Finding the average rate of change of g(z) =4x^2_5 between the points( -2,11) and (2,11

Answer:

A) The actual error |f(0.5) - P3(0.5)| is 0.0340735. The error using Taylor's error formula R₃(0.5) is 0.2917.

B) Te error in the interval [0.5,1.5] is 0.0340735.

C) The approximate integral using P₃(x) is 1/12

D) The real error between the two integrals is 4.687·10^(-3) while the error using \int_{0.5}^{1.5}|R_{3}(x)dx| is 0.018595

Step-by-step explanation:

A) To determine the error we first have to write the third grade Taylor polynomial P3(x):

[tex]P_n(x)=\displaystyle\sum_{k=0}^N \frac{f^{k}(a)}{k!}(x-a)[/tex] with N=3

[tex]P_3(x)=\displaystyle\sum_{k=0}^3 \frac{f^{k}(1)}{k!}(x-1)=(x-1)^2-\frac{1}{2} (x-1)^3[/tex]

The error for the third grade Taylor polynomial R₃(x) is represented as the next term non-written in the polynomial expression near to the point x:

[tex]R_3(x)=\displaystyle \frac{f^{4}(x)}{4!}(x-a)=\frac{1}{24} (\frac{2}{x^3}+\frac{6}{x^4})(x-1)^4[/tex]

Therefore the real error is:

[tex]Err(0.5)=|f(0.5)-P_3(0.5)|=0.0340735[/tex]

The error of the Taylor polynomial R₃(0.5) is:

[tex]R_3(0.5)=\displaystyle \frac{1}{24} (\frac{2}{0.5^3}+\frac{6}{0.5^4})(0.5-1)^4=0.2917[/tex]

B) The error in the interval [0.5,1.5] is the maximum error in that interval.

This is found in the extremes of the intervals. We analyze what happens in X=1.5:

[tex]Err(1.5)=|f(1.5)-P_3(1.5)|=0.01523[/tex]

[tex]R_3(1.5)=\displaystyle \frac{1}{24} (\frac{2}{1.5^3}+\frac{6}{1.5^4})(1.5-1)^4=\frac{1}{216}=4.63\cdot 10^{-3}[/tex]

Both of these errors are smaller than Err(0.5) and R₃(0.5). Therefore the error in this interval is Err[0.5,1.5] is Err(0.5).

C) The approximation of the integral and the real integral is:

[tex]I_f=\displaystyle\int_{0.5}^{1.5} f(x)\, dx=\int_{0.5}^{1.5} (x-1)ln(x)\, dx=0.08802039[/tex]

[tex]I_{p3}=\displaystyle\int_{0.5}^{1.5} P_3(x)\, dx=\int_{0.5}^{1.5} (x-1)^2-0.5(x-1)^3\, dx=1/12=0.0833333[/tex]

D) The error in the integrals is:

[tex]Errint=|I_f-I_{p3}|=4.687\cdot10^{-3}[/tex]

[tex]Errint(R_3)=\displaystyle\int_{0.5}^{1.5} R_3(x)\, dx=\int_{0.5}^{1.5} \frac{1}{24} (\frac{2}{x^3}+\frac{6}{x^4})(x-1)^4\, dx=0.018595[/tex]

Answer:

Solution of the system is (2,1).                                                          

Step-by-step explanation:

We are given the following system of equation:

[tex]3x+4y=10\\x - y = 1[/tex]

We would use the elimination method to solve the following system of equation.

Multiplying the second equation by 4 and adding the two equation we gwt:

[tex]3x + 4y = 10\\4\times (x-y = 1)\\\Rightarrow 4x - 4y = 4\\\text{Adding equations}\\3x + 4y + (4x-4y) = 10 + 4\\7x = 14\\\Rightarrow x = 2\\\text{Substituting value of x in second equation}\\2 - y = 1\\\Rightarrow y = 1[/tex]

Solution of the system is (2,1).

Answer: the solution is (2, 1)

Step-by-step explanation:

The given system of simultaneous equations is given as

3x+4y=10 - - - - - - - - - - - - -1

x−y=1 - - - - - - - - - - - - 2

We would eliminate x by multiplying equation 1 by 1 an equation 2 by 3. It becomes

3x + 4y = 10

3x - 3y = 3

Subtracting, it becomes

7y = 7

Dividing the left hand side and the right hand side of the equation by 7, it becomes

7y/7 = 7/7

y = 1

Substituting y = 1 into equation 2, it becomes

x - 1 = 1

Adding 1 to the left hand side and the right hand side of the equation, it becomes

x - 1 + 1= 1 + 1

x = 2

Answer:

Yes

Step-by-step explanation:

A system in echelon form can have more variables than equations. ... Every linear system with free variables has infinitely many solutions. True, a free variable can take any value, and so there are infinitely many solutions. Any linear system with more variables than equations cannot have a unique solution.

Answer:

The answer to your question is 2.6 in

Step-by-step explanation:

Data

width = ? in

width = 65 mm

Process

1.- To solve this problem, use proportions and cross multiplication

                   1 in ---------------------- 25.4 mm

                   x    ----------------------  65 mm

                   x = (65 x 1) / 25.4

2.- Simplification

                   x = 65 / 25.4

3.- Result

                   x = 2.6 in      

Answer:

D

Step-by-step explanation:

The regression equation is

y=a+bx

Now, we have to estimate a and b.

We know that slope can be determine as

b=byx=r*(sy/sx)

=0.5*(8/40)

b=0.1

So, we have the slope of regression equation is 0.1.

Now the intercept "a" can be estimated as

y=a+bx

a=ybar-bxbar

a=75-0.1*280

a=47

So, we have the intercept of regression equation is 47.

y=a+bx

y=47+0.1x

Final answer:

The least-squares regression line of final-exam scores on pre-exam total scores, given the statistics provided, is y = 47 + 0.1x, which corresponds to option (D).

Explanation:

The question involves identifying the least-squares regression line of final-exam scores on pre-exam total scores in an economics course. Given the correlation coefficient (r = 0.5), the mean and standard deviation of both the pre-exam totals and the final-exam scores, we can use the formula for the regression line, which is:

y = b0 + b1*x

where y is the predicted final exam score, x is the pre-exam total score, b0 is the y-intercept, and b1 is the slope of the line. The slope (b1) is calculated by:

b1 = r * (standard deviation of y / standard deviation of x) = 0.5 * (8 / 40) = 0.1

And the y-intercept (b0) is found using the means of x and y:

b0 = mean of y - b1 * mean of x = 75 - 0.1 * 280 = 47

So, the regression line is y = 47 + 0.1x, which corresponds to option (D).

Answer:

Step-by-step explanation:

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis

change in the value of y = y2 - y1

Change in value of x = x2 -x1

The slope is given as 1/2 and the line passes through (2, - 3)

To determine the intercept, we would substitute x = 2, y = - 3 and m= 1/2 into y = mx + c

y = mx + c. It becomes

- 3 = 1/2 × 2 + c = 1 + c

c = - 3 - 1 = - 4

The equation becomes

y = x/2 - 4

Answer:

cos 38 = 17/c

Step-by-step explanation:

cos 38 = 17/c is the correct statement

Answer:

Step-by-step explanation:

The diagram consists of different boxes if the same sizes. Each box is a square with sides measuring 1 cm. The area of each box would be

1 × 1 = 1cm^2

The area of the parallelogram is the total number of squares that it contains. To determine this, the first step is to count the total number of complete squares. Looking at the diagram, there are 11 complete squares. This means

11 × 1 = 11 square units.

The next step is to count the number of incomplete squares and divide by 2. The number of incomplete squares are 8

8/2 = 4

4 × 1 = 4 square units

The area of the parallelogram would be

11 + 4 = 15 square units

Answer:

27m²

Step-by-step explanation:

surface area is area of all sides

these is two triangles and a rectangle

area of triangle = (base x height) /2 = (3 x 4)/2= 12/2 =6

area of rectangle = length x breadth = 5x3 = 15

surface area = area of triangle + area of triangle + area of rectangle

6 + 6 + 15 = 27m²

Answer:

Step-by-step explanation:

The first step is to rearrange the given quadratic equation so that it will take the form of the standard quadratic equation which is expressed as

ax² + bx + c = 0

The given quadratic equation is expressed as

2 = - x + x² - 4

Rearranging it, we would subtract 2 from the left hand side and the right hand side of the equation. It becomes

2 - 2 = - x + x² - 4 - 2

- x + x² - 6 = 0

x² - x - 6

a = 1

b = - 1

c = - 6

Substituting into the quadratic formula, it becomes

x = [- (- 1) ± √(- 1)² - 4(1)(- 6)] /2(1)

Answer:

10 roses

Step-by-step explanation:

To find the number of roses in the blue vase, we double the number of roses in the red vase. Since the red vase has 5 roses, the blue vase contains 10 roses.

The student's question is about determining the number of roses in the blue vase given certain conditions about the number of flowers in vases. We know that each vase has 12 flowers, the red vase has 7 tulips, and thus 5 roses because the total is 12. The blue vase has twice as many roses as the red vase, so we need to double the number of roses in the red vase to find that number.

Since the red vase has 5 roses, the blue vase will have twice as many, which is 10 roses.

Here's the calculation step by step:

Answer:

f(x) = 0 , when x=(-2) and x=3

Step-by-step explanation:

for a≠0 and b positive then for the function

f(x) = a*(x+2)*(x-3)ᵇ

f(x) = 0 when

1) (x+2)=0 → x=(-2)

2) (x-3)=0 → x=-3

then for x=3 and x=(-2) , f(x)=0

Answer: The change in the value of x would be : [tex]\Delta x=4.5 [/tex]    .

Step-by-step explanation:

We know that if y represents the value of some varying quantity, then as the value of y varies from p to q.

The change in the value of y is given by :-

[tex]\Delta y=q-p[/tex]     [Next value - previous value]

Given : x represents the value of some varying quantity.

So , As the value of x  varies from 3.5 to 8.

The change in the value of x is given by :-

[tex]\Delta x=8-3.5=4.5 [/tex]  

Hence, the change in the value of x would be : [tex]\Delta x=4.5 [/tex]  

Answer:

[tex]volume\ = \pi \frac{625^2}{6}[/tex]

Step-by-step explanation:

See the attached figure.

y₁ = 25x  and  y₂ =x²

The intersection between y₁ and y₂

25x = x²

x² - 25x = 0

x(x-25) = 0

x = 0 or x =25

y = 0 or y =25² = 625

The points of intersection (0,0) and (25,625)

To find the volume of the solid obtained by rotating about the y-axis the region bounded by y₁ and y₂

y₁ = 25x ⇒ x₁ = y/25 ⇒ x₁² = y²/625

y₂ =x² ⇒ x₂ = √y ⇒ x₂² = y

v = ∫A(y) dy = π ∫ (x₂² - x₁²) dy

∴ V =

[tex]\pi \int\limits^{625}_0 {y-\frac{y^2}{625} } \, dy =\pi( \frac{y^2}{2} -\frac{y^3}{3*625} ) =\pi (\frac{625^2}{2} -\frac{625^3}{3*625}) =\pi ( \frac{625^2}{2}-\frac{625^2}{3}) =\pi \frac{625^2}{6}[/tex]

To find the volume of the solid formed by rotating the region bounded by y = 25x and y = x²25 about the y-axis, one needs to calculate the area of a typical slice, then integrate this area over the range of x-values.

To find the volume V of the solid obtained by rotating the region bounded by y = 25x and y = x225 about the y-axis, we can apply the method of slicing.

First, we need to find the area A(x) of a typical slice perpendicular to the x-axis. Here we have two functions, finding the x- values where these functions intersect gives the bounds on the integral for the volume.

We can find the area A(x) = pi*(outer radius)2 - pi*(inner radius)2 = pi[(25x)ˆ2 - (xˆ2 * 25)ˆ2].

To find V, we integrate A(x) over the interval of x-values. This will give us the exact volume of the solid.

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The probability of drawing an ace or a heart from a full deck of cards is approximately 5.88%. The probability of drawing a card that is neither an ace nor a heart is approximately 45.89%.

This question is about probability in Mathematics. A full deck has 52 cards, 4 of which are aces, and 13 of which are hearts. The total number of favorable outcomes are the scenarios where we draw an ace or a heart.

For part a) There are 16 favorable outcomes (4 aces + 12 remaining hearts). For 2 draws, that's 16/52 * 15/51 = 0.0588 or 5.88% chance of drawing an ace or a heart.

For part b) There are 36 non-favorable outcomes (52 total - 16 favorable). The chance of drawing a card that is neither ace nor heart is 36/52 * 35/51 = 0.4589 or 45.89%.

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Answer:

1) The production canister welds have consistently lower burst strengths than the test nozze welds.

2) The production canister weids have much more variable burst strengths.

3) The test nozzle welds data contain 2 outliers.

4) Test nozzle welds have much more variable burst strengths.

5) The production canister welds have much higher burst strengths.

6) The production canister welds data contain 2 outiers.

Step-by-step explanation:

Hello!

The boxplots summarize the information of test nozzle closure welds and production canister nozzle welds.

The boxplot for the test nozzle closure welds shows that the first quartile and second quartile are close to each other but the third quartile is more separated to them, meaning that the data contained in the box is asymmetric, the data seems to have less variability between C₁ and C₂ and more between C₂ and C₃, the box is right-skewed.

The left whisker is larger than the right one, there are no outliers in the sample, due to most of the data being comprehended below C₁, the overall distribution of the data set is left-skewed, with large variability.

The boxplot for production cannister nozzle welds shows that the box is small (the variability of the data set is low) and symmetric, with C₂ in the middle of it and C₁ and C₃ are equidistant to the second quartile.

The whiskers of the box are small but they have almost the same length, showing that there is the same amount of data in them, this adds to the overall symmetry of the data set.

Finally, this data set shows two outliers, these values are far from the box, meaning that they are relatively extreme unusual values in regards to the rest of the sample but their distance to the box seems to be equal wich adds to the conclusion of the symmetrical distribution, with low variability of the data set.

I hope it helps!

Answer:

[tex]\bar X = \frac{52384 +54079 +52762 +54364 +52584+53987 +53108 +51588 +50554 +53987}{10}=52939.7[/tex]

[tex] Mode= 53987[/tex]

[tex] Median = \frac{52762+53108}{2}=52935[/tex]

[tex] Midrange=\frac{50554+54364}{2}=52459[/tex]

Step-by-step explanation:

We can calculate the mean with the following formula:

[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]

And if we replace the values given we got:

[tex]\bar X = \frac{52384 +54079 +52762 +54364 +52584+53987 +53108 +51588 +50554 +53987}{10}=52939.7[/tex]

The mode is the most repeated value and for this case with a frequency of 2 the mode is:

[tex] Mode= 53987[/tex]

In order to find the median we need to order the dataset on increasing way like this:

50554, 51588, 52384  ,52584, 52762

53108, 53987 , 53987 , 54079, 54364

Since we have 10 values an even number the median is calculated from the average between positions 5 and 6 on the data ordered, and we got:

[tex] Median = \frac{52762+53108}{2}=52935[/tex]

The mid range is defined like this:

[tex] Midrange = \frac{Max +Min}{2}[/tex]

And if we replace we got:

[tex] Midrange=\frac{50554+54364}{2}=52459[/tex]

The 'Top 10' list gives insights into the population of college tuitions in a country. The mean, midrange, median, and mode of the data set are $53,138.50, $52,459, $52,584., and  $53,987 respectively.

The "Top 10" list of annual tuition amounts for the most expensive colleges in a country provides insights into the range and distribution of college tuitions within the country's higher education system.

Mean: The mean (average) tuition is calculated by summing up all the tuition amounts and dividing by the total number of colleges (in this case, 10).

Mean = ($52,384 + $54,079 + $52,762 + $54,364 + $52,584 + $53,987 + $53,108 + $51,588 + $50,554 + $53,987) / 10 = $53,138.50 (rounded to two decimal places).

Midrange: The midrange is the average of the minimum and maximum values in the dataset.

Midrange = ($50,554 + $54,364) / 2 = $52,459 (rounded to two decimal places).

Median: The median is the middle value when the data is arranged in ascending or descending order. In this case, the data is already provided in ascending order, so the median is the fifth value.

Median = $52,584.

Mode: The mode is the value that appears most frequently in the dataset.

Mode = $53,987.

In summary, the "Top 10" list of college tuitions in the country indicates that there is a range of tuition amounts, with the mean being around $53,138.50. The midrange falls around $52,459, the median is $52,584, and the mode is $53,987.

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Answer:

2.44491234

Step-by-step explanation:

1 gallon = 3785 cm3

Unit conversion is a way of converting some common units into another without changing their real value. The volume of  9255 cm³ in gallons is equal to 2.4451 gallons.

Unit conversion is a way of converting some common units into another without changing their real value. for, example, 1 centimeter is equal to 10 mm, though the real measurement is still the same the units and numerical values have been changed.

Since 1 gallon is equal to 4546.09 cubic centimeters, therefore, the volume of 9255 cm³ in gallons can be written as,

1 gallons = 3785 cm³

1 cm³ = 1/3785 gallons

The volume of 9255 cm³ in gallons is,

Volume = 9255 × (1/3785 ) gallons

= 2.4451 gallons

Hence, the volume of  9255 cm³ in gallons is equal to 2.4451 gallons.

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Answer:

Step-by-step explanation:

Triangle QRS is a right angle triangle.

From the given right angle triangle

RS represents the hypotenuse of the right angle triangle.

With 30 degrees as the reference angle,

QR represents the adjacent side of the right angle triangle.

QS represents the opposite side of the right angle triangle.

To determine QR, we would apply trigonometric ratio

Cos θ = adjacent side/hypotenuse side. Therefore,

Cos 30 = QR/14

√3/2 = QR/14

QR = 14 × √3/2

QR = 14√3/2 = 7√3/2

Answer:

[tex]P = \left[\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\right][/tex]

[tex]Q = \left[\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{array}\right][/tex]

Step-by-step explanation:

[tex]P*\left[\begin{array}{c}x\\y\\z\end{array}\right]=\left[\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\right] \left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}y\\z\\x\end{array}\right][/tex]

[tex]Q*\left[\begin{array}{c}y\\z\\x\end{array}\right]=\left[\begin{array}{ccc}0&0&1\\1&0&0\\0&1&0\end{array}\right] \left[\begin{array}{c}y\\z\\x\end{array}\right] = \left[\begin{array}{c}x\\y\\z\end{array}\right][/tex]

(a) Growth Rate Constants: 6

(b) Carrying Capacity Constants: 0

Interaction Constants: 3

The species compete since the interaction constant is positive, indicating competition for resources between the two species.

For the given system:

[tex]\[ x' = 6x \][/tex]

[tex]\[ S_{xy} = 32y^2 - 8y - 8 + 3xy - 31 \][/tex]

(a) Growth Rate Constants: The growth rate constant for species [tex]\(x\)[/tex] is [tex]\(6\)[/tex].

(b) Carrying Capacity Constants: There is no explicit term indicating a carrying capacity for either species in the given equations, so the carrying capacity constants are both [tex]\(0\)[/tex].

Interaction Constants: The interaction constant between the two species can be identified from the term [tex]\(S_{xy}\)[/tex], which is [tex]\(3\)[/tex] in this case.

So, the answers are:

(a) Growth Rate Constants: [tex]\(6\)[/tex]

(b) Carrying Capacity Constants: [tex]\(0\)[/tex]

Interaction Constants: [tex]\(3\)[/tex]

The species compete since the interaction constant is positive.

Complete Question :

Population models for two species can report either competition for resources (an rise in one species recede the growth rate in the other) or cooperation for asset (an rise in one species rise the growth rate in the other). For the systems below, identify the boundary as growth rates, carrying capacities, and measures of interconnection between species. Determine if the species compete or cooperate. x' = 6x Sxy = 32y^2 - 8y - 8 + 3xy - 31 (a) Growth Rate Constants: (b) Carrying Capacity Constants: interconnection Constants: Note: Your answers must be numbers (which may include 0). If there is more than one answer, different your answers with commas. Choose One: - The Species Do Not Interact - One species take part, the other Cooperates - Help Both Species Cooperate - Both Species Compete.

Answer: C. If he passes the class, she will pay $100.

A. If she ended up paying $100, he must have passed the class.

Step-by-step explanation:

For a statement to be true, one assumption might imply the other assumption. In this case, A and C fulfills it.

Final answer:

The mathematician's statement makes passing the class a necessary condition for her son to receive $100. If he received the money, he must have passed the class, but not receiving the money does not conclusively prove he did not pass. The statement does not make passing the class a sufficient condition, as there could be other requirements.

Explanation:

The statement made by the mathematician to her son is a classic example of a conditional statement in logic, particularly looking at necessary and sufficient conditions. To analyze the truth of the given statements, we must understand the relationship between passing the class and receiving the $100.

Answer:

Given:

Length of the garden is represented by :

[tex]3y+5[/tex]

The length is increased by 50%.

To find the new length.

Solution:

1) The original length = [tex]3y+5[/tex]

The increase in length will be = [tex]50%[/tex] of [tex]3y+5[/tex]

⇒ [tex]\frac{50}{100}(3y+5)[/tex]

⇒ [tex]\frac{1}{2}(3y+5)[/tex]

The new length  = Original length + Increased length = [tex](3y+5)+ \frac{1}{2}(3y+5)[/tex]

2) Let the original length = 100%

Increase percent of length = 50%

Total percent of length = [tex]100\%+50\%=150\%[/tex]

New length = [tex]150%[/tex] of [tex]3y+5[/tex]

⇒ [tex]\frac{150}{100}(3y+5)[/tex]

⇒ [tex]1.5(3y+5)[/tex]

Thus,  the new length of the garden can be represented by both expression :

[tex](3y+5)+ \frac{1}{2}(3y+5)[/tex] or [tex]1.5(3y+5)[/tex]

Step-by-step explanation:

You need to find six combinations of a and b such that a² + b² = 7.

a = 1, b = √6

a = √2, b = √5

a = √3, b = 2

a = 2, b = √3

a = √5, b = √2

a = √6, b = 1

Answer:

[tex]14\frac{1}{3}[/tex]

Step-by-step explanation:

Let's write this out as an equation. Let the unknown quantity be x:

[tex]x+\frac{2}{3} - \frac{1}{3} (x+\frac{2}{3}) = 10\\\\3x+2-x -\frac{2}{3} = 30\\\\9x+6-3x-2=90\\\\6x= 86\\\\x = \frac{86}{6} =\frac{43}{3} =14\frac{1}{3} \\[/tex]

Answer: the quantity is 9

Step-by-step explanation:

Let x represent the quantity.

A quantity and its 2/3 are added together. The 2/3 of the number is 2/3 × x = 2x/3

The sum of the quantity and its 2/3 would be

x + 2x/3 = (3x + 2x)/3 = 5x/3

From the sum, 1/3 of the sum is subtracted. 1/3 of the sum would be

1/3 × 5x/3 = 5x/9

Subtracting 1/3 of the sum from the sum, it becomes

5x/3 - 5x/9 = (15x - 5x)/9 = 10x/9

If the remainder is 10, it means that

10x/9 = 10

Crossmultiplying

10x = 9 × 10 = 90

x = 90/10

x = 9

Answer:

0.8749

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

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

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

[tex]\mu = 0, \sigma = 1[/tex]

The probability that Z is less than 1.15 is:

This is the pvalue of Z = 1.15, which is 0.8749.

The probability that Z is less than 1.15: P(Z < 1.15) = Φ(1.15) ≈ 0.8749 or 87.49%

For a standard normal distribution with a mean of 0 and a standard deviation of 1, the probability that the random variable Z is less than 1.15 can be calculated using the standard normal cumulative distribution function (CDF).

The standard normal CDF gives the probability that a standard normal random variable takes a value less than or equal to a given value. We can use a standard normal distribution table or a calculator/software to find the value of the CDF at the desired point.

Let's denote the CDF of the standard normal distribution as Φ(z). Then, the probability that Z is less than 1.15 is given by:

P(Z < 1.15) = Φ(1.15)

Using a standard normal distribution table or calculator, we find that Φ(1.15) ≈ 0.8749.

Therefore, the probability that Z is less than 1.15 is approximately 0.8749 or 87.49%.

Answer:

a) Figure attached

b) For this case w have that A =(2.4 km)j, B= (-1.9 km) i , C= (-4.7 km)j

And the final position vector can be calculated adding the 3 vectors like this:

[tex] s = A +B+C[/tex]

[tex] s= (-1.9 km)i +(2.4 -4.7 km) j= (-1.9km)i + (-2.3 km)j[/tex]

We can find the magnitude of s like this:

[tex] |s| = \sqrt{(-1.9)^2 +(-2.3)^2}=2.983[/tex]

And then we can find the angle with this formula:

[tex] \theta = \tan^{-1} (\frac{-2.3 km}{-1.9 km})=50.44 [/tex]

The other possibility is [tex] \theta = 50.44+180 =230.44[/tex]

And since they want the angle measured from East the correct angle would be [tex] \theta = 230.44[/tex]

Step-by-step explanation:

Part a

On the figure attached we have the vectors for the pattern described.

Part b

For this case w have that A =(2.4 km)j, B= (-1.9 km) i , C= (-4.7 km)j

And the final position vector can be calculated adding the 3 vectors like this:

[tex] s = A +B+C[/tex]

[tex] s= (-1.9 km)i +(2.4 -4.7 km) j= (-1.9km)i + (-2.3 km)j[/tex]

We can find the magnitude of s like this:

[tex] |s| = \sqrt{(-1.9)^2 +(-2.3)^2}=2.983[/tex]

And then we can find the angle with this formula:

[tex] \theta = \tan^{-1} (\frac{-2.3 km}{-1.9 km})=50.44 [/tex]

The other possibility is [tex] \theta = 50.44+180 =230.44[/tex]

And since they want the angle measured from East the correct angle would be [tex] \theta = 230.44[/tex]

Answer:

[tex]0.82-2.977\frac{0.048}{\sqrt{15}}=0.783[/tex]    

[tex]0.82+2.977\frac{0.048}{\sqrt{15}}=0.857[/tex]    

So on this case the 99% confidence interval would be given by (0.783;0.857)

So then we can conclude that at 99% of confidence the true mean is between (0.783;0.857) and the specification is satisfied since the value of 0.8 is on the confidence interval

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X= 0.82[/tex] represent the sample mean

[tex]\mu[/tex] population mean (variable of interest)

s=0.048 represent the sample standard deviation

n=15 represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex]   (1)

In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:

[tex]df=n-1=15-1=14[/tex]

Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,14)".And we see that [tex]t_{\alpha/2}=2.977[/tex]

Now we have everything in order to replace into formula (1):

[tex]0.82-2.977\frac{0.048}{\sqrt{15}}=0.783[/tex]    

[tex]0.82+2.977\frac{0.048}{\sqrt{15}}=0.857[/tex]    

So on this case the 99% confidence interval would be given by (0.783;0.857)

So then we can conclude that at 99% of confidence the true mean is between (0.783;0.857) and the specification is satisfied since the value of 0.8 is on the confidence interval

To find a 99% confidence interval for the true average concentration of the active ingredient, we can use the formula: CI = (x - z*(s/sqrt(n)), x + z*(s/sqrt(n))). Plugging in the given values and using the critical value for a 99% confidence level, we get the confidence interval to be (0.799, 0.841) grams per liter.

To find a 99% confidence interval for the true average concentration of the active ingredient, we can use the formula:

CI = (x - z*(s/sqrt(n)), x + z*(s/sqrt(n)))

Where CI is the confidence interval, x is the sample mean, z is the z-score corresponding to the desired confidence level, s is the sample standard deviation, and n is the sample size.

Plugging in the given values: x = 0.82, s = 0.048, and n = 15, and using the critical value for a 99% confidence level (z = 2.576), we get:

CI = (0.82 - 2.576*(0.048/sqrt(15)), 0.82 + 2.576*(0.048/sqrt(15)))

Simplifying the expression, we get the confidence interval to be (0.799, 0.841) grams per liter.

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