Earth is approximately a sphere of radius 6.37 × 106 m. What are (a) its circumference, (b) its surface area, and (c) its volume?

Answer:

95% of monsoon rainfall lies between 688 mm and 1016 mm.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 852 mm

Standard Deviation, σ = 82 mm

We are given that the distribution of monsoon rainfall is a bell shaped distribution that is a normal distribution.

68 ‑ 95 ‑ 99.7 rule

We have to find the monsoon rains fall in the middle 95 % of all years.

By the rule 95% of data lies within two standard deviation of mean.Thus,

[tex]\mu + 2\sigma = 852 + 2(82) = 1016\\\mu - 2\sigma = 852 - 2(82) = 688[/tex]

Thus, 95% of monsoon rainfall lies between 688 mm and 1016 mm.

Using the Empirical Rule, we can determine that 95% of the time, the monsoon rains in India will fall between the amounts of 688 mm and 1016 mm. This is determined by subtracting and adding two standard deviations from the mean amount of rainfall.

The question pertains to the principle of the Normal Distribution curve in statistics, specifically using the 68 - 95 - 99.7 rule. Also known as the Empirical Rule, this principle suggests that for a Normal Distribution: approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Given that the mean monsoon rainfall is 852 mm and standard deviation is 82 mm, we can calculate the range for the middle 95% of all years.

To find this range, we add and subtract two standard deviations from the mean. Therefore for two standard deviations (164 mm, since 82 mm x 2 = 164 mm), our range is: 852 mm - 164 mm = 688 mm and 852 mm + 164 mm = 1016 mm. Therefore, in 95% of all years, the monsoon rainfall in India falls between 688 mm and 1016 mm.

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Answer:

[tex] l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)[/tex]

And then the maximum occurs when [tex] l'(\theta) = 0[/tex], and that is only satisfied if and only if:

[tex] \hat \theta = \bar X[/tex]

Step-by-step explanation:

For this case we have a random sample [tex] X_1 ,X_2,...,X_n[/tex] where [tex]X_i \sim N(\mu=\theta, \sigma)[/tex] where [tex]\sigma[/tex] is fixed. And we want to show that the maximum likehood estimator for [tex]\theta = \bar X[/tex].

The first step is obtain the probability distribution function for the random variable X. For this case each [tex]X_i , i=1,...n[/tex] have the following density function:

[tex] f(x_i | \theta,\sigma^2) = \frac{1}{\sqrt{2\pi}\sigma} exp^{-\frac{(x-\theta)^2}{2\sigma^2}} , -\infty \leq x \leq \infty[/tex]

The likehood function is given by:

[tex] L(\theta) = \prod_{i=1}^n f(x_i)[/tex]

Assuming independence between the random sample, and replacing the density function we have this:

[tex] L(\theta) = (\frac{1}{\sqrt{2\pi \sigma^2}})^n exp (-\frac{1}{2\sigma^2} \sum_{i=1}^n (X_i-\theta)^2)[/tex]

Taking the natural log on btoh sides we got:

[tex] l(\theta) = -\frac{n}{2} ln(\sqrt{2\pi\sigma^2}) - \frac{1}{2\sigma^2} \sum_{i=1}^n (X_i -\theta)^2[/tex]

Now if we take the derivate respect [tex]\theta[/tex] we will see this:

[tex] l'(\theta) = \frac{1}{\sigma^2} \sum_{i=1}^n (X_i -\theta)[/tex]

And then the maximum occurs when [tex] l'(\theta) = 0[/tex], and that is only satisfied if and only if:

[tex] \hat \theta = \bar X[/tex]

Answer: 1. Route ABC is a right triangle.

2. Route CDE ia not a right triangle.

3. Distance HJ= 23.32miles

4. Distance GE = 17

5. Missing length= 35

6. The sides 16, 60, 62 do NOT belong to a right triangle.

Step-by-step explanation:

Going by Pythagoras theorem, for triangle to be proven to be a right triangle, the condition below must be satisfied.

Hypotenuse² = Opposite² + Adjacent²

For question 1,

Hyp =13, opp = 5, Adj is 12

Going by Pythagoras rule.

Since 13²= 5² + 12²

Then triangle ABC is a right triangle.

For question 2,

Using the same Pythagoras theorem to prove,

In triangle CDE,

Hyp= 22, opp= 18, Adj = 14

Since 22² is not = 18² + 14²

then CDE is not a right triangle.

For question 3,

For triangle HIJ, since it is confirmed to be a right triangle, then we use the Pythagoras theorem to calculate the missing side.

Longest side if the triangle= IJ = hypotenuse = 25

HI = 9.

IJ² = HI² + HJ²

HJ²= IJ² - HI²

HJ² = 25² - 9²

HJ² = 625 - 81

HJ= √544

HJ = 23.32miles

For question 4,

FGE is also shown to be a right triangle and the missing side GE is the longest side which is also the hypotenuse.

FG= 8, FE =15

Using the Pythagoras theorem,

Hyp² = FG² + FE²

GE² = 8² + 15²

GE² = 289

GE = √289

GE = 17.

For question 5,

The hypotenuse is given as 37, one side is given as 12, let's call the missing side x

Going by Pythagoras theorem,

37² = 12²+ x²

x²= 37² - 12²

x²= 1225

x=√1225

x=35.

The missing side is 35inches.

For number 6,

The numbers given are 16, 60, 62

To know if three sides belong to a right angle, we simply put them to test using Pythagoras theorem.

It is worthy of note that the longest side is the hypotenuse.

This brings us to the equation to check below that since:

62² Is not = 60² + 16²

Then the side lengths 16, 60, 62 do not belong to a right angle.

Answer:

x^2+y^2+z^2-10x-6y-10z +50 =0

Step-by-step explanation:

Given that a sphere is contained in the first octant

Centre of the sphere is given as (5,3,5)

Since this is contained only in the first octant radius should be at most sufficient to touch any one of the three coordinate planes

When it touches we can get the maximum sphere

We find that y coordinate is the minimum of 3 thus radius can be atmost 3 so that then only it can touch y =0 plane i.e. zx plane without crossing to go to the other octants.

Hence radius =3

Equation of the sphere would be

[tex](x-5)^2 +(y-3)^2+(z-5)^2 = 3^2\\x^2+y^2+z^2-10x-6y-10z +50 =0[/tex]

Answer: The speed that Jesse travels in still water is 6 km/hr.

Step-by-step explanation:

Let the speed that Jesse travels in still water be 'x'.

Distance = 72 km

The trip to the camp area with a 2-km/hr current takes 9 hr less time than the return trip against the current.

Speed of current = 2 km/hr

According to question, we get that

[tex]\dfrac{72}{x-2}-\dfrac{72}{x+2}=9\\\\\dfrac{x+2-(x-2)}{x^2-4}=\dfrac{9}{72}\\\\\dfrac{4}{x^2-4}=\dfrac{1}{8}\\\\32=x^2-4\\\\32+4=x^2\\\\x^2=36\\\\x=\sqrt{36}\\\\x=6[/tex]

Hence, the speed that Jesse travels in still water is 6 km/hr.

Jesse's speed in still water is determined by setting up a system of equations using the distance equals rate times time formula for both downstream and upstream travel. By accounting for the time difference and the current speed, we solve for the variable representing Jesse's speed in still water.

Jesse takes a kayak trip traveling 72 km with and against a current, and we need to find Jesse's speed in still water. Let's denote the speed in still water as v (km/hr) and the current speed as 2 km/hr. The trip downstream increases Jesse's speed to (v + 2) km/hr, and upstream decreases it to (v - 2) km/hr.

Using the distance equals rate times time formula (d = rt), we can write the following equations for the time taken downstream (td) and upstream (tu):

Given that it takes 9 hours less to travel downstream, we have tu = td + 9.

By solving these linear equations, we find the system:

Combining these gives us:

72 / (v + 2) + 9 = 72 / (v - 2)

By solving this equation, we find the value of v, Jesse's speed in still water.

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Answer:

False. See the explanation below.

Step-by-step explanation:

We need to proof if the following statement "If a linear system has four equations and seven variables, then it must have infinitely many solutions." is false.

And the best way to proof that is false is with a counterexample.

Let's assume that we have seven random variables given by [tex]a_1, a_2, a_3, a_4, a_5, a_6, a_7[/tex] and we have the following four equations given by the following system:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 =1[/tex]   (1)

[tex] a_1 +a_2= 0[/tex]    (2)

[tex] a_3 +a_4 +a_5 =1[/tex]   (3)

[tex] a_6 +a_7 =1[/tex]   (4)

As we can see we have system and is inconsistent since equation (1) is not satisfied by equation (2) ,(3) and (4) if we add those equations we got:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 = 0+1+1= 2 \neq 1[/tex]

So then we can have a system of 7 variables and 4 equations inconsistent and with not infinitely solutions for this reason the statement is false.

A linear system can have more variables than equations, this is referred to as an underdetermined system. It's often believed that such systems have infinitely many solutions, but it's not necessarily the case. Such a system could also have no solutions if the system is inconsistent, signifying that not all underdetermined systems yield infinite solutions.

Contrary to the common belief, the statement that a linear system with more variables than equations must have infinitely many solutions is not always true. A system having more variables than equations is termed as underdetermined. Indeed, such systems often have infinite solutions, but not necessarily.

A system could also be inconsistent, meaning there are no solutions. As an example, consider the system of equations where:

Even though there are more variables (3) than equations (4), there are still no solutions as these equations contradict each other.

This emphasizes the point that, for a linear system to have infinitely many solutions, it must have at least one free variable and must be consistent in the first place.

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Answer:

[tex]\frac{\sqrt[3]{90 x^2 y z^2} }{6 y z}[/tex]

Step-by-step explanation:

step 1;-

Given [tex]\frac{\sqrt[3]{5 x^2} }{\sqrt[3]{12 y^2 z} }[/tex]

now you have rationalizing  denominator  (i.e monomial) with

[tex]\frac{\sqrt[3]{5 x^2} }{\sqrt[3]{12 y^2 z} } X \frac{\sqrt[3]{(12 y^2 z)^{2} } }{\sqrt[3]{(12 y^2 z)^2} }[/tex]

By using algebraic formula is

now simplification , we get denominator function

again you have to simplify numerator term

now simplify

        cancelling y and 2 values

we get Final answer

[tex]\frac{\sqrt[3]{90 x^2 y z^2} }{6 y z}[/tex]

Answer:

v = <2, 2√3>

Step-by-step explanation:

Let v be the vector of form <x,y>

Since its determinant is |4|, then:

[tex]x^2 +y^2 =4^2=16[/tex]

If it makes a π/3 angle with the positive x-axis, then the tangent relationship yields:

[tex]tan(\pi/3) = 1.732=\frac{y}{x}\\3x^2=y^2[/tex]

Replacing in the first equation:

[tex]x^2 +3x^2 =16\\x=2\\y=\sqrt{16-4}\\ y=2\sqrt 3[/tex]

Therefore, v can be represented in component form as v = <2, 2√3>.

The vector [tex]v[/tex] that lies in the first quadrant, makes an angle of [tex]\frac{\pi}{3}[/tex] with the positive x-axis, and has a magnitude of [tex]4[/tex] is:

[tex]v = 2i + 2\sqrt{3}j[/tex]

To find the vector v in component form, we start by understanding the relationships between the angle, magnitude, and components of a vector in the Cartesian coordinate system.

Given Data:

Vector Components:
In the first quadrant, the components of vector [tex]v[/tex] can be calculated using the following formulas:

Calculating Components:

For the x-component:
[tex]v_x = 4 \cdot \cos\left(\frac{\pi}{3}\right)[/tex]
The cosine of [tex]\frac{\pi}{3}[/tex] is [tex]\frac{1}{2}[/tex], so:
[tex]v_x = 4 \cdot \frac{1}{2} = 2[/tex]

For the y-component:
[tex]v_y = 4 \cdot \sin\left(\frac{\pi}{3}\right)[/tex]
The sine of [tex]\frac{\pi}{3}[/tex] is [tex]\frac{\sqrt{3}}{2}[/tex], so:
[tex]v_y = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}[/tex]

Resulting Vector:
Thus, the vector [tex]v[/tex] in component form is:
[tex]v = v_x i + v_y j = 2i + 2\sqrt{3} j[/tex]

Answer:

c. Asking people leaving a local election to take part in an exit poll

Step-by-step explanation:

Asking people leaving a local election to take part in an exit poll best represents the highest potential for nonresponse bias in a sampling strategy because of the importance of the local election compared to the exit polls.

It is worthy of note that nonresponse bias occurs when some respondents included in the sample do not respond to the survey. The major difference here is that the error comes from an absence of respondents not the collection of erroneous data. ...

Oftentimes, this form of bias is created by refusals to participate for one reason or another or the inability to reach some respondents.

Answer:

Option c) 0.80        

Step-by-step explanation:

We have to approximate the most possible correlation between spending on tobacco and spending on alcohol.

a) 0.99

This shows almost a perfect straight line relationship between spending on tobacco and spending on alcohol. Thus, this cannot be the right correlation as the relationship between spending on tobacco and spending on alcohol is not so strong.

b)-0.50

This shows a negative relation between spending on tobacco and spending on alcohol which cannot be true as they share a positive relation.

c) 0.80

This correlation shows a strong positive correlation between spending on tobacco and spending on alcohol which is correct because the relationship between spending on tobacco and spending on alcohol is positive

d)0.08

This correlation shows a very weak positive correlation between spending on tobacco and spending on alcohol which cannot be true.

Answer:

The correct answer is B=0.0262

Step-by-step explanation:

91÷3525 = 0.0262

The attached picture below gives a step by step explanation of how I arrived at my answer.

Please let's endeavor to always upload complete questions to avoid wrong answers.

Answer:

a) [tex]Q(t) = 112.8*(0.766)^{t}[/tex]

b) When t = 10, Q = 7.845.

Step-by-step explanation:

The value of a quantity after t years is given by the following formula:

[tex]Q(t) = Q_{0}(1 + r)^{t}[/tex]

In which [tex]Q_{0}[/tex] is the initial quantity and r is the rate that it changes. If it increases, r is positive. If it decreases, r is negative.

a) Write a formula for Q as a function of t.

The initial value of a quantity Q (at year t = 0) is 112.8.

This means that [tex]Q_{0} = 112.8[/tex].

The quantity is decreasing by 23.4% per year.

This means that [tex]r = -0.234[/tex]

So

[tex]Q(t) = 112.8*(1 - 0.234)^{t}[/tex]

[tex]Q(t) = 112.8*(0.766)^{t}[/tex]

b) What is the value of Q when t = 10?

This is Q(10).

[tex]Q(t) = 112.8*(0.766)^{t}[/tex]

[tex]Q(t) = 112.8*(0.766)^{10} = 7.845[/tex]

When t = 10, Q = 7.845.

Answer: In my opinion, I would say b is the correct answer

Step-by-step explanation:

Answer:

a) Parameter

b) Statistic

c) Statistic

d) Parameter

e) Statistic

Step-by-step explanation:

For this case we need to remmber that a parameter describe a population of interest is fixed and not changes , and a statistic is a value that describe the sample size selected and can change between samples.

a. There are 100 senators in the 114th Congress, and 54% of them are Republicans.

The 54% here is a parameter since represent the proportion for all the population of interest on this case.

b. In a 2011 Gallup poll of 1008 adults living in the United States, 11% said they are satisfied with the condition of the national economy.

The 11% here is a statistic since we have a random sample and from this sample we calculate the proportion of interest for this case.

c. A survey of hospital records in 120 hospitals throughout the world shows the mean height of 180 cm for adult males.

The mean height of 180 cm is a statistic since we have a survey not all the population of interest

d. The 59 players on the roster of a championship football team have a mean weight of 248.6 pounds with a standard deviation of 44.6 pounds.

The 44.6 pounds is a parameter since we are interested on all the possible players and we have the info for all of them

e. In a random sample of households in the United States, it is found that 51% of the sampled households have at least one high‑definition television.

The 51% here is a statistic since we have a result from a sample not from the population

Answer:

y=1/6 · ln |x|+c .

Step-by-step explanation:

From Exercise we have the differential equation

4xy dx= (4y6x²) dy.

We calculate the given differential equation, we get

4xy dx= (4y6x²) dy

xy dx=6yx² dy

6 dy=1/x dx

∫ 6 dy=∫ 1/x dx

6y=ln |x|+c

y=1/6 · ln |x|+c

Therefore, we get that the solution of the given differential equation is

y=1/6 · ln |x|+c .

The differential equation presented can be rewritten and solved using an integrating factor. The integrating factor in this case is 1, yielding the solution x^2 y = C.

To solve the given differential equation, we first need to rewrite it in a recognizable form namely, in the form Mdx + Ndy = 0. The given differential equation can be rewritten as 4xy dx + (4y - 6x2 ) dy = 0.

Now, we need to find an integrating factor which is e^∫(M_y - N_x) / N dx. In this case, M = 4xy, N = 4y - 6x2, M_y = 4x and N_x = -12x. Substituting these values into the equation ∫(M_y - N_x) / N dx, we find that the integrating factor is e^0=1.

With the integrating factor being 1, the given differential equation can be rewritten as d(x2y) = 0. Integrating both sides of this equation, we get x2y = C.

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Answer:

[tex]^{13}C_3 \times ^{12}C_2 = 22308[/tex]

Step-by-step explanation:

We have 5 spaces. In our hand.

_ _ _ _ _

a standard deck of 52 cards contains 13 cards for each suit.

so we have 13 spades and 13 diamonds in total. There arrangements don't matter (it doesn't matter if the first card is 9 of spades or the second is 9 of spades, all of these will be counted as one arrangement)

[tex]^{13}C_3[/tex]: to choose the 3 spade card from a total of 13 spades.

[tex]^{12}C_2[/tex]: to choose the 2 diamond cards from a total of 13 diamonds.

and that is it!

we're gonna multiply the two values.

[tex]^{13}C_3 \times ^{12}C_2[/tex]

Answer:

0.140625

Step-by-step explanation:

Given that multiple-choice questions each have four possible answers (a comma b comma c comma d )​, one of which is correct. Assume that you guess the answers to three such questions.

Each question is independent of the other with constant probability

p = Prob for correct guess = 1/4 = 0.25

q = prob for wrong guess = 1-p = 0.75

Hence

[tex]P(CWW)\\= P(C)*P(W)*P(W)[/tex], since each question is independent of the other

=[tex]0.25*0.75*0.75\\= 0.140625[/tex]

Answer:

C makes most sence

Step-by-step explanation:

Answer:

813 seats

Step-by-step explanation:

Given that,

In the auditorium, the number of seats in the 1st row = 21

In the auditorium, the number of seats in the 2nd row = 29

Therefore, the increasing number of seats in each of the row = 8.

According to the question,

The number of seats in a row continues to increase by 8 seats with each additional row. For example, 29, 37, 45 etc.

To find the number of seats in the 100th row, we have to use statistical formula.

As 21 is the total seats in the 1st row, and there is an increase of 8 seats, the formula should be = 21 + (n - 1) × 8

we have to deduct 1 so that we get 99th rows seat numbers as we have to add 21 with that to find the 100th number row.

As the question is to determine the number of seats in the 100th row, therefore, n = 100.

The number of seats in the 100th row = 21 + (100 - 1) × 8 = 21 + 99 × 8

= 21 + 792 = 813 seats.

The dot plot shows the distribution of order weights. There were no orders between 196 and 197 pounds. The most common order weight was 187.5 pounds, and the average weight was closer to 196 pounds.

1. To find the number of orders between 196 and 197 pounds, we need to look at the dot plot. From the given data, there are no orders between 196 and 197 pounds.

2. The most common order weight from the dot plot is 187.5 pounds.

3. To determine if the average weight is closer to 196 or 197 pounds, we need to calculate the mean of the data. The mean weight is calculated as the sum of the weights divided by the total number of weights. In this case, the mean weight is closer to 196 pounds.

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Answer: 5&3/4ths

Step-by-step explanation:

[tex]5+\frac{5}{16} +\frac{7}{16} \\\\5+\frac{5+7}{16} \\\\\5\frac{12}{16}=5\frac{3}{4}[/tex]

Now you know the answer as well as the formula. Hope this helps, have a BLESSED AND WONDERFUL DAY!

- Cutiepatutie ☺❀❤

Answer:

f(14) = 28

Step-by-step explanation:

The function that satisfy the equations is

f(x) = 3x - x

f(10) = 3*10 - 10 = 30 - 10 = 20

f'(x) = 3 - 1 = 2.

Therefore,

f(14) = 3(14) - 14 = 42 - 14 = 28.

f(14) = 28

Answer:  [tex]\dfrac{1}{12}[/tex] or 8.33%

Step-by-step explanation:

The conversion rate is given by :-

Conversion rate =(number of conversions ) ÷( total number of visitors)

As per given , we have

600 people visited your landing page, but only 50 visitors submitted the form..

i.e . Total number of visitors= 600

Number of conversions = 50

Then , the conversion rate  would be:-

Conversion rate = (50) ÷ 600 [tex]=\dfrac{50}{600}=\dfrac{1}{12}[/tex]

Hence, the conversion rate of your form = [tex]\dfrac{1}{12}[/tex]

In percentage , the conversion rate= [tex]\dfrac{1}{12}\times100=8.33\%[/tex]

The conversion rate is calculated by dividing the number of form submissions by the total number of visitors to the page and multiplying by 100. In this case, the conversion rate is 8.33%.

The conversion rate is central to tracking the effectiveness of your landing page. It's calculated by dividing the number of conversions (in this case, form submissions) by the total number of visitors to the page, then multiplying by 100 to get a percentage. In this case, the formula would look like this: (Number of forms submitted / Total visitors) x 100.

Plugging in your numbers, we get: (50 / 600) x 100 = 8.33%. So, the conversion rate of your form was 8.33%.

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Final answer:

A linear system with three equations and five variables does not have to be consistent. The statement 'A linear system with three equations and five variables must be consistent' is false

Explanation:

A linear system with three equations and five variables does not have to be consistent. In fact, it is possible for the system to be inconsistent.

The statement that a linear system with three equations and five variables must be consistent is False. In linear algebra, the consistency of a system depends on whether there are any contradictions among the equations. For a system to be consistent, it must have at least one solution.

For example, consider the system of equations:

x + y + z = 5

2x + 3y + 4z = 10

5x + 2y + 3z = 8

Since there are more variables than equations, there will be infinitely many solutions if the system is consistent. But if the system is inconsistent, there will be no solution.

Therefore, the statement 'A linear system with three equations and five variables must be consistent' is false

Answer:Evaluate the function

k

(

x

)

=

−

x

2

+

6

k

(

x

)

=

-

x

2

+

6

at two different inputs and state the corresponding points.

Step-by-step explanation:

Answer:

  (-1, -5)

Step-by-step explanation:

When you graph the points, you find that the lines intersect at the point (-1, -5). That is the solution to the system of equations.

  (x, y) = (-1, -5)

The solution to a system of linear equations is the point where the lines intersect. Given the data points, we can infer the linear relationship of each equation. However, without more information, we cannot calculate the exact point of intersection.

The solution to a system of linear equations is the point where the two lines intersect. From the provided data for Equation A and Equation B, we can infer their linear relationship by finding the slope and y-intercept.

For Equation A, the points (0, -2) and (2, 4) gives a slope of (4 - -2) / (2 - 0) = 3 and y-intercept of -2. For Equation B, the points (-3,-9) and (-1, -5) gives a slope of (-5 - -9) / (-1 - -3) = 2 and y-intercept of -3.

As the provided slopes and y-intercepts are different for these two equations, they would intersect at a point. To find this point, we set the two equations equal to each other and solve for x and y. However, with the provided data points, we cannot calculate the exact point of intersection without further information.

So, the solution to the system of equations would be the x and y values at the point of intersection of these two lines.

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Answer:

116.45 is the minimum score needed to be stronger than all but 5% of the population.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 100

Standard Deviation, σ = 10

We are given that the distribution of score is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

We have to find the value of x such that the probability is 0.05

P(X > x)  

[tex]P( X > x) = P( z > \displaystyle\frac{x - 100}{10})=0.05[/tex]  

[tex]= 1 -P( z \leq \displaystyle\frac{x - 100}{10})=0.05[/tex]  

[tex]=P( z \leq \displaystyle\frac{x - 100}{10})=0.95 [/tex]  

Calculation the value from standard normal z table, we have,  

[tex]P(z<1.645) = 0.95[/tex]

[tex]\displaystyle\frac{x - 100}{10} = 1.645\\x =116.45[/tex]  

Hence, 116.45 is the minimum score needed to be stronger than all but 5% of the population.

Answer: c. 8!

Step-by-step explanation:

We know , that if we line up n things , then the total number of ways to arrange n things in a line is given by :-

[tex]n![/tex] ( in words :- n factorial)

Therefore , the number of ways 8 cars can be lined up at a toll booth would be 8! .

Hence, the correct answer is c. 8! .

Alternatively , we also use multiplicative principle,

If we line up 8 cars , first we fix one car , then the number of choices for the next place will be 7 , after that we fix second car ,then the number of choices for the next place will be 6 , and so on..

So , the total number of ways to line up 8 cars = 8 x 7 x 6 x 5 x 4 x 3 x 2 x1 = 8!

Hence, the correct answer is c. 8! .

Answer:

The required probability is [tex]\dfrac{5}{12}[/tex].

Step-by-step explanation:

If a fair dice is rolled then total outcomes are

{(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}

We need to find the probability that the second die lands on a higher value than the first.

So, total favorable outcomes are

{(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)}

Formula for probability:

[tex]Probability=\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]

[tex]Probability=\dfrac{15}{36}[/tex]

[tex]Probability=\dfrac{5}{12}[/tex]

Therefore, the required probability is [tex]\dfrac{5}{12}[/tex].

Answer: [tex]Min : $750\ ,\ Q_1= \$800\ ,\ Median : \$925\ ,\ Q_3=\$1050\ ,\ Max: \$1100[/tex]

Step-by-step explanation:

The five -number summary consists of five values :

Minimum value , First quartile [tex](Q_1)[/tex] , Median , Third Quartile [tex](Q_3)[/tex]  , Maximum value.

Given : The Insurance Institute for Highway Safety publishes data on the total damage caused by compact automobiles in a series of controlled, low-speed collisions.

The following costs are for a sample of six cars:

$800, $750, $900, $950, $1100, $1050.

Arrange data in increasing order :

$750,$800, $900, $950, $1050, $1100

Minimum value =  $750

Maximum value = $1100

Median = middle most term

Since , total observation is 6 (even) , so Median = Mean of two middle most values ($900 and  $950).

i.e.  Median[tex]=\dfrac{900+950}{2}=\$925[/tex]

First quartile [tex](Q_1)[/tex] = Median of lower half ($750,$800, $900)

= $800

, Third Quartile [tex](Q_3)[/tex]  = Median of upper half ($950,  $1050, $1100)

= $1050

Hence, the five-number summary of the total damage suffered for this sample of cars will be :

[tex]Min : $750\ ,\ Q_1= \$800\ ,\ Median : \$925\ ,\ Q_3=\$1050\ ,\ Max: \$1100[/tex]