Hence, he sum of a number and [tex]18.4[/tex] is at least [tex]-3.8[/tex] as an inequality is [tex](b+18.4)\geq -3.8[/tex].
What is the inequality?
An inequality is said to be sharp if it cannot be relaxed and still be valid in general.
Here given that,
The sum of a number and [tex]18.4[/tex] is at least [tex]-3.8[/tex] as an inequality.
Let [tex]b[/tex] be the number
As it is mentioned that the sum of [tex]x[/tex] and [tex]18.4[/tex] is at least [tex]-3.8[/tex].
Hence, the sum is greater or equal to [tex]-3.8[/tex]
i.e., [tex](b+18.4)\geq -3.8[/tex]
Hence, he sum of a number and [tex]18.4[/tex] is at least [tex]-3.8[/tex] as an inequality is [tex](b+18.4)\geq -3.8[/tex].
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construct a confidence interval for at the given level of confidence. 374 506 439 491 90% confidence g
The confidence interval for the given data be,
(144.83 , 155.17).
On Subtracting 1 from your sample size, we get
150 – 1 = 149.
On Subtracting confidence level from 1, and then divide by two.
(1 – .95) / 2 = 0.025
Now, df = 149 and α = 0.025
from the table at df = 149 we got 2.262.
Divide your sample standard deviation by the square root of your sample size.
28 / √(150) = 2.28
Now, 2.262 × 2.28 = 5.17
So, the confidence interval be,
(150 - 5.17 , 150 + 5.17) = (144.83 , 155.17)
Hence, the confidence interval for the given ANOVA test to calculate the average typing speed be, (144.83 , 155.17).
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What kind of triangle has angles 60°, 60°, and 60°?
Answer:
equilateral triangle
Step-by-step explanation:
all of the sides to a equilateral triangle equal to 60 degrees.
The area of the Okhotsk Sea is 1.583 × 10^6 km^2. The area of the Indian Ocean is 7.3556 × 10^7 km^2. Find the total area. Write the final answer in scientific notation with the correct number of significant digits.
8.9386 × 10^13 km2
8.939 × 10^13 km2
7.5139 × 10^7 km2
7.514 × 10^7 km2
The total area will be;
⇒ 7.514 × 10⁷ km²
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
The area of the Okhotsk Sea is 1.583 × 10⁶ km².
And, The area of the Indian Ocean is 7.3556 × 10⁷ km².
Now,
Since, The area of the Okhotsk Sea is 1.583 × 10⁶ km².
And, The area of the Indian Ocean is 7.3556 × 10⁷ km².
Hence, The total area = 1.583 × 10⁶ + 7.3556 × 10⁷
= 1.583 × 10⁶ + 73.556 × 10⁶
= 75.14 × 10⁶
= 7.514 × 10⁷ km²
Thus, The total area will be;
⇒ 7.514 × 10⁷ km²
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edit: I did this, but im curious on what other ppl would find
In 2006, in Canada, 7.4 million out of 31.4 million of canadians said they spoke french.
Between 2006 and 2011, Canada's population went up 6%, and the number of canadians speaking french went up 300 000.
Did the pourcentage of french-speaking canadians go up or down between 2006 and 2011?
The percentage of French-speaking Canadians went down between 2006 and 2011.
What is percentage?A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.
In 2006, in Canada, 7.4 million out of 31.4 million of canadians said they spoke french. The percentage will be:
= 7.4/31.4 × 100
= 23.6%
Since Canada's population went up 6%, the bee population will be:
= 31.4 million × (6% × 31.4 million)
= 33,284,000
The number of Canadians speaking french went up 300 000.
The percentage will be:
= 7,700,000/33,284,000 × 100
= 23%
Therefore, the percentage reduced.
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The length of the rectangle is 5 cm less than twice it’s width.the area of the rectangle is 42 cm^2. What is the width of the rectangle
Answer: Could be 7.
Step-by-step explanation:
If f(x) varies directly with x and f(x)= 150 when x=45, find the value of f(x) when
x=11. Round you final answer to the nearest whole number. Work must be shown for
this problem or no credit will be earned.
As x and f(x) vary directly when x = 11 , f(x) = 110/3.
What are ratio and proportion?A ratio is a comparison between two similar quantities in simplest form.
Proportions are of two types one is the direct proportion in which if one quantity is increased by a constant k the other quantity will also be increased by the same constant k and vice versa.
In the case of inverse proportion if one quantity is increased by a constant k the quantity will decrease by the same constant k and vice versa.
Given, x and f(x) varies directly, let f(x) be y.
∴ y = kx.
150 = 45k.
k = 150/45.
k = 30/9.
K = 10/3.
Now, when x = 11.
y = (10/3)×11.
y = 110/3.
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suppose that the cash flow, in dollars , for a pharmacy are normally distributed with an unknown mean and standard deviation. the cash flow of 40 randomly sampled pharmacies are used to estimate the mean of the population. what t-score should be used to find the 80% confidence interval for the population mean?
The t-score should be used to find the 80% confidence interval for the population mean is $1.304
the interval of confidence: This indicates the degree of confidence and ambiguity in a sampling process. It provides us with a range of outcomes, indicating that we would be reasonably certain that our genuine value or variable falls inside the range.
The sample size was 40 pharmacies (n=40). To calculate the degrees of freedom, use the formula: df=n-1=40-1=39.
The scenario specifies an 80% confidence level. So,α=1−CL=1−0.8=0.2 However, we wish to utilize the value for 2, which would be 0.22=0.1. We ought to locate the row for 39 freedom degrees and the column for t0.1 using the table. So the t-score for determining the 80% confidence interval is 1.304. We may alternatively use a calculator and enter 1-α2=0.9 as well as the degrees of freedom into invT, and then enter invT. (0.9,39).
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The star Sirius is more than 4 light years farther from Earth than the star
Proxima Centauri. Write an inequality that represents the distances of Sirius
and Proxima Centauri from Earth.
Answer:
dS > dC + 4 LY
Based on your answer to part b, should kevin increase or decrease his prices toward the end of the season to avoid having unsold produce?
Kevin should decrease his prices toward the end of the season to avoid having unsold produce.
What is price?
The price of a product is the amount that customers are willing to spend. While also taking into account supply costs, seasonal discounts, competitor prices, and retail markup, marketers must connect the price to the product's actual and perceived value.
Consider the following,
Price per Pound (dollars) Quantity Sold (pounds)
1.00 ⇔ 5,700
2.00 ⇔ 4,100
3.00 ⇔ 2,300
Kevin has seen that demand increases when prices decrease.
Hence, Kevin should decrease his prices toward the end of the season to avoid having unsold produce.
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Complete question:
Kevin owns a fruit stand alongside a busy road. For several years, he has kept track of the number of apples that people purchase at different prices. He decides to graph the number of apples sold and the various prices at which he sold them. The graph shows his record of the number of apples sold at different prices.
Fill in the table with the number of apples Kevin sold at each price. Use the graph to estimate approximate quantities of apples.
What is the relationship between the price of apples and the demand for apples among Kevin’s customers?
Based on your answer to part B, should Kevin increase or decrease his prices toward the end of the season to avoid having unsold produce?
Answer: Kevin has seen that demand increases when prices decrease, so he should decrease his prices toward the end of the season to avoid having unsold produce.
Step-by-step explanation: PLATO answer :)
A lottery offers one $900 prize, one $600 prize, three $400 prizes, and four $100 prizes. One thousand tickets are sold at $5 each. Find the expectation if a person buys five tickets. Assume that the player’s ticket is replaced after each draw and that the same ticket can win more than one prize. Round to two decimal places for currency problems.
The expectation if a person buys five tickets is $____.
The Expected value for 4 tickets = $ 7.6
Given - A lottery offers one $900 prize, one $600 Prize, three $400 prizes, and four $100 prizes. One thousand tickets are sold at $5 each.
To find - Find the expectation if a person buys four tickets. Assume that the player's ticket is replaced after each draw and that the same ticket can win more than one prize. Round to two decimal places for currency problems.
Proof -
Given that,
A lottery offers
one $ 900 prize
one $ 600 prize
three $ 400 prizes
Four $ 100 prizes
And
One thousand tickets are sold at $ 5 each
Now,
If the person bought 1000 tickets then the prize he gets is
= 900 + 600 + 3×400 + 4×100
= 900 + 600 + 1200 + 400
= $3100
And
The cost of 1000 tickets = 5×1000 = $ 5000
Now,
Price 1 ticket = $ 3.1
⇒Expectation of 1 ticket = 5 - $ 3.1 = $ 1.9 (Here $5 is price of 1 ticket)
⇒Expected value for 4 tickets = $ 4× -1.9 = $ 7.6
⇒Expected value for 4 tickets = $ 7.6
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HCF and LCM of two numbers are 15 and 180 respectively. If they are in the ratio 3;4,find the numbers
the scores on a standardized test have an average of 1200 with a standard deviation of 60. a sample of 50 scores is selected. what is the probability that the sample mean will be greater than 1205? round your answer to three decimal places.
the scores on a standardized test have an average of 1200 with a standard deviation of 60 and 99.7% of the students scored between (44.67 , 56.33) i.e.
between 44 and 56.
On Subtracting 1 from your sample size, we get
100 – 1 = 99.
On Subtracting confidence level from 1, and then divide by two.
(1 – .997) / 2 = 0.0015
Now, df = 99 and α = 0.0015
from the table at df = 99 we got 2.262.
Divide your sample standard deviation by the square root of your sample size.
28 / √(100) = 2.8
Now, 2.262 × 2.8 = 6.33
So, the confidence interval be,
(100 - 6.33 , 100 + 6.33) = (44.67 , 56.33)
Hence, 99.7% of the students scored between (44.67 , 56.33) i.e.
between 44 and 56.
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(1 point) 5 dice of different colors are rolled, and the number coming up on each die is recorded. how many different outcomes are possible?
The number of total different outcomes are, 7,776.
What is rolling dice?
Every face of a cube known as a dice has a unique number.
By throwing a dice into the air, players can advance in any game by getting a certain number. Typically, this dice or dice has numbers from 1 to 6 written on each face or side of the cube-like shape.
Given:
Discrete 5 dice of different colors are rolled, and the number coming up on each dice is recorded.
Since all the dice are of different colors.
So any combination of the numbers on the dice is unique.
Dice has numbers 1 to 6.
So, for each dice there are 6 possibilities.
Since number coming on a dice is independent to the number coming on the another dice.
So, total different outcomes are = 6 x 6 x 6 x 6 x 6 = 6^5 = 7776.
Hence, the number of total different outcomes are, 7,776.
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a fence must be built to enclose a rectangular area of 20,000square feet. fencing material costs 3 per foot for the twosides facing north and south and 6 per foot for the other twosides. find the cost of the least expensive fence
A fence surrounding the field that is 200 feet long and 100 feet wide will cost $2400 to construct.
What is area?The measurement that indicates the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a shape or planar lamina.
Here,
Area of rectangle=20,000 square feet
Cost of building fence for north and south=$3 per feet
Cost of building fence for east and west=$6 per feet
Let the sides be x and y,
area=x*y
20000=xy
y=20000/x
Perimeter=2l+2b
=2(x+y)
Total cost will be,
=3*2x+6*2y
=6x+12y
C=6x+12*20000/x
C=6x+240000/x
dC/dx=6-240000/x²
dC/dx=0
6=240000/x²
6x²=240000
x²=40000
x=200 feet
y=100 feet
C=6*200+12*100
C=$2400
The cost of building a fence around the field of length 200 feet and width 100 feet is $2400.
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please help fast i dont understand
Does the following relation represent a function? \displaystyle \left\{\left(-12, -21\right), \left(-21, -22\right), \left(22, -12\right), \left(21, 22\right), \left(17, -22\right)\right\}{(−12,−21),(−21,−22),(22,−12),(21,22),(17,−22)} Yes No There is not enough information to answer this question.
Yes, The relation represent a function.
What is Function?
A relation between a set of inputs having one output each is called a function.
Given that;
The points are,
{ (- 12, - 21) , (- 21, - 22) , (22, - 12) , (21, 22) , (17, -22) }
Now,
Since, The points are,
{ (- 12, - 21) , (- 21, - 22) , (22, - 12) , (21, 22) , (17, -22) }
Clearly, All inputs having exactly one output.
Thus, The relation represent a function.
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A college admissions director states that the age of a typical student currently enrolled has increased from 19 years old. In a random sample of 20 students that are currently enrolled, the mean age is found to be 22.9 years. Assume the distribution of ages is normal with the standard deviation of 1.5 years. Use the significance level of 0.05. Which of the following is the appropriate Decision and Conclusion? Reject the null hypothesis. Sample data did not provide evidence to support that the average age of all students currently enrolled has increased from 19 years old. Reject the null hypothesis. Sample data provided evidence to support that the average age of all students currently enrolled has increased from 19 years old. Do not reject the null hypothesis. Sample data provided evidence to support that the average age of all students currently enrolled has increased from 19 years old. None of these Do not reject the null hypothesis. Sample data did not provide evidence to support that the average age of all students currently enrolled has increased from 19 years old.
The required 90% confidence interval for population mean age is (22.35,23.45)
Given,
Sample size, n = 20
Sample mean, μ = 22.9 years
Standard deviation, σ = 1.5 years
Significance level, α = 0.05
We have to find the 90% confidence interval;
Here,
x ± z(critical) σ/√n
Substituting the values;
z(critical) at α₀.₀₅ = ±1.64
22.9 ± 1.64 (1.5/√20)
= 22.9 ± 0.55
= (22.35, 23.45)
That is,
(22.35,23.45) is the required 90% confidence interval for population mean age.
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a statistics teacher started class one day by drawing the names of 10 students out of a hat and asked them to do as many pushups as they could. the 10 randomly selected students averaged 15 pushups per person with a standard deviation of 9 pushups. suppose the distribution of the population of number of pushups that can be done is approximately normal. if we would like to capture the population mean with 95% confidence the margin of error would be .
The true mean number of pushups that can be performed, as determined by the t-distribution, is within a 95% confidence interval of (9, 21).
Since we know the sample standard deviation for this issue, the t-distribution is applied.
Given,
The sample mean, x = 15 .
The sample standard deviation, s = 9 .
The sample size, n = 10 .
As a result, df = 9. Next, we determine the number of degrees of freedom, which is one less than the sample size.
The crucial value for a 95% confidence interval is then determined by looking at the t-table or using a calculator, yielding t = 2.2622.
The margin of error;
M = t × s/n
Then,
M = 2.2622 × 9/√10 = 6
The confidence interval is:
x ± M
Then
x - M = 15 - 6 = 9
x + M = 15 + 6 = 21
Therefore,
The confidence interval is (9, 21).
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The Cartesian coordinate system can be used for more than plotting points and graphing lines. It can also be used to find the distance between points and to determine relationships between figures.
Look up ways that the Cartesian coordinate plane is used in real life. Are any of the examples that you found during your research particularly interesting or surprising?
The real life example of cartesian coordinate system, is given below.
What is cartesian coordinate system?
A Cartesian coordinate system in a plane is a system of coordinates that uniquely identifies each point by a pair of numerical coordinates. These numerical coordinates are the signed distances from two fixed perpendicular oriented lines to the point, measured in the same unit of length.
The real life use of cartesian coordinate system is,
In real life, there are many straightforward situations where the Cartesian coordinate plane of x and y works well.
For instance, you can create a two-dimensional grid representing the room and use the appropriate unit of measurement to plan where to place various pieces of furniture in the room.
Define a location as your starting point and choose one direction to be x and the other (perpendicular) direction to be y. (i.e., the zero coordinate on both axes).
For example, (3, 5) would be 3 meters in the x-direction and 5 meters in the y-direction from your selected (0, 0) point. You can specify any position in the room with two numbers in the format (x, y).
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What is the measure of angle l in parallelogram lmno? 20° 30° 40° 50°
The measure of the angle L in the parallelogram LMNO with the given conditions is equal to 40°.
As given in the question,
In the parallelogram LMNO,
Measure of angle L = (2x)°
Measure of angle O = (4x +60)°
In parallelogram LMNO,
LM is parallel to NO
And LO is the transversal
∠MLO and ∠NOL forms the interior angles
Interiors angles formed by two parallel lines are always supplementary.
m∠L + m∠O = 180°
⇒ (2x)° + (4x + 60)° = 180°
⇒ 6x° = 180° - 60°
⇒ x° = 120°/6
⇒ x° = 20°
Here, m∠L = 2x°
= 2(20°)
=40°
Therefore, the measure of angle L in the parallelogram is equal to 40°.
The complete question is :
What is the measure of angle L in parallelogram LMNO such that measure of angle L is (2x)° and measure of angle O is (4x + 60)° ?
a. 20° b. 30° c. 40° d. 50°
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Answer:
c
Step-by-step explanation:
source: trust me bro
Work with a partner. Supercooling is the process of lowering the temperature
of a liquid or a gas below its freezing point without it becoming a solid. Water
can be supercooled to 86°F below its normal freezing point (32°F) and still
not freeze.
a. Let x represent the temperature of water. Which inequality represents the
temperature at which water can be a liquid or a gas? Explain your reasoning.
Even when water is supercooled to 86 degrees Fahrenheit below its normal freezing point (32 degrees Fahrenheit), it will not freeze. Therefore, x-32≥-86
Define supercooling.The process of lowering a liquid or gas's temperature below its melting point without causing it to solidify is known as supercooling, sometimes known as undercooling. It accomplishes this in the absence of a seed crystal or nucleus around which a crystal structure can form.
Let temperature be x
The temperature at which water turns into a liquid or a gas is represented by inequality is x-32≥-86
The reason is that as water will not freeze even when supercooled to 86°F below its typical freezing point (32°F).
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Finding Angles with Justification
Answer:
m<D = 108
Step-by-step explanation:
m<ACD = 28 degrees. This is because if a transversal cuts through two parallel lines, Alternate Interior Angles are Congruent.
m<CAD = 44 degrees. This is because if a transversal cuts through two parallel lines, Alternate Interior Angles are Congruent. (Same reasoning.)
Now, since the sum of all angles in a triangle is 180 degrees, we can use the two angles that we know as well as angle D to add up to 180 and figure out how much D is. (All of the angles in triangle ADC)
<CAD + <ACD + <D = 180
Substitute the values in.
44 + 28 + <D = 180
72 + <D = 180
Subtract on both sides
<D = 108
The measure of angle D is 108 degrees.
Solve the system of equation.
4x-3y=-9
3x+2y=-11
Answer:
x = 51
y = -82
(51, -82)
Step-by-step explanation:
Look at the attachment
Drag each expression to the correct location in the equation. not all expressions will be used. determine the two rational expressions whose difference completes this equation.
The two found rational expressions with the difference that completes the equation is (x + 2)/(x² - 36) and 1/(x² + 6x).
Explain the term rational expressions?The ratio of 2 polynomials is a rational expression. F can be expressed in the form p/q, where q and p are polynomials, if f is a rational expression.Consider the two rational expressions.
(x + 2)/(x² - 36) ...eq 1
and 1/(x² + 6x) ....eq 2
Subtract the second from the first.
= [(x + 2)/(x² - 36)] - [1/(x² + 6)]
Factorise the denominator of the first equation-
= [(x + 2)/(x + 6)(x - 6)] - [1/(x(x + 6)]
Taking the LCM
= [x(x + 2) - (x - 6)]/[x(x + 6)(x - 6)]
Further simplifying,
= [x² + x + 6]/[x(x-6)(x + 6)]
Thus, the two found rational expressions with the difference that completes the equation is (x + 2)/(x² - 36) and 1/(x² + 6x).
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The complete question is-
Drag each expression to the correct location in the equation. Not all expressions will be used.
Determine the two rational expressions whose difference completes this equation.
x² + x + 6
x(x-6)(x + 6)
help on photo Mathematics
Answer:
B. 48
Step-by-step explanation:
plug in the value of h into the expression and simplify from there
Find the 96th term of the arithmetic sequence -30, -32, -34
The 96th term of the given arithmetic sequence is -220
The nth term of an arithmetic sequenceFrom the question, we are to determine the 96th term of the given arithmetic sequence.
The given arithmetic sequence is
-30, -32, -34
The nth term of an arithmetic sequence is given by
aₙ = a + (n - 1)d
Where a is the first term
and d is the common difference
From the sequence,
a = -30
d = a₂ - a₁ = -32 - (-30)
d = -32 + 30
d = -2
Thus,
The 96th term is
a₉₆ = -30 + (96 -1)-2
a₉₆ = -30 + (95)-2
a₉₆ = -30 + (-190)
a₉₆ = -30 - 190
a₉₆ = -220
Hence, the 96th term is -220
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The table shows information about the
masses of some dogs.
a) Work out the minimum number of dogs
that could have a mass of more than 26 kg
b) Work out the maximum number of dogs
that could have a mass of more than 26 kg
Mass, x (kg)
0≤x≤10
10≤x≤20
20≤x≤30
30≤x≤40
Frequency
4
8
11
5
a) The minimum number of dogs that could have a mass of more than 26 kg is of: 5.
b) The maximum number of dogs that could have a mass of more than 26 kg is of: 16.
What is shown by the frequency table?The frequency table shows the number of times that each value, or values in each range, appears in the data-set.
11 weights are between 20 and 30, hence:
The minimum number of dogs that could have a mass of more than 26 kg is of 5, when all the dogs in the interval 20 ≤ x < 30 have weights that are less than 26 kg, hence only the dogs on the interval 30 ≤ x < 40 will have a mass of more than 26 kg.The maximum number of dogs that could have a mass of more than 26 kg is of 16, when when all the dogs in the interval 20 ≤ x < 30 have weights that are more than 26 kg, along with all the dogs on the interval 30 ≤ x < 40.More can be learned about frequency tables at https://brainly.com/question/16148316
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Simplify the expression.
2.4-5.4n3n+6
...
2.4-5.4n-3n+6=
(Simplify your answer. Use integers or decimals for any numbers in the expression.)
Answer:
8.4-16.2n²
Step-by-step explanation:
2.4-5.4n3n+6
= 2.4-16.2n²+6
= 8.4-16.2n²
in a sample of 41 employees in a certain company, 36 opted to work from home at least one day a week. find the test statistic used to test the claim that fewer than 90% of the company employees opted to work from home at least one day a week at the 5% significance level.
z = -2.114, test statistic used to test the claim that fewer than 90% of the company employees opted to work from home at least one day a week at the 5% significance level.
What is test statistic ?A result of a statistical test of a hypothesis is a number known as the test statistic. It displays how closely the distribution predicted by your observed data fit the null hypothesis of that statistical test.
The p value of your findings, which aids in determining whether to accept or reject your null hypothesis, is calculated using the test statistic.
P = [tex]\frac{x}{n}[/tex]
= [tex]\frac{36}{41}[/tex]
1 - P₀ = 1 - 0.95 = 0.05
Test statistic= z
=P - P₀ / {√P₀ * (1 - P₀)/ n}
=0.8780 - 0.95 /√(0.95 * 0.05) / 41
z = -2.114
Test statistic used to claim that is, z = -2.114 .
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Given f of x is equal to 1 over the quantity x minus 2 end quantity and g of x is equal to the square root of the quantity x plus 2 end quantity comma what is the domain of f (g(x))?
The domain of the given composite f(g(x)) will be (-∞,-√2 - 2)∪(-√2 - 2,√2 - 2)∪(√2 - 2,∞).
What is the range and domain of a function?A function's range is the set of all values that the function accepts, and its domain is the set of all values for which the function is defined.
The domain is for the independent variable while the range is for the dependent variable.
As per the given phrase,
f(x) = 1/(x-2)
g(x) = (x + 2)²
f(g(x)) = 1/((x + 2)² - 2)
The above function will be defines except (x + 2)² - 2 = 0
(x + 2)² = 2
x + 2 = ±√2
x = √2 - 2 and x = -√2 - 2
Thus, the domain will be (-∞,-√2 - 2)∪(-√2 - 2,√2 - 2)∪(√2 - 2,∞).
Hence "The domain of the given composite f(g(x)) will be (-∞,-√2 - 2)∪(-√2 - 2,√2 - 2)∪(√2 - 2,∞)".
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a cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 132 inches. find the dimensions of the package of maximum volume that can be sent. (the cross section is circular.)
The required dimensions of the package for maximum volume are radius = 14, height = 44 inches.
Explain cylindrical shape.A cylinder is round and has a top and base looking like a circle. The top and base are level and consistently a similar size. I figure the most effective way to portray the state of a chamber is to consider a container of soup
According to question:We have,
combined length and girth is 132 inches.
So, Girth = 2πr
If length of cylinder is h,
2πr + h = 132
h = 132- 2πr
the dimensions at which the max volume can be sent
Volume of cylinder; V = πr²h
put 132 - 2πr for h to get
V = πr²(132 - 2πr)
V =132πr² - 2π²r³
Differentiating with respect to r ,
dV/dr = 264πr - 6π²r²
For max volume will be when dV/dr = 0
Then,
264πr - 6π²r² = 0
adding 6π²r² to both sides
264πr = 6π²r²
264/6 = (π²r²)/πr
44 = πr
r =44/π = 14 inches
Radius = 14 inches
Put 44/π for r in h = 132 - 2πr to get;
h = 132 - 2π(44/π)
h = 132 -88
h = 44 inches
Thus, required height is 44 inches.
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