When constructing a confidence interval for a population mean μ from a sample of size 12, the number of degrees of freedom for the critical value tα/2 is ___________________________.
b. Find the critical value tα/2 needed to construct a confidence interval of the given confidence level 90% with sample size 23

Given the costs of 10 college math textbooks, we can calculate that:

a. The median is 70

b. The mean is 70.3

c. The variance is 2.01 and the standard deviation is 1.42

Median, mean, variance, and standard deviation

Median, mean, variance, and standard deviation are very basic but very important concepts of statistics.

Median is the middle value of the data that has been arranged sequentially from the smallest to the largest.

The median for the number of data (n) is odd:

[tex]M_{e} = x_{(\frac{n+1}{2} )}[/tex]

The median for the number of data (n) is even:

[tex]M_{e} = \frac{1}{2} (x_{(\frac{n}{2})} + x_{(\frac{n}{2} +1)} )[/tex]

Mean is the average of all data in a sample group, which is obtained by adding up all the data values, then dividing by the number of samples.

[tex]Mean = \frac{Sum of all data}{size of data (n)}[/tex]

Variance is a value that describes the variation of data, by measuring how far each piece of data is spread from the average of a data set.

[tex]Variance = \frac{sum (x_{i} - mean)^{2}}{n}[/tex]

Standard Deviation is a measure of the spread of observations in a data set relative to their mean. it measures how many observations in a data set differ from the mean and is the square root of the variance.

σ = [tex]\sqrt{variance}[/tex]

To do the problem, we first create a table for the given data.

Then we calculate the median as follows:

[tex]M_{e} = \frac{1}{2} (x_{(\frac{n}{2})} + x_{(\frac{n}{2} +1)} )\\= \frac{1}{2} (x_{5} + x_{6} )\\[/tex]

= 1/2 (70 + 70)

= 70

After that, we calculate the mean as follows:

[tex]Mean = \frac{Sum of all data}{size of data (n)}[/tex]

= (70 + 72 + 71 + 70 + 69 + 73 + 69 + 68 + 70 + 71) / 10

= 703 / 10

= 70.3

Now we calculate the difference between the data and the mean and put it into the table, and find the sum.

Then we can calculate the variance as follows:

[tex]Variance = \frac{sum (x_{i} - mean)^{2}}{n}[/tex]

= 20.1 / 10

= 2.01

Standard deviation can be calculated from the variance:

σ = [tex]\sqrt{variance}[/tex]

= [tex]\sqrt{2.01}[/tex]

= 1.42

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If surface , S is the part of the sphere, x² + y²+ z² =1 ,lies above the cone, x² + y² = z

a)| ⃗rᵩ × ⃗rθ| = ⟨sin²φ cosθ,sin²φsinθ, sinφcosφ⟩

b)∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφdθ,

S

where x = 2π and y = π/4

A surface in space given in Cartesian coordinates as f(x,y,z) = 0, can be parametrized as a vector function with two parameters,

r(u,v)= ⟨r₁(u,v) , r₂(u,v) , r₃(u,v)⟩ , (u,v)∈R²

We have, S is a part of Sphere , x² + y²+ z² =1 , lies above the cone, x² + y² = z and surface S parametrized by following vector equation ,

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ,cosφ⟩ --(1)

⃗rᵩ = ∂r/∂φ =⟨cos φ cosθ,cos φ sinθ,-sin φ⟩

⃗rθ= ∂r/∂θ=⟨- sinφ sinθ , sinφ cosθ ,0⟩

| ⃗rᵩ × ⃗rθ| =| i j k |

|cosφcosθ cosφ sinθ -sinφ |

|- sinφsinθ sinφ cosθ 0 |

= i( 0 + sinφ cosθsinφ) -j(0- (- sinφsinθ)(-sinφ)) + k(sinφ cosθ cosφcosθ - (-sinφ sinθ ) cosφ sinθ)

= (sin²φcosθ )i + (sin²φsinθ)j + (sinφcosφ(sin²θ+cos²θ))k

= sin²φcosθ )i + (sin²φsinθ)j +(sinφcosφ)k

| ⃗rᵩ× ⃗rθ|=⟨sin²φcosθ ,sin²φsinθ,sinφcosφ⟩

b) from part (a) we get,

x = sinφ cosθ

y = sinφsinθ

z = cosφ

are spherical coordinates of S where, 0≤φ≤π/4 and 0≤ θ≤ 2π.

so, ∫∫ z² ds = ₀∫ˣ₀∫ʸcos²φ dφ dθ ,

S

where x = 2π and y = π/4

So,Surface integeral is equals to ₀∫ˣ₀∫ʸcos²φ dφ dθ.

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Complete question:

If Sis the part of the sphere x2 + y2 + x2 = 1 that lies above the cone z= x2 + y2, find the following: a)S can be parametrized by a vector equation

r(θ, φ) = ⟨sinφ cosθ , sinφsinθ , cosφ ⟩ then a)| ⃗rᵩ × ⃗rθ| = ?

b) ∫∫ z² ds = ?

S

Answer:

Step-by-step explanation:

the following standards for variable manufacturing overhead havebeen established for a company that makes only one product:standard hours per unit of output.. 5.6 hoursstandard variable overhead rate. $19.15 per hourthe following data pertain to operations concerning the productfor the last month:actual hours .. 5,100

Answer:

a. compounding by the day: A = $5967.17

b. compounding by the hour: A = $5967.29

c. compounding by the minute: A = $5967.30

d. compounding by the second: A = $5967.30

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]

If the interest is compounding by the day then n = 365.

Given:

Substitute the values into the compound interest formula and solve for A:

[tex]\implies A=4000\left(1+\dfrac{0.04}{365}\right)^{365 \times 10}[/tex]

[tex]\implies A=4000\left(1.00010958...\right)^{3650}[/tex]

[tex]\implies A=4000(1.49179200...)[/tex]

[tex]\implies A=5967.16801...[/tex]

[tex]\implies A=\$5967.17[/tex]

If the interest is compounding by the hour then:

Given:

Substitute the values into the compound interest formula and solve for A:

[tex]\implies A=4000\left(1+\dfrac{0.04}{8760}\right)^{8760 \times 10}[/tex]

[tex]\implies A=4000\left(1.00000456...\right)^{87600}[/tex]

[tex]\implies A=4000\left(1.49182333...\right)[/tex]

[tex]\implies A=5967.29333...[/tex]

[tex]\implies A=\$5967.29[/tex]

If the interest is compounding by the minute then:

Given:

Substitute the values into the compound interest formula and solve for A:

[tex]\implies A=4000\left(1+\dfrac{0.04}{525600}\right)^{525600 \times 10}[/tex]

[tex]\implies A=4000\left(1.00000007...\right)^{5256000}[/tex]

[tex]\implies A=4000(1.49182466...)[/tex]

[tex]\implies A=5967.29867...[/tex]

[tex]\implies A=\$5967.30[/tex]

If the interest is compounding by the second then:

Given:

Substitute the values into the compound interest formula and solve for A:

[tex]\implies A=4000\left(1+\dfrac{0.04}{31536000}\right)^{31536000 \times 10}[/tex]

[tex]\implies A=4000\left(1.00000000...\right)^{315360000}[/tex]

[tex]\implies A=4000\left(1.49182390...\right)[/tex]

[tex]\implies A=5967.29562...[/tex]

[tex]\implies A=\$5967.30[/tex]

The more compounding periods throughout the year, the higher the future value of the investment.  However, the difference between compounding by the day and compounding by the second results in a difference of 13 cents over the year, which is negligible comparatively.

a) The linear equation for R(t) is given as follows: R(t) = -2.1t + 16.

b) The helium reserves in 2015 will be of: 5.5 billion.

c) The reserve will be depleted in the year of: 2018.

The linear function in this problem has the definition in slope-intercept format presented as follows:

R(t) = mt + b.

In which:

Considering the measures in billions, the parameters are given as follows:

m = -2.1, b = 16.

Then the equation is defined as follows:

H(t) = -2.1t + 16.

2015 is five years after 2010, hence the estimate is calculated as follows:

H(5) = -2.1(5) + 16 = 5.5 billion.

The reserve will be depleted in the year t + 2010 + 1, when R(t) = 0, hence:

-2.1t + 16 = 0

2.1t = 16

t = 16/2.1

t = 7.61 years.

Hence the reserves will be depleted during the year of 2018.

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First, multiply [tex]18*5 = 90[/tex]

Then, divide [tex]90 / 6 = 15[/tex]

Therefore, each group has 15 students.

The parameters of the exponential function in this problem are given as follows:

The standard format of an exponential function is given as follows:

y = c(1 - a)^t.

This is the case for a decaying exponential function, and the meaning of each parameter is given as follows:

In this problem, the function is given as follows:

f(x) = 1.1(0.44)^x.

Hence the values for the parameters are given as follows:

Then the percent of change is of -56%, as it is a decaying exponential function with a = 0.56.

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Answer:

1,208.92613

Step-by-step explanation:

this is the answer

Raphael needs to run 3.9 miles on Friday so that he will have a total of 11 1/2 miles for the week.

What is Subtraction ?

An arithmetic operation called subtraction simulates the process of deleting items from a collection. The negative symbol, or, denotes subtraction.

Raphael ran a total of 7 3/5 miles through Monday to Thursday.

We'll convert it from mixed fraction to simple fraction :

7 3/5 miles = [(7×5) + 3] / 5

                   = 38/5 miles.

He wants to have a total of 11 1/2 miles for the week.

Again, we will convert it into simple fraction :

11 1/2 miles = [(11×2) + 1 ] / 2

                  = 23/2 miles.

Let's he needs to run x miles to a total of 23/2 miles.

So, x + 38/5 = 23/2

     x = 23/2 - 38/5

        = [(23×5) - (38×2)] / 10

        =  (115 - 76)/10

        = 39/10 miles = 3.9 miles.

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The true statements about the graph are 1, 2, and 3.

A line of the best fit is a straight line that reduces the distance between it and some data. The line of best fit is used to represent a relationship in a scatter plot with numerous data points.

Given:

The temperature above 52° F affects the sale of hot chocolate at her shop,

As you can see from the graph, the y-axis shows the orders for hot chocolate on a day when the outside temperature was 52° F.

From the graph,

at x = 0, y = 160

Thus, Sheryl gets 160 orders for hot chocolate on a day when the outside temperature is 52° F

At x = 0, y = 160 and at x = 16, y = 128

So the difference is 160-128 / 16 - 0 = 2

Thus, Sheryl gets 2 fewer orders for each degree increase above 52° F in the outside temperature.

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The absolute value equation represented in the graph is

|x + 3|  

Without taking direction into account, absolute value describes how far away from zero a certain number is on the number line.

A number can never have a negative absolute value.

The equation is written in the form below

|x|

The a transformation which involves a translation of 3 units to the left took place.  the transformation rule for 3 units to the left is addition of 3, hence we have the equation

|x + 3|

We can therefore conclude that the absolute value equation of the function is |x + 3|

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According to the given statement the positive exponent is [tex]\frac{16}{625}[/tex].

Exponents are a method of expressing enormous magnitudes in terms of their respective powers. The amount of times some number has already been multiplied in itself is the exponent, so to speak. For instance, the result of multiplying the number 6 on it's own four times is 6 6 6 6. This may be expressed as 64. Here, the exponent and base are 4 and 6, respectively.

The exponent is the exact quantity of times that base would be compounded on its own. As a result, if two powers have the same foundation, they can be multiplied. Whenever two powers are multiplied, exponents are added. If necessary, we can also divide the abilities.

= [tex]( \frac{2}{5} )^{4}[/tex]

= [tex]\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \times\frac{2}{5} \\\\\frac{16}{625}[/tex]

Hence, the required exponent is [tex]\frac{16}{625}[/tex].

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[tex]10+2a < 4[/tex]

Simplify:

[tex]2a+10 < 4[/tex]

Subtract 10 from both sides:

[tex]2a+10-10 < 4-10[/tex]

[tex]2a < -6[/tex]

Divide both sides by 2:

[tex]\dfrac{2a}{2} < \dfrac{-6}{2}[/tex]

[tex]a < -3[/tex]

[tex]\fbox{Second Option}[/tex]

Answer:

-9

Step-by-step explanation:

Given function:

[tex]f(x)=-9x+4[/tex]

Given x-values:

Calculate the value of the function for the two given values of x:

[tex]\begin{aligned}\implies f(x_1)&=-9(-5)+4\\&=45+4\\&=49\end{aligned}[/tex]

[tex]\begin{aligned}\implies f(x_2)&=-9(0)+4\\&=0+4\\&=4\end{aligned}[/tex]

[tex]\boxed{\begin{minipage}{6.3 cm}\underline{Average rate of change of function $f(x)$}\\\\$\dfrac{f(b)-f(a)}{b-a}$\\\\over the interval $a \leq x \leq b$\\\end{minipage}}[/tex]

As -5 < 0:

Therefore:

[tex]\begin{aligned} \implies \textsf{Average rate of change}&=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\\\\&=\dfrac{4-49}{0-(-5)}\\\\&=\dfrac{-45}{5}\\\\&=-9\end{aligned}[/tex]

The correct statement regarding the inverse function of f(x) = 4x^4 is given as follows:

[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]; f^(-1)(x) is not a function.

The function in this problem is defined as follows:

f(x) = 4x^4.

To obtain the inverse of a function y = f(x), first the variables y and x are exchanged, as follows:

x = 4y^4.

Isolating the variable y, we have that:

y^4 = (x/4).

The inverse operation of the fourth power is the fourth root, hence:

[tex]y = \pm \sqrt[4]{\frac{x}{4}}[/tex]

[tex]f^{-1}(x) = \pm \sqrt[4]{\frac{x}{4}}[/tex]

[tex]f^{-1}(x) = \pm \left(\frac{x}{4}\right)^{\frac{1}{4}}[/tex]

The plus/minus symbol means that for each input of x, the inverse function gives two outputs, meaning that there are multiple outputs mapped to each input, and thus the inverse is not a function.

This means that the second statement is correct.

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Step-by-step explanation:

To solve the equation -16(d + 1) = -21 for d, we can first use the distributive property to expand the expression -16(d + 1):

-16(d + 1) = -16d - 16

Then, we can set the two sides of the equation equal to each other and solve for d:

-16d - 16 = -21

-16d = -5

d = 5/16

Therefore, the value of d that satisfies the equation -16(d + 1) = -21 is d = 5/16.

1

The answer is 1 because if u use your brain you'd understand

The cost of the fixed assets will be $64258. The correct option is 1.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the data is, 36,000; - machinery and equipment - 18,600; - spare parts for repair - 608; - raw materials and materials - 7020.

The cost of the fixed assets will be calculated by adding all the values,

Cost = 36000 + 608 + 18600 + 7020

Cost = 62228 ≈ 64258

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Answer:

C) x = 9

Step-by-step explanation:

[tex]\frac{12}{27}[/tex] = [tex]\frac{4}{x}[/tex]

[tex]\frac{12}{27}[/tex] ÷ [tex]\frac{3}{3}[/tex] = [tex]\frac{4}{9}[/tex]  This is the fraction simplified

[tex]\frac{4}{9}[/tex] = [tex]\frac{4}{x}[/tex]  Simplified, I think that it is easier to see that x = 9

Answer:

Step-by-step explanation:

The value of y is unclear given the information provided. However, if BC intersects AD at E, then it stands to reason that y would be equal to the value of x. This is because, if BC intersects AD at E, then the two lines are parallel and therefore have the same value for y. Calculating the internal angles of triangle DEF, we can find the length of BW, CW and CE. Since BD=87 and WE=38, the remaining angle must be 55°. Therefore, using trigonometry and the known side lengths, we can calculate BW = 87 sin 55° = 77.5  , CW = 87cos55° = 48.5  and CE = 38sin55° = 33.2 . As you can see then, these are all easily calculable from the given data. In angle DEF, BD = 87 and WE = 38. Therefore, BW = 87 - 38 = 49, CW = 49 - 38 = 11, and CE = 11 - 1 = 10.

Answer:

y = 1/4x + 1

Step-by-step explanation:

To calculate the number of pieces into which Oscar divided the 4/5 kilogram of clay, we need to divide the total weight of the clay by the weight of each piece. We can do this by first converting the weight of the clay to kilograms and then dividing it by the weight of each piece. The weight of the clay in kilograms is 4/5 * 1 kilogram = 0.8 kilograms. And the weight of each piece is 1/5 * 1 kilogram = 0.2 kilograms. So, to find the number of pieces into which Oscar divided the clay, we divide the total weight of the clay by the weight of each piece, giving us 0.8 kilograms / 0.2 kilograms/piece = <<0.8/0.2=4>>4 pieces. Thus, Oscar divided the 4/5 kilogram of clay into 4 pieces.

The required null hypothesis of the given observation is μ1 = μ2 = μ3.

The null hypothesis is a typical mathematical theory that asserts that there is no statistical relationship and significance between two sets of observed data and measured phenomena for each set of specified, single observable variables.

The null hypothesis can be evaluated to determine whether or not there is a relationship between two measured phenomena, which makes it valuable.

It can let the user know if the outcomes are the product of random chance or deliberate manipulation of a phenomenon.

The null hypothesis, often known as "H-nought," "H-null," or "H-zero," is written as H0 to distinguish it from other hypotheses.

The null hypothesis of the given observation:

μ1 = μ2 = μ3

Therefore, the required null hypothesis of the given observation is μ1 = μ2 = μ3.

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Complete question:
To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below.

Treatment

Observations

A

20

30

25

33

B

22

26

20

28

C

40

30

28

22

The null hypothesis for this ANOVA problem is?

Answer:

6 hundred thousand

Step-by-step explanation:

there is 6 numbers in total.Each number is 5.The next number should be 6.

By applying vertical compression by a factor of 3, then vertical reflection and finally horizontal translation by 1 unit to the right we can transform into required graph.

A transformation is a mathematical operation that changes the position, size, or shape of a geometric object. In other words, it is a way to manipulate the object by moving, scaling, rotating, or reflecting it.

Reflection is a transformation that flips an object over a mirror line, or line of reflection. It creates a mirror image of the object on the other side of the line.

In mathematics, reflection is often represented by a matrix called the reflection matrix. This matrix encodes the rules for reflecting points in a particular direction.

To transform the graph of y = x^2 into y = -3(x+1)^2, we can apply the following transformations:

Vertical stretch or compression by a factor of 3: This transformation stretches or compresses the graph vertically by a factor of 3. To apply this transformation, we need to multiply the y-values of the points on the graph by 3.

Vertical reflection: This transformation reflects the graph across the x-axis. To apply this transformation, we need to multiply the y-values of the points on the graph by -1.

Horizontal translation by 1 unit to the right: This transformation shifts the graph 1 unit to the right. To apply this transformation, we need to add 1 to the x-values of the points on the graph.

Applying these transformations to the graph of y = x^2, we get the graph of y = -3(x+1)^2.

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The maximum dimension of a rectangular microchip she can use is 12 mm and 24 mm.

A rectangular microchip, commonly known as an integrated circuit or chip, resembles a flat rectangle approximately small in size with a slew of wires designated by pins protruding from it.

The maximum dimension of a rectangular microchip can be determined by finding its length and width.

From the information given:

The surface area of the rectangle can be expressed by using the formula:

S = 2(lw + lh + wh)

where;

The surface area then becomes:

864 = 2(2x² + 8x + 4x)

Divide both sides by 2, and we have:

432 = 2x² + 12x

We can now have a quadratic expression as:

2x² + 12x - 432 = 0

x² + 6x - 216 = 0

using the  factorization method;

(x + 18) (x - 12) = 0

x = - 18 or x = 12;

Since x cannot be negative, then x = 12. Therefore, width = 12 mm, and the length = 2(12) = 24 mm. Therefore, we can conclude that the maximum dimension of the microchip she can use is 12 mm and 24 mm.

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Answer:

I went with B. 9 mm and 18 mm   PLATO

Step-by-step explanation:  mad maths skills

The estimated distance traveled by the car is 73.5 mi.

To use the Midpoint Rule, we need to divide the time interval into subintervals and evaluate the average velocity over each subinterval.

The time interval is from 0 seconds to 100 seconds, so we will divide this interval into 5 subintervals of length 20 seconds each. The midpoint of each subinterval is the average time, which we can use to estimate the average velocity over that subinterval.

The midpoints of the subintervals are:

Using the data from the table, we can calculate the average velocity over each subinterval:

We can now use the Midpoint Rule to estimate the distance traveled by the car:

distance = (20/6) * (17 + 53 + 53 + 49.5 + 47)

              = (20/6) * 220.5

              = <<20/6*220.5=73.5>>73.5 mi

So, the estimated distance traveled by the car is 73.5 mi.

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