A sample of 719 times bigger would be needed to estimate the mean per capita income at the 95% level of confidence with an error of at most $0.68.
It is given to us that an economist wants to estimate the mean per capita income (in thousands of dollars) for a major city in California and that he believes that the mean income is $21.2, and the standard deviation is known to be $9.3,
Therefore, here we can make out that:
Number of samples required (n) is given by:
n = (Z²α/2) × б²/e²
It is given to us that, α = 0.05
Therefore, from data provided to us we can get to know that the critical values of Z = ± 1.96
б = 9.3
e = 0.68
when we substitute these values in the above equation, we get:
n = 1.96² × 9.3²/0.68² = 719
Therefore, a sample of 719 times bigger would be needed to estimate the mean per capita income at the 95% level of confidence with an error of at most $0.68.
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how many hours will the patient reach a level of ibuprofen of only 10mg
The patient will reach a level of Ibuprofen of only 10 mg between the 6th and 7th hour after taking the 200 mg dose.
What is logarithm?A logarithm is a mathematical function that measures the relationship between two quantities, by expressing one quantity in terms of another. More specifically, a logarithm is the power to which a base must be raised to produce a given number.
According to question:To model the decay of Ibuprofen in the patient's system, we can use the equation:
A = A0 * [tex]e^(-kt)[/tex]
where A is the amount of Ibuprofen in the patient's system after t hours, A0 is the initial amount of Ibuprofen (200 mg in this case), and k is the decay constant.
To find k, we can use the fact that after one hour, the amount of Ibuprofen in the patient's system has decayed to 136 mg:
136 = 200 * [tex]e^(-k)[/tex]
Solving for k, we get k = ln(5/7) ≈ -0.3365.
Now we can use the equation A = 10 mg and solve for t:
10 = 200 * [tex]e^(-0.3365t)[/tex]
Dividing both sides by 200 and taking the natural logarithm of both sides, we get:
ln(0.05) = -0.3365t
Solving for t, we get t ≈ 6.8 hours.
Therefore, the patient will reach a level of Ibuprofen of only 10 mg between the 6th and 7th hour after taking the 200 mg dose. Answer: C.
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the probability that a patient recovers from a delicate heart operation is 0.9. what is the probability that exactly 5 of the next 7 patients having this operation surviv
The probability that exactly 5 of the next 7 patients survive the operation is approximately 0.123, or 12.3%.
To find the probability that exactly 5 out of the next 7 patients survive the operation, we can use the binomial probability formula, which is:
[tex]P(X=k) = C(n, k) * p^k * (1-p)^(n-k)[/tex]
where:
- P(X=k) represents the probability of exactly k successes (survivals) in n trials (operations)
- C(n, k) is the number of combinations of n items taken k at a time
- n is the number of trials (in this case, 7 patients)
- k is the number of successful outcomes (5 patients surviving)
- p is the probability of a successful outcome (0.9 for a patient surviving the operation)
Applying the formula, we have:
[tex]P(X=5) = C(7, 5) * 0.9^5 * (1-0.9)^(7-5)[/tex]
First, let's calculate the number of combinations C(7, 5):
C(7, 5) = 7! / (5! * (7-5)!)
C(7, 5) = 7! / (5! * 2!)
C(7, 5) = (7 * 6 * 5 * 4 * 3 * 2 * 1) / (5 * 4 * 3 * 2 * 1 * 2 * 1)
C(7, 5) = 21
Now, we can plug this into the binomial probability formula:
[tex]P(X=5) = 21 * 0.9^5 * 0.1^2[/tex]
P(X=5) = 21 * 0.59049 * 0.01
P(X=5) ≈ 0.1232019.
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in 2015, the bureau of labor statistics (bls) announced that of all adult americans, 141.13 million were employed, 16.59 million were unemployed, and 39.65 million were not in the labor force. the adult population is how many million? enter a number rounded to two decimal places.
the adult population in millions in 2015 was 197.37 million
The student question is asking about the adult population in millions, given that the Bureau of Labor Statistics announced that 141.13 million were employed, 16.59 million were unemployed, and 39.65 million were not in the labor force in 2015.
To find the adult population, we need to add the number of employed, unemployed, and not in the labor force adults. This can be done as follows:
Adult population = Employed + Unemployed + Not in the labor force
= 141.13 million + 16.59 million + 39.65 million= 197.37 million
Therefore, the adult population in millions in 2015 was 197.37 million (rounded to two decimal places).
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a study measuring the effects of a new diuretic medication records hourly urine output of subjects. this measure represents which level of measurement? group of answer choices ratio interval nominal ordinal
When measuring the effects of a new diuretic medication, hourly urine output represents the ratio level of measurement.
What is ratio level of measurement?
Ratio level of measurement is a type of measurement that features a true zero point, meaning that the value 0 represents the absence of the characteristic being measured.
Some examples of ratio level measurements include weight, height, length, distance, speed, and many others. In this case, the hourly urine output of subjects is measured to determine the effects of a new diuretic medication.
The hourly urine output is a ratio measurement because it has a true zero point, which is the absence of urine output. In other words, if there is no urine output, the value would be 0, which represents the absence of the characteristic being measured.
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what is the shape of the distribution? the distribution would be non-normal. the distribution is approximately normal. the shape cannot be determined.
For two populations for which μ₁ = 31,σ₁ = 2, μ₂ = 27, and σ₂ = 4. The shape of the distribution is approximately normal. So, the correct choice for answer is option (b). The mean of normal distribution is equals to 4.
In statistics, the collected data distribution shape sometimes normal and sometimes non-normal. In testing of hypothesis, to use the test statistics we have to check whether data is normally or not. Here we have, two populations. In first population : mean μ₁ = 31,
standard deviations,σ₁ = 2,
sample size, n₁ = 49
In case of second sample, mean, μ₂ = 27,
standard deviations, σ₂ = 4.
Sample size, n₂ = 59
Since, population standard deviation are known and sample sizes are greater than 30, therefore the shape of the sampling distribution of the difference of sample means is approximately normal. The shape of the distribution is approximately normal. Mean of the sampling distribution of [tex] \bar x_1 - \bar x_2[/tex] are
[tex]\mu_{ \bar x_1 - \bar x_2} = \mu_1 - \mu_2 [/tex]
= 31 - 27 = 4
Hence, Mean of the distribution is 4.
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Complete question:
Consider two populations for which μ₁ = 31,σ₁ = 2, μ₂ = 27, and σ₂ = 4. Suppose that two independent random samples of sizes n₁ = 49 and n₂ = 59 are selected. Describe the approximate sampling distribution of x₁ bar - x₂ bar (center, spread, and shape). What is the shape of the distribution?
a) The distribution would be non-normal.
b) The distribution is approximately normal.
c) The shape cannot be determined.
What is the mean of the distribution?
help acc i can get part A js not part B
part B: In student A's verification the two trigonometric identities are
i) sin²x+cos²x=1 [ Pythagorean trigonometric identity]
ii) cot x= cos x/ sin x [ Ratio trigonometric identities]
tan x= sin x/ cos x
And the identities appeared in step 3 and 6.
What is trigonometric identities?
Trigonometric Identities are being needed whenever trigonometric functions are involved in an expression. Geometrically, these identities involve trigonometric functions for example sine, cosine, tangent, cotangent of one or more angles.
Here first trigonometric identity showed in step 3
1 is replaced by sin²x+cos²x.
Based on Pythagorean trigonometric identities if Ф is an acute angle of a right angled triangle then sin²Ф+ cos²Ф = 1.
Another trigonometric identity used in step 6
cot x= cos x/ sin x
tan x= sin x/ cos x
These are called ratio trigonometric identities.
Hence, In student A's verification the two trigonometric identities are
i) sin²x+cos²x=1 [ Pythagorean trigonometric identity]
ii) cot x= cos x/ sin x [ Ratio trigonometric identities]
tan x= sin x/ cos x
And the identities appeared in step 3 and 6.
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if we recruit patients into a study and measure their medication adherence 2 weeks after being prescribed a new medication and then we measure their medication adherence agai 6 months later. which statistical test will we run?
The statistical test to run in this scenario is a paired t-test.
A paired t-test is appropriate in this situation because it is used to compare the means of two related groups. In this case, the related groups are the same patients measured at two different time points (2 weeks and 6 months after being prescribed a new medication). The paired t-test assumes that the differences between the two time points are normally distributed.
To conduct the test, we would calculate the mean difference between the two time points and the standard deviation of the differences. We would then calculate the t-statistic and compare it to the critical value from the t-distribution with n-1 degrees of freedom. If the calculated t-value exceeds the critical value, we would reject the null hypothesis that there is no difference between the two time points.
Therefore, a paired t-test is the appropriate statistical test to determine whether there is a significant difference in medication adherence between the two time points
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Justify the last two steps of the proof.
Answer:
STAN BTS
Step-by-step explanation:
Hey! It would be extremely helpful if you would help me out on this question - I am very confused!
Answer:
Square: [tex]A = s^{2}[/tex]
Parallelogram: [tex]A = b x h[/tex]
Trapezoid: [tex]A = \frac{1}{2} xh x (base_{1} +base_{2})[/tex]
Triangle: [tex]A = \frac{1}{2} x b x h[/tex]
Answer: Square:
Parallelogram:
Square:
Parallelogram:
Trapezoid:
Triangle:
Trapezoid:
Triangle:
Step-by-step explanation:
A seed sprouted and grew
2
3
of a foot in 3 months. What was its rate of growth?
Simplify your answer and write it as a proper fraction, mixed number, or whole number.
feet per month
Answer:
[tex]x = 4 \frac{1}{2} [/tex]
PLEASE HELPPP I DON’T UNDERSTAND ITT :(
An equation that models the relationship of the variables for Ethan's job and its days of paid vacation is y(x) = 10 + 3x.
The number of paid vacation days that Ethan has had in the eight years since work began is 30 paid vacation days.
How to find the vacation days ?We are given that Ethan starts with 10 paid vacation days and earns an additional 6 days for every two years, up to a maximum of 6 work weeks (30 days). To find an equation that models this relationship, we can set up a piecewise function as follows:
y(x) = 10 + 3x
To find out how many paid vacation days Ethan has earned after 8 years, we can plug in x = 8 into the piecewise function:
Since 8 > 6, we use the second part of the piecewise function:
y(8) = 30
So, Ethan has earned 30 paid vacation days after working for 8 years at the company.
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the hourly wages of certain group of workers in a large manufacturing company are uniformly distributed on the interval images. what percentage of these workers are making over $25 an hour? g
Approximately 58.33% of the workers in the group are making over $25 an hour. The percentage was obtained by finding the area under the uniform distribution curve above the value of $25.
Since the hourly wages of the workers are uniformly distributed on the interval [20, 32], the probability density function is constant on this interval. The area under the probability density function between any two points a and b gives the probability that a randomly chosen worker's wage will be between a and b. Therefore, the probability that a worker's wage is above $25 is given by the area under the probability density function from 25 to 32, which is (32-25)/(32-20) = 7/12 or approximately 58.33%. Thus, approximately 58.33% of the workers are making over $25 an hour.
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a square pyramid has a surface area of 210 square yards. the length of the base is 7 yards. what is the slant height?
Check the picture below.
so the surface area of the pyramid is really just the area of four triangles with a base of 7 and a height of "h" and a 7x7 square, and we also know that that is 210 yd²
[tex]\stackrel{ \textit{\LARGE Areas} }{\stackrel{ \textit{four triangles} }{4\left[\cfrac{1}{2}(\underset{b}{7})(h) \right]}~~ + ~~\stackrel{ square }{(7)(7)}}~~ = ~~210\implies 14h+49=210 \\\\\\ 14h=161\implies h=\cfrac{161}{14}\implies h=11.5[/tex]
Answer:
the square pyramid's slope height is roughly 5.75 yards.
Step-by-step explanation:
The square pyramid's slope height should be "l."
A square pyramid's surface size is determined by:
Surface area equals base area plus (1/2) * base radius plus slope height.
Given that the base's length is 7 yards and the surface size is 210 yards. The base's size is thus:
Base area equals side times side, or 7 times 7 square yards.
As a result, we have:
210 = 49 + (1/2) * 4 * 7 * l
When we simplify the previous solution, we get:
210 - 49 = 28l
161 = 28l
l = 161/28
Consequently, the square pyramid's slope height is roughly 5.75 yards. (rounded to two decimal places).
A manufacturer sells pens for $0.50 each. If the wholesaler adds a markup of 60% and the retailer adds a markup of
120% of the manufacturer's price, what is the final price of the pen?
The final price of the pen is $1.76 after adding the markup percentages by both the wholesaler and the retailer.
What is final price?The final price is the total amount of money that a customer pays to purchase a product or service. It includes the cost of the product or service plus any additional fees or charges, such as taxes or shipping costs.
According to question:The final price of the pen can be found by adding the markup percentages to the original price of $0.50.
Markup by the wholesaler = 60% of $0.50 = $0.30
Price after markup by wholesaler = $0.50 + $0.30 = $0.80
Markup by the retailer = 120% of $0.80 = $0.96
Final price of the pen = $0.80 + $0.96 = $1.76
Therefore, the final price of the pen is $1.76 after adding the markup percentages by both the wholesaler and the retailer.
For example, if a product costs $50 and there is a 10% sales tax, the final price would be $55 (50 + 5). Similarly, if a service costs $100 and there is a $10 delivery fee, the final price would be $110 (100 + 10).
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What is the standard error of M in statistics?
The standard error of the mean (SEM) in statistics is a measure of the variability or dispersion of a sample mean.
What's standard error of meanThe standard error of M in statistics is the standard deviation of the distribution of sample means. It is a measure of the standard distance between the sample means and the actual population mean.
It indicates the degree of sampling variability of a sample mean. It is a very important tool in inferential statistics that allows the determination of the margin of error in a statistical estimation.
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any one know the answersss?
Answer:
A = 3 1/4 + 2 1/4 = 5 2/4 = 5 1/2
B = 2 1/4 + 1 1/2 = 2 1/4 + 1 2/4 = 3 3/4
C = A + B = 5 1/2 + 3 3/4 = 5 2/4 + 3 3/4 =
8 5/4 = 9 1/4
supplementary angles
Solve for x to make A||B. 13x + 14 A 11x + 28 B
For supplementary angles 13x + 14 = A, 11x + 28= B and A is parallel B, A=B
Therefore x=7
How are parallel lines determined?A and B's corresponding angles are congruent if A and B are parallel. As a result, we may equalize the formulas for the respective angles and find x:
13x + 14 = 11x + 28
2x = 14
x = 7
Consequently, we must fix x equal to 7 in order to make A and B parallel. When we insert this value into the formulas for A and B, we obtain:
A = 13(7) + 14 = 105
B = 11(7) + 28 = 105
The corresponding angles of A and B are now congruent, and A and B are equal and parallel.
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7) -56=7(3+x)
Can someone solve this 2 step equation
Answer:
the answer for the equations is
[tex]x = - 11[/tex]
Please help with this question
The missing frequency in the given table is 24.
What is frequency?
To find the missing frequency, we need to use the information given in the histogram and the frequency table.
We can see from the histogram that the missing class interval is 26-28.
The total frequency for all class intervals is the sum of the frequencies in the table:
110 + 36 + 180 + f = 326 + f
We also know that the total frequency is the number of phones recorded by the manufacturer, so we can assume that the total frequency is a whole number.
Since the sum of the frequencies in the table is 326, the missing frequency must be:
f = total frequency - sum of the other frequencies
f = 350 - 326
f = 24
Therefore, the missing frequency is 24.
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vlad the impaler promotes many of his soldiers into generals (don't ask what happened to the past generals). haley thinks the proportion is about 75%, but wants to estimate the proportion better by sampling soldiers until her margin of error for a 82% confidence interval is less than 0.004. how many soldiers should haley sample?
The sampling soldiers until her margin of error for a 82% confidence interval is less than 0.004, soldiers should Haley sample is 12,724 soldiers.
The degree of inaccuracy in survey findings from random sampling is known as the margin of error in statistics. A wider margin of error in statistics denotes a reduced chance of relying on a survey's or poll's findings, meaning that there will be less trust in the results' ability to accurately reflect a community. It is a very important tool for market research since it shows the degree of assurance that should be placed in survey data by the researchers.
A confidence interval is a measure of how unpredictable a given statistic is. It is typically used in conjunction with the margin of error to indicate how confidently a statistician feels that the findings of an online survey or online poll are worthy of being considered representative of the total population.
In a sample with a number n of people surveyed with a probability of a success of , and a confidence level of , we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi (1-\pi)}{n} }[/tex]
In which, z is the z-score that has a p-value of 1+∝/2.
The margin of error is of:
[tex]M=z\sqrt{\frac{\pi (1-\pi)}{n} }[/tex]
Estimate of the proportion of 64%, hence π = 0.64.
94% confidence level
So , ∝ = 0.94 z is the value of Z that has a p-value of , so z = 1.88
Margin of error of less than 0.008 is wanted, hence, we have to find n when M = 0.008.
[tex]M=z\sqrt{\frac{\pi (1-\pi)}{n} }\\0.008 =1.88\sqrt{\frac{0.64(0.36)}{n} } \\0.008\sqrt{n} = 1.88\sqrt{0.64(0.36)} \\n = 12723.84\\[/tex]
Therefore, Haley should sample 12,724 soldiers.
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The members of a consulting firm rent cars from three rental agencies. It is estimated that 0.36 percent of cars come from agency 1, 0.06 percent of cars come from agency 2, and 0.58 percent of cars come from agency 3. It is also estimated that 0.05 percent of cars from agency 1 need a tune-up, 0.02 percent of cars from agency 2 need a tune-up, and 0.02 percent of cars from agency 3 need a tune-up. Answer the following questions, rounding your answers to two decimal places where appropriate.
(a) What is the probability that a rental car delivered to the firm will need a tune-up?
(b) If a rental car delivered to the firm needs a tune-up, what is the probability that it came from agency 2?
The probability that a rental car delivered to the firm needs a tune-up is 0.0000308, or approximately 0.0031%. The probability that a car comes from agency 2 needs a tune-up is 0.0234, or approximately 2.34%.
What is probability?The probability that an event will occur or a claim will be true is measured by probability theory, a branch of mathematics. The probability of an occurrence is a number from 0 as well as 1, where around 0 represents how likely the occurrence would be to occur while 1 represents certainty. A probability is just a numeric illustration of the possibility that a specific occurrence will take place. Probabilities can also be expressed as percentages ranging between 0% to 100% or even from 0 to 1. the ratio of the number of outcomes to the proportion of occurrence in a whole set of equally probable options that lead to a specific occurrence.
given,
(a) Let's denote the event that a car comes from agency i as Ai and the event that a car needs a tune-up as T. We want to find the probability that a rental car delivered to the firm will need a tune-up, which can be written as P(T).
P(T) = P(T|A1)P(A1) + P(T|A2)P(A2) + P(T|A3)P(A3)
Substituting the given probabilities, we get:
P(T) = 0.05% * 0.36% + 0.02% * 0.06% + 0.02% * 0.58%
P(T) = 0.000018 + 0.0000012 + 0.0000116
P(T) = 0.0000308
Therefore, the probability that a rental car delivered to the firm will need a tune-up is 0.0000308, or approximately 0.0031%.
(b) Let's denote the event that a car comes from agency 2 as A2. We want to find the probability that a car comes from agency 2 given that it needs a tune-up, which can be written as P(A2|T).
P(A2|T) = P(T|A2)P(A2) / P(T)
Substituting the given probabilities, we get:
P(A2|T) = 0.02% * 0.06% / 0.0000308
P(A2|T) = 0.00000072 / 0.0000308
P(A2|T) = 0.0234
Therefore, the probability that a rental car delivered to the firm needs a tune-up and came from agency 2 is 0.0234, or approximately 2.34%.
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someone help please snd asap
Answer:
√9797
The result can be shown in multiple forms.
Exact Form:√9797
Decimal Form:9.84885780…
about 70% of the population prefers coca-cola over pepsi. suppose two people are randomly selected.what is the probability that both prefer coca-cola?
70% of the population prefers coca-cola over pepsi, the probability that both prefer coca-cola is 0.49.
Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means.
Probability is a way to gauge how likely something is to happen. Several things are difficult to forecast with absolute confidence. With it, we can only make predictions about the likelihood of an event happening, or how likely it is. Probability can vary from 0 to 1, with 0 being an impossibility and 1 denoting a certainty. For pupils in Class 10, probability is a crucial subject since it teaches all the fundamental ideas of the subject. One is the probability of every event in a sample space.
70% of the population prefers Coca-Cola over Pepsi,
that is 0.7
The probability of the Pepsi is 30% = 0.3
The probability that both prefers coca cola is 0.7 x 0.7 = 0.49.
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GEOMETRY PLS HELP ASAP
Answer:
a. a = 144, b = 67
Step-by-step explanation:
a + 36 = 180 they are consecutive interior angles, so they add up to 180
a = 180 - 36 = 144
b + 113 = 180 they are consecutive interior angles, so they add up to 180
b = 180 - 113 = 67
I NEED HELP ON THIS ASAP!!!
1. The perimeter is approximately 24.16 units.
2. Determining the height of this triangle was different from the previous activities because we had to first find the length of one side of the triangle and then find the corresponding height that intersects that side at a right angle. In previous activities, the height was either given or easily determined from the given information.
Find the measure of ZL in parallelogram JKLM.
M
14x - 5
15x - 12
L
K
The measure of ZL in parallelogram JKLM is 14x - 5.
What do you mean by Parallelogram ?A parallelogram is a quadrilateral with two pairs of parallel sides. That is, opposite sides are parallel and congruent. Parallelograms have several properties that make them useful in geometry and other fields.
In a parallelogram, opposite sides are parallel and congruent. Therefore, we can set up an equation that relates the measures of the opposite sides of parallelogram JKLM:
KL = MJ
We can use this equation to solve for x, and then find the measure of ZL.
Starting with KL, we have:
KL = 15x - 12
Now, for MJ, we need to notice that MJKL is a rectangle because it has a right angle at K. In a rectangle, opposite sides are congruent. Therefore, we have:
MJ = KL = 15x - 12
Now we can use the given expression for MJ to set up an equation for JL in terms of x:
JL = 14x - 5
We can use the fact that JKLM is a parallelogram to set up another equation that relates JL and ZL:
JL = ZL
Putting the two equations together, we get:
JL = JL
14x - 5 = JL = ZL
Therefore, ZL = 14x - 5.
We can substitute the expression we found earlier for JL into this equation to get:
ZL = 14x - 5 = JL = 14x - 5
Simplifying, we get:
ZL = 14x - 5
Therefore, the measure of ZL in parallelogram JKLM is 14x - 5.
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a researcher conducts a study to identify the relationship of lifestyle choices to the development of chronic diseases. the researcher surveys subjects and identifies diabetes mellitus, coronary artery disease, and renal disease in study subjects. these measures represent which level of measurement? group of answer choices interval ordinal nominal ratio
The measures of diabetes mellitus, coronary artery disease, and renal disease in study subjects represent the nominal level of measurement. This is because these measures are categorical in nature and cannot be ranked or ordered.
Nominal level of measurement is the simplest form of measurement which assigns a name or number to an attribute. The attributes can be classified into different categories or classes without any ranking.
Nominal level of measurement is commonly used in research to represent things like gender, race, marital status, religion, and so on. In this case, the attributes being measured are the different chronic diseases. These are categorical and cannot be ranked.
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Find the volume of the rectangular prism.
Answer:
1.5 × 4.5 × 6 = 40.5 cubic meters
the special two-item character sequence that represents special characters like \n is known as a(n) .
The special two-item character sequence that represents special characters like \n is known as an Escape Sequence. So the option B is correct.
An escape sequence is a sequence of characters that does not represent itself when used inside a character or string literal, but is translated into another character or a sequence of characters. It is used to signal an alternative interpretation of a series of characters. Common examples are the escape sequences used in C/C++ and other programming languages, such as "\n" for newline, "\t" for tab, and "\\" for a backslash. So the option B is correct.
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The complete question is:
The special two-item character sequence that represents special characters like \n is known as a(n) ?
a. backslash code
b. escape sequence
c. Unicode spec
d. literal character
Pls 50 point for each person will mark brainiest for the best answer
Please show me the three common methods for solving quadratic equations
factoring, using the square roots, completing the square and the quadratic formula.
thanks
Step-by-step explanation:
There are several methods to solve a quadratic equation, but the most common ones are:
Factoring:
This method involves factoring the quadratic expression into two linear expressions and setting each factor equal to zero. For example, to solve the equation x^2 + 5x + 6 = 0, you can factor it as (x + 2)(x + 3) = 0, and then set each factor equal to zero to get x = -2 or x = -3.
Using the square roots:
If the quadratic expression can be written in the form of x^2 = a, where a is a constant, then the solutions can be found by taking the square root of both sides. For example, to solve the equation x^2 = 16, you can take the square root of both sides to get x = ±4.
Completing the square and the quadratic formula:
This method involves completing the square on the quadratic expression to obtain an equivalent expression in the form of (x + b)^2 = c, where b and c are constants. The solutions can then be found by taking the square root of both sides and solving for x. Alternatively, you can use the quadratic formula, which is a general formula that gives the solutions of any quadratic equation in the form of ax^2 + bx + c = 0. The quadratic formula is given by x = (-b ± sqrt(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic expression.
Answer:
see below :)
Step-by-step explanation:
The three common methods for solving quadratic equations are:
Factoring Method:
In this method, the quadratic equation is factored into two binomials, and then the values of x that make each binomial equal to zero are found. This method works only if the quadratic equation can be factored. The factored form of the quadratic equation is:
ax^2 + bx + c = 0
The factors of the quadratic equation are (mx + p) and (nx + q), where m, n, p and q are constants. The factored form of the quadratic equation can be written as:
(mx + p)(nx + q) = 0
The values of x that satisfy this equation are:
x = -p/m or x = -q/n
Using the Square Roots Method:
This method is used when the quadratic equation is in the form of:
x^2 = a
The solutions to this equation can be found by taking the square root of both sides:
x = ± √a
Completing the Square and the Quadratic Formula Method:
This method is used to solve any quadratic equation, even if it cannot be factored. The steps involved in this method are:
Step 1: Rewrite the quadratic equation in the form of:
ax^2 + bx + c = 0
Step 2: Complete the square by adding and subtracting a constant term to the quadratic equation:
ax^2 + bx + c + (b/2a)^2 - (b/2a)^2 = 0
Step 3: Simplify the equation by grouping the first three terms and factoring the perfect square trinomial:
a(x + b/2a)^2 + (c - b^2/4a) = 0
Step 4: Solve for x by taking the square root of both sides and using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
This formula gives two solutions for x, one with the plus sign and one with the minus sign.