Exercise 2.2.3: The cardinality of a power set. (a) What is the cardinality of P({1, 2, 3, 4, 5, 6})
Answer:
Cardinality of the power set of the given set = [tex]2^6=64[/tex]
Step-by-step explanation:
Power set is the set of all the possible subsets that can be formed from the given set including the null set and the set itself.
Example set:
{1,2,3}
All the possible subsets of this set:
{}; {1}; {2}; {3}; {1,2,3}; {1,2}; {1,3}; {2,3}
The power set of the above set is written as:
P({ {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} })
Since the no. of elements in the above power set in this example is 8 therefore its cardinality is 8.
Cardinality of the power set of a given set is expressed by a formula: [tex]2^n[/tex]
where n is the cardinality (no. of elements) of the given set whose power set is to be formed for determining cardinality of the power set.
Hence in the given case, we have n = 6.
Matthew participates in a study that is looking at how confident students at SUNY Albany are. The mean score on the scale is 50. The distribution has a standard deviation of 10 and is normally distributed. Matthew scores a 65. What percentage of people could be expected to score the same as Matthew or higher on this scale?
a) 93.32%
b) 6.68%
c) 0.07%
d) 43.32%
Answer:
b) 6.68%
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
The mean score on the scale is 50. The distribution has a standard deviation of 10.
This means that [tex]\mu = 50, \sigma = 10[/tex]
Matthew scores a 65. What percentage of people could be expected to score the same as Matthew or higher on this scale?
The proportion is 1 subtracted by the p-value of Z when X = 65. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{65 - 50}{10}[/tex]
[tex]Z = 1.5[/tex]
[tex]Z = 1.5[/tex] has a p-value of 0.9332.
1 - 0.9332 = 0.0668
0.0668*100% = 6.68%
So the correct answer is given by option b.
A research group at Nike decides to survey NCSU students for their preferences in clothing brands. They divide all students into groups according to the College they belong to (like College of Science, College of Architecture, etc.). Then they take a simple random sample of 50 students from EACH college. What kind of a sample is this
Answer:
Cluster sample
Step-by-step explanation:
i did it before
Two events A and B are _______ if the occurrence of one does not affect the probability of the occurrence of the other.
Answer:
Independent
Step-by-step explanation:
From the word independent, which means being able ot stand alone, that is the absence or presence of one has no impact on the outcome of each phenomenon. Two events A and B are said to be independent, if the occurence of one has no bearing on the probability or chance that B will occur. This means that each event occurs without reliance on the occurence of the other. This is different from mutually exclusive event whereby event A has direct bearing in the probability of the occurence of event B.
Jack and Diane are jogging back and forth along a one-mile path. They started out at 9:00 A.M. from opposite ends of the path. They passed each other in 10 minutes when Diane has gone 1/3 mile. What time will they first meet at one end of the path? You have to assume they keep jogging at the same speeds.
Explain :
Answer:
30 minutes
Step-by-step explanation:
that problem description is imprecise.
I think what is meant here : they each keep jogging at their own same speed.
Diane's speed is 1/3 miles / 10 min.
Jack's speed is 2/3 miles / 10 min.
now, to bring this to regular miles/hour format, we need to find the factor between 10 minutes and an hour (60 minutes) and multiply numerator and denominator (top and bottom of the ratio) by it.
60/10 = 6.
so, we need to multiply both speeds up there by 6/6 to get the miles/hour speeds.
Diane : (1/3 × 6) / hour = 2 miles / hour
Jack : (2/3 × 6) / hour = 4 miles / hour
since Jack is running twice as fast as Diane, she will finish one length in the same time he finishes a round trip (back and forth).
Diane running 1 mile going 2 miles/hour takes her 30 minutes.
Jack running 2 miles (back and forth) going 4 miles/hour will take him also 30 minutes.
so, they will meet at his starting point after 30 minutes.
What are all the values of w such that|-W | = 5?
Hi there!
»»————- ★ ————-««
I believe your answer is:
[tex]w = 5, -5[/tex]
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
Absolute value is simply how far a digit is from zero.The digits '-5' and '5' are 5 away from zero.Therefore:
[tex]w =\pm5[/tex]
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
I need help with this
Answer:
Here,
Angle TUV + Angle NUV=TUV
By substituting the provided or given value ls in the question we obtain,
1+38x+66 Degree= 105x
1+66 Degree= 67x
67 Degree= 67x
1 Degree = x
x = 1 Degree
Therefore
Angle TUN=1+38x=39
Angle NUV=66 Degree
Therefore
1+38x+66 Degree= 105x
=39 Degree+66 Degree= 105 Degree
Therefore
Angle TUV=105 Degree
If ∆ABC is an isosceles triangle and ∆DBE is an equilateral triangle, find each missing
measure.
Answer:
Step-by-step explanation:
The measure for each angle is shown below.
What is Equilateral Triangle?A triangle is said to be equilateral if each of its three sides is the same length. In the well-known Euclidean geometry, an equilateral triangle is also equiangular, meaning that each of its three internal angles is 60 degrees and congruent with the others.
Given:
As, ∆ABC and ∆DBE is an equilateral triangle.
In Equilateral Triangle all the angles are Equal.
So, 4x+ 3= 9x- 7
5x = 50
x= 10
and, <1 = <9 = 4x+ 3= 43
and, <4 = <5 = <6 = 180/ 3= 60
ans, <3 = <8 = 180-60= 120
Also, <2 = < 7 = 180- <1- <3= 17
Learn more about Equilateral Triangle here:
https://brainly.com/question/3461022
#SPJ2
what are the missing numbers ?
A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: (5x+1)(7x−7) where x is measured in meters. Multiply the binomials to find the area of the plot in standard form
Answer:
35x^2 - 28x - 7
Step-by-step explanation:
A box contains 5 orange pencils, 8 yellow pencils, and 4 green pencils.
Two pencils are selected, one at a time, with replacement.
Find the probability that the first pencil is green and the second pencil is yellow.
Express your answer as a decimal, rounded to the nearest hundredth.
Answer:
total pencil = 5 orange pencils + 8 yellow pencils + 4 green pencils
= 17 pencils
P (g n y) = 4/17 + 8/17
= 0.706
Step-by-step explanation:
1. first find the total number of pencils
2. since there is a replacement the demoinator remains the same
3. find the probability of each green and yellow
4. add the two probability
In rural Ireland, a century ago, the students had to form a line. The student at the front of the line would be asked to spell a word. If he spelled it correctly, he was allowed to sit down. If not, he received a whack on the hand with a switch and was sent to the end of the line. Suppose the student could spell correctly 60% of the words in the lesson. What is the probability that the student would be able to sit down before receiving four whacks in the hand?
Answer:
The answer is "0.9102"
Step-by-step explanation:
P(student spell correct) = 0.6
P(student spell incorrect)=1-0.6=0.4
X=the pupil will be allowed to sit down after receiving three slaps on the hand Thus, X would assume Value
X=0 (student sit sans getting whacks)
X=1 (student sit down without receiving 1 whack) (student sit down without receiving 1 whack)
X=2 (student take a seat before receiving 2 whacks)
X=3 (student sit down after receiving 3 punches)
[tex]\to P(X)=0.6 \times 0.4^0 +0.6 \times 0.4 + 0.6 \times 0.4 ^2 + 0.6 \times 0.3^3[/tex]
[tex]=0.6 \times 1+0.24 + 0.6 \times 0.09 + 0.6 \times 0.027\\\\=0.6 +0.24 + 0.054 + 0.0162\\\\=0.9102\\\\[/tex]
Find the area of the irregular figure. Round to the nearest hundredth.
Answer:
16 sq units
Step-by-step explanation:
Which angle is the vertical angle toBEC
Answer:
∠AED
Step-by-step explanation:
Vertical angles are the opposite angles of intersecting lines. ∠BEC and ∠AED are opposite and would therefore also be congruent angles.
Answer:
[tex]\angle BEC=\angle AED [vertical ~angle][/tex]
[tex]\angle AED~vertical~ angle ~to~ \angle BEC[/tex]
[tex]ANSWER:\angle AED[/tex]
-----------------------------
hope it helps...
have a great day!!
i gave someone a Brainliest pls
Answer:
i think the mistake was in step 1
Step-by-step explanation:
Given the functions below, find f(x)+g(x)
CHECK MY ANSWERS PLEASE
The answer is (a)..........
The formula for centripetal acceleration, a, is given by this formula, where v is the velocity of the object and r is the object’s distance from the center of the circular path:
A= V2/R
Solve the formula for r.
Answer:r=v^2/A
Step-by-step explanation: To solve for r means you have to isolate r on one side and put all the other terms on the other. To get r out from under the fraction, multiply both sides by r. This leaves:
A*r=v^2 so to isolate r, divide by A and get:
r=v^2/A.
write 7.263 to 1 decimal place
Answer:
7.3
Step-by-step explanation:
When you round, you look at the number to the right of which you are rounding to.
1 decimal place would be the tenths place.
7.263
So we would look at the 6, in the hundredths place.
6 is larger than 5, so 2 would be bumped up to 3.
7.3.
I hope this helps!
Answer:
7.3
Step-by-step explanation:
rounding up from 7.263 is 7.3
what much is 1/2 - 1/4
Answer:
1/4
Step-by-step explanation:
The answer is 1/4.
1/2 is equivalent to 2/4.
2/4-1/4=1/4
2(2x + 4) + 2(x - 7) = 78. Determine the side lengths of this rectangle.
[tex]2(2x + 4) + 2(x - 7) = 78[/tex]
[tex]4x + 8 + 2x - 14 = 78[/tex]
[tex](4x + 2x) + (8 - 14) = 78[/tex]
[tex]6x - 6 = 78[/tex]
[tex]6x = 78 + 6[/tex]
[tex]6x = 84[/tex]
[tex]x = \frac{84}{6} [/tex]
[tex]x = 14[/tex]
21. Which of the following statements is true?
(1) -18 > -5
(2) -5 > -0.5
(3) -5> 0
(4) -5 > 52
(5) -5 > -18
Answer: (5) -5 > -18
Step-by-step explanation:
The farther the negative number is from 0, the smaller it becomes.Negative numbers will be smaller than positive numbers, since they're smaller than 0.-5 is 5 away from zero, making it larger than -18 since it's closer to 0, while -18 is 18 away from zero.
The list shows the ages of first-year teachers in one school system. What is the mode of the ages? 23, 42, 21, 25, 23, 24, 23, 24, 37, 23, 39, 51, 63, 24, 55
Answer:
La moda es 23
Step-by-step explanation:
23 es el numero que mas se repite es decir la moda
What is the level of measurement for "year of birth"?
Answer:
interval?
Step-by-step explanation:
I'm not sure. I think so....hope its correct :)
is -3 linear pls help
Answer:
Yes it is. Graph will go down as it moves from left to right.
Step-by-step explanation:
find from first principle the derivative of 3x+5/√x
Answer:
[tex]\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}[/tex]
General Formulas and Concepts:
Algebra I
Exponential Rule [Powering]: [tex]\displaystyle (b^m)^n = b^{m \cdot n}[/tex]Exponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex] Exponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Calculus
Derivatives
Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
f(x) = cxⁿ f’(x) = c·nxⁿ⁻¹Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \frac{3x + 5}{\sqrt{x}}[/tex]
Step 2: Differentiate
Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle \frac{3x + 5}{x^\bigg{\frac{1}{2}}}[/tex]Quotient Rule: [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{(x^\bigg{\frac{1}{2}})^2}[/tex]Simplify [Exponential Rule - Powering]: [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})\frac{d}{dx}[3x + 5] - \frac{d}{dx}[x^\bigg{\frac{1}{2}}](3x + 5)}{x}[/tex]Basic Power Rule [Derivative Property - Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx} = \frac{(x^\bigg{\frac{1}{2}})(3x^{1 - 1} + 0) - (\frac{1}{2}x^\bigg{\frac{1}{2} - 1})(3x + 5)}{x}[/tex]Simplify: [tex]\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2}x^\bigg{\frac{-1}{2}})(3x + 5)}{x}[/tex]Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle \frac{d}{dx} = \frac{3x^\bigg{\frac{1}{2}} - (\frac{1}{2x^{\frac{1}{2}}})(3x + 5)}{x}[/tex]Rewrite [Exponential Rule - Root Rewrite]: [tex]\displaystyle \frac{d}{dx} = \frac{3\sqrt{x} - (\frac{1}{2\sqrt{x}})(3x + 5)}{x}[/tex]Simplify [Rationalize]: [tex]\displaystyle \frac{d}{dx} = \frac{3x - 5}{2x^\bigg{\frac{3}{2}}}[/tex]Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
g In the year 2005, the age-adjusted death rate per 100,000 Americans for heart disease was 222.3. In the year 2009, the age-adjusted death rate per 100,000 Americans for heart disease had changed to 213.4. a) Find an exponential model for this data, where t
This question is incomplete, the complete question is;
In the year 2005, the age-adjusted death rate per 100,000 Americans for heart disease was 222.3. In the year 2009, the age-adjusted death rate per 100,000 Americans for heart disease had changed to 213.4.
a) Find an exponential model for this data, where t = 0 corresponds to 1999
Answer:
the exponential model for this data will be; y = 222.3( 0.9898 )^t
Step-by-step explanation:
Given the data in the question;
{ 2005 }, at t = 0, death rate was 222.3
In 2009, { 2009 - 2005 = 4 }, at t = 4 death rate was 213.4
Now, let the exponential equation be;
y = ab^(t)
so at t = 0
222.3 = a × b^(0)
222.3 = a × 1
a = 222.3
at t = 4
213.4 = a × b^(4)
213.4 = 222.3 × b^(4)
b⁴ = 213.4 / 222.3
b = ( 213.4 / 222.3 )^(1/4)
b = 0.9898
y = ab^(t)
Hence, the exponential model for this data will be; y = 222.3( 0.9898 )^t
Suppose a large telephone manufacturer has a problem with excessive customer complaints and consequent returns of the phones for repair or replacement. The manufacturer wants to estimate the magnitude of the problem in order to design a quality control program. How many telephones should be sampled and checked in order to estimate the proportion defective to within 9 percentage points with 89% confidence
Answer:
80 telephones should be sampled
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], 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 [tex]1 - \frac{\alpha}{2}[/tex].
The margin of error is of:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
89% confidence level
So [tex]\alpha = 0.11[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.11}{2} = 0.945[/tex], so [tex]Z = 1.6[/tex].
How many telephones should be sampled and checked in order to estimate the proportion defective to within 9 percentage points with 89% confidence?
n telephones should be sampled, an n is found when M = 0.09. We have no estimate for the proportion, thus we use [tex]\pi = 0.5[/tex]
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
[tex]0.09 = 1.6\sqrt{\frac{0.5*0.5}{n}}[/tex]
[tex]0.09\sqrt{n} = 1.6*0.5[/tex]
[tex]\sqrt{n} = \frac{1.6*0.5}{0.09}[/tex]
[tex](\sqrt{n})^2 = (\frac{1.6*0.5}{0.09})^2[/tex]
[tex]n = 79.01[/tex]
Rounding up(as 79 gives a margin of error slightly above the desired value).
80 telephones should be sampled
A university found that 25% of its students withdraw without completing the introductory statistics course. Assume that 30 students registered for the course.Use Microsoft Excel whenever necessary and answer the following questions:Compute the probability that 2 or fewer will withdraw
Answer:
0.0106 = 1.06% probability that 2 or fewer will withdraw
Step-by-step explanation:
For each student, there are only two possible outcomes. Either they withdraw, or they do not. The probability of an student withdrawing is independent of any other student, which means that the binomial probability distribution is used to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
25% of its students withdraw without completing the introductory statistics course.
This means that [tex]p = 0.25[/tex]
Assume that 30 students registered for the course.
This means that [tex]n = 30[/tex]
Compute the probability that 2 or fewer will withdraw:
This is:
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{30,0}.(0.25)^{0}.(0.75)^{30} = 0.0002[/tex]
[tex]P(X = 1) = C_{30,1}.(0.25)^{1}.(0.75)^{29} = 0.0018[/tex]
[tex]P(X = 2) = C_{30,2}.(0.25)^{2}.(0.75)^{28} = 0.0086[/tex]
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0002 + 0.0018 + 0.0086 = 0.0106[/tex]
0.0106 = 1.06% probability that 2 or fewer will withdraw
Suppose 35.45% of small businesses experience cash flow problems in their first 5 years. A consultant takes a random sample of 530 businesses that have been opened for 5 years or less. What is the probability that between 34.2% and 39.03% of the businesses have experienced cash flow problems?
1) 0.6838
2) 20.3738
3) 0.3162
4) - 11.6695
5) 1.2313
Answer:
1) 0.6838
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
35.45% of small businesses experience cash flow problems in their first 5 years.
This means that [tex]p = 0.3545[/tex]
Sample of 530 businesses
This means that [tex]n = 530[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.3545[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.3545(1-0.3545)}{530}} = 0.0208[/tex]
What is the probability that between 34.2% and 39.03% of the businesses have experienced cash flow problems?
This is the p-value of Z when X = 0.3903 subtracted by the p-value of Z when X = 0.342.
X = 0.3903
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.3903 - 0.3545}{0.0208}[/tex]
[tex]Z = 1.72[/tex]
[tex]Z = 1.72[/tex] has a p-value of 0.9573
X = 0.342
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.342 - 0.3545}{0.0208}[/tex]
[tex]Z = -0.6[/tex]
[tex]Z = -0.6[/tex] has a p-value of 0.27425
0.9573 - 0.2743 = 0.683
With a little bit of rounding, 0.6838, so option 1) is the answer.
The radius of a circle is 6 kilometers. What is the length of a 60° arc.
Answer:
6.284
Step-by-step explanation:
* means multiply
formula is arc length = radius * theta
theta is fancy way of saying
make degrees into radians
you have to make 60 degrees into radians
you do that by
60 * Pi/180 = Pi/3
then you multiply that with the radius
6 * Pi/3 = 2 * Pi = 6.284
