Express the following repeating decimal as a fraction in simplest form.

0.342 repeating (line over entire decimal)

Answer:

Standard deviation.

Step-by-step explanation:

The standard deviation measures how accurate the point estimate is likely to be in estimating a parameter.

The confidence interval measures how accurate the point estimate is likely to be in estimating a parameter.

A confidence interval communicates how accurate our estimate is likely to  be.

The confidence interval is a range of of all plausible values of the random variable under test at a given confidence level which is expressed in percentage such as 98%, 95% and 90% of confidence level.  

The standard deviation is the parameter to signify the  dispersion of data around  the mean value of the data.

Researchers prefer it because    on the basis of the percentage of certainty in the test result of  null hypothesis are accepted or rejected as it includes some chance for errors too. (example  95% sure means 5% not sure) also this gives a range of values and hence good chance to normalize errors.    

An interval estimate is typically preferred over a point estimate because

i) it gives us a sense of accuracy of the point estimate

ii) we know the probability that it contains the parameter (e.g., 95%)

iii) it provides us with more possible parameter values

I only

II only  

both I and II  

all of these

III only

All three statements above are true hence all of these is  the answer.

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Answer:

  129.3% of cost

Step-by-step explanation:

  cost + markup = selling price

  $78.50 + markup = $179.99 . . . . fill in given information

  markup = $101.49

The markup as a percentage of cost is ...

  markup/cost × 100% = $101.49/%78.50 × 100% ≈ 129.3%

__

As a percentage of selling price, the markup is ...

  markup/selling price × 100% = $101.49/$179.99 × 100% ≈ 56.4%

Answer: The sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.

Step-by-step explanation:

The given values : 74.001, 74.003, 74.015, 74.000, 74.005, 74.002, 74.007, and 74.000.

[tex]\text{Mean =}\dfrac{\text{Sum of all values}}{\text{Number of values}}\\\\\Rightarrow\overline{x}=\dfrac{ 592.033}{8}=74.004125\approx74.004[/tex]

The sample standard deviation is given by :-

[tex]\sigma=\sum\sqrt{\dfrac{(x-\overline{x})^2}{n}}\\\\\Rightarrow\ \sigma=\sqrt{\dfrac{0.000177}{8}}=0.00470372193056\approx0.005[/tex]

Hence, the sample mean and sample standard deviation is 74.004 millimeters and 0.005 millimeters respectively.

Answer:

$15.30

30% of 60 is 18

Step-by-step explanation:

To find the maximum amount he can spend on a meal, you have to find how  much he is going to tip.

So to find the tip you multiply 15% by 18 and you get 2.7

Then you subtract 18 by 2.7 to find out how much he can spend on the meal.

18 - 2.7 = 15.30

So he can spend $15.30 on his meal and tip $2.70

To find what percent of 60 is 18, you have to use this equation:

is over of equals percent over 100

So is/of = x/100 We have the x as the percent because that's what you're trying to figure out.

You would put 18 as is because it has the word is before it and put 60 as of because it has of before it.

So 18/60 = x/100

Now you would do Cross Product Property

18*100 = 1800

60*x = 60x

60x = 1800

Now divide 60 by itself and by 1800

1800/60 = 30

x = 30%

Answer:

540

Step-by-step explanation:

we have given E=0.3

σ = 27 hours

100(1-α)%=99%

from here α=0.01

using standard table [tex]Z_\frac{\alpha }{2}=Z_\frac{0.01}{2}=2.58[/tex]

[tex]n=\left ( Z_\frac{\alpha }{2}\times \frac{\sigma }{E} \right )^{2}[/tex] =

[tex]\left ( 2.58\times \frac{27}{3} \right )^{2}[/tex]

n = [tex]23.22^{2}[/tex]

n=539.16

n can not be in fraction so n=540

To obtain a 99% confidence interval with a margin of error of 3 hours, at least 602 components must be sampled.

In order to determine the number of components that must be sampled so that a 99% confidence interval will have a margin of error of 3 hours, we can use the formula:

n = (z * s / E)^2

Where:

n = sample size

z = z-value corresponding to the desired confidence level (in this case, 99% confidence level)

s = population standard deviation

E = margin of error

Plugging in the given values, we have:

n = (2.576 * 27 / 3)^2

n = 601.3696

Rounding up to the nearest whole number, we need to sample at least 602 components.

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Answer:

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

Step-by-step explanation:

We can write the equation of a circle in the form

(x-h)^2 + (y-k)^2 = r^2

Where (h,k) is the center and r is the radius

(x--7)^2 + (y--4)^2 = 6^2

(x+7)^2 + (y+4)^2 = 6^2

or

(x+7)^2 + (y+4)^2 = 36

The answer does not exist.

Note - The statement has typing mistakes. Correct form is presented below:

Let [tex]f(x) = (x-3)^{-2}[/tex]. Find all values of [tex]c[/tex] in (2, 5) such that [tex]f(5) - f(2) = f'(c) \cdot (5-2)[/tex]. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

In this question we should use the Mean Value Theorem, which states that given a secant line between points A and B, there is at least a point C that belongs to the curve whose derivative exists.

We begin by calculating [tex]f(2)[/tex] and [tex]f(5)[/tex]:

[tex]f(2) = (2-3)^{-2}[/tex]

[tex]f(2) = 1[/tex]

[tex]f(5) = (5-3)^{-2}[/tex]

[tex]f(5) = 1[/tex]

And the slope of the derivative is:

[tex]f'(c) = \frac{f(5) - f(2)}{5-2}[/tex]

[tex]f'(c) = 0[/tex]

Now we find the derivative of the function:

[tex]f'(x) = -2\cdot (x-3)^{-3}[/tex]

[tex]-2\cdot (x-3)^{-3} = 0[/tex]

[tex]-2 = 0[/tex] (ABSURD)

Hence, we conclude that the answer does not exist.

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Answer:

18 sqrt(3) in^2

Step-by-step explanation:

If we know the length of a side and the angle

area =  s^2  sin  a

               Since the side length is 6 and the angle a is 60

               =  6^2 sin 60

                 = 36 sin 60

                  = 36 * sqrt(3)/2

                  = 18 sqrt(3)

Answer: $ 3,360

Step-by-step explanation:

Given : The principal amount borrowed for loan : [tex]P=\ \$14,000[/tex]

Time period : [tex]t=4[/tex]

Rate of interest : [tex]r=6\%=0.06[/tex]

The formula to calculate the simple interest is given by :-

[tex]S.I.=P\times r\times t\\\\\Rightatrrow\ S.I.=14000\times4\times0.06\\\\\Rightatrrow\ S.I.=3360[/tex]

Hence, the interest due on the loan = $ 3,360

Final answer:

The sensitivity of the new test for celiac disease is 98.02% (Option C).

Explanation:

The question asks us to calculate the sensitivity of a new test for celiac disease. Sensitivity is the ability of a test to correctly identify those with the disease (true positive rate), and it is calculated as the number of true positives divided by the number of true positives plus the number of false negatives, which is essentially all the actual disease cases.

According to the provided data, the new test for celiac disease has 99 true positive results. To find out the total number of disease cases, we first need to calculate the expected number of people with celiac disease in the sample, which is 1.32% of 7642. That is approximately 100.87, or about 101 people (since we can't have a fraction of a person). Given that, we can assume there are 101 actual cases of celiac disease in the sample.

The sensitivity can be calculated as:

Sensitivity = (True Positives) / (True Positives + False Negatives)

= 99 / 101

= 0.9802 or 98.02%

Therefore, the sensitivity of the test is 98.02%, matching option C).

Answer:

[tex]a) \quad A=\left[\begin{array}{cc}8&5\\5&-1\end{array}\right] \\\\\\b) \quad \{-1+5x, 2+x\}[/tex]

Step-by-step explanation:

To compute the representation matrix A of f with respect the basis {1,x} we first compute

[tex]f(1)=f(1+0x)=(8\cdot 1 + 2 \cdot 0) + (5 \cdot 1 - 0)x=8+5x \\\\f(x)=f(0 + 1\cdot x)=(8 \cdot 0 + 2\cdot 1)+(5 \cdor 0 - 1)x = 2-1[/tex]

The coefficients of the polynomial f(1) gives us the entries of the first column of the matrix A, where the first entry is the coefficient that accompanies the basis element 1 and the second entry is the coefficient that accompanies the basis element x. In a similar way, the coefficients of the polynomial f(x) gives us the the entries of the second column of A. It holds that,

[tex]A=\left[\begin{array}{ccc}8&5\\2&-1\end{array}\right][/tex]

(b) First, note that we are using a one to one correspondence between the basis {1,x} and the basis {(1,0),(0,1)} of [tex]\mathbb{R}^2[/tex].

To compute a basis P1 with respect to which f has a diagonal matrix, we first have to compute the eigenvalues of A. The eigenvalues are the roots of the characteristic polynomial of A, we compute

[tex]0=\det\left[\begin{array}{ccc}8-\lambda & 2\\ 5 & -1-\lambda \end{array}\right]=(8 - \lambda)(-1-\lambda)-18=(\lambda - 9)(\lambda +2)[/tex]

and so the eigenvalues of the matrix A are [tex]\lambda_1=-2 \quad \text{and} \quad \lambda_2=9[/tex].

After we computed the eigenvalues we use the systems of equations

[tex]\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1\\x_2\end{array}\right] = \left[\begin{array}{c}-2x_1\\-2x_2\end{array}\right] \\\\\text{and} \\\\\left[\begin{array}{cc}8&2\\5&-1\end{array}\right]\left[\begin{array}{c}x_1 \\ x_2\end{array}\right]=\left[\begin{array}{cc}9x_1\\2x_1\end{array}\right][/tex]  

to find the basis of the eigenvalues. We find that [tex]v_1=(-1,5 )[/tex] is an eigenvector for the eigenvalue -2 and that [tex]v_2=(2,1)[/tex] is an eigenvector for the eigenvalue 9. Finally, we use the one to one correspondence between the [tex]\mathbb{R}^2[/tex] and the space of liear polynomials to get the basis [tex]P1=\{-1+5x, 2+x\}[/tex] with respect to which f is represented by the diagonal matrix [tex]\left[\begin{array}{ccc}-2&0\\0&9\end{array}\right][/tex]

If [tex]W[/tex] is a random variable representing your winnings from playing the game, then it has support

[tex]W=\begin{cases}43&\text{if you draw something with value at most 3}\\-11&\text{otherwise}\end{cases}[/tex]

There are 52 cards in the deck. Only the 1s, 2s, and 3s fulfill the first condition, so there are 12 ways in which you can win $43. So [tex]W[/tex] has PMF

[tex]P(W=w)=\begin{cases}\frac{12}{52}=\frac3{13}&\text{for }w=43\\1-\frac{12}{52}=\frac{10}{13}&\text{for }w=-11\\0&\text{otherwise}\end{cases}[/tex]

You can expect to win

[tex]E[W]=\displaystyle\sum_ww\,P(W=w)=\frac{43\cdot3}{13}-\frac{11\cdot10}{13}=\boxed{\frac{19}{13}}[/tex]

or about $1.46 per game.

The expected value of the proposition is $7.31.

To calculate the expected value, we need to multiply each possible outcome by its corresponding probability and then sum them up.

The probability of drawing a card with a value of three or less is 12/52 since there are 12 cards with values of three or less in a standard deck of 52 cards. The probability of not drawing a card with a value of three or less is 40/52.

Using these probabilities and the given payoffs, we can calculate the expected value as follows:
Expected Value = (Probability of Winning * Payoff if Win) + (Probability of Losing * Payoff if Lose)
Expected Value = (12/52 * 43) + (40/52 * -11)

Calculating this expression gives us an expected value of $7.31 (rounded to two decimal places).

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Answer:

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

Step-by-step explanation:

It is given that a data set lists earthquake depths. The summary statistics are

[tex]n=300[/tex]

[tex]\overline{x}=5.89km[/tex]

[tex]s=4.44km[/tex]

Level of significance = 0.01

We need to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 5.00.

Null hypothesis [tex]H_0:\mu=5.00km[/tex]

Alternative hypothesis [tex]H_1:\mu\neq 5.00km[/tex]

The formula for z-value is

[tex]z=\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]z=\frac{5.89-5.00}{\frac{4.44}{\sqrt{300}}}[/tex]

[tex]z=\frac{0.89}{0.25634351952}[/tex]

[tex]z=3.4719[/tex]

The p-value for z=3.4719 is 0.000517.

Since the p value is 0.000517, which is less than the significance level 0.01, therefore we reject the null hypothesis and conclude that population mean is not equal to 5.00.

The null and alternative hypotheses are H0: µ = 5.00 km and HA: µ ≠ 5.00 km. A t-test is used to calculate the test statistic, and the p-value is compared to the significance level of 0.01 to either reject or not reject the null hypothesis. The final conclusion is made in consideration of the original claim.

In statistics, hypothesis testing is a tool for inferring whether a particular claim about a population is true. For this question about earthquake depths, we would start by setting our null hypothesis (H0) and our alternative hypothesis (HA).

The null hypothesis would be H0: µ = 5.00 km, and the alternative hypothesis would be HA: µ ≠ 5.00 km.

The test statistic can be calculated using a t test, since we are dealing with a sample mean and we know the sample standard deviation (sequals4.44 km).

The p-value associated with this test statistic would then be calculated, and compared to the significance level of 0.01. If the p-value is less than 0.01, we reject the null hypothesis. If, however, the p-value is greater than 0.01, we cannot reject the null hypothesis.

The final conclusion must be stated in terms of the original claim. If we reject the null hypothesis, we conclude that the evidence supports the claim that the mean earthquake depth is not equal to 5.00 km (supporting the alternative hypothesis). If we do not reject the null hypothesis, we conclude that the evidence does not support the claim that the mean is not 5.00 km. The data does not provide sufficient evidence to support a conclusion that the mean earthquake depth is different than 5.00 km.

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Interpreting the situation, we can conclude that the statement means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

Thus, N(8) is the number of students who got a grade of 8, while N(5) is the number of students who got a grade of 5.

[tex]N(8) > 2N(5)[/tex]

It means that the number of students with a grade of 8 was more than twice the number of students with a grade of 5.

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The statement [tex]\( N(8) > 2 \cdot N(5) \)[/tex] means that the number of students who scored a grade of 8 on the exam is greater than twice the number of students who scored a grade of 5 on the exam.

To understand this, let's break down the notation:

-  N(g)  represents the number of students who scored (g) on the exam.

-  N(8)  is the number of students who scored an 8.

-  N(5)  is the number of students who scored a 5.

The inequality [tex]\( N(8) > 2 \cdot N(5) \)[/tex] compares these two quantities. It states that the count of students with a grade of 8 exceeds two times the count of students with a grade of 5. This indicates that a higher number of students performed better (scoring an 8) than those who scored a 5, with the difference being more than the number of students who scored a 5. In other words, if we were to take the number of students who scored a 5 and double it, there would still be more students who scored an 8. This could be an indicator of the overall performance of the class, suggesting that more students achieved a higher grade than those who scored in the middle range of the grading scale.

Answer: 0.064

Step-by-step explanation:

Binomial probability formula :-

[tex]P(X)=^nC_x \ p^x\ (1-p)^{n-x}[/tex], where P(x) is the probability of getting success in x trials, n is total number of trials and p is the probability of getting succes in each trial.

Given : The proportion of the homes constructed in the Quail Creek area include a security system :  [tex]p=0.40[/tex]

Now, if three homes are selected at random, then the probability all three of the selected homes have a security system is given by :-

[tex]P(3)=^3C_3 \ (0.40)^3\ (1-0.40)^{3-3}\\\\=(0.40)^3=0.064[/tex]

Hence, the probability all three of the selected homes have a security system = 0.064

The probability that all three homes selected at random in the Quail Creek area have a security system is 6.4%.

The probability that all three of the selected homes in the Quail Creek area have a security system, given that 40% of the homes have a security system, can be calculated by using the rule for independent events in probability.

Since each house is selected at random, we can multiply the probability of each house having a security system together:

P(all three homes have a security system) = P(home 1 has security system) × P(home 2 has security system) × P(home 3 has security system)

As every home has a 40% (or 0.40) chance of having a security system:

P(all three) = 0.40 × 0.40 × 0.40 = 0.064

Therefore, there is a 6.4% chance that all three homes selected will have a security system.

Answer: 2.65%

Step-by-step explanation:

Given : The  measurement of the circumference of a circle =  68 centimeters

Possible error : [tex]dC=0.9[/tex] centimeter.

The formula to find the circumference :-

[tex]C=2\pi r\\\\\Rightarrow\ r=\dfrac{C}{2\pi}\\\\\Rightarrow\ r=\dfrac{68}{2\pi}=\dfrac{34}{\pi}[/tex]

Differentiate the formula of circumference w.r.t. r , we get

[tex]dC=2\pi dr\\\\\Rightarrow\ dr=\dfrac{dC}{2\pi}=\dfrac{0.9}{2\pi}=\dfrac{0.45}{\pi}[/tex]

The area of a circle  :-

[tex]A=\pi r^2=\pi(\frac{34}{\pi})^2=\dfrac{1156}{\pi}[/tex]

Differentiate both sides w.r.t r, we get

[tex]dA=\pi(2r)dr\\\\=\pi(2\times\frac{34}{\pi})(\frac{0.45}{\pi})\\\\=\dfrac{30.6}{\pi}[/tex]

The percent error in computing the area of the circle is given by :-

[tex]\dfrac{dA}{A}\times100\\\\\dfrac{\dfrac{30.6}{\pi}}{\dfrac{1156}{\pi}}\times100\\\\=2.64705882353\%\approx 2.65\%[/tex]

To approximate the percent error in computing the area of the circle, calculate the actual area using the given circumference and radius formula. Then find the difference between the actual and estimated areas, and divide by the actual area to get the percent error.

To approximate the percent error in computing the area of the circle, we need to first find the actual area of the circle and then calculate the difference between the actual area and the estimated area. The approximate percent error can be found by dividing this difference by the actual area and multiplying by 100.

The actual area of a circle can be calculated using the formula A = πr^2, where r is the radius. Since the circumference is given as 68 cm, we can find the radius using the formula C = 2πr. Rearranging the formula, we have r = C / (2π). Plugging in the given circumference, we get r = 68 / (2π) = 10.82 cm.

Now we can calculate the actual area: A = π(10.82)^2 = 368.39 cm^2.

The estimated area is given as 4.5 m^2, which is equal to 45000 cm^2 (since 1 m = 100 cm). The difference between the actual and estimated areas is 45000 - 368.39 = 44631.61 cm^2. The percent error can be found by dividing this difference by the actual area (368.39 cm^2) and multiplying by 100:

Percent error = (44631.61 / 368.39) * 100 ≈ 12106.64%.

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Answer: A) 3

Step-by-step explanation:

For example : 1) When we divide 21 by 4 , then the remainder is 1.

Therefore we say that [tex]21\ mod\ 4 =1[/tex]

2) When we divide 10 by 7 , we get 3 as remainder.

Then , we say [tex]10\ mod\ 7=3[/tex]

The given problem : [tex]57\ mod\ 6[/tex]

When we divide 57 by 6 , we get 3 as remainder [as [tex]57=54+3=6(9)+3[/tex]]

Therfeore ,  [tex]57\ mod\ 6=3[/tex]

Hence, A is the correct option.

Answer:

1680 ways

Step-by-step explanation:

We have to select 6 different integers from 1 to 10. It is given that second smallest integer is 3. This means, for the smallest most integer we have only two options i.e. it can be either 1 or 2.

So, the selection of 6 numbers would be like:

{1 or 2, 3, a, b, c ,d}

There are 2 ways to select the smallest digit. Only 1 way to select the second smallest digit. For the rest four digits which are represented by a,b,c,d we have 7 options. This means we can chose 4 digits from 7. Number of ways to chose 4 digits from 7 is calculated as 7P4 i.e. by using permutations.

[tex]7P4 = \frac{7!}{(7-4)!}=840[/tex]

According to the fundamental rule of counting, the total number of ways would be the product of the individual number of ways we calculated above. So,

Total number of ways to pick 6 different integers according to the said criteria would be = 2 x 1 x 840 = 1680 ways

Answer:

[tex]2x+y=3[/tex]

Step-by-step explanation:

Here we aer given a point (2,-1) and a line [tex]2y=x-4[/tex]. We are supposed to find the equation of the line passing through this point and perpendicular to this line.

Let us find the slope of the line perpendicular to [tex]2y=x-4[/tex]

Dividing above equation by 2 we get

[tex]y=\frac{1}{2}x-2[/tex]

Hence we have this equation in slope intercept form and comparing it with

[tex]y=mx+c[/tex] , we get Slope [tex]m = \frac{1}{2}[/tex]

We know that product of slopes of two perpendicular lines in -1

Hence if slope of line perpendicular to [tex]y=\frac{1}{2}x-2[/tex] is m' then

[tex]m\times m' =-1[/tex]

[tex]\frac{1}{2} \times m' =-1[/tex]

[tex]m'=-2[/tex]

Hence the slope of the line we have to find is -2

now we have slope and a point

Hence the equation of the line will be

[tex]\frac{y-(-1)}{x-2}=-2[/tex]

[tex]y+1=-2(x-2)[/tex]

[tex]y+1=-2x+4[/tex]

adding 2x and subtracting  on both sides we get

[tex]2x+y=3[/tex]

Which is our equation asked

Answer:

  13

Step-by-step explanation:

The product of two negative numbers is positive. The absolute value of a number is its magnitude written with a positive sign.

  6 -2(-1) +|-5|

  = 6 + 2 + 5

  = 13

Answer: (a) 19448 ways

(b) 5292 ways

(c) 0.2721

Step-by-step explanation:

(a) 10 county council members be randomly elected out of the 17 candidates in the following ways:

= [tex]^{n}C_{r}[/tex]

= [tex]^{17}C_{10}[/tex]

= [tex]\frac{17!}{10!7!}[/tex]

= 19448 ways

(b) 10 county council members be randomly elected from 17 candidates if 5 must be women and 5 must be men in the following ways:

we know that there are 7 women and 10 men in total, so

= [tex]^{7}C_{5}[/tex] × [tex]^{10}C_{5}[/tex]

= [tex]\frac{7!}{5!2!}[/tex] × [tex]\frac{10!}{5!5!}[/tex]

= 21 × 252

= 5292 ways

(c) Now, the probability that 5 are women and 5 are men are selected:

= [tex]\frac{ ^{7}C_{5} * ^{10}C_{5}}{^{17}C_{10}}[/tex]

= [tex]\frac{5292}{19448}[/tex]

= 0.2721

Answer:

a) reject null hypothesis since p < 0.01

Step-by-step explanation:

Given that the  average assembly time for a Ford Taurus is

[tex]μ = 38 hrs[/tex]

Sample size [tex]n=36[/tex]

[tex]x bar = 37.5\\s=1.2\\SE = 1.2/6 = 0.2[/tex]

Test statistic t = mean diff/se = 0.5/0.2 = 2.5

(Here population std dev not known hence t test is used)

df = 35

p value = 0.008703

a) reject null hypothesis since p < 0.01

Answer:

Mustapha can arrange 15 bouquets per day.

Step-by-step explanation:

Neneh can arrange 20 bouquets per day.

Together Neneh and Mustapha can arrange 35 bouquets per day.

So,  Mustapha can arrange [tex]35-20=15[/tex] bouquets per day.

Therefore, Mustapha’s marginal product is 15 bouquets.

Answer:

68%

Step-by-step explanation:

It is given that the diameters of bearing have a normal distribution.

Mean = u = 1.2 cm

Standard deviation = [tex]\sigma[/tex] = 0.03 cm

We have to find the proportion of values which falls in between 1.17 to 1.23

In order to find this we have to convert these values to z-scores first. The formula to calculate z score is:

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

For 1.17:

[tex]z=\frac{1.17-1.2}{0.03}=-1[/tex]

For 1.23:

[tex]z=\frac{1.23-1.2}{0.03}=1[/tex]

So, we have to tell what proportion of values fall in between z score of -1 and 1. Since the data have normal distribution we can use empirical rule to answer this question.

According to the empirical rule:

68% of the values fall within 1 standard deviation of the mean i.e. 68% of the values fall between the z score of -1 and 1.

Therefore, the answer to this question is 68%

Answer: (2.2, 5.8)

Step-by-step explanation:

The confidence interval for standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),\alpha/2}}} , \sqrt{\dfrac{(n-1)s^2}{\chi^2_{(n-1),1-\alpha/2}}}\right )[/tex]

Given :  Sample size : 16

Mean height : [tex]\mu=67.5[/tex] inches

Standard deviation : [tex]s=3.2[/tex] inches

Significance level : [tex]1-0.99=0.01[/tex]

Using Chi-square distribution table ,

[tex]\chi^2_{(15,0.005)}=32.80[/tex]

[tex]\chi^2_{(15,0.995)}=4.60[/tex]

Then , the 99% confidence interval for the population standard deviation is given by :-

[tex]\left ( \sqrt{\dfrac{(15)(3.2)^2}{32.80}} , \sqrt{\dfrac{(15)(3.2)^2}{4.6}}\right )\\\\=\left ( 2.1640071232,5.77852094812\right )\approx\left ( 2.2,5.8 \right )[/tex]

Answer:  No, the given transformation T is NOT a linear transformation.

Step-by-step explanation:  We are given to determine whether the following transformation T : R² --> R² is a linear transformation or not :

[tex]T(x,y)=(x,y^2).[/tex]

We know that

a transformation T from a vector space U to vector space V is a linear transformation if for [tex]X_1,~X_2[/tex] ∈U and a, b ∈ R

[tex]T(aX_1+bX_2)=aT(X_1)+bT(X_2).[/tex]

So, for (x, y), (x', y') ∈ R², and a, b ∈ R, we have

[tex]T(a(x,y)+b(x',y'))\\\\=T(ax+bx',ay+by')\\\\=(ax+bx',(ay+by')^2)\\\\=(ax+bx',a^2y^2+2abyy'+y'^2)[/tex]

and

[tex]aT(x,y)+bT(x',y')\\\\=a(x,y)+b(x', y'^2)\\\\=(ax+bx',ay+by')\\\\\neq (ax+bx',a^2y^2+2abyy'+y'^2).[/tex]

Therefore, we get

[tex]T(a(x,y)+b(x',y'))\neq aT(x,y)+bT(x',y').[/tex]

Thus, the given transformation T is NOT a linear transformation.

Answer:

see below

Step-by-step explanation:

The horizontal line will have the same y and the y value will be constant

y = -7

The vertical line will have the same x and the x value will be constant

x = -1

The expected value of the game is approximately $0.09375. This is the long-term average value one might expect to gain for each play of the game.

The question involves determining the expected value of a game involving the random selection of marbles. Expected value, in probability, is the long-term average value of repetitions of the experiment. It can be computed using the formula:

Expected Value (E) = ∑ [x * P(x)]

where x represents the outcomes and P(x) is the probability of those outcomes. In this case, our outcomes and their corresponding probabilities are as follows:

Calculating our expected value, we get:

E = $4*(1/32) + $2*(10/32) - $1*(21/32) = $0.09375

This means you can expect to win about $0.09 each time you play the game in the long run.

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Answer:

B'(3, -8)

Step-by-step explanation:

The image is the mirror image of the trapezoid below the x-axis.

Each x-coordinate remains the same. Each y-coordinate becomes the opposite.

B'(3, -8)

Answer:

(3 , -8)

Step-by-step explanation:

The current coordinates of B. are (3,8). The x-axis is the horizontal line that runs across. This means that if the trapezoid were to be reflected, it would end up upside down. When this happens, only the y value changes its sign.

In short, your Y value would become negative, making 8 change to -8

Answer:

a) P=2(L+W)

b)[tex]\frac{dp}{dt}=2\frac{dL}{dt}+2\frac{dW}{dt}[/tex]

c)-2 inch/hour

Step-by-step explanation:

given:

length of the rectangle as L inches

width of the rectangle as W inches

a) The perimeter is defined as the measure of the exterior boundaries

therefore, for the rectangle the perimeter 'P' will be

P= length of AB+BC+CD+DA    (A,B,C and D are marked on the figure attached)

Now from figure

    P= L+W+L+W

           OR

=> P=2L+2W        .....................(1)

b)now dp/dt can be found as by differentiating the equation (1)

[tex]\frac{dP}{dt}=2(\frac{dL}{dt} )+2(\frac{dW}{dt} )[/tex] .............(2)

c)Now it is given for the part c of the question that

L=40 inches

W=104 inches

dL/dt=2 inches/hour

dW/dt= -3 inches/hour    (here the negative sign depicts the decrease in the dimension)

substituting the above values in the equation (2) we get

[tex]\frac{dP}{dt}=2(2)+2(-3)[/tex]

[tex]\frac{dP}{dt}=4-6=-2 inches/hour[/tex]

The formula for the perimeter of a rectangle is P = 2L + 2W. By differentiating this formula, we find that dP/dt = 2(dL/dt) + 2(dW/dt). When the length is increasing at 2 inches per hour and the width is decreasing at 3 inches per hour, the perimeter is changing at a rate of -2 inches per hour.

To find the perimeter of a rectangle, we add the lengths of all four sides of the rectangle. Given that the length is L inches and the width is W inches, the formula for the perimeter is P = 2L + 2W.

To find the rate of change of the perimeter with respect to time, we differentiate the formula with respect to time, using the chain rule. Thus, dP/dt = 2(dL/dt) + 2(dW/dt).

For the specific case where the length is increasing at 2 inches per hour and the width is decreasing at 3 inches per hour, we substitute these values into the formula for the rate of change of the perimeter to find that dP/dt = 2(2) + 2(-3) = -2 inches per hour.

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