An ant is moving on a numbered, horizontal line every second. The number ranges from −[infinity] to [infinity] . It moves to the left integer with a probability of 1/4 and to the right integer with a probability of 3/4. Suppose initially it starts at 0, so what is the probability that after 3 seconds it will be at 1?

Answer:

Step-by-step explanation:

To answer the question, we need to be able to find the number of each type of cone that needs to be sold. For the purpose, it is convenient to define variables that represent those numbers. We have chosen to use the variables x and y to represent the number of small cones and the number of large cones, respectively ("Let" statement in above answer). One could use "s" and "l" as being more mnemonic, but "l" can be confused with a variety of other symbols, so we like not to use it.

The problem statement puts limits on the numbers of small and large cones sold, so our system of inequalities needs to reflect those limits. (We suspect these are not hard limits, but represent historical data. We doubt the shop would decline to sell more, and we can conceive of conditions under which they might sell fewer.)

The heart of the matter is that the profit from each type of cone must total more than $350 (per day). The problem statement requires the shop make a profit, not just break even, so we use the > symbol for profit, instead of ≥.

___

See above for the "Let" statement and the system of inequalities. (This problem does not require we solve them.)

Answer:

15x - 2.

Step-by-step explanation:

3x - 1 + 2x + 1 + 4x - 2 + 4x - 4 + 2x + 4

= 15x - 2.

Answer:

The total length of planting can be calculated by the following steps ;

Step-by-step explanation:

Answer:

[tex]y{x} = \sqrt{7+2Inx}[/tex]

Step-by-step explanation:

[tex]y(x)= 3 + \int\limits^x_e {dx}/ \, ty(t) , x>0}[/tex]

Let say; By y(x)= y(e)  

we have;  

[tex]y(e)= 3 + \int\limits^e_e {dt}/ \, ty= 3+0[/tex]

Using Fundamental Theorem of Calculus and differentiating by Lebiniz Rule:

[tex]y^{1} (x) = 0 + 1/ xy[/tex]

[tex]y^{1} = 1/xy[/tex]  

dy/dx = 1/xy  

[tex]\int\limits {y} \, dxy = \int\limits \, dx/x[/tex]

[tex]y^{2}/2 Inx + C[/tex]

RECALL: y(e) = 3  

[tex](3)^{2} / 2 = In (e) + C[/tex]  

[tex]\frac{9}{2} =In(e)+C[/tex]  

[tex]\frac{9}{2} - 1 = C[/tex]

[tex]\frac{7}{2} = C[/tex]  

[tex]y^{2} / 2 = In x +C[/tex]

[tex]y^{2} / 2 = In x +7/2[/tex]

MULTIPLYING BOTH SIDE BY 2 , TO ELIMINATE THE DENOMINATOR, WE HAVE;

[tex]y^{2} = {7+2Inx}[/tex]  

[tex]y{x} = \sqrt{7+2Inx}[/tex]

Answer:

1. F

observe that [tex](5+2)^2=49 \neq 29=5^2+2^2[/tex]

2. T

Let x and y real numbers.

[tex](x+y)^2=(x+y)(x+y)=x^2+2xy+y^2[/tex]

3. F

Observe that if x=3 and y=2 [tex]\frac{3}{3+2}=\frac{3}{5}\neq \frac{1}{2}[/tex]

4. F

If x=y=3, [tex]3-(3+3)=3-6=-3\neq 3[/tex]

5. F

if x=-1, [tex]\sqrt{-1^2}=\sqrt{1}=1\neq -1[/tex]

6. T

7. F

if x=-1, [tex]\sqrt{-1^2+4}?\sqrt{5}\neq 1=-1+2[/tex]

8. F

If x=1 and y=2, [tex]\frac{1}{1+2}=\frac{1}{3}\neq \frac{3}{2}=\frac{1}{1}+\frac{1}{2}[/tex]

Answer:

a. μ = 9.667 hours

b. σ = 1.972 hours

c. SE = 0.805 hours

Step-by-step explanation:

Sample size (n) = 6

Sample data (xi) = 10, 8, 9, 7, 11, 13

a. Mean time spent in a week for this course by students:

Sample mean is given by:

[tex]\mu = \frac{\sum x_i}{n} \\\mu = \frac{10+9+7+11+13}{6}\\\mu=9.667[/tex]

Mean time spent in a week per student is 9.667 hours

b. Standard deviation of the time spent in a week for this course by students:

Standard deviation is given by:

[tex]\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{n}}\\\sigma = \sqrt{\frac{(10- 9.667)^2+(8- 9.667)^2+(9- 9.667)^2+(7- 9.667)^2+(11- 9.667)^2+(13- 9.667)^2}{6}}\\\sigma =1.972[/tex]

c. Standard error of the estimated mean time spent in a week for this course by students:

Standard error is given by:

[tex]SE = \frac{\sigma}{\sqrt n}\\SE = \frac{1.972}{\sqrt 6}\\SE=0.805[/tex]

To find the probability, use the Central Limit Theorem to calculate the z-scores for the lower and upper limits of the sample mean. Look up these z-scores in the standard normal distribution table to find the probabilities. Subtract the probability for the lower z-score from the probability for the higher z-score to find the probability that the mean of the sample falls between the two values.

To find the probability that the mean of a sample will be between 94.8 inches and 95.8 inches, we can use the Central Limit Theorem. We start by calculating the z-scores for these values using the formula: z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. With the given values, the z-score for 94.8 inches is -3.85 and the z-score for 95.8 inches is 1.92.

Next, we look up these z-scores in the standard normal distribution table to find the corresponding probabilities. The probability for a z-score of -3.85 is approximately 0.00005 and the probability for a z-score of 1.92 is approximately 0.97128. To find the probability that the mean of the sample falls between these two values, we subtract the probability for the lower z-score from the probability for the higher z-score: 0.97128 - 0.00005 = 0.97123.

Therefore, the probability that the mean of the sample will be between 94.8 inches and 95.8 inches is approximately 0.97123, or 97.12%.

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Answer:

[293.21;372.28]

Step-by-step explanation:

Hello!

You are asked to construct a 90% Confidence Interval for the mean duration of the foreclosure process in Boston.

The Study variable is X: duration of foreclosure in Boston. X≈N(μ;σ²)

Sample n=8

To study the population sample you can use either a Z or a t-statistic. Since the sample is less than 10, I'll choose a Student t-statistic, because it is more potent with small samples than the Z, but either one is a good choice.

X[bar]= 332.75

S= 59.01

[tex]X[bar] ± t_{n-1; 1-\alpha /2}*\frac{S}{\sqrt{n} }[/tex]

[tex]332.75 ± 1.895*\frac{59,01}{\sqrt{8} }[/tex]

[293.21;372.28]

With a confidence level of 90% you'd expect the interval [293.21;372.28] to contain the population mean of the duration of the foreclosure process in Boston.

I hope you have a SUPER day!

Answer:

The volume is decreasing at a rate 20 cubic meters per hour.

Step-by-step explanation:

We are given the following in the question:

Rate of change of radius =

[tex]\displaystyle\frac{dr}{dt} = 1 \text{ meter per hour}[/tex]

Rate of change of height =

[tex]\displaystyle\frac{dr}{dt} = -4 \text{ meter per hour}[/tex]

At an instant,

radius, r = 5 meters

Height, h = 8 meters

Volume of cylinder , V=

[tex]\pi r^2 h[/tex]

where r is the radius of cylinder and h is the height of cylinder.

Rate of change of volume of cylinder =

[tex]\displaystyle\frac{dV}{dt} = \frac{d(\pi r^2 h)}{dt} = \pi\Big(2r\frac{dr}{dt}h + r^2\frac{dh}{dt}\Big)[/tex]

Putting the value, we get,

[tex]\displaystyle\frac{dV}{dt} = \pi\Big(2(5)(1)(8) + (5)^2(-4)\Big) = -20\text{ cubic meters per hour}[/tex]

Thus, the volume is decreasing at a rate 20 cubic meters per hour.

The rate of change of the volume at that particular instant is - 62.8 m^3/h.

First, we can write the dimensions as:

Where t is the time in hours.

Then the volume of the cylinder will be:

V = pi*R^2*H = 3.14*(5m + 1m/h*t)^2*(8m - 4 m/h*t)

To get the rate of change, we need to differentiate it with respect to t, we will get:

V' = 3.14*(2*(5m + 1m/h*t)*(1m/h)*(8m - 4 m/h*t) + (5m + 1m/h*t)^2*(-4m/h))

V' = 3.14*( (2 m/h)*(5m + 1m/h*t)*(8m - 4 m/h*t) - (4m/h)*(5m + 1m/h*t)^2)

V' = 3.14*(2m/h)*( (5m + 1m/h*t)*(8m - 4 m/h*t) - 2*(5m + 1m/h*t)^2)

As you can see the rate of change depends on t, but we want the rate of change at this instant, then we use t = 0, replacing that on the above equation we get:

V'(0) = 3.14*(2m/h)*( (5m + 1m/h*0)*(8m - 4 m/h*0) - 2*(5m + 1m/h*0)^2)

V'(0) = 3.14*(2m/h)*( (5m)*(8m ) - 2*(5m)^2)=  -62.8 m^3/h

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Answer:

95% Confidence interval for taxi fare:  ($20.5,$24.2)

Step-by-step explanation:

We are given the following data set: for fares:

$22.10, $23.25, $21.35, $24.50, $21.90, $20.75, and $22.65

Formula:

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{156.5}{7} = 22.35[/tex]

95% Confidence interval:

[tex]\bar{x} \pm z_{critical}\frac{\sigma}{\sqrt{n}}[/tex]

Putting the values, we get,

[tex]z_{critical}\text{ at}~\alpha_{0.05} = 1.96[/tex]

[tex]22.35 \pm 1.96(\frac{2.5}{\sqrt{7}} ) = 22.35 \pm 1.85 = (20.5,24.2)[/tex]

The 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston is  $20.5031 and $24.2111 dollars.

To calculate the 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston, we'll use the sample data provided and the given standard deviation of $2.50.

1. Calculate the sample mean:

  First, find the mean of the sample fares:

  [tex]\[ \bar{x} = \frac{22.10 + 23.25 + 21.35 + 24.50 + 21.90 + 20.75 + 22.65}{7} = \frac{156.5}{7} = 22.3571 \][/tex]

2. Calculate the standard error of the mean (SEM):

  Since the population standard deviation [tex](\(\sigma\))[/tex] is known and the sample size (n) is 7:

  [tex]\[ SEM = \frac{\sigma}{\sqrt{n}} = \frac{2.50}{\sqrt{7}} \approx 0.946 \][/tex]

3. Determine the margin of error (ME):

  For a 95% confidence level, find the critical value (z*) from the standard normal distribution table, which is approximately 1.96.

  [tex]\[ ME = z^* \times SEM = 1.96 \times 0.946 \approx 1.854 \][/tex]

4. Calculate the confidence interval:

  [tex]\[ \text{Confidence interval} = \bar{x} \pm ME = 22.3571 \pm 1.854 \][/tex]

  [tex]\[ \text{Confidence interval} = (20.5031, 24.2111) \][/tex]

The 95% confidence interval suggests that we are 95% confident that the true population mean taxi fare lies between $20.5031 and $24.2111. This interval takes into account the variability in John's recent taxi fares and provides a range within which we expect the true mean fare to fall.

The 95% confidence interval for the population mean taxi fare from Logan Airport to downtown Boston is approximately  $20.5031 and $24.2111 dollars.

Final answer:

The probability that a randomly selected man from the survey group favors raising the legal drinking age is 5/16, or 0.3125.

Explanation:

The question asks for the probability that a man chosen at random from the survey group favors raising the legal drinking age from 18 to 21. To solve this, we need to look at the numbers provided. We know that of the 540 people who favored raising the age, 390 were female. This means that 540 - 390 = 150 were male. Similarly, from the 460 who opposed, 130 were female, leading to 460 - 130 = 330 males who opposed.

Now, we calculate the total number of men in the survey, which is 150 men who favored raising the age plus 330 men who opposed, giving us a total of 150 + 330 = 480 men. The probability that a randomly selected man favors raising the age is then the number of men who favor it divided by the total number of men, which is 150/480.

To further simplify, we divide both numerator and denominator by 30, yielding an answer of 5/16. Therefore, the probability that a man favors raising the drinking age is 5/16, or approximately 0.3125.

Answer:

93.03%

Step-by-step explanation:

Population mean (μ) = 66.1 inches

Standard deviation (σ) = 2.7 inches

The z-score for a given 'X' value is:

[tex]z = \frac{X- \mu}{\sigma}[/tex]

For X = 62 inches

[tex]z = \frac{62- 66.1}{2.7}\\z=-1.5185[/tex]

A z-score of -1.5185 corresponds to the 6.44-th percentile of a normal distribution.

For X = 62 inches

[tex]z = \frac{73- 66.1}{2.7}\\z=2.5555[/tex]

A z-score of 2.5555 corresponds to the 99.47-th percentile of a normal distribution.

The total percentage of men eligible for the military is the percentage within those two values, therefore:

[tex]E = 99.47-6.44\\E=93.03 \%[/tex]

93.03%  this​ country's men are eligible for the military based on height.

Final answer:

To determine the percentage of men eligible for the military based on height, we need to calculate the proportion of men whose heights fall between 62 inches and 73 inches. First, convert the height values to z-scores using the formula z = (x - mean) / standard deviation. Then, look up the corresponding z-scores in the standard normal distribution table to find the proportion of men within the desired height range.

Explanation:

To determine the percentage of men eligible for the military based on height, we need to calculate the proportion of men whose heights fall between 62 inches and 73 inches.

First, we convert the height values to z-scores using the formula z = (x - mean) / standard deviation.

Then, we look up the corresponding z-scores in the standard normal distribution table to find the proportion of men within the desired height range. We can subtract this proportion from 1 to find the percentage of men who are eligible for the military based on height.

Answer:

A is right

you can reject

Step-by-step explanation:

Final answer:

The hypothesis testing involves comparing two independent sample proportions to check if their difference is statistically significant. The provided p-value of 0.0417 is compared with the significance level (0.02, and because the p-value is higher, we do not reject the null hypothesis, indicating there is not sufficient evidence to suggest a significant difference between the two proportions.(Option b)

Explanation:

The student's question involves testing the equality of two population proportions to determine if there is a significant difference between them. This entails conducting a hypothesis test for two independent sample proportions. To perform this test, one would typically use a test statistic that follows a standard normal distribution under the null hypothesis, which is based on the differences between the sample proportions.

Steps for the hypothesis test:

State the null hypothesis (H0: p1 = p2) and the alternative hypothesis (Ha: p1 ≠ p2).

Calculate the test statistic using the sample data.

Compare the test statistic to the critical value or use the p-value approach to make a decision.

Since the student already provided the sample sizes (n1 = n2 = 80) and the number of successes (x1 = 57 and x2 = 46), the test statistic can be computed using these values. Without the specific calculation here, we instead refer to the result that would come from using the test statistic to obtain the p-value.

Given the provided information that the p-value is 0.0417, we compare this with α = 0.02. Because the p-value is greater than α, we do not reject the null hypothesis (No significant difference between proportions).

The final conclusion is option B: There is not sufficient evidence to reject the null hypothesis that (p1−p2) = 0.

Answer:

0.9641 or 96.41%

Step-by-step explanation:

Mean career duration (μ) = 88 weeks

Standard deviation (σ) = 20

The z-score for any given career duration 'X' is defined as:  

[tex]z=\frac{X-\mu}{\sigma}[/tex]  

In this problem, we want to know what is the probability that the professor's son's next career lasts more than a year. Assuming that a year has 52 weeks, the equivalent z-score for a 1-year career is:

[tex]z=\frac{52-88}{20}\\z=-1.8\\[/tex]

According to a z-score table, a z-score of -1.8 is at the 3.59-th percentile, therefore, the likelihood that this career lasts more than a year is given by:

[tex]P(X>52) = 1-0.0359\\P(X>52) = 0.9641\ or\ 96.41\%[/tex]

Answer:

0.9641 or 96.41%

Step-by-step explanation:

Answer:

sigma should be used

Step-by-step explanation:

Given that The mean length of time in jail from the survey was four years with a standard deviation of 1.9 years.

The above given is for sample of 27 size.

For hypothesis test to compare mean of sample with population we can use either population std dev or sample std dev.

But once population std deviation is given, we use only that as that would be more reliable.

So here we can use population std deviation 1.4 only.

If population std deviation is used we can use normality and do Z test

Answer:

There is no evidence that the spirituality category proportions are different for natural and social scientists.

Step-by-step explanation:

To solve this question we must perform a Chi square test calculating the expected values of the observed behavior.

We start by completing the table given by adding the totals:

Observed    Very   Moderate   Slightly   Not at all      Total

Natural Sc    54           161            195            216          626

Social Sc      55           220          239           242         756

Total            109           381           434           458         1382

Now, the chi square value is calculated with the following formula:

[tex]x^{2}[/tex]=∑∑[tex]\frac{O_{ij}-E_{ij} }{E_{ij} }[/tex]

Where:

[tex]O_{ij}[/tex]: Observed value (the ones we have in our table)

[tex]E_{ij}[/tex]: Expected value

The expected value of every observation (ij) is calculated as it follows:

[tex]E_{ij}=\frac{n_{i}c_{j} }{N}[/tex]

Where,

[tex]n_{i}[/tex]: marginal total by rows

[tex]c_{j}[/tex]: marginal total by columns

N: Total of observations

Now, for the expected observations we obtain the following table:

Expected     Very     Moderate   Slightly     Not at all      Total

Natural Sc    49.37      172.58       196.59      207.46        626

Social Sc      59.63      208.42      237.41      250.54        756

Total            109           381           434           458              1382

Having the expected values we now can calculate [tex]x^{2}[/tex] by first calculating [tex]\frac{O_{ij}-E_{ij} }{E_{ij} }[/tex] for each observation:

Chi sq           Very     Moderate   Slightly     Not at all      Total

Natural Sc    0.434      0.777         0.013        0.352           1.575

Social Sc      0.359      0.643        0.011         0.291            1.304

Total             0.793      1.420         0.024       0.643           2.879

[tex]x^{2}=2.879[/tex]

We may consider our null hypothesis as it follows:

[tex]H_{0}:[/tex] The degree of spirituality category proportions are the same for natural and social scientists.

To prove this we have:

[tex]P(x^{2}\geq 2.879)[/tex]

α=0.01

α: significance level  

Calculating this value with a chi square table (or with statistical software like R) we obtain:

P-value=0.4105

Because the p-value is larger than α the null hypothesis is accepted. Which means we cannot say that the spirituality category proportions are different.

Answer:

There is no sufficient evidence to suggest that the relaxation exercise slowed the brain waves

Step-by-step explanation:

Given that in a  experiment on relaxation techniques, subject's brain signals were measured before and after the relaxation exercises with the following results:

The population is normally distributed

This is a paired t test since sample size is very small and same population under two different conditions studied.

[tex]H_0: \bar x-\bar y =0\\H_a: \bar x >\bar y[/tex]

(Right tailed test)

Alpha = 5%

x y Diff

32 26 6

38 36 2

66 59 7

49 52 -3

29 24 5

Mean 3.4

Std dev 4.037325848

Mean difference = [tex]3.40[/tex]

n =5

Std deviation for difference = [tex]4.037326[/tex]

Test statistic t = mean difference / std dev for difference

= [tex]1.883[/tex]

p value =0.132

Since p >0.05 we accept null hypothesis.

There is no sufficient evidence to suggest that the relaxation exercise slowed the brain waves

Paired-sample t-test suggests there's a significant effect on slowing down the brain waves due to the relaxation exercises.

The subject of this question involves conducting a paired-sample t-test in a study on relaxation techniques. The t-test will allow us to determine whether there is a significant difference between the brain signals of subjects before and after engaging in relaxation exercises.

First, we compute the paired differences (Before - After) and calculate the mean and standard deviation of this list.
The differences are: 6, 2, 7, -3, 5. Hence, the mean difference (µd) = 3.4 and standard deviation (Sd) = 3.346.
Now, with t = (µd / (Sd/√n)) = (3.4 / (3.346/√5)) = 3.023 which is the t-value and n-1 = 4 (degrees of freedom).

Check a t-value chart for a two-tailed test (since we do not know if the relaxation exercises will increase or decrease brain activity) with degree of freedom = 4 and α=0.05. If our t-value is beyond the critical value in the table, we have a significant result. If it falls within that range, we do not. In this case, it's beyond so we can conclude the relaxation exercises have a significant effect on slowing down the brain waves.

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Answer:

20.14

Step-by-step explanation:

Given that the consumption rate of​ electricity, in​ kW/h has​ increased, on​ average, at 3.5​% per year.

So if initial consumption is C after one year it would be

[tex](1+0.035)C[/tex] and after 2 years [tex](1+0.035)^2C[/tex]

and so on

In general after n years

[tex]C(n) = 1.035^n C\\[/tex]

When C(n) is twice C we have

[tex]2=1.035^n\\log 2= n log 1.035\\n = 20.14[/tex]

Between 20 and 21 years

Hence The electric utities will need to double their generating capacity in about ___20.14__ years.

There is a system glitch during the process of adding my answer here so i create a document file for the answer, you find it in the attachment below.

Answer:

mean = 70000

SD = 15239

X= 95000

Z  = (X-mean)/ SD

   = (95000-70000)/15239

   = 1.64

Now from Z-Table

% of employees = 100(1-.9495) = 5.02%

Answer:

  290.6 backpacks per year

Step-by-step explanation:

The average of 5 numbers is 1/5 of their sum:

  average = (278 +310 +320 +242 +303)/5 = 290.6

The average number of backpacks given out in the last 5 years is 290.6, about 291.

Answer:

  15, 16

Step-by-step explanation:

Let x represent the smaller number. Then x+1 is the larger, and their product is x(x+1). Their sum is (x +(x+1)) = 2x+1. So, the required relation is ...

  x(x+1) -(2x+1) = 209

Expressed in the form f(x) = 0, we can write this as ...

  x(x+1) -(2x+1) -209 = 0

Graphing this shows x=15 to be the positive solution.

The numbers are 15 and 16.

To find the two consecutive positive integers, we can set up an equation using the given information. Solving the equation, we find that the integers are 14 and 15.

To solve this problem, let's assume the two consecutive positive integers are x and x+1. The product of these integers is x(x+1) and their sum is x + (x+1). We are given that the product is greater than their sum by 209, so we can set up the equation:

x(x+1) = x + (x+1) + 209

Expanding the equation and simplifying, we get x^2 + x = 210.

This is a quadratic equation that can be solved by factoring or using the quadratic formula. Let's solve it by factoring:

x^2 + x - 210 = 0

(x+15)(x-14) = 0

So, the two possible values for x are -15 and 14, but since we are dealing with positive integers, the value of x is 14. Therefore, the two consecutive positive integers are 14 and 15.

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Answer:

[tex]\displaystyle\frac{15\sqrt{3}}{4}-90\sqrt{146}[/tex]

Step-by-step explanation:

The line integral with respect to arc length of the function f(x, y, z) = xy2 along the parametrized curve that is the line segment from (1, 1, 1) to (2, 2, 2) followed by the line segment from (2, 2, 2) to (−9, 6, 5) equals the sum of the line integral of f along each path separately.

Let  

[tex]C_1,C_2[/tex]  

be the two paths.

Recall that if we parametrize a path C as [tex](r_1(t),r_2(t),r_3(t))[/tex] with the parameter t varying on some interval [a,b], then the line integral with respect to arc length of a function f is

[tex]\displaystyle\int_{C}f(x,y,z)ds=\displaystyle\int_{a}^{b}f(r_1,r_2,r_3)\sqrt{(r'_1)^2+(r'_2)^2+(r'_3)^2}dt[/tex]

Given any two points P, Q we can parametrize the line segment from P to Q as

r(t) = tQ + (1-t)P with 0≤ t≤ 1

The parametrization of the line segment from (1,1,1) to (2,2,2) is

r(t) = t(2,2,2) + (1-t)(1,1,1) = (1+t, 1+t, 1+t)

r'(t) = (1,1,1)

and  

[tex]\displaystyle\int_{C_1}f(x,y,z)ds=\displaystyle\int_{0}^{1}f(1+t,1+t,1+t)\sqrt{3}dt=\\\\=\sqrt{3}\displaystyle\int_{0}^{1}(1+t)(1+t)^2dt=\sqrt{3}\displaystyle\int_{0}^{1}(1+t)^3dt=\displaystyle\frac{15\sqrt{3}}{4}[/tex]

The parametrization of the line segment from (2,2,2) to  

(-9,6,5) is

r(t) = t(-9,6,5) + (1-t)(2,2,2) = (2-11t, 2+4t, 2+3t)  

r'(t) = (-11,4,3)

and  

[tex]\displaystyle\int_{C_2}f(x,y,z)ds=\displaystyle\int_{0}^{1}f(2-11t,2+4t,2+3t)\sqrt{146}dt=\\\\=\sqrt{146}\displaystyle\int_{0}^{1}(2-11t)(2+4t)^2dt=-90\sqrt{146}[/tex]

Hence

[tex]\displaystyle\int_{C}f(x,y,z)ds=\displaystyle\int_{C_1}f(x,y,z)ds+\displaystyle\int_{C_2}f(x,y,z)ds=\\\\=\boxed{\displaystyle\frac{15\sqrt{3}}{4}-90\sqrt{146}}[/tex]

The line integral with respect to the arc length of the function is mathematically given as

= −1080.97

Question Parameter(s):

The line integral with respect to arc length of the function f(x, y, z) = xy2

the line segment from (1, 1, 1) to (2, 2, 2)

the line segment from (2, 2, 2) to (−9, 6, 5).

Generally, the equation for the line integral   is mathematically given as

[tex]\int_C f(x,y,z)ds= \int_a^ b f(r_1,r_2,r_3) *\sqrt{(r'_1)^2+(r'_2)^2+(r'_3)^2}dt[/tex]

Therefore

[tex]\int_{C_1}f(x,y,z)ds=int_{0}^{1}f(1+t,1+t,1+t)\sqrt{3}dt\\\\\int_{C_1}f(x,y,z)ds=\sqrt{3} * \int_{0}^{1}(1+t)(1+t)^2dt\\\\[/tex]

[tex]\int_{C_1}f(x,y,z)ds=\sqrt{3} \int_{0}^{1}(1+t)^3dt\\\\\int_{C_1}f(x,y,z)ds=6.495[/tex]

r(t) = t(-9,6,5) + (1-t)(2,2,2)

r(t) = (-11,4,3)

then,

[tex]\displaystyle\int_{C_2}f(x,y,z)ds\\=\displaystyle\int_{0}^{1}f(2-11t,2+4t,2+3t)\sqrt{146}dt\\\\=-90\sqrt{146}[/tex]

thus,

[tex]\displaystyle\int_{C}f(x,y,z)ds=\displaystyle\int_{C_1}f(x,y,z)ds+\displaystyle\int_{C_2}f(x,y,z)ds=\\\\={\displaystyle\frac{15\sqrt{3}}{4}-90\sqrt{146}}[/tex]

In conclusion, considring the (2,2,2) segment

[tex]\displaystyle\int_{C}f(x,y,z)ds=\displaystyle\int_{C_1}f(x,y,z)ds+\displaystyle\int_{C_2}f(x,y,z)ds=\\\\={\displaystyle\frac{15\sqrt{3}}{4}-90\sqrt{146}}[/tex]

= −1080.97

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Answer:

(0.4062, 0.5098)

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

Z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex]

For this problem, we have that:

356 dies were examined by an inspection probe and 163 of these passed the probe. This means that [tex]n = 365[/tex] and [tex]\pi = \frac{163}{356} = 0.458[/tex]

Assuming a stable process, calculate a 95% (two-sided) confidence interval for the proportion of all dies that pass the probe.

So [tex]\alpha[/tex] = 0.05, z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[tex], so [tex]z = 1.96[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.458 - 1.96\sqrt{\frac{0.458*0.542}{356}} = 0.4062[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.458 + 1.96\sqrt{\frac{0.458*0.542}{356}} = 0.5098[/tex]

The correct answer is

(0.4062, 0.5098)

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My nickname - Lovely

Answer:

Step-by-step explanation:

Hello!

You have the hypothesis that the average weight loss for rubber after an abrasion test is less than 3.5 mg. To test this a large sample of pieces of rubber were sampled and subjected to the abrasion test.

With the given information you must test whether the researcher's hypothesis is sustained or not.

The study variable is,

X: Weight loss of rubber cured in a certain way after being subjected to the abrasion test. (mg)

There is no information about the variable distribution, but since it is said that the sample is a "large number" I'll take it as if it is bigger than 30 and apply the Central Limit Theorem to use the approximation of the sample mean to normal. This way I can use the Z-statistic for the test.

Symbolically the statistic hypothesis is:

H₀: μ ≥ 3.5

H₁: μ < 3.5

α: 0.05 (since is not listed, I'll choose one of the most common signification levels)

You have a one-tailed critical region, this means the p-value will also be one-tailed to the left of the distribution (i.e. →-∞)

The formula of the statistic is:

Z= X[bar] - μ ≈ N(0;1)

        δ/√n

To calculate the statistic you have to use the information given.

The sample mean X[bar]= 3.4 mg

Upper bond of 95% CI= 3.45 mg

The basic structure of a CI for the mean is

"estimator" ± "margin of error"

Upper bound is "estimator" + "margin of error"

Using the formula:

Ub= X[bar] + d ⇒ 3.45= 3.4 + d

⇒ d= 3.45 - 3.4 = 0.05

Where d is the margin of error

d= [tex]Z_{1-\alpha /2}[/tex] * (δ/√n)

d= [tex]Z_{0.975}[/tex] * (δ/√n)

d/[tex]Z_{0.975}[/tex]= (δ/√n)

(δ/√n)= 0.05/ 1.96 = 0.0255

(δ/√n) is the denominator in the formula, corresponds to the standard deviation of the distribution.

Now you have all values and can calculate the statistic under the null hypothesis:

Z= 3.4 - 3.5 = -3.92

       0.0255

And the p-value:

P(Z ≤ -3.92) = 0.000044 ⇒ My Z- table goes up to P(Z ≤ -3.00) = 0.001, so using strictly the table I can say that the probability is less than 0.001.

To calculate the exact probability I've used a statistic program.

p-value < 0.001

I hope it helps!

Answer:

Option B.

Step-by-step explanation:

The given table is:

Number of Lease Offices :  0           1         2       3        4         5

Probability                         : 5/18     1/4     1/9    1/18     2/9      1/12

The expected probability is

Expected probability = [tex]\sum_{i=0}^5 x_{i}p(x_i)[/tex]

Expected probability = [tex]0p(0)+1P(1)+2P(2)+3P(3)+4P(4)+5P(5)[/tex]

Expected probability = [tex]0\cdot (\frac{5}{18})+1\cdot (\frac{1}{4})+2\cdot (\frac{1}{9})+3\cdot (\frac{1}{18})+4\cdot (\frac{2}{9})+5\cdot (\frac{1}{12})=\frac{35}{18}[/tex]

It is given that the yearly lease = $12,000.

The yearly leases for the whole building in a given year is

Yearly leases = [tex]\frac{35}{18}\times 12000=23333.3333333\approx 23333.33[/tex]

Therefore, the correct option is B.

The expected yearly amount that John can collect from the rental of the multi-office building, taking into account the provided probability distribution for the number of offices that might be rented, is approximately $23,333.33.

The subject of this problem involves the concept of expected value in mathematics, particularly in statistics. The expected value, denoted as E(X), is a way of summarizing a probability distribution in terms of an average value. In this case, John is looking to understand the expected amount of income in rent he will receive in a year from this multi-office building based on the provided probability distribution.

First, we have to calculate the expected value of the number of leased offices. This can be done by multiplying each possible outcome by its probability and then summing up these products: E(X) = (0*5/18) + (1*1/4) + (2*1/9) + (3*1/18) + (4*2/9) + (5*1/12). After calculation, we get E(X) approximatively to be 1.9444.

However, we need to find the expected yearly amount in dollars that John could collect. Given that each lease is $12,000 per year, we multiply this lease value by the expected number of leased offices to give us: E($X) = E(X) * $12,000 which gives the value $23,333.34.

Therefore, the correct answer is (b) E(X) = $23,333.33 per year, which is the closest option.

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The test statistic is determined for the random sample conducted by rhe psychologist and the values are:

a) t ≈ 2.12

b) P-value ≈ P(T > 2.12)

c) Do not reject H0 because the P-value is less than the α = 0.05 level of significance.

d) It is not possible to make a definitive conclusion about the IQs of children born to mothers who listen to Mozart.

Given data:

To test whether the children in the study have higher IQs,conduct a one-sample t-test.

a) To conduct the t-test, set up the null and alternative hypotheses:

Null hypothesis (H0):

The population mean IQ of the children is equal to 100.

Alternative hypothesis (Ha):

The population mean IQ of the children is greater than 100.

b)

Calculate the P-value:

Determine the P-value using the t-distribution with the sample mean, sample standard deviation, and sample size.

P-value = P(T > t), where T follows a t-distribution with (n-1) degrees of freedom.

Using the given information, calculate the t-value:

t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

t = (104.1 - 100) / (15 / sqrt(20))

t ≈ 2.12

Next, we need to find the P-value associated with this t-value using the t-distribution.

For a one-tailed test, the P-value is the probability of observing a t-value greater than 2.12.

P-value ≈ P(T > 2.12)

c)

State the conclusion for the test:

To make a conclusion, we compare the P-value to the significance level (α = 0.05).

Since the P-value is not provided, we cannot make a definitive conclusion. However, based on the options given, the correct answer would be:

c. Do not reject H0 because the P-value is less than the α = 0.05 level of significance.

d)

State the conclusion in the context of the problem:

Based on the incomplete information provided, we cannot make a definitive conclusion about the IQs of children born to mothers who listen to Mozart.

To learn more about p-value, refer:

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To determine if there is evidence that the children have higher IQs, a p-value is calculated from the sample using a t-test. If the p-value is less than the significance level of 0.05, the null hypothesis is rejected, indicating evidence of higher IQs; otherwise, there is insufficient evidence.

The question asks whether the children have higher IQs than the normal mean of 100, given a sample mean IQ of 104.1 with a standard deviation of 15 from a sample of 20 children.

To calculate the p-value for a one-sample t-test:

The exact calculation is not provided here, but once you have the p-value:

We would either conclude that there is or is not sufficient evidence at the α=0.05 level to suggest that mothers who listen to Mozart have children with higher IQs, depending on whether the p-value is less than or greater than 0.05.

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Answer:

the graph is in the attachment.

the coordinates of the centroid : (2/3,2/3)

Step-by-step explanation:

by using these sketch the region.

I have uploaded the region bounded in the attachment. You may refer it. The region shaded with grey is the required region.

it can be easily identified that the formed region is a triangle

(1,2) , (0,0) , (1,0)

( See the graph. the three intersection points of the lines are the three vertices of the triangle)

G = [tex](\frac{a+c+e}{3}  , \frac{b+d+f}{3} )[/tex]

[tex](\frac{1+0+1}{3}, \frac{2+0+0}{3}  ) = (\frac{2}{3}, \frac{2}{3} )[/tex]

this point is marked as G in the graph uploaded.

The centroid of the region bounded by the curves y = 2x, y = 0, and x = 1 has the exact coordinates (2/3, 2/3) found through the application of the centroid formula integrating over the bounded region.

The question asks for a sketch and calculation of the centroid of the region bounded by the curves given by the equations y = 2x, y = 0, and x = 1. First, we must plot these equations on a graph. The line y = 2x starts at the origin and goes upward diagonally, and it is bounded by the line y = 0 (the x-axis) and the vertical line x = 1.

To find the exact coordinates of the centroid, we use the centroid formula for x-coordinate (̇x) and y-coordinate (̇y), which are ̇x = (1/A)∫xdb and ̇y = (1/(2A))∫ydb where db is the differential area and A is the total area. In this case, A is the area under the curve from x=0 to x=1, which is integral from 0 to 1 of 2x dx, yielding an area of 1. The centroid x-coordinate is ̇x = (1/1)∫0¹x(2x)dx = 2/3. The centroid y-coordinate is ̇y = (1/(2*1))∫0¹(2x)dbx = 2/3.

The centroid of the region, therefore, has the coordinates (x, y) = (2/3, 2/3).

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Answer:

3(x-12)/(x+2)

Step-by-step explanation:

volume = area *height

2x+ 9x-8x-36 = area * (x+2)

area = (2x+ 9x-8x-36)/(x+2)

=(3x-36)/(x+2)

=3(x-12)/(x+2)

Answer:

2x^2+5x-18

Step-by-step explanation:

Answer:

110

lol

Step-by-step explanation: