The total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax is approximately $219.50.
To calculate the total cost, we first need to find the amount of sales tax. We do this by multiplying the cost of the item by the sales tax rate:
$200.00 x 0.0975 = $19.50
Then, we add the sales tax amount to the cost of the item to get the total cost:
$200.00 + $19.50 = $219.50
Therefore, the total cost of the online purchase of $200.00 in Dallas, including the 9.75% sales tax, is $219.50.
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Complete Question:
If the City of Dallas has a 9.75% sales tax on all online purchases, what is the total cost when you buy an item online that costs $200.00?
Why is it important to learn math? Give three reasons
Answer: here
Step-by-step explanation:
Math is used in various careers such as construction. You will need to know the dimensions to make sure whatever it is being constructed is stable.
Math can help you financially. It can help you budget and prevent you from going into debt.
Math can help you understand real world problems better even if you may not use everything that you learn.
Answer:
Mathematics provides an effective way of building mental discipline and encourages logical reasoning and mental rigor. In addition, mathematical knowledge plays a crucial role in understanding the contents of other school subjects such as science, social studies, and even music and art. It gives us a way to understand patterns, to quantify relationships, and to predict the future. Math helps us understand the world — and we use the world to understand math. The world is interconnected. Everyday math shows these connections and possibilities.
Step-by-step explanation:
1.we use mathematics for our daily living
2.from day care of school we started our mathematics lesson
3.mathematics is the most important specially when your in business 1.our daily living needs simple mathematics to calculate. example; I buy a goods amounted 100 pesos then I have 500 pesos in my pocket so i am sure 400 pesos is my change.
2.From the very beginning or the early stage of our education we have to start learning mathematics so the as we grow older math is in our brain system.
3.In business we know that math is very important because we calculate the capital, expenses, equals the profit.
Joseph has a bag filled with 2 red, 4 green, 10 yellow, and 9 purple marbles. Determine P(not yellow) when choosing one marble from the bag.
8%
24%
40%
60%
Answer:
The correct is 60%
Step-by-step explanation:
The total number of marbles in the bag is:
2 + 4 + 10 + 9 = 25
The probability of selecting a yellow marble from the bag is:
P(yellow) = 10/25 = 2/5
Therefore, the probability of not selecting a yellow marble (i.e., selecting a red, green, or purple marble) is:
P(not yellow) = 1 - P(yellow) = 1 - 2/5 = 3/5 = 60%
Therefore, the answer is 60%.
PLEASE DO 7-10 I WILL GIVE BRAINLEST!!!!!
Part A: Create your own experiment with 5 or more possible outcomes. (2 points)
Part B: Create the sample space for your experiment in Part A. Explain how you determined the sample space. (2 points)
Experiment: "Favorite Cake Flavor"
Hypothesis: Different people have different favorite cakes flavors.
Now, Data collection enables us to identify the cakes flavors both popular and unpopular amongst users.
Furthermore, data analysis based on factors such as age, gender, and location allows for determining if disparities exist between consumer flavor preferences.
The findings of this study would be beneficial to cakes manufacturers and retailers, as they could enhance comprehension of preferred buyer choices and eventuate their products accordingly.
Conclusively, improved knowledge of preferred flavor profiles would facilitate refining marketing methodologies to attract a broader target audience.
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Timothy had 30 dollars to spend on 3 gifts. He spent 10 9/10 dollars on gift A and 3 3/5 on gift B. How much money did he have left for gift C?
The amount of money left for gift c is 15 5/10 dollars
How to calculate the amount of money that is left for gift C?Timothy had 30 dollars to spend on three gifts
He spent 10 9/10 dollars on gift A
He spent 3 3/5 dollars on gift B
The amount spent on gift C can be calculated as follows
10 9/10 + 3 3/5
= 109/10 + 18/5
= 545 + 180/50
= 725/50
= 145/10
30 - 145/10
= 30/1 - 145/10
= 300-145/10
= 155/10
= 15 5/10
Hence the amount of Money left for gift C is 15 5/10 dollars
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what expression is equivalent to
1/5(n+45)
The equivalent expression of the fraction 1/{5(n+45)} as,
1/{5(n+45)} = 1/ (5n + 225)
Equivalent expression of 1/{5(n+45)} can be expressed as,
By applying distributive property of addition in the denominator of the given fraction 1/{5(n+45)} , we get
5(n + 45) = 5·n + 5·45
⇒ 5(n + 45) = 5n + 225
The numerator of fraction 1/{5(n+45)} can be written in equivalent expression as,
1 = 1·1 = 1
Thus, we can write the equivalent expression of the fraction 1/{5(n+45)} as,
1/{5(n+45)} = 1/ (5n + 225)
Equivalent expressions are defined as expressions which work in same way even though they look different from each other.
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what is the following formula? group of answer choices the total sum of squares of y the variance of y times z the population correlation coefficient the sample correlation coefficient
The variables and gives a measure of the strength of the linear relationship that is independent of the units in which the variables are measured.
The formula is the population correlation coefficient, denoted by the Greek letter rho (ρ).
The population correlation coefficient is a measure of the strength and direction of the linear relationship between two variables, denoted by X and Y. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 0 indicates no linear relationship, and 1 indicates a perfect positive linear relationship.
The formula for the population correlation coefficient is:
ρ = Cov(X,Y) / (σX * σY
where Cov(X,Y) is the covariance between X and Y, and σX and σY are the standard deviations of X and Y, respectively.
The numerator, Cov(X,Y), measures the degree to which X and Y vary together. It is positive when above-average values of X tend to be associated with above-average values of Y, and negative when above-average values of X tend to be associated with below-average values of Y.
The denominator, σX * σY, is a normalizing factor that puts the covariance on a standardized scale. It represents the spread of X and Y around their respective means.
By dividing the covariance by the product of the standard deviations, the population correlation coefficient removes the scale effects of the variables and gives a measure of the strength of the linear relationship that is independent of the units in which the variables are measured.
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Nine people sit in chairs in a room. In how many ways can three of these people be chosen to stand up?
Number of ways to choose three of these people to stand up is,
⇒ 84
We have to given that;
Nine people sit in chairs in a room.
And, We have to choose three of these people to stand up.
Hence, Number of ways to choose three of these people to stand up is,
⇒ ⁹C₃
⇒ 9! / 3! (9 - 3)!
⇒ 9! / 3! 6!
⇒ 9×8×7 / 3×2
⇒ 84
Thus, Number of ways to choose three of these people to stand up is,
⇒ 84
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Answer: 504
Step-by-step explanation: i took the quiz
find the midpoint of A and B where A has the coordinates (-7,1) and B has coordinates (3,-5)
Answer:
the midpoint would be (-2,-2)
Step-by-step explanation:
Determine any data values that are missing from the table, assuming that the data represent a linear function..
1
2
6
10
a. 6
b. 15
Please select the best answer from the choices provided
OA
OB
c. 16
d. 14
OD
Mark this and return
Save and Exit
Next
Submit
The missing value is 14, and option d is correct.
How to solveConsider the data table:
x y
1 6
2 10
3 __
Data represent a linear function.
To find:
The missing value.
Solution:
Let the missing value be p.
Slope Formula:
m= y2-y1/x2-x1
Data represent a linear function. So, the slope always remains the same.
10-6/2-1 = p -10/3-2
4= p- 10
Adding 10 on both sides, we get
p = 14
Therefore, the missing value is 14, and option d is correct.
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The double-entry principle in the balance of payments How will the following transactions affect the U.S. balance of payments? A Saudi Arabian oil company purchases $1 million worth of U.S. government bonds from a U.S. bank, paying from its account held with the bank. This transaction will appear as a credit in the account and as a debit In the __________ account. financial capital current Arielle, a French tourist, stays at a hotel In San Francisco and pays $400 for it with her debit card issued by a French bank. This transaction will appear as a credit in the _______ account and as a debit in the _________ account. Frank, a U.S. resident, receives yen 500, 000 in dividend payments on shares that he holds in a Japanese company, which are deposited in his account in a Japanese bank. This transaction will appear as a credit in the __________ account and as a debit in the ___________ account.
The double-entry principle in the balance of payments states that every transaction in the balance of payments is recorded as both a credit and a debit.
In the first transaction, the Saudi Arabian oil company's purchase of $1 million worth of U.S. government bonds from a U.S. bank will appear as a credit in the financial capital account and as a debit in the current account.
In the second transaction, Arielle's payment of $400 for her hotel stay in San Francisco using her French bank-issued debit card will appear as a credit in the current account and as a debit in the financial capital account.
In the third transaction, Frank's receipt of yen 500,000 in dividend payments on his shares in a Japanese company, which are deposited in his account in a Japanese bank, will appear as a credit in the financial capital account and as a debit in the current account.
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PLEASEE HELP MEEEEEE
The total surface area of the square pyramid is 585 in²
How to solve an equation?An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.
The area of a figure is the amount of space it occupies in its two dimensional state.
To get the surface area:
Area of base = 15 in * 15 in = 225 in²
Area of one triangle lateral face = (1/2) * 15 in * 12 in = 90 in²
Area of four triangle lateral face = 4 * 90 in² = 360 in²
Total surface area = 225 in² + 360 in² = 585 in²
The total surface area is 585 in²
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If C(q) =6q' +11q+124 is the total cost (in dollars) of producing q units, find the rate of change of the marginal cost with respect to the production level when q=4 units. A. 144 dollars per unit per unit B. 443 dollars per unit per unit C. 299 dollars per unit per unit D. 552 dollars per unit per unit
To find the rate of change of the marginal cost with respect to the production level, we need to take the derivative of the total cost function, C(q), with respect to q.
C'(q) = 6 + 11
C'(q) = 17
This gives us the marginal cost (MC) function:
MC(q) = 17
Therefore, the rate of change of the marginal cost with respect to the production level is constant at 17 dollars per unit.
To find the rate of change of the marginal cost when q=4 units, we simply plug in q=4 into the MC function:
MC(4) = 17
So the answer is not listed among the choices provided. Therefore, the correct answer is not A, B, C, or D.
In summary, the rate of change of the marginal cost with respect to the production level is constant at 17 dollars per unit. When q=4 units, the marginal cost is also 17 dollars per unit.
To find the rate of change of the marginal cost with respect to the production level when q=4 units, given the total cost function C(q) = 6q^3 + 11q + 124, we need to follow these steps:
Step 1: Find the derivative of the total cost function with respect to q. This will give us the marginal cost function.
C'(q) = d(C(q))/dq = d(6q^3 + 11q + 124)/dq
Step 2: Calculate the derivative.
C'(q) = 18q^2 + 11
Step 3: Substitute q=4 into the marginal cost function to find the rate of change at this production level.
C'(4) = 18(4^2) + 11
Step 4: Evaluate the expression.
C'(4) = 18(16) + 11 = 288 + 11 = 299
The rate of change of the marginal cost with respect to the production level when q=4 units is 299 dollars per unit per unit (Option C).
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Help me please will mark brainiest if you explain how
The graph is attached.
Given is a triangle ABC, we need to find its coordinates if it is reflected over y = -x,
The rule of reflection over y = -x is,
(x, y) = (-x, -y)
So,
A = (-5, -4)
B = (1, -4)
C = (-1, -5)
After reflection,
A' = (5, 4)
B' = (-1, 4)
C' = (1, 5)
Hence the points after reflection are A' = (5, 4), B' = (-1, 4) and C' = (1, 5)
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Use the following notations: r = radius of a circle, v = linear velocity, w = angular velocity. Find the missing quantity. Round to the nearest tenth if necessary. V= 1235 m/min, r = 65 m, w = ?
The angular velocity of the circle is approximately 19 m/s.
In order to find the missing quantity, we can use the relationship between linear velocity and angular velocity in a circle. The linear velocity of a point on the edge of a circle is the product of the radius and the angular velocity. This can be expressed as:
v = r * w
where v is the linear velocity, r is the radius, and w is the angular velocity.
To find the value of w, we can rearrange this equation to solve for w:
w = v / r
Substituting the given values of v and r, we get:
w = 1235 m/min / 65 m
w = 19 m/s
It's important to note that units must be consistent when using formulas to solve problems. In this case, we converted the given linear velocity from m/min to m/s before plugging it into the formula. Also, we rounded the answer to the nearest tenth, as instructed, since the given values were rounded to the nearest unit.
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Which event will have a sample space of S = {h, t}?
Flipping a fair, two-sided coin
Rolling a six-sided die
Spinning a spinner with three sections
Choosing a tile from a pair of tiles, one with the letter A and one with the letter B
The event that have a sample space of h and t would be Flipping a fair, two-sided coin. That is option A.
How to determine the sample space for the event?To calculate the possibility of selecting an outcome form an event the formula that should be used is given as follows;
Probability = possible outcome/sample space.
For a coin, the possible sample space = H and T when flipped only once.
Therefore, the event that have a sample space of h and t would be Flipping a fair, two-sided coin.
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A person was driving their car on an interstate highway and a rock was kicked up and cracked their windshield on the passenger side.
The driver wondered if the rock was equally likely to strike any where on the windshield, what the probability was that it would have cracked the windshield in his line of site on the windshield. Determine this probability, provided that the windshield is a rectangle with the dimensions 28 inches by 54 inches and his line of site through the windshield is a rectangle with the dimensions 30 inches by 24 inches.
The probability, provided that the windshield is a rectangle with the dimensions 28 inches by 54 inches and his line of site through the windshield is a rectangle with the dimensions 30 inches by 24 inches is 0.476
How to calculate the probabilityContinuous Probability is used for this information. Probability = Area of line of sight / total area of windshield
Probability = (30*24)/(28*54)
Probability = 0.476
The probability will be 0.476.
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Find the next four terms in the arithmetic sequence 1/4, 3/4, 5/4
Answer:
7/4, 9/4, 11/4, 13/4
Step-by-step explanation:
+ In an arithmetic sequence, to find the pattern you must subtract a term from the term after it to find the common difference.
3/4 subtracted from 5/4 is 2/4
1/4 subtracted from 3/4 is 2/4
+ So the common difference is 2/4 (Aka 1/2, but you want to keep the same denominator)
+ Therefore, between each new term, you add 2/4
1/4, 3/4, 5/4, 7/4, 9/4, 11/4, 13/4... and so on
D'Angelo is reviowing potential sources for his research writing assignment. One of the articles covers his topic, but from a very different point of view, one that he is not reallyconsidoring in his thesis. While the articie.is very interesting, and seems at first glance to apply to his topic, D'Angelo is woried that this article is outside the scope of his ownpaper What aspect of fthe C.RAAP method is D'Angelo using to consider this article?O. PurposeO. RelevanceO. CurencyO. Authority
D'Angelo is reviewing potential sources for his research writing assignment and is considering an article with a different point of view. In this situation, he is using the Relevance aspect of the CRAAP method to determine if the article is appropriate for his paper.
The CRAAP method stands for Currency, Relevance, Authority, Accuracy, and Purpose, and helps evaluate the credibility and reliability of sources.
In this scenario, D'Angelo is reviewing an article that covers his topic, but from a different point of view that he is not considering in his thesis. While the article may be interesting, it may not be directly relevant to his research paper if it does not contribute to his thesis statement. D'Angelo is using the relevance aspect of the CRAAP method to evaluate whether or not to include this article in his research writing assignment. By doing so, he is ensuring that he uses sources that are directly relevant to his thesis and will help support his arguments.
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(Note click on Question to enlarge) It is known that n!=nx(n-1)x(n-2)x...x
1. Find the number of "0" at the end of 2022!
To find the number of zeros at the end of 2022!, we need to count the number of factors of 10 in its prime factorization. Since 10 = 2 x 5, we need to find the number of pairs of 2's and 5's that multiply to make 10.
First, we can count the number of factors of 5 in 2022!. There are 404 multiples of 5, 80 multiples of 25, and 16 multiples of 125. Therefore, there are 404 + 80 + 16 = 500 factors of 5.
Next, we need to count the number of factors of 2 in 2022!. This is equivalent to counting the number of multiples of 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024 that are less than or equal to 2022. We can simplify this by noticing that each power of 2 is a multiple of the previous power of 2. So we only need to count the multiples of 2, 4, and 8, and then multiply by the number of each power of 2 that divides into 2022.
There are 1011 multiples of 2, 505 multiples of 4, and 252 multiples of 8. The highest power of 2 that divides into 2022 is 2^1, so we only need to consider the factors of 2. Therefore, there are 1011 + 505 + 252 = 1768 factors of 2.
Since we need to find the number of pairs of 2's and 5's, the number of zeros at the end of 2022! is equal to the minimum of the number of factors of 2 and the number of factors of 5, which is 500. Therefore, there are 500 zeros at the end of 2022!.
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place the steps for determining the geometry of a covalently bonded species in the correct order. start with the first step at the top of the list.
1. Draw the Lewis structure of the species, including all valence electrons.
2. Use the electron pair repulsion theory (VSEPR theory) to determine the electron pair geometry.
3. Identify the molecular geometry based on the arrangement of atoms around the central atom and Label the bond angles between the atoms.
Here are the steps for determining the geometry of a covalently bonded species in the correct order:
1. Determine the central atom: Identify the central atom in the molecule or ion, which is usually the least electronegative element or the one with the highest bonding capacity.
2. Count the number of electron domains: Calculate the total number of electron domains surrounding the central atom. This includes both bonding and non-bonding electron pairs.
3. Identify the electron domain geometry: Based on the number of electron domains, identify the corresponding electron domain geometry using the VSEPR (Valence Shell Electron Pair Repulsion) theory.
4. Determine the molecular geometry: Considering the positions of the bonded atoms only (ignoring non-bonding electron pairs), identify the molecular geometry of the species.
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The mean life of one make of keyboard is typically 6 years with a standard deviation of 1.3 years. If the manufacturer wishes to replace no more than 5 per cent of keyboards under guarantee, how long should the guarantee be, to the nearest month?
The make of one keyboard guarantee should be for about 58 months, or approximately 5 years.
We can use the normal distribution to model the lifetime of the keyboards. Let X be the random variable representing the lifetime of the keyboards, and let μ = 6 and σ = 1.3 be the mean and standard deviation, respectively.
To find the length of guarantee such that no more than 5% of the keyboards need to be replaced under guarantee, we need to find the value x such that P(X < x) = 0.05. We can standardize the variable by subtracting the mean and dividing by the standard deviation:
Z = (X - μ) / σ
Using the standard normal distribution table or a calculator with a normal distribution function, we find that the z-score corresponding to a cumulative probability of 0.05 is approximately -1.645.
Therefore,
-1.645 = (x - 6) / 1.3
Solving for x, we get:
x = -1.645 * 1.3 + 6 = 4.83
So the length of guarantee should be approximately 4.83 years. Converting this to months, we get:
4.83 years * 12 months/year ≈ 58 months
Therefore, the guarantee should be for about 58 months, or approximately 5 years.
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Cuantos litros requiere para recorrer 120 km?
The number of liters it would take to cover 120 km is 8 liters.
How to find the number of liters ?Looking at the graph that shows the liters consumed per kilometer, or rather the number of kilometers per liter, we see that each liter enables to car to go for 15 km.
This means that if we should want to go 120 km, the number of liters needed would be:
= Distance to cover / Kilometers per liter
Distance to cover = 120 km
Kilometers per liter = 15 km
The liters needed are:
= 120 / 15
= 8 liters
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Find the Taylor polynomial of degree 3 for sin(x), for x near 0:
P3(x) = ?
Approximate sin(x) with P3(x) to simplify the ratio:
(sin x)/x= ?
Using this, conclude the limit:
limx0 (sin x) / x = ?
If anyone helps me, I will reward points
which of the following successfully switches the integer values of the variables x and y? temp may be used as a temporary variable if necessary. x
A web server is software and hardware that uses HTTP (Hypertext Transfer Protocol) and other protocols to respond to client requests made over the World Wide Web. The main job of a web server is to display website content by storing, processing, and delivering web pages to users.
To successfully switch the values of the variables x and y using a temporary variable called "temp," you can follow these steps:
1. Assign the value of x to the temporary variable: temp = x;
2. Assign the value of y to the variable x: x = y;
3. Assign the value stored in the temporary variable (temp) to the variable y: y = temp; By following these steps, the values of the variables x and y will be switched using the temporary variable "temp."The purpose of the break statement is to break out of a loop early. For example, if the following code asks you to input an integer number, x, If x is divisible by 5, the break statement is executed, and this causes the exit from the loop.The break statement ends the loop immediately when it is encountered. Its syntax is broken. The break statement is almost always used with an if-else statement inside the loop.
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Write an essay about"i realized that this was my moment to shine
At some point in our lives, we come across a moment that presents an opportunity to showcase our talents and abilities. This moment, often referred to as "our moment to shine," can be a turning point that propels us to greater heights of success and achievement. For me, such a moment came when I least expected it, and it changed the course of my life.
It was during my senior year of high school when I got the chance to compete in a regional public speaking competition. I had always been interested in public speaking and had participated in a few contests in the past, but this was different. This competition was going to be fierce, with participants from some of the most prestigious schools in the region. The pressure was high, and the stakes were even higher.
As the day of the competition approached, I found myself getting more and more nervous. I had prepared extensively, spending countless hours rehearsing my speech and polishing my delivery. But still, the thought of standing in front of a panel of judges and a large audience was daunting.
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Is the random variable discrete or continuous?
The number of passengers in a passenger vehicle on a highway at rush hour
The random variable is discrete because it can only take on integer values.
You cannot have a fraction or decimal number of passengers in a vehicle. For example, a vehicle can have 1, 2, 3, or 4 passengers, but it cannot have 2.5 passengers. Discrete variables have a countable number of possible values and can be listed and counted. In contrast, continuous variables can take on any value within a range and are not limited to specific values. Examples of continuous variables include time, weight, and height.
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8.25 Tossing a coin 10,000 times! Ô-P = 1.34 1000 n ZE SE n=10,000 times and obtained 5067 heads 5067 a) P 0.5067 p = 0.5 SE = P(1-p) Pvalue = 2P(Z > 1.34) = 2P(Ž <-1.34) = 0.1802 The data do not provide evidence that the coin is biased b) m = 2*SE z' = 1.96 Cl = $m = (0.4969,0.5165)
it is 95% confident that the true proportion of heads in a large number of tosses of this coin lies between 0.4969 and 0.5165.
a) The proportion of heads obtained in the experiment is
[tex]\frac{5067}{10000} = 0.5067[/tex]. The null hypothesis is that the coin is unbiased, which means the probability of getting a head is 0.5. The standard error (SE) of the proportion is given by
[tex]SE = \sqrt{\frac{p(1-p)}{n} } = \sqrt{\frac{(0.5)(0.5)}{10000} } = 0.005[/tex]
The test statistic is [tex]Z=\frac{(P-0.5)}{SE} = \frac{0.5067-0.5}{0.005} = 1.34[/tex]. The one-tailed p-value for this test is P(Z > 1.34) = P(Z < -1.34) = 0.0901 (assuming a significance level of 0.1). Since this p-value is greater than the significance level, we fail to reject the null hypothesis and conclude that the data do not provide evidence that the coin is biased.
b) The margin of error (m) for a 95% confidence interval is given by m = 2SE = 0.01. The critical value for a 95% confidence interval is z= 1.96. Therefore, the 95% confidence interval for the true proportion of heads is given by P +/- mz' = 0.5067 +/- 0.01*1.96 = (0.4969, 0.5165).
In summary, we can say that based on the given information, the data do not provide evidence that the coin is biased and we are 95% confident that the true proportion of heads in a large number of tosses of this coin lies between 0.4969 and 0.5165.
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ILL GIVE BRAINLEIST THIS WAS DUE YESTERDAY!! 5. Use the following information to answer the questions.
.
A survey asked 75 people if they wanted a later school day start time.
.
45 people were students, and the rest were teachers.
.
50 people voted yes for the later start.
• 30 students voted yes for the later start.
.
a) Use this information to complete the frequency table. (5 points: 1 point for
each cell that was not given above)
Students
Teachers
Total
Vote YES for
later start
Vote NO for later
start
Total
b) Use the completed table from Part a. What percentage of the people surveyed
were teachers? (2 points)
From the table shown below, there are 40% of the people from the survey that were teachers.
What is the table?Through the use of given information, the numbers can be used to fill in the frequency table as follows:
Vote YES for later start Vote NO for later start Total
Students 30 15 45
Teachers 20 10 30
Total 50 25 75
b) To be able to know the percentage of people that were surveyed and who were teachers, we have to divide the number of teachers by the total number of people that were said to have been surveyed:
Percentage of teachers
= 30 /75 x 100%
= 40%
Therefore, based on the above, 40% of the people were surveyed and they were teachers.
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Solve the particular solution of xyy' = x2 + y2 , when x=1, y=1. O a. x/y = - Inx + 1 O b. none of these O c. y= x(Inx+1) O d. y/x = y(Inx+1)
The particular solution of the differential equation is given by:
y/x = sqrt(0.7652), or y = x sqrt(0.7652), which is equivalent to option (c).
We can rewrite the given differential equation as:
xyy' = x^2 + y^2
Dividing both sides by y^2, we get:
x/y + y'/y = x/y * (x/y + 1/y')
Letting v = y/x, we have:
y' = v + xv'
Substituting this in the above equation, we get:
x/y + v + xv'/y = x/y * (x/y + 1/(v + xv'/y))
Simplifying this equation, we get:
v' = -(v/x) * [(1+v^2)^(1/2) - v]
This is a separable differential equation. We can separate the variables and integrate:
(1+v^2)^(-1/2) dv = -dx/x
Integrating both sides, we get:
ln|x| - ln|x0| = -arcsin(v)
where x0 is the initial value of x, which is given to be 1. Simplifying this equation, we get:
ln|x| = -arcsin(v) = -arcsin(y/x)
Taking exponentials of both sides, we get:
|x| = e^(-arcsin(y/x))
Squaring both sides, we get:
x^2 = e^(-2arcsin(y/x))
Simplifying this equation, we get:
y/x = sqrt((e^(2arcsin(y/x))-1)/(e^(2arcsin(y/x))+1))
Using the initial condition y = 1 when x = 1, we get:
y/x = sqrt((e^(2arcsin1)-1)/(e^(2arcsin1)+1)) = sqrt((e^(pi/2)-1)/(e^(pi/2)+1))
Simplifying this expression, we get:
y/x = sqrt((e^(pi/2)-1)/(e^(pi/2)+1)) = sqrt((2.71828^(pi/2)-1)/(2.71828^(pi/2)+1)) = sqrt(0.7652)
Therefore, the particular solution of the differential equation is given by:
y/x = sqrt(0.7652), or y = x sqrt(0.7652), which is equivalent to option (c).
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