[tex]\dfrac{\mathrm dx}{\mathrm dt}=-5y[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dt}=-5x\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}=-5\dfrac{\mathrm dx}{\mathrm dt}[/tex]
[tex]\implies\dfrac{\mathrm d^2y}{\mathrm dt^2}-25y=0[/tex]
This ODE is linear in [tex]y(t)[/tex] with the characteristic equation and roots
[tex]r^2-25=0\implies r=\pm5[/tex]
so that
[tex]y(t)=C_1e^{5t}+C_2e^{-5t}[/tex]
Then
[tex]\dfrac{\mathrm dx}{\mathrm dt}=-5C_1e^{5t}-5C_2e^{-5t}[/tex]
[tex]\implies x(t)=-C_1e^{5t}+C_2e^{-5t}[/tex]
Given that [tex]x(0)=1[/tex] and [tex]y(0)=3[/tex], we find
[tex]\begin{cases}1=-C_1+C_2\\3=C_1+C_2\end{cases}\implies C_1=1,C_2=2[/tex]
and the particular solution to this system is
[tex]\begin{cases}x(t)=-e^{5t}+2e^{-5t}\\y(t)=e^{5t}+2e^{-5t}\end{cases}[/tex]
The value of x and y would be [tex]x(t) =-e^{5t}+ 2e^{-5t}\\\\[/tex] and [tex]y(t) = e^{5t}+ 2e^{-5t}[/tex].
What is a differential equation?An equation containing derivatives of a variable with respect to some other variable quantity is called differential equations.
The derivatives might be of any order, some terms might contain product of derivatives and the variable itself, or with derivatives themselves. They can also be for multiple variables.
we have the following differential equations
[tex]\dfrac{dx}{dt} =-5y\\\\\dfrac{dy}{dt}=-5x[/tex]
by differentiating the second equation we have
[tex]\dfrac{d^2y}{dt^2}=-5\dfrac{dx}{dt}\\\\\dfrac{d^2y}{dt^2}-25y = 0[/tex]
[tex]r^2 - 25 = 0\\\\r = \pm5[/tex]
[tex]y(t) = C_1e^{5t}+ C_2e^{-5t}[/tex]
and by using the characteristic polynomial
[tex]x(t) =-e^{5t}+ 2e^{-5t}\\\\y(t) = e^{5t}+ 2e^{-5t}[/tex]
Learn more about differential equations;
https://brainly.com/question/15345308
#SPJ2
A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, what is the probability that 13 or more ounces will be dispensed in a given glass? .3333 .5000 .6666 .1666
Answer:
I guess they rounded, but the answer is .5000 or 50% chance
Step-by-step explanation:
Count the total chances. There is a chance it will be 10oz, 11oz, 12oz, 13oz, 14oz, 15oz, or 16oz. The question asked 13 or more. That is 13, 14, 15, and 16.
That is a 4/7 chance or probability it will be more than 13oz. Divide 4 and 7 to get your answer.
0.57 or 57%
: Planning a summer study schedule About Susan plans on studying four different subjects (Math, Science, French, and Social Studies) over the course of the summer. There are 100 days in her summer break and each day she will study one of the four subjects. A schedule consists of a plan for which subject she will study on each day (a) How many ways are there for her to plan her schedule if there are no restrictions on the number of days she studies each of the four subjects? (b) How many ways are there for her to plan her schedule if she decides that the number of days she studies each subject will be the same?
Answer:
Part (A): The total number of ways are [tex]4^{100}[/tex]
Part (B): The total number of ways are [tex]1.6122075076\times10^{57}[/tex]
Step-by-step explanation:
Consider the provided information.
Part(A) a) How many ways are there for her to plan her schedule if there are no restrictions on the number of days she studies each of the four subjects?
She plans on studying four different subjects.
That means she has 4 choices for each day also there is no restriction on the number of days.
Hence, the total number of ways are:
[tex]4\times 4\times4\times4\times......4=4^{100}[/tex]
Part (B) How many ways are there for her to plan her schedule if she decides that the number of days she studies each subject will be the same?
She wants that the number of day she studies each subject will be the same that means she studies for 25 days each subject.
Therefore, the number of ways are:
[tex]^{100}C_{25}\times ^{75}C_{25}\times ^{50}C_{25}\times ^{25}C_{25}=1.6122075076\times10^{57}[/tex]
In the case of no restrictions, there are 4^100 possible study schedules. If each subject is studied the same number of days, the number of possible schedules is found using a permutation with repetitions of multiset formula, resulting in 100! / (25! * 25! * 25! * 25!).
Explanation:This question focuses on combinatorics which is a branch of mathematics that deals with combinations of objects belonging to a finite set. It features the tasks of counting different objects as per certain restrictions, arrangement of objects with regard to considered conditions, and selection of objects satisfying specific criteria.
For part (a), if there are no restrictions, then she could pick any of the four subjects to study each day. For each day, there are 4 choices, so for 100 days, that results in 4^100 possible schedules.
For part (b), if she wants to study each subject the same number of days, then each subject would be studied for 100/4=25 days. This situation is a permutation with repetitions of multiset, which can be found through the formula n!/(r1! * r2! * ... * rk!) where n is total number of items, and r1, r2, ..., rk are numbers of same elements of multiset. Using this equation, the number of ways would be 100! / (25! * 25! * 25! * 25!).
Learn more about Combinatorics here:https://brainly.com/question/31293479
#SPJ3
Personnel selection. Suppose that 7 female and 5 male applicants have been successfully screened for 5 positions. If the 5 positions are filled at random from the 12 âfinalists, what is the probability of selecting â
(A) 3 females and 2â males? â
(B) 4 females and 1â male? â
(C) 5â females?
â(D) At least 4â females?
The question involves calculating the probabilities of different combinations of females and males being selected for positions using the hypergeometric distribution, a concept in mathematics related to probability without replacement from two distinct groups.
Explanation:The question is related to the concept of probability in mathematics, specifically the use of the hypergeometric distribution, which is appropriate when sampling without replacement from a finite population divided into two groups. To calculate the probability of various combinations of female and male applicants selected for the positions, we need to use the hypergeometric probability formula:
For part (A), calculating the probability of selecting 3 females and 2 males:For part (B), calculating the probability of selecting 4 females and 1 male:For part (C), calculating the probability of selecting 5 females:For part (D), calculating the probability of selecting at least 4 females (which includes scenarios of selecting 4 or 5 females):These calculations involve combinations (denoted as C(n, k)) and the formula for hypergeometric probability, which is:
Hypergeometric Probability = [C(group1 size, group1 subset size) * C(group2 size, group2 subset size)] / C(total population size, total subset size)
Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) to (−1, 0), followed by the line segment from (−1, 0) to (2, −π). Evaluate this line integral in two ways:
Direct computation:
Parameterize the top part of the circle [tex]x^2+y^2=1[/tex] by
[tex]\vec r(t)=(x(t),y(t))=(\cos t,\sin t)[/tex]
with [tex]0\le t\le\pi[/tex], and the line segment by
[tex]\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)[/tex]
with [tex]0\le t\le1[/tex]. Then
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)[/tex]
[tex]=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt[/tex]
[tex]=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}[/tex]
Using the fundamental theorem of calculus:
The integral can be written as
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}[/tex]
If there happens to be a scalar function [tex]f[/tex] such that [tex]\vec F=\nabla f[/tex], then [tex]\vec F[/tex] is conservative and the integral is path-independent, so we only need to worry about the value of [tex]f[/tex] at the path's endpoints.
This requires
[tex]\dfrac{\partial f}{\partial x}=\sin x\implies f(x,y)=-\cos x+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=\cos y=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\sin y+C[/tex]
So we have
[tex]f(x,y)=-\cos x+\sin y+C[/tex]
which means [tex]\vec F[/tex] is indeed conservative. By the fundamental theorem, we have
[tex]\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=f(2,-\pi)-f(1,0)=-\cos2-(-\cos1)=\boxed{\cos1-\cos2}[/tex]
The line integral of (sin x dx + cos y dy) over curve C is calculated using parameterization and direct computation for the circle part and the line segment part, or alternative methods such as polar coordinates or contour integration.
The question involves evaluating the line integral of the function (sin x dx + cos y dy) along a given curve C, which in this case consists of two parts: the upper semicircle of x2 + y2 = 1 and a line segment from (−1, 0) to (2, −π). To evaluate this, we use two methods: direct computation and using Green's theorem when applicable. For the semicircle, we parameterize it by setting x = cos(θ), y = sin(θ), where θ ranges from 0 to π. The line segment can be parameterized by a straight line equation derived from the two points. The integral along the curve is computed separately for each piece, and the results are added together to find the integral over the entire curve. To evaluate the line integral in an alternate method, one might use tricks such as transforming to polar coordinates or employing complex analysis techniques such as contour integration, depending on the nature of the integral.
USA Today reported that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice. Suppose Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet. Does this indicate that the population proportion of consumers loyal to Chevrolet is more than 47%? Use alpha = 0.01.
Answer: No, It does not indicates that the population proportion of consumers loyal to Chevrolet is more than 47%.
Step-by-step explanation:
Let p denotes the proportion of consumers loyal to Chevrolet.
As per given , we have
[tex]H_0: p=0.47\\\\ H_a: p>0.47[/tex]
Since the alternative hypothesis [tex](H_a)[/tex] is right-tailed so the test would be a right-tailed test.
Also , it is given that ,Chevrolet did a study of a random sample of 1006 Chevrolet owners and found that 490 said they would buy another Chevrolet.
i.e. n = 1006
x= 490
[tex]\hat{p}=\dfrac{490}{1006}=0.487[/tex]
Test statistic :
[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
, where p=population proportion.
[tex]\hat{p}[/tex]= sample proportion
n= sample size.
i.e. [tex]z=\dfrac{0.487-0.47}{\sqrt{\dfrac{0.47(1-0.47)}{1006}}}=1.08[/tex]
P-value (for right-tailed test)= P(z>1.08)=1-P(z≤1.08) [∵P(Z>z)=1-P(Z≤z)]
=1- 0.8599=0.1401 [By z-value table.]
Decision : Since p-value (0.14) is greater than the significance level ([tex]\alpha=0.01[/tex]) , it means we are failed to reject the null hypothesis.
Conclusion : We have sufficient evidence to support the claim that about 47% of the general consumer population in the United States is loyal to the automobile manufacturer of their choice.
Hence, it does not indicates that the population proportion of consumers loyal to Chevrolet is more than 47%.
Final answer:
There's sufficient evidence to support the claim that about 47% of the general consumer population in the United States is loyal to their chosen automobile manufacturer. Thus, it doesn't indicate that the population proportion of consumers loyal to Chevrolet is more than 47%.
Explanation:
Given:
Null hypothesis (H0): The proportion of consumers loyal to Chevrolet is 47%.
Alternative hypothesis (Ha): The proportion of consumers loyal to Chevrolet is greater than 47%.
Data:
Chevrolet conducted a study of a random sample of 1006 Chevrolet owners.
Among them, 490 said they would buy another Chevrolet.
Calculations:
Sample proportion [tex](\hat{p}) = 490/1006 ≈ 0.487[/tex]
Test statistic (z) = [tex](0.487 - 0.47) / \sqrt((0.47 * (1 - 0.47)) / 1006) \approx 1.08[/tex]
P-value calculation:
Since it's a right-tailed test, P-value = P(z > 1.08) = 1 - P(z ≤ 1.08).
Look up the value in the z-table to find P(z ≤ 1.08) ≈ 0.8599.
P-value ≈ 1 - 0.8599 ≈ 0.1401.
Decision:
Since the P-value (0.14) is greater than the significance level (α = 0.01), we fail to reject the null hypothesis.
There's sufficient evidence to support the claim that about 47% of the general consumer population in the United States is loyal to their chosen automobile manufacturer.
Thus, it doesn't indicate that the population proportion of consumers loyal to Chevrolet is more than 47%.
In a sample of 24 spools of wire, the average diameter was found to be 3.16mm with a variance of 0.13. Give a point estimate for the population standard deviation of the diameter of the spools of wire. Round your answer to two decimal places, if necessary.
Answer: 0.36
Step-by-step explanation:
Given : Sample size of spools of wire : n= 24
The sample variance = [tex]s^2=0.13[/tex]
The the sample standard deviation should be the square root of the sample variance.
i.e. Sample standard deviation = [tex]s=\sqrt{0.13}=0.360555127546approx0.36[/tex]
Also, the sample standard deviation(s) is the best estimate of population standard deviation [tex](\sigma)[/tex].
Therefore , a point estimate for the population standard deviation [tex](\sigma)[/tex] of the diameter of the spools of wire. =s= 0.36
Hence, a point estimate for the population standard deviation of the diameter of the spools of wire= 0.36
Some sources report that the weights of full-term newborn babies in a certain town have a mean of 8 pounds and a standard deviation of 0.6 pounds and are normally distributed. a. What is the probability that one newborn baby will have a weight within 0.6 pounds of the meaning dash that is, between 7.4 and 8.6 pounds, or within one standard deviation of the mean
Answer:
0.682 is the probability that one newborn baby will have a weight within one standard deviation of the mean.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 8 pounds
Standard Deviation, σ = 0.6 pounds
We are given that the distribution of weights of full-term newborn babies is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(weight between 7.4 and 8.6 pounds)
[tex]P(7.4 \leq x \leq 8.6) = P(\displaystyle\frac{7.4 - 8}{0.6} \leq z \leq \displaystyle\frac{8.6-8}{0.6}) = P(-1 \leq z \leq 1)\\\\= P(z \leq 1) - P(z < -1)\\= 0.841 - 0.159 = 0.682 = 68.2\%[/tex]
[tex]P(7.4 \leq x \leq 8.6) = 6.82\%[/tex]
This could also be found with the empirical formula.
0.682 is the probability that one newborn baby will have a weight within 0.6 pounds of the mean.
A music professor says that his school's new test for "creative musical analysis" can't be trusted because the test counts and discordant string of pitches as creative. The music professor is attacking the test's:
A) confidence level.
B) margin of error.
C) validity.
D) reliability.
Answer:
C i think i already answerd this
Step-by-step explanation:
The music professor attacking the test is validity. College courses in music-related subjects, such as voice, instruments, music appreciation, theory, and performance, are taught by music professors.
What is meant by music professor ?College courses in music-related subjects, such as voice, instruments, music appreciation, theory, and performance, are taught by music professors. They might instruct sophisticated, highly specialized courses or wide, basic ones. They design curricula, teach classes, and evaluate students' work.
Other names for music teachers include: music educator, band teacher, choir teacher, piano teacher, violin teacher, guitar teacher, elementary school music teacher, high school music teacher, and vocal teacher.
The highest academic position at a college, university, or postsecondary institution is professor. Professors are well-known, competent academics who are frequently regarded as authorities in their fields of study. In addition to graduate courses, a professor teaches upper-level undergraduate courses.
To learn more about music professor refer to:
https://brainly.com/question/16462757
#SPJ2
The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. An article reports the following summary data on intake for a sample of males age 65−74 years: n = 114, x = 11.2, and s = 6.58. Does this data indicate that average daily zinc intake in the population of all males age 65−74 falls below the recommended allowance? (Use α = 0.05.) State the appropriate null and alternative hypotheses.
Answer:
Step-by-step explanation:
Hello!
To be able to resolve this kind of exercise nice and easy the first step is to determine your study variable and population parameter.
The variable is
X: daily dietary allowance for zinc for a male 50 years old or older. (mg/day)
The parameter of interest is the population mean. (μ)
You need to test if the average zinc intake is bellowed the recommended allowance, symbolically μ < 15. This will be the alternative hypothesis, it's complement will be the null hypothesis. (easy tip to detect the null hypothesis fast, it always carries the = sign)
The hypothesis is:
H₀: μ ≥ 15
H₁: μ < 15
α: 0,05
Now you have no information about the variable distribution. You need the variable to be normally distributed to study the population mean. Since the sample is large enough, you can apply the Central Limit Theorem and approximate the distribution of the sample mean to normal, this way, you can use the approximate Z to make the test.
X[bar]≈N(μ;σ²/n)
Z= X[bar]-μ ≈N(0;1)
√(σ²/n)
Z= 11.2 - 15 = -6.17
6.58/√114
The critical region of this test is one-tailed (left) the critical value is:
[tex]Z_{\alpha } = Z_{0.05} = -1.64[/tex]
If Z ≤ -1.64, you reject the null hypothesis.
If Z > -1.64, you support the null hypothesis
The calculated value is less than the critical value, so the decision is to reject the null hypothesis.
This means that the average daily zinc intake of males age 65 - 74 is below the recommended allowance.
I hope it helps!
A technician compares repair costs for two types of microwave ovens (type I and type II). He believes that the repair cost for type I ovens is greater than the repair cost for type II ovens. A sample of 60 type I ovens has a mean repair cost of $85.79, with a standard deviation of $15.13. A sample of 56 type II ovens has a mean repair cost of $78.67, with a standard deviation of $17.84. Conduct a hypothesis test of the technician's claim at the 0.1 level of significance. Let μ1 be the true mean repair cost for type I ovens and μ2 be the true mean repair cost for type II ovens.Step 1 of 4: State the null and alternative hypotheses for the test.Step 2 of 4: Compute the value of the test statistic. Round your answer to two decimal places.Step 3 of 4: Determine the decision rule for rejecting the null hypothesis H0. Round the numerical portion of your answer to three decimal places.Step 4 of 4: Make the decision for the hypothesis test. Reject or Fail to Reject Null Hypothesis
Answer:
Since p value <0.1 accept the claim that oven I repair costs are more
Step-by-step explanation:
The data given for two types of ovens are summarised below:
Group Group One Group Two
Mean 85.7900 78.6700
SD 15.1300 17.8400
SEM 1.9533 2.3840
N 60 56
Alpha = 10%
[tex]H_0: \mu_1 - \mu_2 =0\\H_a: \mu_1 - \mu_2> 0[/tex]
(Right tailed test)
The mean of Group One minus Group Two equals 7.1200
df = 114
standard error of difference = 3.065
t = 2.3234
p value = 0.0219
If p value <0.10 reject null hypothesis
4) Since p value <0.1 accept the claim that oven I repair costs are more
11. Jillian needs to make a group of 5 people. She has 10 Democrats, 13 Republicans, and 7 Independents to choose from. (a) What is the probability that the group of 5 people will have 2 Democrats, 2 Republicans, and 1 Independent
Answer:
0.1724 or 17.24%
Step-by-step explanation:
There are
10+13+7 = 30 people
so there are C(30,5) combinations of 30 elements taken 5 at a time ways to select groups of 5 people.
[tex]C(30,5)=\displaystyle\binom{30}{5}=\frac{30!}{5!25!}=142,506[/tex]
There are C(10,2) combinations of 10 elements taken 2 at a time ways to select the 2 Democrats
[tex]C(10,2)=\displaystyle\binom{10}{2}=\frac{10!}{2!8!}=45[/tex]
There are C(13,2) combinations of 13 elements taken 2 at a time ways to select the 2 Republicans
[tex]C(13,2)=\displaystyle\binom{13}{2}=\frac{13!}{2!11!}=78[/tex]
There are C(7,1) combinations of 7 elements taken 1 at a time ways to select the Independent
[tex]C(7,1)=\displaystyle\binom{7}{1}=\frac{7!}{1!6!}=7[/tex]
By the Fundamental Principle of Counting, there are
45*78*7 = 24,570
ways of making the group of 5 people with 2 Democrats, 2 Republicans and 1 independent, and the probability of forming the group is
[tex]\displaystyle\frac{24,570}{142,506}=0.1724[/tex] or 17.24%
The Ball Corporation's beverage can manufacturing plant in Fort Atkinson, Wisconsin, uses a metal supplier that provides metal with a known thickness standard deviation σ = .000507 mm. Assume a random sample of 53 sheets of metal resulted in an x⎯⎯ = .3333 mm. Calculate the 90 percent confidence interval for the true mean metal thickness. (Round your answers to 4 decimal places.) The 90% confidence interval is from to
Answer: (0.3332, 0.33341)
Step-by-step explanation:
Formula to find the confidence interval for population mean[tex](\mu)[/tex] :
[tex]\overline{x}\pm z^*\dfrac{\sigma}{\sqrt{n}}[/tex]
, where n= Sample size
[tex]\overline{x}[/tex] = sample mean.
[tex]z^*[/tex] = Critical z-value (two-tailed)
[tex]\sigma[/tex] = population standard deviation.
As per given , we have
n= 53
[tex]\sigma=0.000507\ mm [/tex]
[tex]\overline{x}=0.3333\ mm[/tex]
The critical values for 90% confidence interval : [tex]z^*=\pm1.645[/tex]
Now , the 90 percent confidence interval for the true mean metal thickness:
[tex]0.3333\pm (1.645)\dfrac{0.000507}{\sqrt{53}}\\\\=0.3333\pm(1.645)(0.0000696)\approx0.3333\pm0.00011456\\\\=(0.3333-0.00011456,\ 0.3333+0.00011456)\\\\=(0.33318544,\ 0.33341456)\approx(0.3332,\ 0.3334)[/tex]
Hence, the 90 percent confidence interval for the true mean metal thickness. : (0.3332, 0.3334)
McPhee Company manufactures rugs in the cutting and assembly process. Rugs are manufactured in 70-rug batch sizes. The cutting time is 14 minutes per rug. The assembly time is 24 minutes per rug. It takes 18 minutes to move a batch of rugs from cutting to assembly. What is the value-added lead time?
Answer:38 min
Step-by-step explanation:
Given
Cutting time [tex]t_1=14 min[/tex]
Assembly Line time [tex]t_2=24 min[/tex]
[tex]t_3=18[/tex] to move a batch of rugs from cutting to assembly
Value-added refers to a process or phase in a system that turns raw materials or work in progress into much more desirable goods and services for downstream consumers.
thus value added time is [tex]t_1+t_2=14+24 =38 min[/tex]
2log4x(x^3)=5log2x(x)
Good evening ,
Answer:
x=2
Step-by-step explanation:
Look at the photo below for the details.
:)
Flip a coin 100 times. We want the chance of getting exactly 50 heads. What is the exact probability, correct to six decimal places? What is the Normal approximation of the probability, to six decimal places?
Answer: a) 0.079589 b) 0.079656
Step-by-step explanation:
Since we have given that
Number of times a coin is flipped = 100 times
Number of times he get exactly head = 50
Probability of getting head = [tex]\dfrac{1}{2}[/tex]
We will use "Binomial distribution":
Probability would be
[tex]^{100}C_{50}(\dfrac{1}{2})^{50}(\dfrac{1}{2})^50\\\\=0.079589[/tex]
Using "Normal approximation":
n = 100
p = 0.5
So, mean = [tex]np=100\times 0.5=50[/tex]
Standard deviation is given by
[tex]\sqrt{np(1-p)}\\\\=\sqrt{50(1.05)}\\\\=\sqrt{50\times 0.5}\\\\=\sqrt{25}\\\\=5[/tex]
So,
[tex]P(X<x)=P(Z<\dfrac{\bar{x}-\mu}{\sigma})\\\\So, P(X=50)=P(49.5<X<50.5)\\\\=P(\dfrac{49.5-50}{5}<Z<\dfrac{50.5-50}{5})\\\\=P(-0.1<Z<0.1)\\\\=P(Z<0.1)-P(Z<-0.1)\\\\=0.539828-0.460172\\\\=0.079656[/tex]
Hence, a) 0.079589 b) 0.079656
10 kids are randomly grouped into an A team with five kids and a B team with five kids. Each grouping is equally likely.
(a) What is the size of the sample space?
(b) There are two kids in the group, Alex and his best friend Jose. What is the probability that Alex and Jose end up on the same team?
The size of the sample space for grouping 10 kids into two teams is 252. The probability of Alex and Jose being on the same team is 4/9.
To calculate the sample space for randomly grouping 10 kids into two teams of five, we consider the number of ways to choose 5 kids out of 10, since choosing the first team automatically determines the second. The size of the sample space is given by the combination formula C(n, k) = n! / (k!(n - k)!), where n! represents the factorial of n. Therefore, the sample space size is C(10, 5) = 10! / (5!5!) = 252.
To find the probability that Alex and Jose end up on the same team, we need to consider two scenarios: both in team A or both in team B. Since the order they are picked doesn't matter, they are one unit, and we need to choose 3 additional members for their team from the remaining 8 kids, we use the combination formula again: C(8, 3).
There are C(8, 3) ways for them to be on the same team and this can happen in two different teams. Thus, the probability that they are on the same team is (2 * C(8, 3)) / C(10, 5).
Calculating this gives us (2 * 56) / 252, which simplifies to 112/252, and reduces to 4/9 when simplified. So, the probability is 4/9.
In a random sample of 27 people, the mean commute time to work was 32.8 minutes and the standard deviation was 7.2 minutes. Assume the population is normally distributed and use a t-distribution to construct a 99 % confidence interval for the population mean mu . What is the margin of error of mu ? Interpret the results.
The confidence interval for the population mean ___ , ___
(Round to one decimal place as needed.)
A) Using the given sample statistics, find the left endpoint of the interval using technology, rounding to one decimal place.
B) Using the given sample statistics, find the right endpoint of the interval using technology, rounding to one deci?mal place.
C) Using the formulas and values for the left and/or rightendpoints, find the margin of error from technology, rounding to two decimal places.
Final answer:
To calculate the 99% confidence interval for the population mean commute time, we use the formula CI = x± t * (s/√n), where x is the sample mean, t is the t-value, s is the sample standard deviation, and n is the sample size. By substituting the given values into the formula, we find that the confidence interval is approximately (29.4, 36.2) and the margin of error is 1.7. This means we are 99% confident that the population mean commute time lies within the interval, and the margin of error represents the possible deviation of the sample mean from the population mean.
Explanation:
To construct a 99% confidence interval for the population mean μ, we can use the formula:
CI = x± t * (s/√n)
Where CI is the confidence interval, x is the sample mean, t is the t-value, s is the sample standard deviation, and n is the sample size. In this case, x = 32.8, s = 7.2, and n = 27.
To find the t-value, we need to determine the degrees of freedom (df), which is equal to n - 1. So, df = 27 - 1 = 26. Looking up the t-value for a 99% confidence level with 26 degrees of freedom in a t-table or using calculator software, we find it to be approximately 2.787.
Substituting these values into the formula, we get:
CI = 32.8 ± 2.787 * (7.2/√27)
Calculating this, we find that the confidence interval is approximately (29.4, 36.2).
The margin of error, also known as the error bound for a population mean (EBM), is half the width of the confidence interval. So, the margin of error in this case is (36.2 - 32.8)/2 = 1.7.
Interpreting the results, we can say that we are 99% confident that the population mean μ lies within the range of 29.4 to 36.2 minutes. Furthermore, the margin of error of 1.7 indicates how much the sample mean could vary from the population mean, taking into account the variability in the data.
Maria ate 1/3of a pie.Her sister,rebecca,ate 1/5of that.what fraction of the whole pie did rebecca eat?
Answer: the fraction of the whole pie that Rebecca ate is 1/15
Step-by-step explanation:
Let x represent the size or area of the whole pie. Maria ate 1/3 of the pie. This means that the amount of pie that maria ate is 1/3 × x = x/3
Her sister,Rebecca ate 1/5 of that. This means that Rebecca ate 1/5 of the amount that Maria ate. The amount that Rebecca ate will be expressed as 1/5 × x/3 = x/15
To determine the fraction of the whole pie that Rebecca ate, we will divide the amount that Rebecca ate by the size of the whole pie. It becomes
x/15/x = x/15 × 1/x
= 1/15
Which of the following expression is equivalent to 6^−7? A 1/(−6)^−7 B 1/6^7 C (−6^)−7 D 1/6^−7
Answer:B
Step-by-step explanation: The answer is B. 1/6^7. Your trying to find the reciprocal of 6^-7. The way you do that is put a 1 over the 6 with a power of -7 like this (6^-7)/(1) and that's how you get your answer!
[tex]\huge\text{Hey there!}[/tex]
[tex]\huge\textbf{This equation should be somewhat}\\\huge\textbf{easy because it has like THREE steps}\\\huge\textbf{to solve it.}[/tex]
[tex]\huge\textbf{But....}[/tex]
[tex]\huge\textbf{Your equation should look like:}[/tex]
[tex]\mathsf{6^{-7}}[/tex]
[tex]\huge\textbf{Now, let's get to the steps:}[/tex]
[tex]\mathsf{6^{-7}}\\\\\mathsf{\approx \dfrac{1}{6\times6\times6\times6\times6\times6\times6}}\\\\\mathsf{\approx \dfrac{1}{6^7}}\\\\\mathsf{\approx \dfrac{1}{279,936}}[/tex]
[tex]\huge\textbf{Therefore, the closet answer to your}\\\huge\textbf{question should be:}[/tex]
[tex]\huge\boxed{\frak{Option\ B. \rm{\dfrac{1}{6^7}}}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
Listed below are the lead concentrations in mug/g measured in different traditional medicines. Use a 0.01 significance level to test the claim that the mean lead concentration for all such medicines is less than 17 mug/g. 13 21 3.5 18.5 21.5 8.5 15.5 19 9 5.5
There is no significant difference in mean lead concentration for the medicines.
Hypothesis means an assumption or a claim which is needed to be tested.
To test the following hypothesis:
[tex]Null\ hypothesis,H_0: \mu=17\\Alternative\ Hypothesis,\ H_1: \mu < 17[/tex]
For the given data to test the hypothesis, t statistics can be used as follows:
[tex]t=\dfrac{\bar{X}-\mu}{\sigma}[/tex]
Given, Mean, μ=17
We are required to sample mean and σ in the excel table shown
below:
t=(17-13.05)/6.23
t=0.634
Using excel function to calculate p value of the t statistics as follows:
=T.DIST(0.634,9,1)
p=0.729
Since, p value is greater than level of significance 0.01. We fail to reject the null hypothesis and conclude that there is no significant difference in mean.
Learn more about Hypothesis testing, here:
brainly.com/question/33445215
#SPJ4
To test the claim, calculate the mean and standard deviation of your sample data, set up your null and alternative hypotheses, calculate your test statistic t, reference a t-distribution table and compare your calculated t with the critical t-value.
Explanation:To test the claim that the mean lead concentration for all such medicines is less than 17 mug/g, we'll use a one-sample t-test because we have one sample present and we're comparing it to a known mean. Here are the steps:
First, calculate the mean and standard deviation of your sample data. For the given data set, you'll sum up all the values and then divide by the number of values to find the mean. Then, use the formula for standard deviation.Second, set up your null (H0) and alternative (H1) hypotheses: H0: The mean is greater than or equal to 17 mug/g. H1: The mean is less than 17 mug/g.Next, you need to calculate your test statistic t, using the formula t = (sample mean - hypothesized mean)/(standard deviation/square root of sample size (n)).Referencing a t-distribution table, check what the critical t-values is at 0.01 significance level with degree of freedom = n-1.If the calculated t is less than the critical t-value, we reject the null hypothesis. If it's greater, we do not reject the null hypothesis.Remember, rejecting the null hypothesis supports our claim that the mean concentration is less than 17 mug/g. Not rejecting the null hypothesis means that our data does not support the claim that the mean concentration is less than 17 mug/g, but it isn't proof that the concentration exactly equals to or exceeds 17 mug/g.
Learn more about Hypothesis Testing here:https://brainly.com/question/31665727
#SPJ11
Yahto and Nora want to find out if the quadrilateral formed by connecting points J, K, L, and M is a rectangle.
Yahto’s plan:
• Use the distance formula to find the lengths of the sides.
• Then see if the side lengths of opposite sides are the same.
Nora’s plan:
• Find the slopes of the four sides.
• See if the slopes of adjacent sides are negative reciprocals of each other.
Which student’s plan will work?
A. Both plans are correct.
B. Neither plan is correct.
C. Only Yahto’s plan is correct.
D. Only Nora’s plan is correct.
Answer:
B. Neither plan is correct
Step-by-step explanation:
Yahto has the right idea in that opposite sides of a rectangle are the same length. However, that is also true of a parallelogram that is not a rectangle. The condition Yahto is looking for is necessary, but not sufficient.
__
Nora's plan will discover if adjacent sides are perpendicular to each other, provided that the slopes are defined in each case. Her plan will not work in the event there are vertical sides with undefined slope. Any quadrilateral in which all adjacent pairs of sides form right angles will be a rectangle.
_____
Strictly speaking, neither plan is completely correct. Yahto can discover rectangles that Nora cannot, and Nora can determine quadrilaterals are not rectangles when Yahto would improperly classify them.
_____
Comment on showing a quadrilateral is a rectangle
My favorite plan is to show the diagonals are the same length and have the same midpoint: J+L = K+M; ║J-L║ = ║K-M║.
Both Yahto and Nora's plans make sense, but only Nora's plan will always correctly determine if a quadrilateral is a rectangle. This is because finding negative reciprocal slopes confirms perpendicular, or right, angles which are necessary for a shape to be a rectangle.
Explanation:Both Yahto and Nora have good strategies, but only Nora’s plan will always lead to the correct identification of a rectangle. A rectangle is defined as a quadrilateral with four right angles. While having equal length of opposite sides is a property of rectangles, it is not exclusive to rectangles. It can also be true for other quadrilaterals, such as parallelograms and rhombuses.
On the other hand, Nora's plan relies on finding negative reciprocal vertices, which is in line with a property of rectangles – adjacent sides are perpendicular. In a coordinate plane, if two lines are perpendicular, the slopes of the lines are negative reciprocals of each other. Hence, if these conditions are met, it would confirm that the quadrilateral is a rectangle.
Learn more about Rectangle Identification here:https://brainly.com/question/32976228
#SPJ12
An ANOVA procedure for a two-factor factorial experiment produced the following: a = 6, b = 2, r = 2, SSA = 1.05, SSB = 16.67, SSAB = .60, and SST = 94.52. What is the value of the test statistic for determining whether there is a main effect for factor A? a. 2.63 b. .02 c. The test statistic cannot be computed because SSE is not given. d. .03
Answer:
Which is the output of the formula =AND(12>6;6>3;3>9)?
A.
TRUE
B.
FALSE
C.
12
D.
9
The test statistic for the main effect for factor A cannot be computed from the provided information because the mean square for error (MSE) is not given, which is required for the F statistic calculation.
Explanation:The test statistic for determining whether there is a main effect for factor A is calculated as follows.
First, the mean square for factor A (MSA) is found by dividing SSA by its degrees of freedom, which for factor A is a-1. In this case, with a = 6, degrees of freedom for A is 6-1, which is 5.
Therefore, MSA = SSA / dfA = 1.05 / 5 = 0.21.
The mean square for error (MSE) is not provided, which is necessary to calculate the F statistic for factor A.
To obtain the test statistic for factor A, we need to divide MSA by MSE. Without SSE or MSE, the test statistic cannot be computed. Hence, the correct answer is c.
The test statistic cannot be computed because SSE is not given.
Help me please and explain
Answer:
its 114
Step-by-step explanation:
180-66=114 so that's it
An article describes a factorial experiment designed to determine factors in a high-energy electron beam process that affect hardness in metals. The first factor is the travel speed in mm/s (three levels: 10, 20, 30). The second fact g
The subject of this question is Physics and the grade level is High School. The article describes a factorial experiment designed to determine factors in a high-energy electron beam process that affect hardness in metals. The experiment focuses on two factors: travel speed and beam current. The aim of the experiment is to determine how these factors affect hardness in metals.
Explanation:In the given article, the subject of the question is physics, specifically the experimentation involving a high-energy electron beam process that affects hardness in metals. The experiment focuses on two factors: travel speed and beam current. The travel speed is measured in mm/s and has three levels: 10, 20, and 30. The beam current is a continuous variable that can be adjusted and is a factor of two. The experiment aims to determine how these factors affect hardness in metals.
what is the slope of the line that passes through the points (-2, 5) and (1, 4)
Answer:
-1/3
Step-by-step explanation:
rise / run = slope
4-5/ 1-(-2)
= -1/3
. In the 1980s, it was generally believed that congenital abnormalities affected about 5% of the nation’s children. Some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of abnormalities. A recent study examined 384 children and found that 46 of them showed signs of an abnormality. Is this strong evidence that the risk has increased?
a. Define the parameter and state the hypotheses.
b. Define the sampling distribution (mean and standard deviation).
c. Perform the test and calculate P-value
d. State your conclusion.
e. Explain what the p-value means in this context.
Answer:
Reject null hypothesis
Step-by-step explanation:
a) [tex]H_0: p =0.05\\H_a: p >0.05[/tex]
(Right tailed test at 5% level)
p = risk proportion
Sample proportion = [tex]\frac{46}{384} \\\\=0.1198[/tex]
Std error = [tex]\sqrt{\frac{pq}{n} } \\=\sqrt{\frac{0.05(0.95)}{384} } \\=0.0111[/tex]
p difference = [tex]0.1198-0.05=0.0698[/tex]
b) Sampling proportion is Normal with mean = 0.05 and std error = 0.0698
c) Z = test statistic = p diff/std error
= 6.29
p value <0.05
d) Since p < alpha we reject null hypothesis.
e) The probability that null hypothesis is rejected when true is negligible =p value
Answer:
Step-by-step explanation:
Reject null hypothesis
Consider the sequence −8, −4, 0, 4, 8, 12, ellipsis. Select True or False for each statement. A recursive rule for the sequence is f(1) = −8; f(n) = −4 (n − 1) for all n ≥ 2. A True B False An explicit rule for the sequence is f(n) = −8 + 4 (n − 1). A True B False The tenth term is 28.
Answer:
A recursive rule for the sequence is f(1) = -8; f(n) = -4 (n – 1) for all n ≥ 2 is "FALSE"
An explicit rule for the sequence is f(n) = -8 + 4 (n – 1) is "TRUE"
The tenth term is 28 is "TRUE"
Step-by-step explanation:
Statement (1)
While the first part [f(1) = –8] is TRUE, the second part [f(n) = –4 (n – 1) for all n ≥ 2] would only be true if the sequence ends at the second term.
Check: Since the fifth term of the sequence is 8, then f(5) = 8
From the statement,
f(5) = –4 (5 – 1)
f(5) = –4 × 4 = –16
:. f(5) ≠ 8
Statement (2)
f(n) = –8 + 4 (n – 1) is TRUE
Check: The fifth term of the sequence is 8 [f(5) = 8]
From the statement,
f(5) = –8 + 4 (5 – 1)
f(5) = –8 + 4 (4)
f(5) = –8 + 16 = 8
:. f(5) = 8
Statement (3)
f(10) = 28 is TRUE
Since the explicit rule is TRUE, use to confirm if f(10) = 28:
f(10) = –8 + 4 (10 – 1)
f(10) = –8 + 4 (9)
f(10) = –8 + 36
f(10) = 28
:. f(10) = 28
Suppose Clay, a grocery store owner, is monitoring the rate at which customers enter his store. After watching customers enter for several weeks, he determines that the amount of time in between customer arrivals follows an exponential distribution with mean 25
What is the 70th percentile for the amount of time between customers entering Clay's store? Round your answer to the nearest two decimal places. 70th percentile
Answer:
70th percentile for the amount of time between customers entering is 30.10
Step-by-step explanation:
given data
mean = 25
to find out
What is the 70th percentile for the amount of time between customers entering Clay's store
solution
we know that here mean is
mean = [tex]\frac{1}{\lambda}[/tex] ..................1
so here 25 = [tex]\frac{1}{\lambda}[/tex]
and we consider time value corresponding to 70th percentile = x
so we can say
P(X < x) = 1 - [tex]e^{- \lambda *x}[/tex]
P(X < x) = 70 %
1 - [tex]e^{\frac{-x}{25}}[/tex] = 70 %
1 - [tex]e^{\frac{-x}{25} }[/tex] = 0.70
[tex]e^{\frac{- x}{25} }[/tex] = 0.30
take ln both side
[tex]\frac{-x}{25}[/tex] = ln 0.30
[tex]\frac{x}{25}[/tex] = 1.203973
x = 30.10
70th percentile for the amount of time between customers entering is 30.10
The 70th percentile for the amount of time between customers entering Clay's store is 30
How to determine the 70th percentile?The mean of an exponential distribution is:
[tex]E(x) = \frac 1\lambda[/tex]
The mean is 25.
So, we have:
[tex]\frac 1\lambda = 25[/tex]
Solve for [tex]\lambda[/tex]
[tex]\lambda = \frac 1{25}[/tex]
[tex]\lambda = 0.04[/tex]
An exponential function is represented as:
[tex]P(x < x) = 1 - e^{-\lambda x}[/tex]
For the 70th percentile, we have:
[tex]1 - e^{-\lambda x} = 0.70[/tex]
This gives
[tex]e^{-\lambda x} = 1 - 0.70[/tex]
[tex]e^{-\lambda x} = 0.30[/tex]
Substitute [tex]\lambda = 0.04[/tex]
[tex]e^{-0.04x} = 0.30[/tex]
Take the natural logarithm of both sides
[tex]-0.04x = \ln(0.30)[/tex]
This gives
[tex]-0.04x = -1.20[/tex]
Divide both sides by -0.04
[tex]x = 30[/tex]
Hence, the 70th percentile for the amount of time between customers entering Clay's store is 30
Read more about probability distribution at:
https://brainly.com/question/4079902
A statistician uses Chebyshev's Theorem to estimate that at least 15 % of a population lies between the values 9 and 20. Use this information to find the values of the population mean, μ , and the population standard deviation σ.
Answer:
\mu = 14.5\\
\sigma = 5.071\\
k = 1.084
Step-by-step explanation:
given that a statistician uses Chebyshev's Theorem to estimate that at least 15 % of a population lies between the values 9 and 20.
i.e. his findings with respect to probability are
[tex]P(9<x<20) \geq 0.15\\P(|x-14.5|<5.5) \geq 0.15[/tex]
Recall Chebyshev's inequality that
[tex]P(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}\\P(|X-\mu |\leq k\sigma )\geq 1-{\frac {1}{k^{2}}}\\[/tex]
Comparing with the Ii equation which is appropriate here we find that
[tex]\mu =14.5[/tex]
Next what we find is
[tex]k\sigma = 5.5\\1-\frac{1}{k^2} =0.15\\\frac{1}{k^2}=0.85\\k=1.084\\1.084 (\sigma) = 5.5\\\sigma = 5.071[/tex]
Thus from the given information we find that
[tex]\mu = 14.5\\\sigma = 5.071\\k = 1.084[/tex]
In a study of red/green color blindness, 550 men and 2400 women are randomly selected and tested. Among the men, 48 have red/green color blindness. Among the women, 5 have red/green color blindness. Test the claim that men have a higher rate of red/green color blindness than women. Also, construct a 98% confidence interval for the difference between color blindness rates of men and women. Does there appear to be a significant difference?
Answer: 98% confidence interval is (0.06,0.11).
Step-by-step explanation:
Since we have given that
Number of men = 550
Number of women = 2400
Number of men have color blindness = 48
Number of women have color blindness = 5
So, [tex]p_M=\dfrac{48}{550}=0.087\\\\p_F=\dfrac{5}{2400}=0.0021[/tex]
So, at 98% confidence interval, z = 2.326
so, interval would be
[tex](p_M-P_F)\pm z\sqrt{\dfrac{p_M(1-p_M)}{n_M}+\dfrac{p_F(1-p_F)}{n_F}}\\\\=(0.087-0.0021)\pm 2.326\sqrt{\dfrac{0.087\times 0.912}{550}+\dfrac{0.0021\times 0.9979}{2400}}\\\\=0.0849\pm 2.326\times 0.012\\\\=(0.0849-0.027912, 0.0849+0.027912)\\\\=(0.06,0.11)[/tex]
Hence, 98% confidence interval is (0.06,0.11).
No, there does not appears to be a significant difference.