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Related Questions
the area of the parallelogram is 40 in2. one base of the parallelogram is 5 in long. the other base is 10 in long find its 2 heights
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Step-by-step explanation:
Area of parallelogram = base × height or say b×h
Given: h=8 inches
Area=120 sq.inches
⇒120=8×b
⇒120/8=b
⇒120/8=b = 15
Answer = 15 inches b
Thomas bought 120 whistles, 168 yo-yos and 192 tops . He packed an equal amount of items in each bag.
a) What is the maximum number of bag that he can get?
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Answer:
To find the maximum number of bags that Thomas can pack, we need to find the greatest common divisor (GCD) of 120, 168, and 192.
Prime factorizing the three numbers:
120 = 2^3 x 3 x 5
168 = 2^3 x 3 x 7
192 = 2^6 x 3
The GCD is the product of the common prime factors with the lowest exponents, which is 2^3 x 3 = 24.
So, Thomas can pack the items into 24 bags, each containing an equal number of whistles, yo-yos, and tops.
Answer:
Step-by-step explanation:
To find the maximum number of bags that Thomas can pack, we need to find the greatest common divisor (GCD) of 120, 168, and 192.
We can start by finding the prime factorization of each number:
120 = 2^3 × 3 × 5
168 = 2^3 × 3 × 7
192 = 2^6 × 3
Then we can find the GCD by taking the product of the smallest power of each common prime factor:
GCD = 2^3 × 3 = 24
Therefore, Thomas can pack a maximum of 24 bags.
Test the conjecture. 1) A rectangle has an area of 54 square units. A scale factor of is applied to the rectangle to create a scaled figure. What is the area of the scaled figure?
9 units 6 units
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Answer:
Step-by-step explanation:
yr face add more points NOW
In Biology class, Marissa is viewing cells with a microscope. Cell W
-7
measures 1.8 x 10 microns in diameter and Cell S measures
-5
7.2 x 10 microns in diameter. How many times larger is the bigger
cell than the smaller cell?
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The number of times larger Cell S is compared to Cell W is 4 x 10²,
How many times larger is the bigger cell?
Scientific notation is used to compress large numbers into smaller numbers. In order to write a number in scientific notation, the number is written as a decimal number, between 1 and 10 and multiplied by a power of 10.
When the power of the scientific notation is negative, it means that the number is less than 1. 0.01 would be written as 1 x [tex]10^{-2}[/tex]. Cell S is larger than Cell W.
(7.2 x [tex]10^{-5}[/tex]) ÷ (1.8 x [tex]10^{-7}[/tex])
(7.2 / 1.8) x [tex]10^{-5--7}[/tex]
4 x 10²
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CAN SOMEONE HELP WITH THIS QUESTION?✨
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Step-by-step explanation:
it was not clear if an average change rate would be sufficient, or if you needed an immediate change rate (as I also don't know if you covered derivatives already or not).
so, it would be helpful, if you could put a message to an answer that was not giving you what you need.
so, here now an answer for an immediate change rate (hopefully that is what you need) :
we have a right-angled triangle.
the direct line of sight (the direct distance between police and red car) is the Hypotenuse (the baseline opposite of the 90° angle).
the 50 ft and 180 ft are the legs.
Pythagoras gives us the length of the Hypotenuse :
Hypotenuse² = 50² + 180² = 2500 + 32400 = 34,900
Hypotenuse = sqrt(34900) = 186.8154169... ft
in general terms let's say x is the distance of the cop to the road, y is the distance on the road to the crossing point with the distance cop to road, and z is the line of sight distance between the red car and the cop (the Hypotenuse).
x² + y² = z²
now, the first derivative of distance is the change of distance = speed.
then dy/dt (= y') is how fast the car is traveling down the road. dx/dt (= x') is how fast the cop is traveling toward the road. and dz/dt (= z') is how fast the distance between the cop and the car is changing.
now, we take the derivative of our equation
x² + y² = z² with respect to time, variable by variable :
d(x² + y² = z²)/dt =
dx²/dx × dx/dt + dy²/dy × dy/dt = dz²/dz × dz/dt
that gives us the equation
2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)
x(dx/dt) + y(dy/dt) = z(dz/dt)
from the problem we know x (50 ft), y (180 ft), dz/dt (85 ft/s). we calculated z (the Hypotenuse = sqrt(34900), and since the cop is not moving, we know dx/dt = 0.
and we get
50ft×0ft/s + 180ft×(y') = sqrt(34900)ft×(85)ft/s
we solve for y' (the speed of the car on the road)
y' = sqrt(34900)×85/180 = 88.21839132... ft/s
≈ 88.22 ft/s
and now here the difference for an average change rate over the unrevealed of 1 second :
the radar measured the change of the distance (Hypotenuse) from 1 second ago to now.
so, 1 second ago, the distance was
186.8154169... + 85 = 271.8154169... ft
the 50 ft leg stays the same, but the 180 ft leg was (again via Pythagoras)
271.8154169...² = 50² + leg²
leg² = 271.8154169...² - 50² = 71,383.62088...
leg = 267.1771339... ft
so, the red car traveled
267.1771339... - 180 = 87.1771339... ft/s
as you can see, it is close, but there has to be a difference, as the average change rate is only an approximation to the immediate change rate.
The points (-2, -2) and (5,5) are the endpoints of the diameter of a circle. Find the length of the radius of the circle.
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Answer:
Step-by-step explanation:
[tex]diameter=\sqrt{(5+2)^2+(5+2)^2} \\=\sqrt{49+49} \\=\sqrt{98} \\=7\sqrt{2} \\radius=\frac{7\sqrt{2} }{2} \\\approx 4.95[/tex]
A quadrilateral has two angles that measure 235° and 40°. The other two angles are in a ratio of 5:12. What are the measures of those two angles?
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Answer: Let's denote the two unknown angles as x and y.
We know that the sum of the angles in any quadrilateral is 360°, so we can set up an equation using this fact:
235° + 40° + x + y = 360°
Simplifying this equation, we get:
x + y = 85° (equation 1)
We also know that the other two angles are in a ratio of 5:12. This means that:
x/y = 5/12
Multiplying both sides by y, we get:
x = (5/12)y (equation 2)
Now we can substitute equation 2 into equation 1 and solve for y:
(5/12)y + y = 85°
(17/12)y = 85°
y = (12/17) * 85°
y = 60°
Substituting y = 60° into equation 2, we can solve for x:
x = (5/12) * 60°
x = 25°
Therefore, the two angles that are in a ratio of 5:12 measure 25° and 60°, respectively.
Step-by-step explanation:
A sequence has the nth term rule T(n) = 32 - 3n
What is the value of the first negative term in this sequence?
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[tex]T(11) = -1[/tex] represents the initial negative term inside the series.
What do the concepts positive and negative in logic mean?A phrase that acknowledges a characteristic or trait in anything is said to be positive. Negative terms are those that downplay some aspect or characteristic of something. The positive or negativizes of a phrase is termed its quality.
What does "positive" and "negative" mean?Positive simply denotes excellent or the polar opposite of negative. For instance, you're less likely to receive favorable feedback on your scorecard if you've a good mindset toward your assignments. It might be difficult to keep track of all the different definitions of the word positive.
T(n) is negative when [tex]32 - 3n < 0[/tex], which can be rearranged as:
[tex]3n > 32[/tex]
[tex]n > 32/3[/tex]
Since [tex]n[/tex] is an integer, the smallest value of n that satisfies this inequality is [tex]n = 11[/tex].
Therefore, the [tex]11th[/tex] term of the sequence is:
[tex]T(11) = 32 - 3(11) = -1[/tex]
So the first negative term in the sequence is [tex]T(11) = -1.[/tex]
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h=p√m2+n find the value of h when p = 3, n=20 , m=6
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Answer:
We can substitute the given values of p, m, and n into the formula for h:
h = p√(m^2 + n)
h = 3√(6^2 + 20)
h = 3√(36 + 20)
h = 3√56
We can simplify this by factoring 56 into its prime factors:
h = 3√(2^3 × 7)
h = 3 × √(2^2 × 7) × √2
h = 3 × 2√7
Therefore, when p = 3, n = 20, and m = 6, the value of h is 6√7 or approximately 13.42.
16) Critique the sample method and the survey used by the guild. You’ll want to use some of the following words in your critique: sample, bias, random, response-bias, undercoverage, non-response, voluntary response.
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Answer:
The critique of the sample method and survey used by the guild is as follows:
Sample: The sample used in the survey is not representative of the entire population. The survey was conducted only among the guild members, and hence it does not represent the views of non-members. This type of bias is called undercoverage bias.Bias: The survey seems to suffer from selection bias, where the participants who responded may not represent the entire population. This type of bias is known as response bias. It is possible that the members who chose to participate in the survey have different views from those who did not respond. Furthermore, the survey did not specify how the participants were selected, and hence there may be other types of bias present in the survey.Randomness: The survey did not use random sampling, which is a fundamental principle of survey research. A random sample ensures that every member of the population has an equal chance of being selected, which increases the representativeness of the sample.Non-Response: The survey did not address the issue of non-response bias, where participants who choose not to respond may have different views from those who respond. The survey did not specify the response rate, and hence it is unclear how many members chose to participate in the survey.Voluntary response: The survey seems to be a voluntary response survey, where members chose to participate voluntarily. This type of survey is known to suffer from bias because participants who respond are usually more motivated or have stronger opinions than those who do not respond.In summary, the survey used by the guild suffers from various types of bias, which may affect the validity and generalizability of the results. To improve the validity and reliability of the survey, the guild should use random sampling and address issues such as non-response and response bias.
Why is the following not a probability? model?
LOADING...
Click the icon to view the data table.
Determine why it is not a probability model. Choose the correct answer below.
A.
This is not a probability model because at least one probability is greater than 0.
B.
This is not a probability model because at least one probability is greater than 1.
C.
This is not a probability model because the sum of the probabilities is not 1.
D.
This is not a probability model because at least one probability is less than 0
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The correct answer is C. This is not a probability model because the sum of the probabilities is not 1.Therefore, the sum of the probabilities must be equal to 1 for a set of probabilities to be considered a probability model.
What is probability?Probability is a measure of the probability of an event, expressed as a number between 0 and 1.
For example, the probability of rolling a 6 on a fair six-sided die is 1/6 or approximately 0.17.
In order for a set of probabilities to represent a probability model, the sum of all the probabilities must be equal to 1. This ensures that there is a 100% chance that one of the possible outcomes will occur.
If the sum of the probabilities is less than 1, then there is a possibility that none of the outcomes will occur.
If the sum of the probabilities is greater than 1, then there is a possibility that more than one outcome will occur, which is not possible in a probability model.
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A test consists of 30 multiple choice questions, each with five possible answers, only one of which is correct. find the mean and the standard deviation of the number of correct answers. round the answers to the nearest hundredth.
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The mean and standard deviation for 30 multiple-choice questions for the number of correct answers is Mean(μ) = 6 and Standard deviation(σ) = 2.19.
What is mean?
A collection of values are averaged to form the mean. The total points were divided by the total scores. When population samples are tiny, the mean is sensitive to extreme scores. For example, if two students in a class of 20 earned significantly higher than the rest, the mean will be skewed higher than the other students' scores might suggest. When using means, larger sample sizes are preferable.
Define standard deviation.
In statistics, a measure of variability known as the standard deviation ( standard deviation ) is frequently used. It demonstrates how different things are from the norm. (mean). When the SD is low, the data tend to be close to the mean, while when it is high, the data are dispersed over a wide variety of values.
Given: number of questions(n) = 30, p = [tex]\frac{1}{5}[/tex], q = [tex]\frac{4}{5}[/tex]
Mean μ = n*p
= 30 *[tex]\frac{1}{5}[/tex]
μ = 6
Standard deviation σ = [tex]\sqrt{n*p*q}[/tex]
= [tex]\sqrt{30*\frac{1}{5}*\frac{4}{5}}[/tex]
= [tex]\sqrt{4.8}[/tex]
σ = 2.19
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Three randomly selected children are surveyed. The ages of the children are 2, 4, and 12. Assume that samples of size n=2 are randomly selected
with replacement from the population of 2, 4, and 12. Listed below are the nine different samples. Complete parts (a) through (d).
2,2 2,4 2,12 4,2 4,4 4,12 12,2 12,4 12,12
a. Find the value of the population variance o²
Σ(x-1)
N
The formula for the population variance is o²=
where u is the population mean and N is the population size.
While either technology or the formula can be used to find the population variance, in this exercise, use technology. Determine the population
variance.
4
(Round to three decimal places as needed.)
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Three youngsters are interviewed at random. Population variation is around [tex]18.67[/tex].
What is population standard deviation vs mean difference?The standard deviation refers to the square base of variance, which is the average of the squares departures from the mean. Both metrics capture distributional variability, although they use different measurement units: The units used to indicate standard deviation are the same as the values' original ones.
We compute population variance for what reason?In statistics, population standard deviation is a crucial indicator of dispersion. further reading. Statisticians compute variance to determine how order to overcome the drawbacks in a data gathering interact to one another. By calculating the population variance, one may also compute the dispersion in relation to the population means.
we need to first find the population mean [tex]u[/tex]
[tex]u = (2 + 4 + 12)/3 = 6[/tex]
To calculate the population variance
[tex]= [(-4)^{2} + (-2)^{2} + 6^{2} ]/3[/tex]
[tex]= (16 + 4 + 36)/3[/tex]
[tex]= 56/3[/tex]
[tex]= 18.67[/tex] (rounded to two decimal places)
Therefore, the population variance is approximately [tex]18.67[/tex].
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A photographer needs a frame for an 5x7 inch picture, such that the total area is 80in^2.Calculate the width of the frame.
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Answer:
The width of the frame is 10 inches.
5x7 = 35
80-35 = 45
45/7 = 6.4
5 + 6.4 = 11.4
11.4 rounded down = 11
11 - 5 = 10
Width of the frame = 10 inches
Need help please!
Thank you so much!!
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The graph repeats itself every 32 units hοrizοntally. [ The graph is attached belοw ]
What is an interactive widget in graphs:An interactive widget in a graph is a tοοl that allοws the user tο manipulate different elements οf the graph using their mοuse οr tοuch screen.
Interactive widgets can be used tο explοre the behaviοr οf a functiοn οr dataset in a mοre dynamic and engaging way than a static graph.
Sοme cοmmοn examples οf interactive widgets in graphs include sliders, buttοns, checkbοxes, and drοpdοwn menus.
Here we have
y = -5 cοs (π/16 x) - 4
Represents a periοdic functiοn with a cοsine wave shape that is shifted dοwnwards by 4 units.
The cοsine functiοn οscillates between -1 and 1, sο multiplying it by -5 stretches the wave vertically by a factοr οf 5 and reflects it acrοss the x-axis.
The cοefficient οf x, π/16, cοntrοls the periοd οf the functiοn, which is the distance between twο cοnsecutive peaks οr trοughs οf the wave.
Specifically, the periοd οf this functiοn is 32 units, since the cοsine functiοn has a periοd οf 2π and 2π/(π/16) = 32.
Therefοre,
The graph repeats itself every 32 units hοrizοntally. [ The graph is attached belοw ]
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A spinner with 10 equally sized slices has 5 red slices, 3 yellow slices, and 2 blue slices. Ann spun the dial 25 times. It landed on red 12 times, landed on yellow 10 times, and landed on blue 3 times. From Ann's results, compute the experimental probability of landing on blue or yellow
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Answer:
0.600 or 600, 0.500 or 500, select option 2
Step-by-step explanation:
Please help I-readyyyyyy
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10/3= 3.4
I think but I’m not sure as there are no coordinates
You can find slope on your graph by picking two points and finding rise (y axis) over run (x axis)
Here we see a rise up of +1 and a run to the right of + 3.
y over x = 1/3 slope
We can also pick two coordinates and use slope formula to find slope.
Coordinates
(0, 2) and (3, 3)
y 2 - y1 over x2 over x1
3 - 2 over 3 - 0 = 1/3 slope
I attached a graph to help
I need help with this
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Angle PLE is measured at 90°. The options supplied do not contain the solution.
What measurement of an angle does not equal 90?The acute angle is defined as being less than 90 degrees. Right angles are 90 degrees in length. Angles that are obtuse have a greater angle than 90 degrees. Discover the various sorts of angles and examples of each.
As complementary angles, ZPLA and ZELA, we can infer that:
ZPLA + ZELA = 90°
Using the following expressions in place of ZPLA and ZELA, we obtain:
5x-2 + x+8 = 90
When we simplify the equation, we obtain:
6x + 6 = 90
6 is subtracted from both sides to yield:
6x = 84
Dividing by 6, we get:
x = 14
With the knowledge of x, we can determine the dimensions of ZPLA and ZELA:
ZPLA = 5x-2 = 5(14)-2 = 68°
ZELA = x+8 = 14+8 = 22°
Therefore, we can find the measure of angle PLE by subtracting the measures of ZPLA and ZELA from 180°:
PLE = 180 - ZPLA - ZELA
PLE = 180 - 68 - 22
PLE = 90
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Find g. Write your answer as a whole number or a decimal. Do not round.
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The value of length of side g using the similar triangles is found as 20 ft.
Explain about the similar triangles?Triangles that are similar to one another in terms of shape, angle measurements, and proportion are said to be similar.If the single difference between two triangles is their size and perhaps the requirement to rotate or flip one of them, then they are similar.In the given figures:
DC || EA
So,
∠D = ∠A
∠C = ∠E
By Angle -Angle similarity both triangles are similar.
Thus,
Taking the ratios of their side, it will be also equal.
EA / DC = EB / BC
5 / 10 = g / 10
g = 10*10 / 5
g = 100 / 5
g = 20
Thus, the value of length of side g using the similar triangles is found as 20 ft.
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Macy of New York sold LeeCo. of Chicago office equipment with a $6,300 list price. Sale terms were 3/10, n/30 FOB New York. Macy agreed to prepay the $40 freight. LeeCo. pays the invoice within the discount period. What does LeeCo. pay Macy?
Answers
The amount that LeeCo pays Macy for the office equipment at the $6,300 list price, sales terms of 3/10, n/30 FOB with payment made within the discount window, is $6,111.
What is a cash discount?A cash discount refers to a reduction in the price of an item due to payment within the discount period.
A cash discount incentivizes the customer to make prompt payments.
The list price of the equipment = $6,300
Sales terms: 3/10, n/30 FOB
Prepaid freight = $40
Cash discount = $189 ($6,300 x 3%)
Payment after the discount = $6,111 ($6,300 - $189)
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properties of the rectangle, rhombus, and square - practice determine if the following statements answers
Answers
1. The diagonals are equal. Rectangle
2. All sides are equal, and one angle is 60°. Rhombus
3. All sides are equal, and one angle is 90°. Square
4. It has all the properties of parallelogram, rectangle, and rhombus. Square
5. It is an equilateral parallelogram. Rhombus
A rectangle is a four-sided figure with two sets of parallel sides, with each side being a different length. The opposite sides of a rectangle are always equal in length, so the angles of a rectangle are all 90 degrees. A rectangle can also be referred to as a quadrilateral.
A rhombus is a four-sided figure with all sides the same length. The angles of a rhombus are not all 90 degrees, but the opposite sides of a rhombus are equal in length. A rhombus can also be referred to as a diamond.
A square is a four-sided figure with all sides being the same length and all angles being 90 degrees. A square can also be referred to as a regular quadrilateral.
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The complete question is:
Identify whether the following statements describe a rectangle, rhombus or square.
1. The diagonals are equal. ____________
2. All sides are equal, and one angle is 60°. ____________
3. All sides are equal, and one angle is 90°. ____________
4. It has all the properties of parallelogram, rectangle, and rhombus. ____________
5. It is an equilateral parallelogram. ____________
In a school district, 57% favor a charted school for grades K to 5. A random sample of 300 are surveyed and proportion of those who favor charter school is found. Let it be X. What is the probability that less than 50% will favor the charter school? Assume central limit theorem conditions apply.
Answers
To find the probability that less than 50% will favor the charter school, we need to standardize the sample proportion and use the standard normal distribution.
z = (X - µ) / σ
z = (0.50 - 0.57) / 0.035
z = -2.00
Using a standard normal distribution table or calculator, we can find that the probability of getting a z-score less than -2.00 is approximately 0.0228.
Therefore, the probability that less than 50% will favor the charter school is approximately 0.0228 or 2.28%.
Imagine we have a simple linear model, with one X predicting one Y, where R-squared is equal to .81. What was the correlation between X and Y?A) .81 (or maybe -.81)B) There is not enough information to tell.C) .66 (or maybe -.66)D) .90 (or maybe -.90)
Answers
The correlation between X and Y can be calculated using the formula r = SQRT(R-squared).The correlation coefficient is a measure of the strength of the linear relationship between two variables and can range from -1 to 1
In this case, the R-squared value is 0.81, so the correlation between X and Y is r = SQRT(0.81) = 0.9 (or -0.9 depending on the direction of the relationship).The correlation between X and Y can be calculated using the formula r = SQRT(R-squared). The correlation coefficient is a measure of the strength of the linear relationship between two variables and can range from -1 to 1, where -1 is a perfectly negative linear relationship, 0 is no linear relationship, and 1 is a perfectly positive linear relationship. In this case, the correlation between X and Y was 0.9, indicating a strong linear relationship between the two variables.
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For which of the following conditions is it not appropriate to assume that the sampling distribution of the sample mean is approximately normal? А. A random sample of 8 taken from a normally distributed population B. A random sample of 50 taken from a normally distributed population C. A random sample of 10 taken from a population dintribution that is skewed to the right D. A random sample of 75 taken from a population distribution that is skewed to the left E. A random sample of 100 taken from a population that is uniform
Answers
The conditions in which is it not appropriate to assume that the sampling distribution of the sample mean is approximately normal : (C) A random sample of 10 taken from a population distribution that is skewed to the right.
In statistics, the normal or Gaussian distribution is a continuous probability distribution for real-valued random variables.
The normal distribution is important in statistics and is often used in the natural and social sciences to represent real-valued random variables whose distribution is unknown. Their importance is partly due to the central limit theorem. It states that in some cases the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable - whose distribution converges to a normal distribution as the size of the l sample increases.
Now,
If we look at the options given below, we see that the random samples in options A and B are normally distributed, so their sample means will be approximately normally distributed.
Similarly, option E indicates that the population is uniform, so the sample mean will also be approximately normal.
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Students at a virtual school are allowed to sign up for one math class each year. The numbers of students signing up for various math classes for the next school year are given in the following table:
Grade Geometry Algebra I Pre-Calculus AP Statistics Total
A student calculated the joint relative trequency of 10th grade students in geometry as being 71.4%. What did the student actually calculate and what is the correct answer?
A) The student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20.7%.
B) The student calculated the conditional relative frequenc for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 29%.
C) The student calculated the marginal relative frequency for 10th grade students in geometry. The correct value of the joint relative trequency of 10th grade students in geometry is 20.7%.
D) The student calculated the marginal relative frequency for 10th grade students in geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 29%
Answers
Option C) The student calculated the marginal relative frequency for 10th grade students in geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20.7%.
How to calculate a relative frequency?Relative frequency is a measure of the proportion or percentage of times an event occurs in a given sample. It is calculated by dividing the frequency of an event by the total number of events in the sample. The formula for relative frequency is given as follows:
Relative frequency = Frequency of an event / Total number of events in the sample
For the joint relative frequency of 10th grade students in geometry, the parameters are given as follows:
Frequency of 10th grade students in geometry: 20.7%.Total number of events in the sample: 100% of students.Hence the joint relative frequency is obtained as follows:
20.7/100 x 100%= 20.7%.
Hence option C is correct.
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hellpppppppppppp. need it for math and am just confused
Answers
Step-by-step explanation:
this is solution which is given above in photo
PLEASE ANSWER THIS QUESTION, 20 POINTS!!
Answers
Answer:
∠1 = 50
∠2 = 50
∠3 = 80
∠4 = 130
∠5 = 130
Step-by-step explanation:
∠1 = 180 - 130 = 50
∠2 = ∠1 = 50
∠3 = 180 - ∠1 - ∠2 = 180 - 50 - 50 = 80
∠4 = 180 - ∠2 = 180 - 50 = 130
∠5 = ∠4 = 130
what is the largest integer $n$ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$?
Answers
the largest integer [tex]n $ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$ is $\boxed{62}$.[/tex]
To find the largest integer[tex]n $ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$[/tex], we need to count how many factors of 3 are in the product of the odd integers from 1 to 99.
One way to do this is to factor each odd integer into its prime factors and count how many factors of 3 are present. However, this would be quite tedious and time-consuming.
A quicker approach is to use the fact that every third odd integer is a multiple of 3. Thus, we can count how many multiples of 3 are present in the product of the odd integers from 1 to 99.
Let [tex]$m$[/tex] be the number of multiples of 3 in the range from 1 to 99. Then we have:
[tex]m = \left\lfloor \frac{99}{3} \right\rfloor = 33[/tex]
This is because there are 33 multiples of 3 in the range from 1 to 99 (namely, 3, 6, 9, ..., 96, 99).
Each multiple of 3 contributes at least one factor of 3 to the product of the odd integers. However, some multiples of 3 contribute two or more factors of 3, depending on how many factors of 3 they contain.
To count how many multiples of 3 contribute two or more factors of 3, we need to count how many multiples of 9, 27, and 81 are present in the range from 1 to 99.
There are [tex]$\left\lfloor \frac{99}{9} \right\rfloor = 11$[/tex]multiples of 9, namely 9, 18, 27, ..., 81, 90, 99. Each multiple of 9 contributes at least two factors of 3 to the product of the odd integers.
There are [tex]$\left\lfloor \frac{99}{27} \right\rfloor = 3$[/tex] multiples of 27, namely 27, 54, 81. Each multiple of 27 contributes at least three factors of 3 to the product of the odd integers.
There is only one multiple of 81 in the range from 1 to 99, namely 81, which contributes at least four factors of 3 to the product of the odd integers.
Thus, the total number of factors of 3 in the product of the odd integers from 1 to 99 is:
[tex]n = m + 2\times\text{number of multiples of 9} + 3\times\text{number of multiples of 27} + 4\times\text{number of multiples of 81}[/tex]
[tex]n = 33 + 2\times 11 + 3\times 3 + 4\times 1 = 62[/tex]
Therefore, [tex]the $ largest integer $n$ such that $3^n$ is a factor of $1 \times 3 \times 5 \times \dots \times 97 \times 99$ is $\boxed{62}$.[/tex]
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a. show that the properties of a probability distribution for a discrete random variable are satisfied.
Answers
To show that the properties of a probability distribution for a discrete random variable are satisfied, we need to verify that the probability distribution function (PDF) satisfies, The PDF must be non-negative for all values of the random variable and The sum of the probabilities for all possible values of the random variable must be equal to 1.
Let X be a discrete random variable taking values x1, x2, ..., xn, and let P(X) be the probability distribution function for X, such that P(X = xi) = pi for all i.
The PDF must be non-negative for all values of the random variable. This means that for any value xi of the random variable, the probability pi must be greater than or equal to zero. That is,
pi ≥ 0 for all i.
This property ensures that probabilities are never negative, which is a necessary condition for a valid probability distribution.
The sum of the probabilities for all possible values of the random variable must be equal to 1. That is,
∑ pi = 1 for all i.
This property ensures that the total probability of all possible outcomes is equal to 1, which is a necessary condition for a valid probability distribution.
Therefore, if the PDF satisfies these two properties, we can conclude that it represents a valid probability distribution for the given discrete random variable.
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here are the factors of sixteen and twenty. click the factors that are common to both numbers. (choose 3)
Answers
The common factors of sixteen and twenty are 1, 2, and 4.
The factors of sixteen are 1, 2, 4, 8, and 16. The factors of twenty are 1, 2, 4, 5, 10, and 20. The factors that are common to both numbers are 1, 2, and 4.
To calculate the factors of sixteen, we can start by dividing sixteen by two until we cannot divide any further. We can start with sixteen divided by two, which equals eight. Eight divided by two equals four. Four divided by two equals two. Two divided by two equals one. As we can see, the factors of sixteen are 1, 2, 4, 8, and 16
To calculate the factors of twenty, we can start by dividing twenty by two until we cannot divide any further. We can start with twenty divided by two, which equals ten. Ten divided by two equals five. Five divided by two equals two. Two divided by two equals one. As we can see, the factors of twenty are 1, 2, 4, 5, 10, and 20.The common factors of sixteen and twenty are 1, 2, and 4. This can be determined by comparing the two lists of factors. As we can see, 1, 2, and 4 are present in both lists.
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Complete question
What are the common factors of sixteen and twenty ?
