Answer and Explanation:
Given : A lab technician is tested for her consistency by taking multiple measurements of cholesterol levels from the same blood sample.
The target accuracy is a variance in measurements of 1.2 or less. If the lab technician takes 16 measurements and the variance of the measurements in the sample is 2.2.
To find :
1) Does this provide enough evidence to reject the claim that the lab technician’s accuracy is within the target accuracy?
2)Compute the value of the appropriate test statistic ?
Solution :
1) n=16 number of sample
The target accuracy is a variance in measurements of 1.2 or less i.e. [tex]\sigma_1^2 =1.2[/tex]
The variance of the measurements in the sample is 2.2 i.e. [tex]\sigma_2^2=2.2[/tex]
According to question,
We state the null and alternative hypotheses,
Null hypothesis [tex]H_o : \text{var}^2 \geq 1.2[/tex]
Alternative hypothesis [tex]H_a : \text{var}^2<1.2[/tex]
We claim the alternative hypothesis.
2) Compute the value of the appropriate test statistic.
Using Chi-square,
[tex]\chi =\frac{(n-1)\sigma_2^2}{\sigma_1^2}[/tex]
[tex]\chi =\frac{(16-1)(2.2)}{(1.2)}[/tex]
[tex]\chi =\frac{(15)(2.2)}{1.2}[/tex]
[tex]\chi =\frac{33}{1.2}[/tex]
[tex]\chi =27.5[/tex]
Therefore, The value of the appropriate test statistic is 27.5.
To determine if the lab technician's accuracy is within the target accuracy, a one-sample variance test can be performed using the chi-square statistic.
Explanation:To determine if the lab technician's accuracy is within the target accuracy, we can perform a hypothesis test. We will use a one-sample variance test to compare the sample variance to the target variance.
The appropriate test statistic for a one-sample variance test is the chi-square statistic. The chi-square statistic is calculated by taking the sample variance and dividing it by the target variance, then multiplying by the degrees of freedom.
In this case, we have 16 measurements and a target variance of 1.2. The sample variance is 2.2. We can calculate the test statistic using the formula chi-square = (n-1) * (sample variance / target variance). Plugging in the values, we get chi-square = (16-1) * (2.2 / 1.2) = 29.67.
Learn more about Hypothesis testing here:https://brainly.com/question/34171008
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Determine the truth values of these statements
a) The product of x 2 and x 3 is x 6 .
b) 2π + 5π = 7π
c) x 2 > 0 for any real number.
d) The integer 315 − 8 is even.
e) The sum of two odd integers is even.
f) √ 2 ∈ Z
g) −1 ∈/ Z +
h) π ∈ Q
Answer:
a) False b) True c) False d ) False e) True f) False g) False h) False
Step-by-step explanation:
a) False
x² × x³ = [tex]x^5[/tex]
When we multiply two exponential number with same base, their powers add up.
b) True
2π + 5π = π(2+5) = 7π
The coefficients of π are added together.
c) False
x² ≥ 0. For any value of x, negative or positive x² is always positive. But for x = 0, x² = 0×0 = 0
d) False
315 - 8 = 307, which is clearly an odd number.
e)True
The sum of all integers is always even.
Let m and n be two odd integers.
Thus, they can be expressed as m = 2r + 1 and n = 2s +1, where r and s are even integers.
m + n = 2r +2s + 2, which is clearly even.
f) False
Since √2 is an irrational number. It cannot belong to z, which is collection of all integer number.
√2 ∉ z
g) False
Since -1 is a negative integer, it cannot belong to [tex]z^+[/tex], as it is collection of all positive integers.
h) False
π cannot belong to Q because Q is a collection of all rational numbers and π is not a rational number. The decimal expansion of π is non- terminating that is it does not end.
Use mathematical induction to prove that for each integer n ≥ 4, 5^n ≥ 2^2n+1 + 100. (it is 5 to power of n and 2 to the power of 2n+1)
Step-by-step explanation:
The statement to be proved using mathematical induction is:
"For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]We will begin the proof showing that the base case is satisfied (n=4).
[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex].
Then, 1 is true for n=4.
Now we will assume that the statement holds for some arbitrary natural number [tex]n\geq 4[/tex] and prove that then, the statement holds for n+1. Observe that
[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]
With this the inductive step has been proven and then, our statement is true,
For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]
3. Write a recursive algorithm of the sequence t(1)=1 and t(n)=n2 t(n-1) as a function.
Answer:
int t(int n){
if(n == 1)
return 1;
else
return n*n*t(n-1);
}
Step-by-step explanation:
A recursive function is a function that calls itself.
I am going to give you an example of this algorithm in the C language of programming.
int t(int n){
if(n == 1)
return 1;
else
return n*n*t(n-1);
}
The function is named t. In the else clause, the function calls itself, so it is recursive.
In the following ordinary annuity, the interest is compounded with each payment, and the payment is made at the end of the compounding period. Find the amount of time needed for the sinking fund to reach the given accumulated amount. (Round your answer to two decimal places.) $255 monthly at 5.8% to accumulate $25,000.
Answer:
Ans. the amount of time needed for the sinking fund to reach $25,000 if invested $255/month at 5.8% compounded monthly (Effective monthly=0.4833%) is 80.45 months.
Step-by-step explanation:
Hi, first we need to transform that 5.8% compounded monthly into an effective monthly rate, that is as follows.
[tex]r(EffectiveMonthly)=\frac{r(Comp.Monthly)}{12} =\frac{0.058}{12} =0.00483[/tex]
That means that our effective monthly rate is 0.483%
Now, we need to solve for "n" the following formula.
[tex]FutureValue=\frac{A((1+r)^{n}-1) }{r}[/tex]
Let´s start solving
[tex]25,000=\frac{255((1.00483)^{n}-1) }{0.00483}[/tex]
[tex]\frac{25,000*0.00483}{255} =1.00483^{n} -1\\[/tex]
[tex]0.47352941=1.00483^{n} -1[/tex]
[tex]1+0.47352941=1.00483^{n[/tex]
[tex]1.47352941=1.00483^{n}[/tex]
[tex]Ln(1.47352941)=n*Ln(1.00483)[/tex]
[tex]\frac{Ln(1.47352941)}{Ln(1.00483)} =n=80.45[/tex]
This means that it will take 80.45 months to reach $25,000 with an annuity of $255 at a rate of 5.8% compounded monthly (0.4833% effective monthly).
Best of luck.
PLZ HELP I BEG DUE IN 30 MIN!!! 30 POINTS!!!!
The total distance from A to B is 5 ( -3 to 2 = 5).
Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.
Answer:
A to B is 5 ( -3 to 2 = 5).//!!//Then you are gonna be Using the ratio 1/5, split the distance in to 1/5th's, point K would be at -2.
In △ABC, CD is an altitude, such that AD = BC. Find AC, if AB = 3 cm, and CD = 3 cm.
PLEASE ANSWER !!!!!
Final answer:
To find the length of AC in the right-angled isosceles triangle △ADC with altitude CD and AD equals to BC, we use the Pythagorean theorem, substituting known lengths to solve for AC, which is √11.25 cm.
Explanation:
The student is asking to find the length of side AC in a triangle △ABC where CD is an altitude and AD is equal to BC, with given lengths AB = 3 cm and CD = 3 cm.
To solve for AC, we can use the Pythagorean theorem which states that for a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Since CD is an altitude and AD = BC, △ADC is a right-angled isosceles triangle.
Using the Pythagorean theorem in △ADC, we have:
AD2 + CD2 = AC2
Since AD = BC and CD = 3 cm, let's assume AD = x cm. Then,
x2 + 32 = AC2
x2 + 9 = AC2
But, AB = 3 cm, and AB = AD + DB = x + x = 2x,
Therefore, x = AB/2 = 1.5 cm. Substituting this value back into the previous equation:
(1.5)2 + 9 = AC2 = 2.25 + 9 = 11.25
AC = √11.25 cm
Therefore, the length of AC is √11.25 cm.
How does remote work relate to taking an online class or being an online student (fully online or hybrid)?
Answer:
Answered
Step-by-step explanation:
Online courses are those classes which are delivered entirely online. Students study via web cam, chat rooms and smart boards. Whereas hybrid learning is a combination of both online learning and traditional in class learning. Remote work is working away from the work place at your own comfort and choice of location.
Taxes reduce your paycheck by 22% each month. In an Excel spreadsheet, the salary earned for a month stored in cell A35. Write an Excel formula that would calculate the dollar amount of taxes.
a.
=A35*1.22
b.
=A35/0.22
c.
=A35*0.22
d.
=0.22/A35
e.
=A35/22
f.
None of the above.
Answer: Option 'c' is correct.
Step-by-step explanation:
Since we have given that
The salary earned for a month stored in = A35
Rate of tax reduces by = 22%
We need to remove the % sign by dividing 22 by 100 and it becomes 0.22.
So, the dollar amount of taxes is given by
[tex]Taxes=A35\times \dfrac{22}{100}\\\\Taxes=A35\times 0.22[/tex]
Hence, Option 'c' is correct.
How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?
Answer:
The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.
Step-by-step explanation:
To find : How much heat (Btu) is produced by a 150-W light bulb that is on for 20-hours?
Solution :
A 150-W light bulb is on for 20-hours.
The heat produced by bulb is given by,
[tex]H=150\times 20[/tex]
[tex]H=3000\ W-hr[/tex]
We know that,
[tex]1\ \text{W-hr}=3.4\ \text{BTU}[/tex]
Converting W-hr into BTU,
[tex]3000\ \text{W-hr}=3000\times 3.4\ \text{BTU}[/tex]
[tex]3000\ \text{W-hr}=10200\ \text{BTU}[/tex]
Therefore, The heat is produced by a 150-W light bulb that is on for 20-hours is 10200 BTU.
Enrollment at ELAC decreased by 5%, or 600 people, the year. How many people were enrolled last year?
Answer:
12,000.
Step-by-step explanation:
Let x be the number of people that were enrolled last year.
We have been given that enrollment at ELAC decreased by 5%, or 600 people, the year. We are asked to find the number of people that were enrolled last year.
We can set as equation such that 5% of x equals 600.
[tex]\frac{5}{100}\cdot x=600[/tex]
[tex]0.05x=600[/tex]
[tex]\frac{0.05x}{0.05}=\frac{600}{0.05}[/tex]
[tex]x=12,000[/tex]
Therefore, 12,000 people were enrolled last year.
Given any integer m≥2, is it possible to find a sequence of m−1 consecutive positive integers none of which is prime? Explain your answer.
7
4
Final Answer:
No, it is not possible to find a sequence of m−1 consecutive positive integers none of which is prime for any integer m≥2.
Explanation:
In order to determine whether it is possible to find a sequence of m−1 consecutive positive integers none of which is prime, we need to consider the properties of prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Therefore, if we take any m−1 consecutive positive integers, at least one of them will be divisible by a prime number. This means that it is not possible to find a sequence of m−1 consecutive positive integers none of which is prime.
For example, let’s consider the sequence of 4 consecutive positive integers: 10, 11, 12, and 13. Among these numbers, 11 and 13 are prime. Even if we consider larger sequences, we will always encounter at least one prime number within the consecutive positive integers.
In conclusion, due to the nature of prime numbers and their distribution among positive integers, it is not possible to find a sequence of m−1 consecutive positive integers none of which is prime for any integer m≥2.
Given that the number 8881 is not a prime number, prove that it has a prime factor that is at most 89.
Answer with Step-by-step explanation:
We are given that a number 8881 is not a prime number
We have to prove that it has given a prime factor that is at most 89.
In order to prove that given number has highest prime factor is 89 we will find the prime factorization of given number.
[tex]8881=83\times 107[/tex]
Therefore, 8881 is not a prime number and it has two factors 83 and 107.
83 is a prime factor of 8881 which is less than 89.We have to find a prime factor of 8881 which is atmost 89.
Therefore, 83 is that prime factor .
Hence, 8881 has a prime factor that is at most 89.
5. (6 marks) Use mathematical induction to prove that for each integer n ≥ 4, 5^n ≥ 2^2n+1 + 100.
(please take +100 into considersation since previous solution didnt )
Step-by-step explanation:
We will prove by mathematical induction that, for every natural [tex]n\geq 4[/tex],
[tex]5^n\geq 2^{2n+1}+100[/tex]
We will prove our base case, when n=4, to be true.
Base case:
[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex]
Inductive hypothesis:
Given a natural [tex]n\geq 4[/tex],
[tex]5^n\geq 2^{2n+1}+100[/tex]
Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.
Inductive step:
[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=\\=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]
With this we have proved our statement to be true for n+1.
In conlusion, for every natural [tex]n\geq4[/tex].
[tex]5^n\geq 2^{2n+1}+100[/tex]
If f(x)=7/x^2, then what is the area enclosed by the graph of the function, the horizontal axis, and vertical lines at x=3 and x=4
Answer:
[tex]Area=\frac{7}{12}[/tex]
Step-by-step explanation:
[tex]Area=\int\limits^a_b {f(x)} \, dx =\int\limits^4_3 {\frac{7}{x^{2}} } \, dx =-7*\frac{1}{x}=-7(1/4-1/3)=\frac{7}{12}[/tex]
Sketch these four lines y = 2x+1, y =-x, and x = 0 and x = 2. Then use integrals to find the area of the region bounded by these lines. Finally, check your answer by computing this area using simple geometry.
Answer:
Area = 8
Step-by-step explanation:
A skecth is given in the attached file, there are two extra lines used to calculate the area with simple geometry:
We must use a double integral to obtain the area:
[tex]\int\limits^2_0 {\int\limits^b_a \, dy } \, dx[/tex]
Where
b stands for y=2x+1
a stands for y=-x
Carring out the integrals we find the area:
[tex]\int\limits^2_0 {(2x+1 - (-x))} \, dx = \int\limits^2_0 {3x+1} \, dx = (3x^{2}/2+x) \left \{ {{2} \atop {0}} \right.\\ A =( 3*2^{2}/2) + 2 =8[/tex]
Geometrically we can divide the area bounded by this lines as two triangles and a rectangle from the figure and the intersection of these lines we kno that the three figures have a base of 2. The heigth of the rectangle is 1 and for the triangles we have 4 for the upper triangle and 2 for the lower.
Therefore:
[tex]A = A_{upperT}+ A_{rect}+ A_{lowerT}[/tex]
and
[tex]A_{upperT}=2*4/2=4\\A_{rect}=2*1=2\\ A_{lowerT}=2*2/2=2[/tex]
Summing the four areas we have:
A=8
Greeting!
A worker performs a repetitive assembly task at a workbench to assemble products. Each product consists of 25 components. Various hand tools are used in the task. The standard time for the work cycle is 7.45 min, based on using a PFD allowance factor of 15%. If the worker completes 75 product units during an 8-hour shift, determine the number of standard hours accomplished.
Answer:
9.3125 hours
Step-by-step explanation:
Given:
Number of components consisting in a product = 25
Standard time for work cycle = 7.45 minutes
or
standard time for work cycle = [tex]\frac{\textup{7.45}}{\textup{60}}[/tex] hours
Number of units completed = 75
Now,
The number of standard hours
= Number of units completed × standard time for the work cycle
= [tex]75\times\frac{\textup{7.45}}{\textup{60}}[/tex] hours
or
The number of standard hours = 9.3125 hours
Answer:
The answer is 9.3125 hours :)
Step-by-step explanation:
Give the first three non-zero terms of the Taylor series for f(x) = tan(x) about x 0· Use this to approximate tan(1) and give an upper bound on the error in this approximation
Answer:
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]
Approximate error = 0.4426
Step-by-step explanation:
f(x)=tanx, a=0
Maclaurin series formula used is given below
[tex]f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(0)x^{n}}{n!}=f(0)+f'(0)x+\dfrac{f''(0)}{2!}x^{2}+\dfrac{f'''(0)}{3!}x^{3}+....[/tex]
f(x)=tanx
f(0)=tan0=0
[tex]f'(x)=sec^{2}x\\f'(0)=sec^{2}0=1\\f''(x)=2sec^{2}xtanx\\f''(0)=2sec^{2}0tan0=0\\f'''(x)=-4sec^{2}x+6sec^{4}x\\f'''(0)=-4sec^{2}0+6sec^{4}0=-4+6=2\\[/tex]
[tex]f''''(x)=-8(2sec^{2}xtan^{2}x+sec^{4}x)+24(4sec^{4}xtan^{2}x)+sec^{6})\\f''''(0)=-8(0+1)+24(0+1)=-8+24=16\\[/tex]
[tex]f(x)=0+x+0+\dfrac{2x^{3}}{3!}+\dfrac{16x^{4}}{4!}\\[/tex]
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}\\[/tex]
Hence, the Taylor series for f(x)=tanx is given by
[tex]f(x)=x+\dfrac{x^{3}}{3}+\dfrac{2x^{4}}{3}....[/tex]
Maclaurin series upper bound error formula used is given as
R_n(x)=|f(x)-T_n(x)|
R_3(x)=|tanx-T_3(x)|
[tex]R_3(x)=|tanx-x-\dfrac{x^{3}}{3}-\dfrac{2x^{4}}{3}|[/tex]
Plugging this value x=1
[tex]R_3(x)=|tan(1)-1-\dfrac{1}{3}-\dfrac{2}{3}|\\[/tex]
R_3(x)=|1.5574-1-0.333-0.666|
R_3(x)=|-0.4426|=0.4426
Hence, upper bound on the error approximation
tan(1)=0.4426
Let V be the set of pairs (x; y) of real numbers and let the eld F be the
real number set. Dene the addition and scalar multiplication as follows:
(x1; y1) + (x2; y2) = (x1 + x2; 0)
c(x; y) = (cx; 0):
Is V , with these operations, a vector space? Explain.
To prove that V is a vector space we must prove that the sum define on it satisfy conmutativiy, asociativity and existence of the neutral element and inverses. Also, the scalar multiplication define on V must satisfy distributivity propertie with respect to the sum and viceversa, and an asosiativity too in the sense that [tex]x(y\cdot v)= (xy)\cdot v[/tex] for [tex]x,y\in \mathbb{R}[/tex] and [tex]v\in V[/tex]. One can prove with this that the neutral element for the sum is unique. But with your operations you have two neutral elements for [tex](1;2)[/tex]
[tex](1;2)+(-1;3)=(0;0)[/tex]
and
[tex](1;5)+(-1;11)=(0;0)[/tex]
So, you dont have a vector space.
Final answer:
The set V, with its defined addition and scalar multiplication operations, does not fulfill essential vector space properties such as the existence of an additive identity, presence of additive inverses, and correct scalar multiplication effects on components. Therefore, V is not a vector space.
Explanation:
To determine if a set V, defined with specific addition and scalar multiplication operations, is a vector space, it must satisfy several properties commonly defined in linear algebra.
For V to be considered a vector space, the addition operation must be associative and commutative, there must be an additive identity (zero vector), each vector must have an additive inverse, scalar multiplication must be associative, there must be a multiplicative identity (1), and both operations must distribute over vector addition and scalar addition.
The defined operations on V are (x₁; y₁) + (x₂; y₂) = (x₁ + x₂; 0) for vector addition and c(x; y) = (cx; 0) for scalar multiplication. These operations fail to satisfy several vector space properties, including:
The existence of an additive identity that affects both components, since (x₁; y₁) + (0; 0) should equal (x₁; y₁), but according to the given addition rule, it equals (x₁; 0).The presence of additive inverses, as there is no pair (x₂; y₂) such that (x₁; y₁) + (x₂; y₂) equals the zero vector (0; 0).The scalar multiplication does not adequately affect the y-component; it should leave it y unchanged, i.e., c(x; y) = (cx; cy), but the given rule yields (cx; 0).Due to these inadequacies, V does not meet the criteria for a vector space under the provided operations.
YP=+8+10+12+...+ 106 and Q=2+4+6+8+ ... 104 are sums of arithmetic sequences, determine which is greater, Por Q, and by how much
Final answer:
The sum of sequence P (YP) is greater than the sum of sequence Q (Q) by 94, with P summing to 2850 and Q summing to 2756.
Explanation:
To determine which sum is greater between P and Q, we need to first understand that P and Q represent the sum of arithmetic sequences. The first sequence, P, begins with 8 and increments by 2, ending at 106. The second sequence, Q, starts at 2 and increments by 2, ending at 104.
To find out the sums, we can use the formula for the sum of an arithmetic sequence: S = n/2 (a1 + an), where S is the sum, n is the number of terms, a1 is the first term, and an is the last term. For the sequence P, the first term a1 is 8, and the last term an is 106. To find n, the number of terms, we can use the formula n = (an - a1) / d + 1, where d is the common difference, which is 2 in this case.
Applying the formula to the sequence P, we get:
The number of terms in P: n(P) = (106 - 8) / 2 + 1 = 50
Sum of P: S(P) = 50/2 x (8 + 106) = 25 x 114 = 2850
For the sequence Q:
The number of terms in Q: n(Q) = (104 - 2) / 2 + 1 = 52
Sum of Q: S(Q) = 52/2 x (2 + 104) = 26 x 106 = 2756
Comparing these sums, we can see that the sum of sequence P, 2850, is greater than the sum of sequence Q, 2756. The difference between the sums is 2850 - 2756 = 94.
Therefore, P is greater than Q by 94.
A bird feeder has a diameter of 3 inches and is composed of a cylinder and a cone. A diagram of the feeder is shown below.
What is the volume of this bird feeder, to the nearest tenth of cubic inch?
The total volume of the bird feeder is 47.1 cubic inches.
The total volume of the bird feeder.
Given:
Diameter (D) of the bird feeder = 3 inchesHeight of the cone (hcone) = 2 inchesHeight of the cylinder (hcylinder) = 6 inchesCalculate the Radius (r):So, the total volume of the bird feeder is 47.1 cubic inches.
A consumer products company is formulating a new shampoo and is interested in foam height (in mm). Foam height is approximately normally distributed and has a standard deviation of 20 mm. The company wishes to test H0: μ = 175 mm versus H1: μ > 175 mm, using a random sample of n = 10 samples.(a) Find P-value if the sample average is = 185? Round your final answer to 3 decimal places.(b) What is the probability of type II error if the true mean foam height is 200 mm and we assume that α = 0.05? Round your intermediate answer to 1 decimal place. Round the final answer to 4 decimal places.(c) What is the power of the test from part (b)? Round your final answer to 4 decimal places.
Answer:
a) 0.057
b) 0.5234
c) 0.4766
Step-by-step explanation:
a)
To find the p-value if the sample average is 185, we first compute the z-score associated to this value, we use the formula
[tex]z=\frac{\bar x-\mu}{\sigma/\sqrt N}[/tex]
where
[tex]\bar x=mean\; of\;the \;sample[/tex]
[tex]\mu=mean\; established\; in\; H_0[/tex]
[tex]\sigma=standard \; deviation[/tex]
N = size of the sample.
So,
[tex]z=\frac{185-175}{20/\sqrt {10}}=1.5811[/tex]
[tex]\boxed {z=1.5811}[/tex]
As the sample suggests that the real mean could be greater than the established in the null hypothesis, then we are interested in the area under the normal curve to the right of 1.5811 and this would be your p-value.
We compute the area of the normal curve for values to the right of 1.5811 either with a table or with a computer and find that this area is equal to 0.0569 = 0.057 rounded to 3 decimals.
So the p-value is
[tex]\boxed {p=0.057}[/tex]
b)
Since the z-score associated to an α value of 0.05 is 1.64 and the z-score of the alternative hypothesis is 1.5811 which is less than 1.64 (z critical), we cannot reject the null, so we are making a Type II error since 175 is not the true mean.
We can compute the probability of such an error following the next steps:
Step 1
Compute [tex]\bar x_{critical}[/tex]
[tex]1.64=z_{critical}=\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}[/tex]
[tex]\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}=\frac{\bar x_{critical}-175}{6.3245}=1.64\Rightarrow \bar x_{critical}=185.3721[/tex]
So we would make a Type II error if our sample mean is less than 185.3721.
Step 2
Compute the probability that your sample mean is less than 185.3711
[tex]P(\bar x < 185.3711)=P(z< \frac{185.3711-185}{6.3245})=P(z<0.0586)=0.5234[/tex]
So, the probability of making a Type II error is 0.5234 = 52.34%
c)
The power of a hypothesis test is 1 minus the probability of a Type II error. So, the power of the test is
1 - 0.5234 = 0.4766
The p-value is 0.014, indicating strong evidence against the null hypothesis. The probability of a type II error is 0.0907, and the power of the test is 0.9093.
Explanation:To find the p-value, we need to determine the probability of observing a sample average of 185 or higher, given that the true mean foam height is 175. Since the sample size is small, we'll use the t-distribution instead of the normal distribution. With a sample size of 10, the degrees of freedom is 9. Using the t-distribution table or a calculator, we find the p-value to be 0.014 (rounded to 3 decimal places).
The probability of a type II error is the probability of failing to reject the null hypothesis when it is actually false. In this case, the null hypothesis is μ = 175, but the true mean foam height is 200. We assume α = 0.05, so the critical value is 1.645 (from the t-distribution table for a one-tailed test). Using the formula for the standard error of the sample mean, σ/√n, we can calculate the standard deviation of the sample mean to be 20/√10 = 6.32 (rounded to 2 decimal places). The difference between the critical value and the true mean is (200 - 175)/6.32 = 3.96 (rounded to 2 decimal places). Using a t-distribution table or calculator to find the area to the right of 3.96 with 9 degrees of freedom, we find the probability of a type II error to be 0.0907 (rounded to 4 decimal places).
The power of the test is 1 minus the probability of a type II error. So the power of the test is 1 - 0.0907 = 0.9093 (rounded to 4 decimal places).
Learn more about Hypothesis testing here:https://brainly.com/question/34171008
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An item normally $15.99 is listed as being on sale for 30% off its original price, what must you pay?
A discount store promises that all the items it sells are 40% of their normal asking retail price. If one buys shoes that normally retail for $60.99 what is the price you would expect to pay?
Describe how you would answer each question
Then rewrite the percent off problem as a percent of problem.
Answer: a. You must pay $11.19 for the item.
b.The price you would expect to pay would be $36.59
Step-by-step explanation:
Hi, for first the question you need to calculate the 30 percent of the price of the item, and then subtract that result to the original price.
So: $15.99 × 0,30 = $4.797
$15.99 - $4.797 = $11.19
You must pay $11.19 for the item.
Question 2:It´s a similar resolution, first you calculate the 40 percent of the retail price, and then subtract that result to the retail price.
So:
$60.99 × 0,40= $24.396
$60.99 - $24.396 = $36.59
The price you would expect to pay would be $36.59
Let A be the set represented by the bitstring 01011011100, let B be the set represented by the bitstring 10110111010. Find the bitstrings representing Ac, AUB, AnB, and A-E.
Answer:
Ac = 10100100011
[tex]A \cup B = 11111111110[/tex]
[tex]A \cap B = 00010011000[/tex]
[tex]A - B = 100100111010[/tex]
Step-by-step explanation:
All these operations are bitwise operations.
Ac is the complement of a. So where we have a bit 0, the complement is 1. Where we have a bit 1, the complement is 0. So
A = 01011011100
Ac = 10100100011
The second operation is the union between A and B. This is bitwise(bit 1 of A with bit 1 of B, bit 2 of A with bit 2 of B,...). The union operation is only 0 when it is between two zeros. So:
A = 01011011100
B = 10110111010
[tex]A \cup B = 11111111110[/tex]
The third operation is the intersection between A and B. Again, bitwise. The intersection is only 1 when it is between two bits that are 1. So
A = 01011110100
B = 10110111010
[tex]A \cap B = 00010011000[/tex]
The last operation is the bitwise subtraction between A and B. We start from the least significant bit(the last one). And we have to take care of the borrow operator also, similarly to a decimal subtraction.
We can only borrow from a previous bit 1, and this bit is set to 0
0-0 is 0 with no borrow
1-0 is 1 with no borrow
1 with borrow - 0 is 1 with borrow
1 with borrow - 1 is 1 with no borrow
0-1 is 1 with no borrow
0 with borrow -1 is 0 with no borrow
1-1 is 0 with no borrow
If we arrive at the first bit(the most significant) with a borrow, we must add a 1 at the front of the answer. So
A = 01011110100
B = 10110111010
[tex]A - B = 100100111010[/tex]
Use the binomial theorem to compute (2x-1)^5
Answer:
The expended form of the provided expression is: [tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]
Step-by-step explanation:
Consider the provided expression.
[tex](2x-1)^5[/tex]
The binomial theorem:
[tex](a+b)^{n}=\sum _{r=0}^{n}{n \choose r}a^{n-r}b^r[/tex]
Where,
[tex]{n \choose r}= ^nC_r =\frac{n!}{(n-r)!r!}[/tex]
Now by using the above formula.
[tex]\frac{5!}{0!\left(5-0\right)!}\left(2x\right)^5\left(-1\right)^0+\frac{5!}{1!\left(5-1\right)!}\left(2x\right)^4\left(-1\right)^1+\frac{5!}{2!\left(5-2\right)!}\left(2x\right)^3\left(-1\right)^2+\frac{5!}{3!\left(5-3\right)!}\left(2x\right)^2\left(-1\right)^3+\frac{5!}{4!\left(5-4\right)!}\left(2x\right)^1\left(-1\right)^4+\frac{5!}{5!\left(5-5\right)!}\left(2x\right)^0\left(-1\right)^5[/tex]
[tex]2^5\cdot \:1\cdot \:1\cdot \:x^5-1\cdot \frac{2^4\cdot \:5x^4}{1!}+1\cdot \frac{2^3\cdot \:20x^3}{2!}-1\cdot \frac{2^2\cdot \:20x^2}{2!}+1\cdot \frac{5\cdot \:2x}{1!}+1\cdot \frac{\left(-1\right)^5}{\left(5-5\right)!}[/tex]
[tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]
Hence, the expended form of the provided expression is: [tex]32x^5-80x^4+80x^3-40x^2+10x-1[/tex]
What of the following basic rules is NOT true about geometry?
A. A straight line intersecting two parallel lines produces corresponding equal angles
B. Similar triangles have the same sides
C. Similar triangles have the same angles
D. The sum of the interior angles of any triangle equals 180°
E. None of the above
Answer:
A. A straight line intersecting two parallel lines produces corresponding equal angles.
Step-by-step explanation:
If two straight lines are parallel, and a straight line intersects them, then we get alternate angles equal, exterior angle equal to the opposite interior angle on the same side and the interior angles on the same sides equal to two right angles.
Hence, here the rule that is not true is : A. A straight line intersecting two parallel lines produces corresponding equal angles.
Prove that the trajectory of a projectile is parabolic, having the form y = ax + bx2. To obtain this expression, solve the equation x = v0xt for t and substitute it into the expression for y = v0yt − 1 2 gt2. (These equations describe the x and y positions of a projectile that starts at the origin.) You should obtain an equation of the form y = ax + bx2 where a and b are constants.
Answer: y = v₀tgθx - gx²/2v₀²cos²θ
a = v₀tgθ
b = -g/2v₀²cos²θ
Step-by-step explanation:
x = v₀ₓt
y = v₀y.t - g.t²/2
x = v₀.cosθt → t = x/v₀.cosθ
y = v₀y.t - g.t²/2
v₀y = v₀.senθ
y = v₀senθ.x/v₀cosθ - g/2.(x/v₀cosθ)²
y = v₀.tgθ.x - gx²/2v₀²cos²θ
a = v₀tgθ → constant because v₀ and θ do not change
b = - g/2v₀²cos²θ → constant because v₀, g and θ do not change
Final answer:
The trajectory of a projectile is shown to be parabolic by substituting time from the x-direction motion equation into the y-direction motion equation and rearranging, yielding a parabolic form y = ax + bx², with constants determined by initial velocity and gravity.
Explanation:
To prove that the trajectory of a projectile is parabolic, we start with the equations of motion in the x and y directions for a projectile that starts at the origin. For the x-direction, we have x = v_0x t, where v_0x is the initial velocity component in the x-direction and t is the time.
In the y-direction, the equation is y = v_0y t - (1/2) g t², where v_0y is the initial velocity component in the y-direction and g is the acceleration due to gravity.
To find t from the x equation: t = x / v_0x. Substituting this into the y equation yields: y = v_0y (x / v_0x) - (1/2) g (x / v_0x)^2.
Now, simplifying we get: y = (v_0y / v_0x) x - (g/2 v_0x^2) x^2, which is of the form y = ax + bx². Here, a = v_0y/v_0x and b = -g/2 v_0x²are constants depending on the initial velocity components and acceleration due to gravity. The equation y = ax + bx² represents a parabolic path, confirming that the projectile's trajectory is indeed a parabola.
You deposit $10,000 into a bank account at 2% annual interest. How long will it take for the $10,000 to compound to $30,000?
N= I/Y= PV= PMT= FV= P/Y=
Answer: time = 55.48 years
Explanation:
Given:
Principal amount = $10000
Interest rate = 2% p.a
Amount = $30000
We can evaluate the time taken using the following formula:
[tex]Amount=Principal(1+\frac{r}{100})^{t}[/tex]
[tex]30000 = 10000\times(1+\frac{2}{100})^{t}[/tex]
Solving the above equation, we get
[tex]1.02^{t} = 3[/tex]
Now taking log on both sides, we get;
[tex]t\ log(1.02) = log(3)[/tex]
time = 55.48 years
Find all optimal solutions to the following LP using the Simplex Algorithm:
maxz = x1 + 2x2 + 3x3
s.t.
x1 + 2x2 + 3x3 ≤ 10
x1 + x2 ≤ 5
x1 ≤ 1
x1,x2,x3 ≥ 0
Answer:
z=10
x1=0
x2=0
x3=3.33
Step-by-step explanation:
First Step convert your constraints in standard equations
so we have
x1 + 2x2 + 3x3+x4 = 10
x1 + x2 +x5= 5
x1 +x6= 1
Now we pass it all to the simplex table
Remember that we choose the column with the most negative value
Pivot Element=3
Divide all elements on Pivot Line by Pivot Element
Line x5= 0*Pivot Line +Line x5
Line x6= 0*Pivot Line+ Line X6
Line Z= 3* Pivot Line + Line Z
We finish when all the elements from the line Z are positive
Hence we have that x3=3.33 and x1=0, x2=0 and the max of z is 10
Answer:
z=10
x1=0
x2=0
x3=3.33
Step-by-step explanation:
Find the cube root of 10 upto 5 signaficant figures by newton raphson method
Answer: The cube root of 10 is 2.1544 using an Xo value of -0.003723
Step-by-step explanation: The Newton-Raphson is a root finding method and its formula is NR: X=Xo-(f(x)/f'(x). Once you have the equation you also need to find the derivative of that equation before applying the formula. Since the problem stated that X =10, the method was applied to find the best root in order to find the cube root of 10 up to 5 significant figures. The best method is to use a software like Excel that helps you calculate those iterations faster. The root finding for this example was -0.003723.
Give a power series representation for the function f(x) x^3/(1 + 9x^2)
Recall that for [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n\ge0}x^n[/tex]
Replace [tex]x[/tex] with [tex]-9x^2[/tex] and we get
[tex]\displaystyle\frac1{1-(-9x^2)}=\sum_{n\ge0}(-9x^2)^n=\sum_{n\ge0}(-9)^nx^{2n}[/tex]
Lastly, multiply this by [tex]x^3[/tex], so that
[tex]\boxed{f(x)=\displaystyle\sum_{n\ge0}(-9)^nx^{2n+3}}[/tex]