
[tex]y=ln(x^2-1)[/tex]
[tex]e^y=x^2-1[/tex]
[tex]\sqrt{e^y+1}=x^2-1[/tex] notice that [tex]e^y+1[/tex] is always positive, so the range of f1(x) is all real numbers.
Be aware that although point number two of the problem mentioned two equations, f1(x)=f2(x) by logarithmic properties, so their domains and ranges are the same.
2B. Graph of f1(x) is attached. Because f1(x)=f2(x) both functions plot equal.
A father wishes to give his son P200, 000 ten years from now. What amount should he invest if it will earn interest at 10% compounded quarterly during the first five years and 12% compounded annually during the next five years? A. P68,757.82 B. P62,852.23 C. P69,256.82 D. P67,238.54
Answer:
C. P69,256.82
Step-by-step explanation:
We know that,
The amount formula in compound interest is,
[tex]A=P(1+\frac{r_1}{n_1})^{n_1t_1} (1+\frac{r_2}{n_2})^{n_2t_2}.......[/tex]
Where, P is the principal amount,
[tex]r_1, r_2....[/tex] are the annual rate for the different periods,
[tex]t_1, t_2,.....[/tex] are the number of year for different periods,
[tex]n_1, n_2, n_3...[/tex] are the number of periods,
Given,
A = P 200,000,
[tex]r_1=10%=0.1[/tex], [tex]n_1=4[/tex], [tex]t_1=5[/tex],[tex]r_2=12%=0.12[/tex], [tex]n_2=1[/tex], [tex]t_2=5[/tex]
Thus, by the above formula the final amount would be,
[tex]200000=P(1+\frac{0.1}{4})^{4\times 5}(1+\frac{0.12}{1})^{1\times 5}[/tex]
[tex]200000=P(1+0.025)^{20}(1+0.12)^5[/tex]
[tex]200000=P(1.025)^{20}(1.12)^5[/tex]
[tex]\implies P=69,256.824\approx 69,256.82[/tex]
Option C is correct.
The correct option is C. P69,256.82. The father should invest approximately P69,256.82 today to ensure his son receives P200,000 in ten years.
To determine the amount the father needs to invest today to give his son P200,000 ten years from now, we will break the problem into two phases due to different interest rates and compounding periods.
Phase 1: First Five Years (10% compounded quarterly)
→ Future Value (FV) needed after 10 years: P200,000
→ Future Value (FV) after first five years at 12% annual interest for the next five years:
Let's use the formula for compound interest to calculate amount required after the first five years.
Here,
→ n is the number of times the interest is compounded per year
→ t is time in years.
→ [tex]FV = PV*(1 + r/n)^{(nt)[/tex]
After the first five years, the amount needs to grow at 12% compounded annually for 5 years to reach P200,000.
We can calculate the present value (PV) at the end of the first five years needed to achieve P200,000 after next 5 years.
→ P200,000 = [tex]PV * (1 + 0.12/1)^{(1*5)[/tex]
→ P200,000 = [tex]PV * (1.12)^5[/tex]
Calculating PV:
→ PV = P200,000 / (1.7623)
≈ P113,477.57
Phase 2: First Five Years Investment Calculation
Now, we need to find out the amount the father should invest today to reach P113,477.57 after five years with 10% interest compounded quarterly.
Here,
→ r is the quarterly rate,
→ nt is the total number of quarters.
We use the same compound interest formula:
→ [tex]FV = PV * (1 + r/n)^{(nt)[/tex]
→ P113,477.57 = [tex]PV * (1 + 0.10/4)^{(4*5)[/tex]
→ P113,477.57 = [tex]PV * (1.025)^{20[/tex]
→ [tex](1.025)^{20} \approx 1.6386[/tex]
Calculating PV:
→ PV = P113,477.57 / 1.6386
≈ P69,256.82
So, the father needs to invest approximately P69,256.82 today.
Suppose that the inflation rate is 2.5% and the real terminal value of an investment is expected to be $20,000 in 2 years. Calculate the nominal terminal value of the investment at the end of year 2.
Answer:
$21012.50
Step-by-step explanation:
$20,000 today will be inflated to $20,000·(1.025)^2 ≈ $21012.50 in 2 years. We presume this is the nominal terminal value you want.
2. Let a, b, cE Z such that ged(a, c)d for some integer d. Prove that if a | bc then a | bd. [3
Answer with explanation:
It is given that, a, b and c belong to the set of integers.
→ gcd(a,c)=d
→GCD=Greatest Common Divisor
→The greatest number which divides both a and c is d.
It means d divides a, and d divides c.
a=d k, for some integer k.-------(1)
c= d m, for some integer m.-------(2)
Now, it is given that, a divides bc.
So,→ 'a' will divide "bdm".--------[using 2, as c=d m]
It shows that, a divides bd, that is a| bd.
Hence proved
4x^2 y+8xy'+y=x, y(1)= 9, y'(1)=25
Answer with explanation:
[tex]\rightarrow 4x^2y+8x y'+y=x\\\\\rightarrow 8xy'+y(1+4x^2)=x\\\\\rightarrow y'+y\times\frac{1+4x^2}{8x}=\frac{1}{8}[/tex]
--------------------------------------------------------Dividing both sides by 8 x
This Integration is of the form ⇒y'+p y=q,which is Linear differential equation.
Integrating Factor
[tex]=e^{\int \frac{1+4x^2}{8x} dx}\\\\e^{\log x^{\frac{1}{8}+\frac{x^2}{2}}\\\\=x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}[/tex]
Multiplying both sides by Integrating Factor
[tex]x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}\times [y'+y\times\frac{1+4x^2}{8x}]=\frac{1}{8}\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}\\\\ \text{Integrating both sides}\\\\y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=\frac{1}{8}\int {x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}} \, dx \\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=\int {x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}} \, dx\\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=-[x^{\frac{9}{8}}]\times\frac{ \Gamma(0.5625, -x^2)}{(-x^2)^{\frac{9}{16}}}\\\\8y\times x^{\frac{1}{8}}\times e^{\frac{x^2}{2}}=(-1)^{\frac{-1}{8}}[ \Gamma(0.5625, -x^2)]+C-----(1)[/tex]
When , x=1, gives , y=9.
Evaluate the value of C and substitute in the equation 1.
A) In 2000, the population of a country was approximately 5.82 million and by 2040 it is projected to grow to 9 million. Use the exponential growth model A=A0e^kt, in which t is the number of years after 2000 and A0 is in millions, to find an exponential growth function that models the data.
B) By which year will the population be 15 million?
Answer:
By 2086
Step-by-step explanation:
The provided equation is:
[tex]A=A0*e^{k*t}[/tex] , where:
A=total of population after t years
A0=initial population
k= rate of growth
t= time in years
Given information:
The final population will be 15 million, then A=15.
We start in 2000 with a 5.82 million population, then A0=5.82.
Missing information:
Although k is not given, we can calculate k by using the following statement, from 2000 to 2040 (within 40 years) population is proyected to grow to 9 million, which means a passage from 5.8 to 9 million (3.2 million increament).
Then we can use the same expression to calculate k:
[tex]A=A0*e^{k*t}[/tex]
[tex]9=5.8*e^{40*k}[/tex]
[tex]ln(9/5.8)/40=k[/tex]
[tex]0.010984166494596147=k[/tex]
[tex]0.011=k[/tex]
Now that we have k=0.011, we can find the time (t) by which population will be 15 million:
[tex]A=A0*e^{k*t}[/tex]
[tex]15=5.8*e^{0.011t}[/tex]
[tex]ln(15/5.8)/0.011=t[/tex]
[tex]86.38111668634878=t[/tex]
[tex]86.38=t[/tex]
Because the starting year is 2000, and we need 86.38 years for increasing the population from 5.8 to 15 million, then by 2086 the population will be 15 million.
Find the directional derivative of f(x,y,z)=2z2x+y3f(x,y,z)=2z2x+y3 at the point (−1,4,3)(−1,4,3) in the direction of the vector 15–√i+25–√j15i+25j. (Use symbolic notation and fractions where needed.)
[tex]f(x,y,z)=2z^2x+y^3[/tex]
[tex]f[/tex] has gradient
[tex]\nabla f(x,y,z)=2z^2\,\vec\imath+3y^2\,\vec\jmath+4xz\,\vec k[/tex]
which at the point (-1, 4, 3) has a value of
[tex]\nabla f(-1,4,3)=18\,\vec\imath+48\,\vec\jmath-12\,\vec k[/tex]
I'm not sure what the given direction vector is supposed to be, but my best guess is that it's intended to say [tex]\vec u=15\,\vec\imath+25\,\vec\jmath[/tex], in which case we have
[tex]\|\vec u\|=\sqrt{15^2+25^2}=5\sqrt{34}[/tex]
Then the derivative of [tex]f[/tex] at (-1, 4, 3) in the direction of [tex]\vec u[/tex] is
[tex]D_{\vec u}f(-1,4,3)=\nabla f(-1,4,3)\cdot\dfrac{\vec u}{\|\vec u\|}=\boxed{\dfrac{294}{\sqrt{34}}}[/tex]
To find the directional derivative, we need to find the gradient of the function and dot product it with the given direction vector.
Explanation:To find the directional derivative, we need to find the gradient of the function and dot product it with the given direction vector. The gradient of the function f(x, y, z) = 2z^2x + y^3 is (∂f/∂x, ∂f/∂y, ∂f/∂z) = (4zx, 3y^2, 4xz). The directional derivative in the direction of the vector (15-√i + 25-√j) is given by the dot product of the gradient and the direction vector: (4(-1)(15-√) + 3(4^2)(25-√) + 4(3)(-1)(15-√))/√((15-√)^2 + (25-√)^2). Simplifying this expression gives the directional derivative.
Learn more about Directional derivative here:https://brainly.com/question/32589894
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If f(x) = 2x - 1 and g(x) = x^2 - 2, find [g · f](x)
please show me how to do this
Answer:
(x² -7)/(2x + 1)
Step-by-step explanation:
f(x) = 2x+1 and g(x) = x² -7
thus: (g/f)(x) = g(x)/f(x) = x² -7/2x + 1
The answer is:
[tex](g \circ f)(x)=4x^{2}-4x-1[/tex]
Why?To solve the problem, we need to remember that composing functions means evaluate a function into another different function.
Also, we need to remember how to solve the following notable product:
[tex](a-b)^{2}=a^{2}-2ab+b^{2}[/tex]
We have that:
[tex](g \circ f)(x)=g(f(x))[/tex]
Now, we are given the equations:
[tex]f(x)=2x-1\\g(x)=x^{2}-2[/tex]
So, composing we have:
[tex](g \circ f)(x)=g(f(x))[/tex]
[tex](g \circ f)(x)=(2x-1)^{2}-2[/tex]
Now, we have to solve the notable product:
[tex](g \circ f)(x)=((2x)^{2}-2(2x*1)+1^{2})-2[/tex]
[tex](g \circ f)(x)=4x^{2}-4x+1-2[/tex]
Hence, we have that:
[tex](g \circ f)(x)=4x^{2}-4x-1[/tex]
Have a nice day!