Answer: MUb / Pb ∠ MUd / Pd
Step-by-step explanation:
First let us define the parameters given;
we have that :
Marginal utility of from book MUb = 44
Marginal utility of from DVD MUd = 1212
Cost of DVD Pd = $ 66
Cost of Book Pb = $ 33
Using,
⇒ MUb / Pb = 44 / 33 = 1.33
⇒ MUd / Pd = 1212 / 66 = 18.36
From this we can see that;
MUb / Pb ∠ MUd / Pd
The question asked is if he is maximizing his utility?
From the law of equi-marginal utility, a consumer will maximize utility only when the utility derived from every unit of spending on goods and services is the same. That is to say that for this to happen,
MUb / Pb = MUd /Pd.
But from the question solved, this is not true, so from the answer gotten The consumer is not maximizing his utility because;
MUb / Pb ∠ MUd / Pd
cheers i hope this helps.
An urn contains n black balls and n white balls. Three balls are chosen from the urn at random and without replacement. What is the value of n if the probability is 1/12 that all three balls are white
Answer:
The value of n is 5.
Step-by-step explanation:
It is provided that in there are n black and n white balls in an urn.
Three balls are selected at random without replacement.
And if all the three balls are white the probability is [tex]\frac{1}{12}[/tex]
The probability of selecting 3 white balls without replacement is:
[tex]P(3\ white\ balls)=\frac{n}{2n}\times \frac{n-1}{2n-1}\times \frac{n-2}{2n-2} \\\frac{1}{12}=\frac{n(n-1)(n-2)}{2n(2n-1)(2n-2)} \\\frac{1}{12}=\frac{(n-1)(n-2)}{2(2n-1)(2n-2)} \\2(4n^{2}-6n+2)=12(n^{2}-3n+2)\\8n^{2}-12n+4=12n^{2}-36n+24\\4n^{2}-24n+20=0\\n^{2}-6n+5=0\\[/tex]
Solve the resultant equation using factorization as follows:
[tex]n^{2}-6n+5=0\\n^{2}-5n-n+5=0\\n(n-5)-1(n+5)=0\\(n-1)(n-5)=0[/tex]
So the value of n is either 1 or 5.
Since 3 white balls are selected the value of n cannot be 1.
So the value of n is 5.
Check:
[tex]P(3\ white\ balls)=\frac{n}{2n}\times \frac{n-1}{2n-1}\times \frac{n-2}{2n-2} \\=\frac{5}{2\times5}\times \frac{5-1}{(2\times5)-1}\times \frac{5-2}{(2\times5)-2}\\=\frac{1}{2}\times\frac{4}{9}\times\frac{3}{8} \\=\frac{1}{12}[/tex]
Thus, the value of n is 5.
A fast food restaurant processes on average 5000 pounds of hamburger per week. The observed inventory level of raw meat, over a long period of time, averages 2500 pounds. What is the average time spent by a pound of meat in production (in weeks)
Answer:
0.5 week
Step-by-step explanation:
The time spent by a pound of meat in this system is given by the average number of pounds of meat on inventory (2500 pounds) divided by the average weekly meat consumption (5000 pounds per week).
The time, in weeks, is:
[tex]t=\frac{2500\ pounds}{5000\ pounds/week} \\t= 0.5\ week[/tex]
The average time spent by a pound of meat in production is 0.5 week.
Suppose we have a test for a certain disease. If a person has the disease,the test has 95% chance to be positive. If a person is healthy, the test has 5% chanceto give a false positive result. We assume that a person has 10% chance to have thedisease. Given that the test for a person is positive, what is the probability that theperson has the disease?
Answer:
P ( disease | positive ) = 0.68
Step-by-step explanation:
This is a classic question of Bayes' Theorem, which calculates probability in a scenario where one event has already occured.
In this case, we have the following data:
P ( disease ) = 0.10
P (no disease) = 0.90
P ( positive | disease ) = 0.95
P ( positive | no disease) = 0.05
Formula to use:
[tex]P ( disease\ |\ positive ) = \frac{P(disease)\ P(positive\ |\ disease)}{P(disease)\ P(positive\ |\ disease) + P(no\ disease)\ P(positive\ |\ no\ disease)}[/tex]We substitute the values from our data in this formula:
[tex]\frac{(0.10) \times (0.95)}{(0.10)\times(0.95) + (0.90)\times(0.05)} \\\\\frac{0.095}{0.095+0.045}\\\\\frac{0.095}{0.14}\\ \\\\\=0.68[/tex] (Rounded off to two decimal places)
Hence, the probability of a person testing positive once they have the disease is 68%.
seven people are seated in a row. They all got up and sit down again in random order. What is the probability that the two originally seated at the two end are no longer at the ends
Answer:
P(A&B) = 0.4
Explanation:
Because it is a random process and there are no special constraints the probability for everybody is the same, the probability of choosing a particular site is 1/7, the person originally seated in chair number seven has 5/7 chance of not seating in chair number six and seven, the same goes for the person originally seated in chair number six; Because we want the probability of the two events happening, we want the probability of the intersection of the two events, and because the selection of a chair change the probability for the others (Dependents events) the probability P(A&B) = P(A) * P(B/A) where P(A) is 5/7 and the probability of choosing the right chair after the event A is 4/7, therefore, P(A&B) = 4/7*5/7 = 0.4.
If the events were independent the probability would be 0.51.
Andy is always looking for ways to make money fast. Lately, he has been trying to make money by gambling. Here is the game he is considering playing: The game costs $2 to play. He draws a card from a deck. If he gets a number card (2-10), he wins nothing. For any face card ( jack, queen or king), he wins $3. For any ace, he wins $5, and he wins an extra $20 if he draws the ace of clubs.
Create a probability model and find Andy's expected profit per game.
Final answer:
Andy's expected profit per game is $1.73 when playing the described gambling game.
Explanation:
To find Andy's expected profit per game, we need to calculate the probability and corresponding profit for each possible outcome. Let's create a probability model:
- Andy draws a number card (2-10): This has a probability of 36/52 since there are 36 number cards in a standard deck of 52 cards. The profit for this outcome is $0.
- Andy draws a face card: This has a probability of 12/52 since there are 12 face cards in a standard deck. The profit for this outcome is $3 if the coin lands on heads, $2 if it lands on tails.
- Andy draws an ace: This has a probability of 4/52 since there are 4 aces in a standard deck. The profit for this outcome is $5, except if he draws the ace of clubs, in which case the profit is $25.
To find the expected profit, we multiply each probability by the corresponding profit and sum them up:
(36/52) * $0 + (12/52) * (($3 * 1/2) + ($2 * 1/2)) + (4/52) * (($5 * 3/4) + ($25 * 1/4)) = $0 + $0.69 + $1.04 = $1.73
Therefore, the expected profit per game for Andy is $1.73.
Research suggests that children who eat hot breakfast at home perform better at school. Many argue that not only hot breakfast but also parental care of children before they go to school has an impact on children’s performance. In this case, parental care is moderating variable.TrueFalse
The given statement is true, and the further discussion can be defined as follows:
Moderating variable:The moderating variable is the variable that can alter the relationship between independent and dependent variables by increasing, weakening, canceling, or otherwise altering it. The direction of this association can be influenced by moderator factors.According to research, kids who have a good breakfast at home perform better in school. Several say that not only a hot breakfast but also parental care of children before they leave for school affects their performance. Parental involvement is a mitigating factor in this scenario.That's why the given statement is "true".
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Final answer:
True. Parental care is considered a moderating variable that can influence the relationship between eating a hot breakfast and a child's school performance.
Explanation:
Research indicates that children who eat a hot breakfast at home tend to perform better in school. However, this outcome is not solely based on the meal itself, but also on the parental care provided before the child goes to school. In this context, parental care acts as a moderating variable, potentially influencing the relationship between having a hot breakfast and a child's school performance. A moderating variable is one that affects the strength or direction of the relationship between an independent variable (hot breakfast) and a dependent variable (school performance). Therefore, the statement is True.
On a map of Texas, the
distance between Houston
and Austin is 2 3/4 inches. The
scale on the map is
1 inch = 50 miles. What is
the actual distance between
Houston and Austin? will mark brainest can u show ur work if not the answer is ok ty please help me been on this a hour
1 inch on the map = 50 miles on the Earth.
A certain trip on the map is 2-3/4 inches.
-- first inch = 50 miles
-- second inch = another 50 miles
-- 3/4 inch = 3/4 of 50 miles (37.5 miles)
Total:
Here's an equation;
1 map-inch = 50 real-miles
Multiply each side by 2-3/4 :
2-3/4 inches = (2-3/4) x (50 miles)
2-3/4 map-inches = 137.5 real-miles
The actual distance between Houston and Austin is 137.5 miles.
We are given that;
The distance between Houston and Austin = 2 3/4 inches
Now,
To find the actual distance between Houston and Austin, we need to multiply the map distance by the scale factor.
The map distance is 2 3/4 inches, which is equivalent to 11/4 inches. The scale factor is 1 inch = 50 miles, which means that every inch on the map corresponds to 50 miles in reality. So, we have:
11/4 x 50 = (11 x 50) / 4
= 550 / 4
= 137.5
Therefore, by unit conversion answer will be 137.5 miles.
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According to records in a large hospital, the birth weights of newborns have a symmetric and bell-shaped relative frequency distribution with mean 6.8 pounds and standard deviation 0.5 Approximately what proportion of babies are born with birth weight under 6.3 pounds?
Answer:
15.9% of babies are born with birth weight under 6.3 pounds.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 6.8 pounds
Standard Deviation, σ = 0.5
We are given that the distribution of birth weights is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(birth weight under 6.3 pounds)
P(x < 6.3)
[tex]P( x < 6.3) = P( z < \displaystyle\frac{6.3 - 6.8}{0.5}) = P(z < -1)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < -1) = 0.159 = 15.9\%[/tex]
15.9% of babies are born with birth weight under 6.3 pounds.
To find the proportion of babies born with a weight less than 6.3 pounds, we calculate the Z-score which results in -1. This Z-score corresponds to about 16% of the population in a standard normal distribution. Therefore, roughly 16% of babies are born weighing less than 6.3 pounds.
Explanation:To solve this problem, you can use the properties of a normal distribution. In a normal distribution, scores fall within one standard deviation of the mean approximately 68% of the time, within two standard deviations about 95% of the time, and within three standard deviations about 99.7% of the time.
In this scenario, we would find the Z-score, a measure that describes a value's relationship to the mean of a group of values. The formula for the Z-score is (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
With a mean (μ) = 6.8 pounds, standard deviation (σ) = 0.5 pounds, and X = 6.3 pounds, the calculation for the Z-score would be (6.3-6.8)/0.5 = -1. This means that 6.3 pounds is one standard deviation below the mean. Referring to the standard normal distribution table, a Z-score of -1 corresponds to approximately 16% or 0.16 of the population. Therefore, approximately 16% of babies are born with a weight of less than 6.3 pounds.
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ests for tuberculosis like all other diagnostic tests are not perfect. QFT-G is one of such tests for tuberculosis. Suppose that for the population of adults that is taking the test, 5% have tuberculosis. The test correctly identifies 74.6% of the time adults with a tuberculosis and correctly identifies those without tuberculosis 76.53% of the time. Suppose that POS stands for the test gives a positive result and S means that the adult really has tuberculosis. Represent the "76.53%" using notation. Group of answer choices P(S) P(Sc) P(POSc | Sc) P( POSc | C)
The 76.53% percentage, which represents the rate at which the QFT-G test correctly identifies those without tuberculosis, can be represented using notation as P(POSc | Sc). This is a conditional probability noting the likelihood of a negative test result when the individual does not have tuberculosis.
Explanation:In the context of probabilities and statistics, you've asked about the interpretation of the 76.53% correctly identified as non-tuberculosis afflicted individuals in terms of notation. Based on the notation you provided and the description of the problem, the 76.53% would be represented as P(POSc | Sc).
This can literally be translated as the probability that the QFT-G test will result as negative (i.e., no tuberculosis, or POSc), given that the person is indeed not afflicted with tuberculosis (i.e., Sc). This is a conditional probability, expressing how likely we are to get a negative test result, given that the person doesn't really have tuberculosis.
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A concrete beam may fail either by shear (S) or flexure (F). Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let X = the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of X.
The Possible outcomes and associated values in the sample space along with X are SSS (X = 3), SSF, SFS, FSS, SFF (X = 1), FSF, FFS,
FFF (X = 0).
We have,
Let's list all the possible outcomes in the sample space when three concrete beams are selected and their failure types (shear or flexure) are determined.
For each outcome, we'll also determine the value of X, which represents the number of beams that failed by shear.
Let S represent shear failure and F represent flexure failure.
Possible outcomes and associated values of X:
SSS (All three beams failed by shear)
X = 3
SSF (Two beams failed by shear, one by flexure)
X = 2
SFS (Two beams failed by shear, one by flexure)
X = 2
FSS (Two beams failed by shear, one by flexure)
X = 2
SFF (One beam failed by shear, two by flexure)
X = 1
FSF (One beam failed by shear, two by flexure)
X = 1
FFS (One beam failed by shear, two by flexure)
X = 1
FFF (All three beams failed by flexure)
X = 0
These are the eight possible outcomes in the sample space along with the associated values of X, representing the number of beams that failed by shear in each outcome.
Thus,
Possible outcomes and associated values of X: SSS (X = 3), SSF, SFS, FSS, SFF (X = 1), FSF, FFS, FFF (X = 0).
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The outcomes in the sample space for three concrete beams failing are: (S,S,S) with X = 3, (S,S,F),(S,F,S),(F,S,S) with X = 2, (S,F,F),(F,F,S),(F,S,F) with X = 1, and (F,F,F) with X = 0. S represents a shear failure, F a flexure failure, and X the number of shear failures.
Explanation:The sample space for this problem includes all possible outcomes for the three concrete beams that can fail. The possible outcomes are:
(S,S,S) for 3 shear failures with X = 3.(S,S,F),(S,F,S),(F,S,S) for 2 shear failures with X = 2.(S,F,F),(F,F,S),(F,S,F) for 1 shear failure with X = 1.(F,F,F) for no shear failures with X = 0.Here, S represents a shear failure and F represents a flexure failure. The number specified by X in each scenario represents how many beams among the three randomly selected ones failed by shear.
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Find all values of m the for which the function y=emx is a solution of the given differential equation. ( NOTE : If there is more than one value for m write the answers in a comma separated list.) (1) y′′+3y′−4y=0, (2) y′′′+2y′′−3y′=0
Answer:
1) m=[1,4]
2) m=[-3,0,1]
Step-by-step explanation:
for y= e^(m*x) , then
y′=m*e^(m*x)
y′′=m²*e^(m*x)
y′′′=m³*e^(m*x)
thus
1) y′′+3y′−4y=0
m²*e^(m*x) + 3*m*e^(m*x) - 4*e^(m*x) =0
e^(m*x) *(m²+3*m-4) = 0 → m²+3*m-4 =0
m= [-3±√(9-4*1*(-4)] /2 → m₁=-4 , m₂=1
thus m=[1,4]
2) y′′′+2y′′−3y′=0
m³*e^(m*x) + 2*m²*e^(m*x) - 3*m*e^(m*x) =0
e^(m*x) *(m³+2*m²-3m) = 0 → m³+2*m²-3m=0
m³+2*m²-3m= m*(m²+2*m-3)=0
m=0
or
m= [-2±√(4-4*1*(-3)] /2 → m₁=-3 , m₂=1
thus m=[-3,0,1]
In clinIcal study, volunteers are tested for a gene that has been found to increase the risk for a disease. The probability that a person carries the gene is 0.1.
a) what is the probability four or more people will have to be tested before two with the gene are detected?b) How many people are expected to be tested before two with gene are detected?
Answer:
(a) P (X ≥ 4) = 0.972
(b) E (X) = 20
Step-by-step explanation:
Let X = number of people tested to detect the presence of gene in 2.
Then the random variable X follows a Negative binomial distribution with parameters r (number of success) and p probability of success.
The probability distribution function of X is:
[tex]f(x)={x-1\choose r-1}p^{r}(1-p)^{x-r}[/tex]
Given: r = 2 and p = 0.1
(a)
Compute the probability that four or more people will have to be tested before two with the gene are detected as follows:
P (X ≥ 4) = 1 - P (X = 3) - P (X = 2)
[tex]=1-[{3-1\choose 2-1}(0.1)^{2}(1-0.1)^{3-2}]-[{2-1\choose 2-1}(0.1)^{2}(1-0.1)^{2-2}]\\=1-0.018-0.01\\=0.972[/tex]
Thus, the probability that four or more people will have to be tested before two with the gene are detected is 0.972.
(b)
The expected value of a negative binomial random variable X is:
[tex]E(X)=\frac{r}{p}[/tex]
The expected number of people to be tested before two with gene are detected is:
[tex]E(X)=\frac{r}{p}=\frac{2}{0.1}=20[/tex]
Thus, the expected number of people to be tested before two with gene are detected is 20.
The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwisestate that there is no solution. Use x1, x2, x3 as variables.
Answer:
The augmented matrix has been given in the attachment
Step-by-step explanation:
The steps for the determination of INCONSISTENCY are as shown in the attachment.
In a West Texas school district the school year began on August 1 and lasted until May 31. On August 1 a Soft Drink company installed soda machines in the school cafeteria. It found that after t months the machines generated income at a rate of f(t) = 300t/2t2 + 8 dollars per month. Find the total income, $Tscc, produced during the second semester beginning on January 1.
Answer:
$95.78
Step-by-step explanation:
f(t) = 300t / (2t² + 8)
t = 0 corresponds to the beginning of August. t = 1 corresponds to the end of August. t = 2 corresponds to the end of September. So on and so forth. So the second semester is from t = 5 to t = 10.
$T₂ = ∫₅¹⁰ 300t / (2t² + 8) dt
$T₂ = ∫₅¹⁰ 150t / (t² + 4) dt
$T₂ = 75 ∫₅¹⁰ 2t / (t² + 4) dt
$T₂ = 75 ln(t² + 4) |₅¹⁰
$T₂ = 75 ln(104) − 75 ln(29)
$T₂ ≈ 95.78
Which car traveled the farthest on 1 gallon of gas? SEE THE PICTURE
Answer:
Car A. would travel the farthest
Answer:
Step-by-step explanation:
i need help to show the work
What is the surface area of the figure
240
48
192
Answer:
[tex]240cm^2[/tex]
Step-by-step explanation:
The area of the rectangular face with dimension 8 by 8 is [tex]8*8=64cm^2[/tex]
The area of the rectangular face with dimension 10 by 8 is [tex]10*8=80cm^2[/tex]
The area of the rectangular face with dimension 6 by 8 is [tex]6*8=48cm^2[/tex]
The area of the two rectangular faces is [tex]2*\frac{1}{2}*8*6=48cm^2[/tex]
The total surface area is [tex]64+80+48+48=240cm^2[/tex]
2 video games and 3 DVDs cost $90.00. 1 video game and 2 DVDs cost $49.00. What is the cost of a DVD? What is the cost of a video game?
Answer: the cost of a DVD is $8 and the cost of a video game is $33
Step-by-step explanation:
Let x represent the cost of a video game.
Let y represent the cost of a DVD.
2 video games and 3 DVDs cost $90.00. This is expressed as
2x + 3y = 90 - - - - - - - - - - - 1
1 video game and 2 DVDs cost $49.00. This is expressed as
x + 2y = 49 - - - - - - - - - - - 2
Multiplying equation 1 by 1 and equation 2 by 2, it becomes
2x + 3y = 90 - - - - - - - - - - - -3
2x + 4y = 98 - - - - - - - - - - - - 4
Subtracting equation 4 from equation 3, it becomes
3y - 4y = 90 - 98
- y = - 8
y = 8
Substituting y = 8 into equation 2, it becomes
x + 2 × 8 = 49
x + 16 = 49
x = 49 - 16
x = 33
By solving a system of linear equations, it is determined that the cost of a video game is $33 and the cost of a DVD is $8.
The question involves solving a system of linear equations to determine the cost of a video game and a DVD.
Let's denote the cost of one video game as V and the cost of one DVD as D. Based on the given information, we can set up the following two equations:
2V + 3D = 90V + 2D = 49To solve for D, we can multiply the second equation by 2 and subtract it from the first equation:
2V + 3D - (2V + 4D) = 90 - 98
This simplifies to -D = -8, which means D = $8.
Now that we know the cost of a DVD, we can substitute it back into the second equation:
V + 2(8) = 49
V + 16 = 49
V = 49 - 16
V = $33.
Therefore, the cost of a video game is $33, and the cost of a DVD is $8.
Fifty pro-football rookies were rated on a scale of 1 to 5, based on performance at a training camp as well as on past performance. A ranking of 1 indicated a poor prospect whereas a ranking of 5 indicated an excellent prospect. The following frequency distribution was constructed.
Rating Frequency
1 4
2 10
3 14
4 18
5 4
a-1. How many of the rookies received a rating of 4 or better?
a-2. How many of the rookies received a rating of 2 or worse?
b-1. Construct the corresponding relative frequency distribution. (Round your answers to 2 decimal places.)
b-2. What percent received a rating of 5?
Answer:
(a-1) 22 rookies receiving a rating of 4 or better.
(a-2) 14 rookies received a rating of 2 or worse.
(b-1) Constructed below in explanation.
(b-2) 8% of total rookies received a rating of 5.
Step-by-step explanation:
We are provided the rating of Fifty pro-football rookies on a scale of 1 to 5 based on performance at a training camp as well as on past performance.
The frequency distribution constructed is given below:
Rating Frequency
1 4
2 10 where ranking of 1 indicate a poor prospect whereas
3 14 ranking of 5 indicate an excellent prospect.
4 18
5 4
(a-1) Rookies receiving a rating of 4 or better = Rating of 4 + Rating of 5
So, by seeing the frequency distribution 18 rookies received a rating of
4 and 4 rookies received a rating of 5.
Hence, Rookies receiving a rating of 4 or better = 18 + 4 = 22 rookies.
(a-2) Number of rookies received a rating of 2 or worse = Rating of 2 +
Rating of 1
So, by seeing the frequency distribution 10 rookies received a rating of
2 and 4 rookies received a rating of 1.
Hence, Rookies receiving a rating of 2 or worse = 10 + 4 = 14 rookies.
(b-1) Relative Frequency is calculated as = Each frequency value /
Total Frequency
Rating Frequency(f) Relative Frequency
1 4 4 / 50 = 0.08
2 10 10 / 50 = 0.2
3 14 14 / 50 = 0.28
4 18 18 / 50 = 0.36
5 4 4 / 50 = 0.08
[tex]\sum f[/tex]= 50
Hence, this is the required relative frequency distribution.
(b-2) To calculate what percent received a rating of 5 is given by equation :
x% of 50 = 4 {Here 4 because 4 rookies received rating of 5}
x = [tex]\frac{4*100}{50}[/tex] = 8% .
Therefore, 8% of total rookies received a rating of 5.
22 rookies received a performance rating of 4 or better, and there are 14 rookies that received a rating of 2 or worse. The relative frequency distribution rounded to 2 decimal places for ratings 1, 2, 3, 4, and 5 is 0.08, 0.2, 0.28, 0.36, and 0.08 respectively. Finally, 8% of rookies received a rating of 5.
Explanation:To answer question a-1, we add the frequencies of the ratings 4 and 5 together. So 18 (the number of rookies who received a 4) plus 4 (the number of rookies who received a 5) equals 22. Therefore, 22 rookies received a rating of 4 or better.
For question a-2, we add the frequencies of the ratings 1 and 2 together. So 4 (the number of rookies who received a 1) plus 10 (the number of rookies who received a 2) equals 14. Thus, 14 rookies received a rating of 2 or worse.
Next, for question b-1, we find the relative frequency distribution by dividing each frequency by the total number of players (50). So, the relative frequency for 1 would be 4/50 ≈ 0.08, for 2 it's 10/50=0.2, for 3 it's 14/50 ≈ 0.28, for 4 it's 18/50 = 0.36, and for 5 it's 4/50 ≈ 0.08. These are all rounded to 2 decimal places.
Finally, for question b-2, to find the percent of players who received a rating of 5, we take the frequency of 5 which is 4, and divide it by the total number of rookies, which is 50, then multiply by 100 to convert it to a percentage (4/50 * 100). The answer is 8%.
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Which statements are true about the graph of f(x)=sec(x) ?
Select each correct answer.
The correct statements regarding the secant function are as follows:
The graph has a vertical asymptote at [tex]x = \frac{\pi}{2}[/tex].The graph has a vertical asymptote at [tex]x = 3\frac{\pi}{2}[/tex].The graph will go through point [tex]\left(\frac{\pi}{4}, \sqrt{2}\right)[/tex]The graph will go through point [tex]\left(\frac{\pi}{3}, 2\right)[/tex]What is the secant function?It is one divided by the cosine, that is:
[tex]\sec{x} = \frac{1}{\cos{x}}[/tex]
The vertical asymptotes are when the cosine is 0, that is, [tex]x = k\frac{\pi}{2}, k = 1, 2, ...[/tex]
As for the values of the function:
[tex]\sec{\left(\frac{\pi}{4}\right)} = \frac{1}{\cos{\left(\frac{\pi}{4}\right)}} = \sqrt{2}[/tex]
[tex]\sec{\left(\frac{\pi}{3}\right)} = \frac{1}{\cos{\left(\frac{\pi}{3}\right)}} = 2[/tex]
You can learn more about the secant function at https://brainly.com/question/14961055
The volume of the cone when x = 3 is 18. Which equation can be used to represent the volume of the cone, V(x)?
Answer: the last option is the correct answer.
Step-by-step explanation:
When x = 3, height = 2 × 3 = 6
When x = 3, base = π × 3² = 9π
Volume = 1/3 × 6 × 9π = 18π
Therefore,
The formula for determining the volume of the cone is
Volume = 1/3 × height × area of base
Height is f(x) = 2x
Area of base is g(x) = πx²
Therefore, the equation that can be used to the volume of the cone, V(x) would be
1/3(f.g)(x)
Final answer:
The student needs to find an equation V(x) representing the volume of a cone. Using the volume formula V(x) = kx² and the given volume of 18 when x is 3, the constant k can be calculated. Thus, the equation for the volume is V(x) = 2x².
Explanation:
The student is asking for an equation that represents the volume of a cone, V(x), based on a given volume when x equals 3. The general formula for the volume of a cone is V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. Since the volume is given as a quadratic function of x, and we know that when x equals 3 the volume is 18, a possible representation for the volume as a function of x could be V(x) = kx², where k is a constant that we would solve for using the given volume at x = 3.
To find the constant k, we substitute x with 3 and V(x) with 18 in the equation V(x) = kx² and solve for k. Therefore, the equation becomes 18 = k(3²), which simplifies to 18 = 9k. Solving for k gives us k = 2, so the equation for the volume of the cone as a function of x is V(x) = 2x².
Let X1 and X2 be independent random variables with mean μand variance σ².
Suppose that we have 2 estimators of μ:
θ₁^ = (X1+X2)/2
θ₂^ = (X1+3X2)/4
a) Are both estimators unbiased estimators ofμ?
b) What is the variance of each estimator?
Answer:
a) [tex] E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_1[/tex] is an unbiased estimator of [tex]\mu[/tex]
[tex] E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_2[/tex] is an unbiased estimator of [tex]\mu[/tex]
b) [tex] Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2} [/tex]
[tex] Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8} [/tex]
Step-by-step explanation:
For this case we know that we have two random variables:
[tex] X_1 , X_2[/tex] both with mean [tex]\mu = \mu[/tex] and variance [tex] \sigma^2[/tex]
And we define the following estimators:
[tex] \hat \theta_1 = \frac{X_1 + X_2}{2}[/tex]
[tex] \hat \theta_2 = \frac{X_1 + 3X_2}{4}[/tex]
Part a
In order to see if both estimators are unbiased we need to proof if the expected value of the estimators are equal to the real value of the parameter:
[tex] E(\hat \theta_i) = \mu , i = 1,2 [/tex]
So let's find the expected values for each estimator:
[tex] E(\hat \theta_1) = E(\frac{X_1 +X_2}{2})[/tex]
Using properties of expected value we have this:
[tex] E(\hat \theta_1) =\frac{1}{2} [E(X_1) +E(X_2)]= \frac{1}{2} [\mu + \mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_1[/tex] is an unbiased estimator of [tex]\mu[/tex]
For the second estimator we have:
[tex]E(\hat \theta_2) = E(\frac{X_1 + 3X_2}{4})[/tex]
Using properties of expected value we have this:
[tex] E(\hat \theta_2) =\frac{1}{4} [E(X_1) +3E(X_2)]= \frac{1}{4} [\mu + 3\mu] = \mu[/tex]
So then we conclude that [tex] \hat \theta_2[/tex] is an unbiased estimator of [tex]\mu[/tex]
Part b
For the variance we need to remember this property: If a is a constant and X a random variable then:
[tex] Var(aX) = a^2 Var(X)[/tex]
For the first estimator we have:
[tex] Var(\hat \theta_1) = Var(\frac{X_1 +X_2}{2})[/tex]
[tex] Var(\hat \theta_1) =\frac{1}{4} Var(X_1 +X_2)=\frac{1}{4} [Var(X_1) + Var(X_2) + 2 Cov (X_1 , X_2)] [/tex]
Since both random variables are independent we know that [tex] Cov(X_1, X_2 ) = 0[/tex] so then we have:
[tex] Var(\hat \theta_1) =\frac{1}{4} [\sigma^2 + \sigma^2 ] =\frac{\sigma^2}{2} [/tex]
For the second estimator we have:
[tex] Var(\hat \theta_2) = Var(\frac{X_1 +3X_2}{4})[/tex]
[tex] Var(\hat \theta_2) =\frac{1}{16} Var(X_1 +3X_2)=\frac{1}{4} [Var(X_1) + Var(3X_2) + 2 Cov (X_1 , 3X_2)] [/tex]
Since both random variables are independent we know that [tex] Cov(X_1, X_2 ) = 0[/tex] so then we have:
[tex] Var(\hat \theta_2) =\frac{1}{16} [\sigma^2 + 9\sigma^2 ] =\frac{5\sigma^2}{8} [/tex]
Both θ₁^ and θ₂^ are unbiased estimators of μ. The variance of θ₁^ is σ² / 2, while the variance of θ₂^ is 5σ² / 8.
Let's analyze the provided estimators of the mean (">").
a) Unbiased Estimators
An estimator heta is unbiased if E[θ] = μ. We have:
θ₁^ = (X₁ + X₂) / 2
The expected value of θ₁^ is:
E[θ₁^] = E[(X₁ + X₂) / 2] = (E[X₁] + E[X₂]) / 2 = (μ + μ) / 2 = μ
Thus, θ₁^ is an unbiased estimator of μ.
θ₂^ = (X₁ + 3X₂) / 4
The expected value of θ₂^ is:
E[θ₂^] = E[(X₁ + 3X₂) / 4] = (E[X₁] + 3E[X₂]) / 4 = (μ + 3μ) / 4 = 4μ / 4 = μ
Thus, θ₂^ is also an unbiased estimator of μ.
b) Variance of Each Estimator
The variance of an estimator θ is given by Var(θ). Considering the given variances of X₁ and X₂ (both σ²):
For θ₁^:Var(θ₁^) = Var[(X₁ + X₂) / 2] = (1/2)²Var(X₁) + (1/2)²Var(X₂) = (1/4)σ² + (1/4)σ² = σ² / 2
For θ₂^:Var(θ₂^) = Var[(X₁ + 3X₂) / 4] = (1/4)²Var(X₁) + (3/4)²Var(X₂) = (1/16)σ² + (9/16)σ² = (1/16 + 9/16)σ² = (10/16)σ² = 5σ² / 8
Thus, the variance of θ₁^ is σ² / 2 and the variance of θ₂^ is 5σ² / 8.
The diameters of bearings used in an aircraft landing gear assembly have a standard deviation of ???? = 0.0020 cm. A random sample of 15 bearings has an average diameter of 8.2535 cm. Please (a) test the hypothesis that the mean diameter is 8.2500 cm using a two-sided alternative and ???? = 0.05; (b) find P-value for the test; and (c) construct a 95% two-sided confidence interval on the mean diameter.
Answer:
(a) We conclude after testing that mean diameter is 8.2500 cm.
(b) P-value of test = 2 x 0.0005% = 1 x [tex]10^{-5}[/tex] .
(c) 95% confidence interval on the mean diameter = [8.2525 , 8.2545]
Step-by-step explanation:
We are given with the population standard deviation, [tex]\sigma[/tex] = 0.0020 cm
Sample Mean, Xbar = 8.2535 cm and Sample size, n = 15
(a) Let Null Hypothesis, [tex]H_0[/tex] : Mean Diameter, [tex]\mu[/tex] = 8.2500 cm
Alternate Hypothesis, [tex]H_1[/tex] : Mean Diameter,[tex]\mu[/tex] [tex]\neq[/tex] 8.2500 cm{Given two sided}
So, Test Statistics for testing this hypothesis is given by;
[tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] follows Standard Normal distribution
After putting each value, Test Statistics = [tex]\frac{8.2535-8.2500}{\frac{0.0020}{\sqrt{15} } }[/tex] = 6.778
Now we are given with the level of significance of 5% and at this level of significance our z score has a value of 1.96 as it is two sided alternative.
Since our test statistics does not lie in the rejection region{value smaller than 1.96} as 6.778>1.96 so we have sufficient evidence to accept null hypothesis and conclude that the mean diameter is 8.2500 cm.
(b) P-value is the exact % where test statistics lie.
For calculating P-value , our test statistics has a value of 6.778
So, P(Z > 6.778) = Since in the Z table the highest value for test statistics is given as 4.4172 and our test statistics has value higher than this so we conclude that P - value is smaller than 2 x 0.0005% { Here 2 is multiplied with the % value of 4.4172 because of two sided alternative hypothesis}
Hence P-value of test = 2 x 0.0005% = 1 x [tex]10^{-5}[/tex] .
(c) For constructing Two-sided confidence interval we know that:
Probability(-1.96 < N(0,1) < 1.96) = 0.95 { This indicates that at 5% level of significance our Z score will lie between area of -1.96 to 1.96.
P(-1.96 < [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < 1.96) = 0.95
P([tex]-1.96\frac{\sigma}{\sqrt{n} }[/tex] < [tex]Xbar - \mu[/tex] < [tex]1.96\frac{\sigma}{\sqrt{n} }[/tex] ) = 0.95
P([tex]-Xbar-1.96\frac{\sigma}{\sqrt{n} }[/tex] < [tex]-\mu[/tex] < [tex]1.96\frac{\sigma}{\sqrt{n} }-Xbar[/tex] ) = 0.95
P([tex]Xbar-1.96\frac{\sigma}{\sqrt{n} }[/tex] < [tex]\mu[/tex] < [tex]Xbar+1.96\frac{\sigma}{\sqrt{n} }[/tex]) = 0.95
So, 95% confidence interval for [tex]\mu[/tex] = [[tex]Xbar-1.96\frac{\sigma}{\sqrt{n} }[/tex] , [tex]Xbar+1.96\frac{\sigma}{\sqrt{n} }[/tex]]
= [[tex]8.2535-1.96\frac{0.0020}{\sqrt{15} }[/tex] , [tex]8.2535+1.96\frac{0.0020}{\sqrt{15} }[/tex]]
= [8.2525 , 8.2545]
Here [tex]\mu[/tex] = mean diameter.
Therefore, 95% two-sided confidence interval on the mean diameter
= [8.2525 , 8.2545] .
Answer:
a
b
c
Step-by-step explanation:
decided to purchase a Toyota 4Runner for $25,635. you have promised your daughter that the SUV will be hers when the car is worth $10,000. According to the car dealer, the SUV will depreciate in the value approximately $3,000 a year. Write a linear equation in which y represents the total value of the car and x represents the age of the car.
Ansoogawer:
Step-by-step explanation:
Answer:
y = -3000x + 25,635
Step-by-step explanation:
well if the inital value of the car is $25,635 this means that when
x = 0 y = 25,635
(0 , 25635)
this will be our first point
Now if you tell us that in 1 year depreciate in value $3,000
this means thaw when
x = 1 y = 25,635 - 3,000
x = 1 y = 22,635
(1, 22635)
Now that we have 2 points we can have the equation
First we take the slope as follows
m = (y2 - y1) / (x2 - x1)
m = (22,635 - 25,635) / (1 - 0)
m = -3000 / 1
m = -3000
after calculating the slope we have to replace it in the following formula
(y - y1) = m (x - x1)
y - 25,635 = -3000 ( x - 0)
y - 25,635 = -3000x
y = -3000x + 25,635
Finally we replace the value of y by 10000
10,000 = -3000x + 25,635
10,000 - 25,635 = -3000x
-15,635 = -3000x
-15,635/-3000 = x
5.21167 = x years
These are the years it would take for the value to be 10,000
to know the days we simply multiply by 365
5.21167 * 365 = 1902.26 days
A new extended-life light bulb has an average life of 750 hours, with a standard deviation of 50 hours. If the life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?
Answer:
99.7% of the distribution will be between 600 hours and 900 hours.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 750
Standard deviation = 50
What percent of the distribution will be between 600 hours and 900 hours?
600 = 750 - 3*50
600 is 3 standard deviations below the mean
900 = 750 + 3*50
900 is 3 standard deviations above the mean
By the Empirical Rule, 99.7% of the distribution will be between 600 hours and 900 hours.
A set S of strings of characters is defined recursively by 1. a and b belong to S. 2. If x belongs to S, so does xb. Which of the following strings belong to S? a. a b. ab c. aba d. aaab e. bbbbb
Answer:
a) a
b) ab
e) bbbbb
Step-by-step explanation:
We are given the following in the question:
[tex]a, b \in S[/tex]
[tex]x \in S \Rightarrow xb \in S[/tex]
a) a
It is given that [tex]a \in S[/tex]
b) ab
[tex]\text{If }a \in S\\\Rightarrow ab \in S[/tex]
Thus, ab belongs to S.
c) aba
This does not belong to S because we cannot find x for which xb belongs to S.
d) aaab
This does not belong to S because we cannot find x for which xb belongs to S.
e) bbbbb
[tex]\text{If }b \in S\Rightarrow bb \in S\\\text{If }bb \in S\Rightarrow bbb \in S\\\text{If }bbb \in S\Rightarrow bbbb \in S\\\text{If }bbbb \in S\Rightarrow bbbbb \in S[/tex]
Thus, bbbbb belongs to S.
Find k. HELP ME PLEASE PLEASE
Answer:
8
Step-by-step explanation:
Sin 30 = k/16
k = 16 x sin 30
k = 16 x (0.5) = 8
A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each. What is the probability that both groups will have the same number of men?
Answer:
P(A) = 400/924 = 100/231 or 0.4329
Step-by-step explanation:
The probability that both groups will have the same number of men P(A);
For the two groups to have the same number of men they must include 3 men and 3 women in each group.
P(A) = Number of possible selections of 3 men from 6 and 3 women from 6 into each of the two groups N(S) ÷ total number of possible selections of members into the two groups N(T).
P(A) = N(S)/N(T)
Since order is not important, we will use combination.
N(S) = 6C3 × 6C3 = 20 × 20 = 400
N(T) = 12C6 = 924
P(A) = 400/924 = 100/231 or 0.4329
The probability that both groups will have the same number of men is 0.43.
When dividing the group into two equal-sized groups of 6 each, the total number of ways to do this is represented by the binomial coefficient C(12, 6), which is equal to 924.
Now, to ensure that both groups have the same number of men, we need to consider the ways in which we can choose 3 men from the 6 available men, which is represented by C(6, 3), and similarly, we can choose 3 women from the 6 available women, which is also represented by C(6, 3).
The total number of ways to choose 3 men and 3 women for both groups is C(6, 3) * C(6, 3) = 20 * 20 = 400.
Since there are 2 equally likely outcomes (either both groups have the same number of men or they don't), the probability is 400/924, which simplifies to 0.43.
In summary, the probability that both groups will have the same number of men is 0.43 because there are 400 ways to select 3 men and 3 women for each group out of a total of 924 possible ways to divide the 12 people into two groups.
This results in a 0.43 probability for the desired outcome.
for such more questions on probability
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Ruby has $0.86 worth of pennies and nickels. She has 4 more nickels than pennies. Determine the number of pennies and the number of nickels that Ruby has.
Answer:
15 Nickels, 11 Pennies
Step-by-step explanation:
Simplify your life and take out the decimals
5*N + P = 86
P + 4 = N (4 more nickels than pennies)
By substitution of the second eq into the first: 5*(P+4) + P = 86
5*P + 20 + P = 86
6P = 66
P = 11, so N = 4 + 11 = 15
Answer:Ruby has 11 pennies and 15 nickels.
Step-by-step explanation:
The worth of a penny is 1 cent. Converting to dollars, it becomes
1/100 = $0.01
The worth of a nickel is 5 cents. Converting to dollars, it becomes
5/100 = $0.05
Let x represent the number of pemnies that Ruby has.
Let y represent the number of nickels that Ruby has.
She has 4 more nickels than pennies. This means that
y = x + 4
Ruby has $0.86 worth of pennies and nickels. This means that
0.01x + 0.05y = 0.86 - - - - - - - - - - - 1
Substituting y = x + 4 into equation 1, it becomes
0.01x + 0.05(x + 4) = 0.86
0.01x + 0.05x + 0.2 = 0.86
0.06x = 0.86 - 0.2 = 0.66
x = 0.66/0.06
x = 11
y = x + 4 = 11 + 4
y = 15
A factory makes rectangular sheets of cardboard, each with an area 2 1/2 square feet. Each sheet of cardboard can be cut into smaller pieces of cardboard measuring 1 1/6 square feet. How many smaller pieces of cardboard does each sheet of cardboard provide?
Answer: each sheet of cardboard provides 2 pieces of smaller pieces of cardboard
Step-by-step explanation:
The area of each rectangular sheet of cardboard that the factory makes is 2 1/2 square feet. Converting
2 1/2 square feet to improper fraction, it becomes 5/2 square feet.
Each sheet of cardboard can be cut into smaller pieces of cardboard measuring 1 1/6 square feet. Converting
1 1/6 square feet to improper fraction, it becomes 7/6 square feet.
Therefore, the number of smaller pieces of cardboard that each sheet of cardboard provides is
5/2 ÷ 7/6 = 5/2 × 6/7 = 15/7
= 2 1/7 pieces
answer is 15 smaller pieces Step-by-step explanation:
The Nielsen Media Research Company uses people meters to record the viewing habits of about 5000 households, and today those meters will be used to determine the proportion of households tuned to CBS Evening News.
Answer:
Cross-sectional study.
Step-by-step explanation:
- In a cross-sectional study, data are observed, measured, and collected at one point in time.
- In a prospective (or longitudinal) study, data are collected in the future from
groups sharing common factors.
- In a retrospective (or case-control) study, data are collected from the past by going
back in tirme (through exanmination of records, interviews, arıd so on).
Hope this Helps!!