Given are the numbers of 30 test scores in a math class. 56, 69, 76, 84, 102, 69, 78, 92, 79, 80, 66, 78, 90, 99, 45, 76, 77, 80, 57, 57, 85, 88, 55, 86, 99, 97, 87, 75, 65, 70 a. Create a histogram to represent this data. (3points) b. Find the mean, median, mode and standard deviation. (3 points) c. Determine the five-number summary and draw a boxplot. (4 points)
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a) Histogram is attached. b) Mean is 75.8, median is 78.5, mode is 57, Standard Deviation is 15.56. c) Five- line summary is Minimum: 45, Q1: 66, Median: 78.5, Q3: 87, Maximum: 102.
a. To create a histogram for the given data, we need to divide the range of scores into intervals (bins) and count the frequency of scores falling within each interval.
Scores Frequency
40-50 |
50-60 |||
60-70 ||||||||
70-80 |||||
80-90 |||||
90-100 ||||||
100-110 |
b) Calculate the mean:
To find the mean, we sum up all the test scores and divide by the total number of scores.
Mean = (56 + 69 + 76 + 84 + 102 + 69 + 78 + 92 + 79 + 80 + 66 + 78 + 90 + 99 + 45 + 76 + 77 + 80 + 57 + 57 + 85 + 88 + 55 + 86 + 99 + 97 + 87 + 75 + 65 + 70) / 30
Mean = 2274 / 30
Mean = 75.8
Therefore, the mean of the test scores is 75.8.
Calculate the median:
To find the median, we need to arrange the scores in ascending order and then determine the middle value.
Arranging the scores in ascending order: 45, 55, 56, 57, 57, 65, 66, 69, 69, 70, 75, 76, 76, 77, 78, 78, 79, 80, 80, 84, 85, 86, 87, 88, 90, 92, 97, 99, 99, 102
Since we have 30 scores, the median will be the average of the 15th and 16th values, which are 78 and 79.
Median = (78 + 79) / 2
Median = 78.5
Therefore, the median of the test scores is 78.5.
Calculate the mode:
The mode is the value(s) that appear(s) most frequently in the data set.
The mode of the given test scores is 57 because it appears twice, which is more frequently than any other value.
Therefore, the mode of the test scores is 57.
Calculate the standard deviation:
The standard deviation measures the spread of the data around the mean.
To calculate the standard deviation, we can use the following formula:
Standard Deviation = [tex]\sqrt{(xi-mean)^{2}/n }[/tex]
Where xi represents the sum, xi is each individual score, mean is the mean we calculated earlier, and n is the total number of scores.
Using this formula, we can calculate the standard deviation as follows:
Standard Deviation = [tex]\sqrt{((56-75.8)^{2}+(69-75.8)^{2}........+(70-75.8)^{2})/30}[/tex]
After calculating all the differences and summing them up, we divide by 30 and take the square root.
Standard Deviation ≈ 15.56
Therefore, the standard deviation of the test scores is approximately 15.56.
c) Calculate the five-number summary and draw a boxplot:
The five-number summary consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values of the data set.
Minimum: 45
Q1: 66
Median: 78.5
Q3: 87
Maximum: 102
To draw a boxplot, we use these five values. The box represents the interquartile range (Q3 - Q1), with a line inside representing the median. The whiskers extend to the minimum and maximum values.
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Related Questions
X 1 A probability density function of a random variable is given by f(x) = on the interval [2, 8]. Find the expected value, the variance, 18 9 and the standard deviation. The expected value is u (Roun
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The expected value is `10X/3`, the variance is `20X/27`, and the standard deviation is `[2 sqrt(5X/27)]/3`.
Given: A probability density function of a random variable is given by `f(x) = X/18` on the interval `[2, 8]`.
We have to find the expected value, the variance, and the standard deviation.
So, `f(x) = X/18` on the interval `[2, 8]`.
To find the expected value, we have to use the formula:
`u = int(x*f(x)) dx`.
Here, `int` means the integration of `x*f(x)` over the interval `[2, 8]`.
So, `u = int(x*f(x)) dx
= int(x*X/18) dx` over the interval `[2, 8]`
=`X/18 int(x) dx` over the interval `[2, 8]`
=`X/18 [(x^2)/2]` over the interval `[2, 8]`
=`X/18 [(8^2 - 2^2)/2]`=`X/18 [60]`
=`10X/3`.
Therefore, the expected value is `10X/3`.
To find the variance, we have to use the formula:
`sigma^2 = int((x-u)^2 * f(x)) dx`.
Here, `int` means the integration of `(x-u)^2 * f(x)` over the interval `[2, 8]`.
So, `sigma^2 = int((x-u)^2 * f(x)) dx
= int((x-(10X/3))^2 * X/18) dx` over the interval `[2, 8]`
=`X/18 int((x-(10X/3))^2) dx` over the interval `[2, 8]`
=`X/18 int(x^2 - (20/3) x + (100/9)) dx` over the interval `[2, 8]`
=`X/18 [(x^3/3) - (10/3) (x^2/2) + (100/9) x]` over the interval `[2, 8]`
=`X/54 [(8^3 - 2^3) - (10/3) (8^2 - 2^2) + (100/9) (8 - 2)]`
=`X/54 [1240]`
=`20X/27`.
Therefore, the variance is `20X/27`.
To find the standard deviation, we have to use the formula: `sigma = sqrt(sigma^2)`.
So, `sigma = sqrt(sigma^2) = sqrt(20X/27) = sqrt[4*5X/27] = [2 sqrt(5X/27)]/3`.
Therefore, the standard deviation is `[2 sqrt(5X/27)]/3`.
Hence, the expected value is `10X/3`, the variance is `20X/27`, and the standard deviation is `[2 sqrt(5X/27)]/3`.
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The central limit theorem is basic to the concept of statistical inference, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population. O True False
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The given statement that states “The central limit theorem is basic to the concept of statistical inference, because it permits us to draw conclusions about the population based strictly on sample data, and without having any knowledge about the distribution of the underlying population” is a true statement.
The central limit theorem plays an important role in the field of statistical inference.Statistical inference is the act of utilizing data from a random sample to deduce the features of an overall population. The foundation of statistical inference lies on the presumption that samples will vary to some extent from each other, whereas the population is considered to be constant.
The Central Limit Theorem (CLT) states that the average of a large number of independent random variables drawn from the same population is distributed around the true population mean and its deviation decreases as the sample size increases. This theorem's importance to the concept of statistical inference cannot be overstated because it allows us to draw conclusions about the population based solely on sample data, without having any information about the underlying population's distribution.
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4. Two people who prepare tax forms are working in a store at a local mall. Each has a chair next to his desk where customers can sit and wait. Customers arrive at rate X = 2 but will go away if there is already someone sitting in the chair waiting. Suppose that the server i requires an exponential amount of time with rate p = i + 2 (i = 1,2) and that when both servers are free an arriving customer is equally likely to choose either one. a) Formulate a Markov chain model (i.e., identify the parameters 's and qij's) for this system with state space (0,1,2,12,3} where the first four states indicate the servers that are busy while the last indicates that there is a total of three customers in the system: one at each server and one waiting. b) Set-up the balance equations (exit rates = entry rates) needed to solve for the proportion of time the Markov chain is in the different states.
Answers
The equations for the sum of probabilities are s0 + s1 + s2 + s12 + s3 = 1
a) Formulation of Markov Chain Model:
A Markov Chain model can be represented in the form of a state transition diagram. The system in question has 5 states with a state space of (0, 1, 2, 12, 3). The first four states (0, 1, 2, 12) indicate the servers that are busy, while the last state (3) indicates that there are three customers in the system: one at each server and one waiting.
The parameters for the model are as follows:
Parameter s: The probability of the system being in state i. It represents the proportion of time the system is in state i.
Parameter qij: The rate of transition from state i to state j. It represents the probability that the system will make a transition from state i to state j per unit time.
State Space (0, 1, 2, 12, 3):
State 0: Both servers are free.
State 1: Server 1 is busy, and the customer is waiting at server 2.
State 2: Server 2 is busy, and the customer is waiting at server 1.
State 12: Both servers are busy with one customer waiting.
State 3: Both servers are busy with no customer waiting.
We can conclude that the qij's will be as mentioned below:
q10 = λ1 = i + 2
q20 = λ2 = i + 2
q31 = μ1 + λ1
q32 = μ2 + λ2
q43 = 2μ1
q34 = 2μ2
q54 = λ1/2
q45 = λ2/2
q52 = μ1 + λ1/2
q53 = μ2 + λ2/2
b) Balance Equations:
For any Markov Chain model, the sum of transition rates leaving a state is equal to the sum of transition rates entering that state. Therefore, we can write the balance equation for each state as follows:
State 0: s0q10 = s1q01
State 1: s1q12 + s1q10 = s0q01 + s1q21
State 2: s2q21 + s2q20 = s0q02 + s2q12
State 12: s1q21 + s2q20 + s12q34 = s12q43 + s12q52
State 3: s12q43 = s3q34
The sum of probabilities must be equal to 1. Thus, we can write the equations for the sum of probabilities as follows:
s0 + s1 + s2 + s12 + s3 = 1
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A company plans to spend $8,000 in year 2 and $10,000 in year 4. At an interest rate of 10% per year, compounded semiannually. The equation that represents the equivalent annual worth A in years I through 4 is: A) A-8,000(P/F, 10%,2) * ( A/P, 10%, 4)+ 10,000(A/F, 10%,4) B) A 8,000(P/F, 10.25%,2) * (A/P, 10%, 4)+ 10,000 (A/F, 10.25%,4) C) A-8,000(P/F,5%,2) * (A/P,5%, 4) + 10,000(A/F,5%,4) D) A-8,000(P/F, 10.25%,4)*(A/P, 10.25%, 8) + 10,000(A/F, 10.25%,8)
E) A-8,000(A/P, 10 %,4) + 10,000(A/F,10%,4)
Answers
Option (B) is the correct answer.
A company plans to spend $8,000 in year 2 and $10,000 in year 4. At an interest rate of 10% per year, compounded semiannually. The equation that represents the equivalent annual worth A in years I through 4 is given by option (B).That is;Option (B) represents the equation that represents the equivalent annual worth A in years I through 4.
The formula for equivalent annual worth is given by;A = PW (A/P, i, n) + F(A/F, i, n)where,PW = Present WorthF = Future WorthA/P = Present Worth FactorA/F = Future Worth Factori = interest raten = number of yearsSo,A = 8000(P/A, 10/2,2) + 10000(F/A, 10/2,4)A = 8000(3.1051) + 10000(0.6848)A = 24,840.80 + 6848A = $31,688.80The equation that represents the equivalent annual worth A in years I through 4 is represented by the option (B);A = 8000(P/A, 10.25/2,2) + 10000(F/A, 10.25/2,4).
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sinus arrest resembles normal sinus rhythm except for one distinguishing characteristic, which is _________________.
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Sinus arrest resembles normal sinus rhythm except for one distinguishing characteristic, which is that there is an absence of P waves in the sinus arrest.
Sinus arrest is a condition where the normal functioning of the sinoatrial (SA) node is interrupted. This causes a delay or pause in the heart's electrical impulses, which prevents the heart from beating for a brief period.
The absence of P waves is caused by the interruption of the electrical activity in the sinoatrial node that generates the impulse for the atria to contract.
As a result, the atria and ventricles can stop beating for several seconds, leading to a temporary loss of consciousness, weakness, dizziness, or fainting. Sinus arrest is caused by several factors, including hypoxia, electrolyte imbalance, medication toxicity, vagal stimulation, and cardiac diseases such as heart block or ischemia.
Treatment for sinus arrest depends on the underlying cause, and it may involve lifestyle changes, medication, pacemaker insertion, or other medical interventions.
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During the Middle Ages, Henry Ill of England made an Assize of Arms in 1252. It obligated all freemen between the ages
of 15 and 60 In England to swear an oath to bear arms in the service of the king. Citizens were ordered to practice archery on
Sundays and holidays. One of the weapons that were used was the longbow. A skilled longbowman could release 10-12 arrows
per minute with a range of about 200 yards. Young boys were used to run additional arrows to the archers. Longbows were a
very beneficial weapon because since they were made of wood they could be easily crafted during times of war. Longbowman
were used to both defend their castles and on the battlefield.
Question 1. The path of the arrow is modeled by the equation h(t) =-16t^2 + 80t + 4 where t is the time in seconds and h(t) is the height of the arrow at time t. What type of vertex (minimum or maximum), would this quadratic function create? Explain, using any method, how you found your answer
Question 2. Find the coordinates of the vertex algebraically
Question 3. Construct an appropriate table of values to model this function. Choose an appropriate domain
T. h(t)
Answers
Maximum vertex due to the negative coefficient of the t^2 term (-16), indicating a downward-opening parabolic graph.
Vertex coordinates: (2.5, 104)
Construct a table with values of t ranging from 0 to 5 seconds to model the function's height.
Question 1: The quadratic function h(t) = [tex]-16t^2 + 80t + 4[/tex] represents the height of the arrow at time t. To determine the type of vertex it creates, we can analyze the coefficient of the t^2 term (-16). Since the coefficient is negative, the parabolic graph opens downward. This means the function has a maximum vertex.
Question 2: To find the coordinates of the vertex algebraically, we can use the formula x = -b / (2a), where a and b are the coefficients of the quadratic function. In this case, a = -16 and b = 80. Plugging these values into the formula
, we get x = [tex]-80 / (2*(-16)) = -80 / (-32) = 2.5.[/tex] To find the corresponding y-coordinate, we substitute x = 2.5 into the function: h(2.5) = [tex]-16*(2.5)^2 + 80*(2.5) + 4 = -100 + 200 + 4 = 104.[/tex] Therefore, the coordinates of the vertex are (2.5, 104).
Question 3: To construct an appropriate table of values, we can choose values of t within a suitable domain.
Since we are dealing with the height of the arrow, a reasonable domain would be the time interval in which the arrow is in flight. Let's consider t values ranging from 0 to 5 seconds. Using these values, we can compute the corresponding h(t) values by substituting them into the function
h(t) = [tex]-16t^2 + 80t + 4.[/tex]
The resulting table would provide a representation of the arrow's height at different points in time, allowing us to analyze its trajectory and behavior.
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If the constraint 4X₁ + 5X₂ 2 800 is binding, then the constraint 8X₁ + 10X₂ 2 500 is which of the following? O binding O infeasible O redundant O limiting
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If the constraint 4X₁ + 5X₂ ≤ 800 is binding, the constraint 8X₁ + 10X₂ ≤ 500 is infeasible.
Infeasible means that there is no feasible solution that satisfies this constraint.
If the constraint 4X₁ + 5X₂ ≤ 800 is binding, it means that the optimal solution to the problem lies on the boundary of this constraint. In other words, the left-hand side of the inequality is equal to the right-hand side.
Now, let's consider the constraint 8X₁ + 10X₂ ≤ 500. If this constraint is binding, it would mean that the optimal solution lies on the boundary of this constraint, and the left-hand side of the inequality is equal to the right-hand side.
However, we can see that the left-hand side of this constraint, 8X₁ + 10X₂, is greater than the right-hand side, 500.
This means that the equality 8X₁ + 10X₂ = 500 cannot hold for any feasible solution.
Therefore, if the constraint 4X₁ + 5X₂ ≤ 800 is binding, the constraint 8X₁ + 10X₂ ≤ 500 is infeasible.
Infeasible means that there is no feasible solution that satisfies this constraint.
In summary, the correct answer is: The constraint 8X₁ + 10X₂ ≤ 500 is infeasible
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Which of the following is NOT a problem modular division is used for? answer choices. o Finding remainder. o Patterns in programs. o Finding mode.
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The modular division or remainder operator or modulus operator is not used for finding the mode.
Modular division, also known as the remainder operator or modulus operator, is a mathematical operation that calculates the remainder when one number is divided by another. It is commonly used in various applications, but it is not directly related to finding the mode.
A. Finding remainder: Modular division is used to find the remainder when dividing one number by another. For example, in the expression 17 % 5, the remainder is 2.
B. Patterns in programs: Modular division is often used to identify patterns in programs. It can be used to determine if a number is even or odd, or to cyclically repeat a sequence of values. It is useful for tasks such as generating repetitive patterns or implementing modular arithmetic.
C. Finding mode: The mode is the value that appears most frequently in a set of data. Finding the mode involves determining the frequency of each value, which is different from the concept of modular division. The mode can be found by counting occurrences or using statistical methods, but it does not involve modular division.
Therefore, finding the mode is not a problem that modular division is typically used for.
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The owner of a moving company wants to predict labor hours, based on the number of cubic feet moved. A total of 34 observations were made. An analysis of variance of these data showed that b1=0.0505 and Sb1=0.0032. a. At the 0.05 level of significance, is there evidence of a linear relationship between the number of cubic feet moved and labor hours? b. Construct a 95% confidence interval estimate of the population slope, β1. Choose one of the following:
-0.044≤β1≤0.0592. There is no evidence of a linear relationship between the number of cubic feet moved and labor hours.
0.0434≤β1≤0.0698. There is no evidence of a linear relationship between the number of cubic feet moved and labor hours.
0.044≤β1≤0.057. There is evidence of a linear relationship between the number of cubic feet moved and labor hours.
-0.0434≤β1≤0.057. There is evidence of a linear relationship between the number of cubic feet moved and labor hours.
Answers
a. there is evidence of a linear relationship between the number of cubic feet moved and labor hours. b. the correct answer is: 0.044 ≤ β1 ≤ 0.057. There is evidence of a linear relationship between the number of cubic feet moved and labor hours.
a. To determine whether there is evidence of a linear relationship between the number of cubic feet moved and labor hours, we need to perform a hypothesis test. The null hypothesis (H0) states that there is no linear relationship, while the alternative hypothesis (H1) states that there is a linear relationship.
H0: β1 = 0 (no linear relationship)
H1: β1 ≠ 0 (linear relationship exists)
To test this, we can use the t-test and compare the calculated t-value to the critical t-value at the desired level of significance. Since the sample size is small (34 observations), we need to use the t-distribution.
The calculated t-value can be obtained using the formula t = b1 / Sb1, where b1 is the estimated slope and Sb1 is the standard error of the slope estimate.
Given that b1 = 0.0505 and Sb1 = 0.0032, we can calculate the t-value:
t = 0.0505 / 0.0032 ≈ 15.7813
At the 0.05 level of significance, the critical t-value for a two-tailed test with (n - 2) degrees of freedom (where n is the sample size) is approximately 2.028.
Since the calculated t-value (15.7813) is much larger than the critical t-value (2.028), we reject the null hypothesis. Therefore, there is evidence of a linear relationship between the number of cubic feet moved and labor hours.
b. To construct a 95% confidence interval estimate of the population slope (β1), we can use the formula:
β1 ± t*(Sb1)
where β1 is the estimated slope, t* is the critical t-value at the desired confidence level, and Sb1 is the standard error of the slope estimate.
At a 95% confidence level, the critical t-value for a two-tailed test with (n - 2) degrees of freedom is approximately 2.032.
Plugging in the values:
β1 ± t*(Sb1) = 0.0505 ± 2.032*(0.0032)
Calculating the confidence interval:
0.0505 ± 2.032*0.0032 = 0.0505 ± 0.0065
Therefore, the 95% confidence interval estimate of the population slope (β1) is:
0.044 ≤ β1 ≤ 0.057
Hence, the correct answer is: 0.044 ≤ β1 ≤ 0.057. There is evidence of a linear relationship between the number of cubic feet moved and labor hours.
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Find the largest degree of x that can be factored out of all the terms.
a. 1
b. 2
c. 3
d. 4
Answers
The largest degree of x that can be factored out of all the terms is 1.
In this problem, we are asked to determine the largest degree of x that can be factored out of all the terms. To solve this, we need to look at the terms and identify the common factors of x. The options provided are 1, 2, 3, and 4.
If we look at the given terms, there is no variable x present in any of them. Therefore, we cannot factor out any powers of x from the terms. In other words, the degree of x in each term is 0. Hence, the largest degree of x that can be factored out of all the terms is 1, as x^1 is equivalent to x.
Factoring is a process in algebra where we break down an expression into its factors. It involves finding common factors and removing them from each term. By factoring, we can simplify expressions and solve equations more easily.
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A population of meerkats grows according to the logistic differential equation dP =-0.002P2 +6P. dt a.) Find lim P(t). Explain the meaning of this value in the context of the problem. t-> b.) What is the population of the meerkats when it is growing the fastest? 4. A termite population grows according to the logistic differential equation dP = KP -0.0001P2. If the carrying capacity is 2000, what is the value of the dt constant k? (A) 0.01 (B) 0.02 (C) 0.1 (D) 0.2
Answers
Given logistic differential equation of population of meerkats, dP/dt = -0.002P^2 + 6P,Let us solve the differential prism equation for dP/dt to find the population of the meerkats when it is growing the fastest:At maximum, dP/dt = 0
Therefore, 0 = -0.002P^2 + 6PPutting 0 on one side,6P = 0.002P^2Divide both sides by P,6 = 0.002PTherefore, P = 3000 (population of meerkats when it is growing the fastest)Now, let us find the limit P(t) as t approaches infinity; that is, when the population stops growinglim P(t) = limit as t approaches infinity of the population P(t)Solving the logistic differential equation for P(t) by separation of variables,We get,∫(1/(K - P) dP) = ∫(-0.002 dt)Solving the integration,log(K - P) = -0.002t + C,where C is the constant of integration.At t = 0, P = P0
Then, C = log(K - P0)Therefore,log(K - P) = -0.002t + log(K - P0)log((K - P)/(K - P0)) = -0.002tTaking the antilog of both sides of the equation,(K - P)/(K - P0) = e^(-0.002t)Therefore, K - P = (K - P0) e^(-0.002t)Solving for P,We get,P = K - (K - P0) e^(-0.002t)As t approaches infinity, e^(-0.002t) approaches 0Hence, P approaches KTherefore, lim P(t) = K = 2000The value of the dt constant k for the logistic differential equation of the termite population dP/dt = KP - 0.0001P^2 with carrying capacity K = 2000 is given by dP/dt = KP - 0.0001P^2Given, K = 2000Also, dP/dt = KP - 0.0001P^2,So, dP/dt = K (1 - 0.0001(P/K)^2) = KP (1 - (P/20,000)^2)Therefore, the value of the constant k is 0.02 (option B).
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suppose that p is the proposition ""it is not snowing."" which of the following propositions would be equivalent to not-p?
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There are various ways to represent the negation of a statement, or proposition p. One possible method is to use the symbol ~. Therefore, the negation of proposition p would be represented as ~p.
Similarly, if the proposition p is defined as "it is not snowing," then the negation of p, or not-p, would be represented as ~p or "it is snowing." This is because the negation of "it is not snowing" is "it is snowing."Thus, the equivalent proposition of not-p is "it is snowing." In summary, the negation of any statement p is a proposition that is the opposite of p. The negation of "it is not snowing" would be "it is snowing."Long explanation:For any statement p, there are different ways to express its negation or opposite. One way is to use the logical negation symbol, which is ~.
If p is the proposition "it is not snowing," then the negation of p or not-p would be represented as ~p or "it is snowing." This is because the negation of a negative statement is a positive statement. For instance, if we negate the statement "I am not happy," the result would be "I am happy." Likewise, if we negate the statement "it is not snowing," we would obtain "it is snowing."Therefore, the equivalent proposition to not-p is "it is snowing." This is because not-p is the negation of p, which means that it is the opposite of p. Since p is "it is not snowing," then not-p would be "it is snowing."
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3. A random sample of 149 scores for a university exam are given in the table. Score, x 0≤x≤ 20 20 < x≤ 40 40 < x≤ 60 60 < x≤ 80 80 < x≤ 100 21 Frequency 14 32 43 39 a. Find the unbiased e
Answers
The unbiased estimate of the population mean is 13.78.The unbiased estimate of the population mean can be found using the formula:
$\overline{x} = \frac{\sum{x}}{n}$,
where $\overline{x}$ is the sample mean,
$\sum{x}$ is the sum of the sample scores, and n is the sample size.
Here, we are given the frequency distribution of the sample scores, so we first need to calculate the midpoint for each class interval.
The midpoint is found by adding the lower and upper bounds of each class interval and dividing by 2.
Using this information, we can construct a table of the frequency distribution with the class midpoints as shown below.
Score, x
FrequencyMidpoint (x)014.5 (0+29)/22114.523.5 (20+39)/234032.5 (40+59)/246039.5 (60+79)/25390.5 (80+99)/2
We can then calculate the sample mean as:$$\overline{x}=\frac{\sum{x}}{n}$$$$=\frac{(14)(14.5)+(32)(23.5)+(43)(32.5)+(39)(39.5)+(21)(90.5)}{149}$$$$=\frac{2051.5}{149}$$$$=13.78$$
Therefore, the unbiased estimate of the population mean is 13.78.
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Let X and Y be two independent random variables. Suppose that X ~ Unif({0, 1,...,n}) and Y~ Ber(p), i.e. P(X= k) = 1 n+1' ke {0, 1,..., n}, and P(Y= 1) = 1 - P(Y= 0) = p. (a) [3 pts] Find the pmf and
Answers
The pmf and joint pmf of X and Y are given by:
P(X = k) = 1 / (n + 1), for k = 0, 1, ..., n
P(Y = 0) = 1 - p
P(Y = 1) = p
P(X = k, Y = 0) = (1 / (n + 1)) * (1 - p), for k = 0, 1, ..., n
P(X = k, Y = 1) = (1 / (n + 1)) * p, for k = 0, 1, ..., n
To find the probability mass function (pmf) and joint pmf of random variables X and Y, we need to consider their individual probability distributions.
Since X follows a discrete uniform distribution over the set {0, 1, ..., n}, the probability of X taking any specific value k is given by:
P(X = k) = 1 / (n + 1), for k = 0, 1, ..., n
On the other hand, Y follows a Bernoulli distribution with parameter p. The pmf of Y is:
P(Y = 0) = 1 - p
P(Y = 1) = p
Now, to find the joint pmf of X and Y, we assume that X and Y are independent random variables. Therefore, their joint pmf is simply the product of their individual pmfs:
P(X = k, Y = 0) = P(X = k) .P(Y = 0) = (1 / (n + 1)). (1 - p), for k = 0, 1, ..., n
P(X = k, Y = 1) = P(X = k) . P(Y = 1) = (1 / (n + 1)) . p, for k = 0, 1, ..., n
Note that for each value of k, we have two possible outcomes for Y (0 or 1) since Y is independent of X.
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help I will upvote
Consider the joint probability distribution given by 1 f(xy) (x + y) .where x = 0,1,2,3 ********* 30 a. Find the following: i. Marginal distribution of X [3 Marks] ii. Expectation of X [3marks] iii. V
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The marginal distribution of X in the given joint probability distribution can be calculated by summing the probabilities over all possible values of X.
To find the marginal distribution of X, we need to sum the joint probabilities for each value of X and Y. Given that X can take the values 0, 1, 2, 3, we can calculate the marginal distribution as follows:
P(X = 0) = f(0Y)(0 + Y) for Y = 0, 1, 2, 3, ..., 30
P(X = 1) = f(1Y)(1 + Y) for Y = 0, 1, 2, 3, ..., 30
P(X = 2) = f(2Y)(2 + Y) for Y = 0, 1, 2, 3, ..., 30
P(X = 3) = f(3Y)(3 + Y) for Y = 0, 1, 2, 3, ..., 30
The marginal distribution of X is a probability distribution that represents the probabilities of each value of X.
To calculate the expectation of X, we multiply each value of X by its corresponding probability and sum them up:
E(X) = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2) + 3 * P(X = 3)
Finally, to calculate the variance of X, we need to subtract the square of the expectation of X from the expectation of the square of X:
Var(X) = E(X²) - (E(X))²
Where E(X²) can be calculated as:
E(X²) = 0² * P(X = 0) + 1² * P(X = 1) + 2² * P(X = 2) + 3² * P(X = 3)
This gives us the variance of X.
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For what value(s) of k will the relation not be a function?
#1: A = {(k^2, 16), (4k, 32)}
#2: B = {(k^2-5k, 10), (k+7, 4)}
#3: R = {(k^3-5k^2+3k, -5), (-k, 4)}
#4: S = {(|k+1|+2, 4), (8, 7)}
Answers
A relation is a function if and only if every element in the domain has exactly one corresponding element in the range.
For a relation to be a function, there should be no two elements in the domain that are mapped to the same element in the range. If this happens, then the relation is not a function.#1: A = {(k², 16), (4k, 32)}For A to be a function, no two elements in the domain should map to the same element in the range. If k² = 4k, then k = 0 or k = 4, which means that (0, 16) and (16, 32) will be distinct elements. A is a function for all values of k.#2: B = {(k²-5k, 10), (k+7, 4)}For B to be a function, no two elements in the domain should map to the same element in the range.
If k² - 5k = k + 7, then k² - 6k - 7 = 0, which means that k = -1 or k = 7. (-1, 10) and (7, 4) are two distinct elements in B. Therefore, B is a function for all values of k.#3: R = {(k³-5k²+3k, -5), (-k, 4)}For R to be a function, no two elements in the domain should map to the same element in the range. If k³ - 5k² + 3k = -k, then k⁴ - 5k³ + 4k² + k = 0, which means that k = -1, k = 0, or k = 1. (-1, 4) and (0, -5) are two distinct elements in R. But, (1, -5) and (-1, 4) map to the same element in the range. Therefore, R is not a function when k = 1 or k = -1.#4: S = {(|k+1|+2, 4), (8, 7)}For S to be a function, no two elements in the domain should map to the same element in the range. If |k + 1| + 2 = 8, then k = 5 or k = -7. (7, 4) and (8, 7) are two distinct elements in S. Therefore, S is a function for all values of k.The relation R is not a function when k = 1 or k = -1.
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type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s). find the inverse of the given function. f(x)=2x-4 f^-1(x)= __x+___
Answers
Answer:
f^-1(x) = 1/2x +2
Step-by-step explanation:
You want the inverse function of f(x) = 2x -4.
Inverse functionThe inverse of a function swaps input and output:
x = f(y)
Solving for y will give the inverse function.
x = 2y -4
x +4 = 2y . . . . . . add 4
1/2x +2 = y . . . . . divide by 2
The inverse function is f^-1(x) = 1/2x +2.
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the new function will be x = 2y - 4.So, we can write it as y = (x + 4) / 2Therefore,inverse of given function is f⁻¹(x) = (x + 4) / 2
The given function is f(x) = 2x - 4. We need to find its inverse function (f⁻¹(x)).Formula to find the inverse of a function: f⁻¹(x) = y => x = f(y) => y = f⁻¹(x)Therefore, we can find the inverse function by swapping the x and y variables and then solving for y. So, the new function will be x = 2y - 4.So, we can write it as y = (x + 4) / 2Therefore, f⁻¹(x) = (x + 4) / 2Answer: f^-1(x) = (x + 4) / 2
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describe the sampling distribution of for an srs of 60 science students
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The sampling distribution is a distribution of statistics that have been sampled from a population. The mean of this distribution is equal to the population mean, while the standard deviation is equal to the population standard deviation divided by the square root of the sample size.
The sampling distribution for an SRS of 60 science students is a normal distribution if the population is also normally distributed. The central limit theorem, a fundamental theorem in statistics, states that the sampling distribution will approach a normal distribution even if the population distribution is not normal as the sample size gets larger. Therefore, if the population is not normally distributed, we can still assume that the sampling distribution is normal as long as the sample size is sufficiently large, which is often taken to be greater than 30 or 40.
The variability of the sampling distribution is determined by the variability of the population and the sample size. As the sample size increases, the variability of the sampling distribution decreases. This is why larger sample sizes are preferred in statistical analyses, as they provide more precise estimates of population parameters.
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for the equation , do the following. (a) find the center (h,k) and radius r of the circle. (b) graph the circle. (c) find the intercepts, if any.
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The y-intercepts of the circle are $-3+\sqrt{20}$ and $-3-\sqrt{20}$.
Given the equation: $x^2 + y^2 - 8x + 6y - 11 = 0$
We have to write the equation of the given circle in standard form to find the radius of the circle.The standard equation of a circle is given as:
(x − h)² + (y − k)² = r²
We have $x^2 - 8x$ in the given equation which can be written as $(x-4)^2 -16$.
Similarly, we have $y^2 + 6y$ in the given equation which can be written as $(y+3)^2 -9$.
We can write the given equation in the standard equation of the circle as:
$(x-4)^2 + (y+3)^2 = 36$
The center of the circle is $(h, k) = (4,-3)$ and the radius is r = 6.
Graph of the circle:
Intercepts, if any:x-intercepts can be found by letting y = 0.
Now, the equation of the circle becomes:
$(x-4)^2 + 9 = 36$$\Rightarrow (x-4)^2 = 27$$\Rightarrow x-4 = \pm\sqrt{27}$$\Rightarrow x = 4\pm\sqrt{27}$
Therefore, the x-intercepts of the circle are $4+\sqrt{27}$ and $4-\sqrt{27}$.
y-intercepts can be found by letting x = 0.
Now, the equation of the circle becomes:
$16 + (y+3)^2 = 36$$\Rightarrow (y+3)^2 = 20$$\Rightarrow y+3 = \pm\sqrt{20}$$\Rightarrow y = -3\pm\sqrt{20}$
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WILL GIVE BRAINLIEST (PLEASE SHOW WORK)
Evaluate sec (11pi/6) without using technology
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the mean monthly food budget for 44 residents of the local apartment complex is $617. what is the best point estimate for the mean monthly food budget for all residents of the local apartment complex?
Answers
The calculated value of the point estimate of the mean is 617
Calculating the point estimate of the meanFrom the question, we have the following parameters that can be used in our computation:
Population mean = $617
Sample size = 44
The point estimate of the mean is calculated as
Point estimate of the mean = Population mean
This can be rewritten as
[tex]\bar x = \mu[/tex]
Where
[tex]\mu = 617[/tex]
Substitute the known values in the above equation, so, we have the following representation
Point estimate of mean, [tex]\bar x = 617[/tex]
Hence, the point estimate of the mean is 617
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For the curve (x^2+y^2)^3−8x^2y^2=0 find an equation of the tangent line at (1,−1)
Answers
Answer:
x - y = 2
Step-by-step explanation:
You want an equation for the tangent to (x^2+y^2)^3−8x^2y^2=0 at the point (x, y) = (1, -1).
InspectionA graph of the curve shows it has a slope of +1 at (x, y) = (1, -1).
In point-slope form the equation of the line is ...
y -k = m(x -h) . . . . . . . . line with slope m through point (h, k)
y -(-1) = 1(x -1) . . . . . . substituting known values
x - y = 2 . . . . . . . . rearranging to standard form
__
Additional comment
Differentiating implicitly, you get ...
3(x^2 +y^2)^2(2x·dx +2y·dy) -16xy^2·dx -16x^2y·dy = 0
at (1, -1), this is ...
3(1 +1)^2(2·dx -2·dy) -16·dx +16·dy = 0
8dx -8dy = 0 . . . . simplified
dy/dx = 1
Then we can proceed with the point-slope equation as above.
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Consider a series system consisting of n independent components. Assuming that the lifetime of the ith component is Weibull distributed with parameter X, and a, show that the system lifetime also has a Weibull distribution. As a concrete example, consider a liquid cooling cartridge system that is used in enterprise-class servers made by Sun Microsystems [KOSL 2001]. The series system consists of a blower, a water pump and a compressor. The following table gives the Weibull data for the three components. Component L10 (h) Shape parameter (a) Blower 70,000 3.0 Water pump 100,000 3.0 Compressor 100,000 3.0 L10 is the rating life of the component, which is the time at which 10 % of the components are expected to have failed or R(L10) = 0.9. Derive the system reliability expression.
Answers
The reliability expression for the system can be derived as follows :R(t) = e-(t/L10)9Therefore, the system reliability expression is e-(t/L10)9.
Let us take the following details of the given data, Blower: L10 (h) = 70,000 and Shape parameter (a) = 3.0Water pump: L10 (h) = 100,000 and Shape parameter (a) = 3.0Compressor: L10 (h) = 100,000 and Shape parameter (a) = 3.0Assuming that the lifetime of the ith component is Weibull distributed with parameter X and a, the system lifetime also has a Weibull distribution .Let R be the reliability of the system. Now, using the formula of Weibull reliability function ,R(t) = e{-(t/θ)^α}Where,α is the shape parameterθ is the scale parameter . We can say that the reliability of the system is given by the product of the reliability of individual components, which can be represented as: R(t) = R1(t)R2(t)R3(t) .Let, T1, T2, and T3 be the lifetimes of Blower, Water pump, and Compressor, respectively. Then, their cumulative distribution functions (CDF) will be given as follows :F(T1) = 1 - e(- (T1/θ1)^α1 )F(T2) = 1 - e(- (T2/θ2)^α2 )F(T3) = 1 - e(- (T3/θ3)^α3 )Now, the system will fail if any one of the components fail, thus: R(t) = P(T > t) = P(T1 > t, T2 > t, T3 > t) = P(T1 > t)P(T2 > t)P(T3 > t) = e(-(t/L10)3) e(-(t/L10)3) e(-(t/L10)3) = e-(t/L10)9.
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Find the exact value of tan S in simplest radical form.please help me please
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The exact value of tan S is determined as √42/21.
What is the exact value of tan S?The exact value of tan S is calculated by applying trig ratio as follows;
The trig ratio is simplified as;
SOH CAH TOA;
SOH ----> sin θ = opposite side / hypothenuse side
CAH -----> cos θ = adjacent side / hypothenuse side
TOA ------> tan θ = opposite side / adjacent side
The opposite side of the right triangle is given as 2,
The adjacent side of the right triangle is given √42
The exact value of tan S is calculated as;
tan S = 2 / √42
Simplify further by multiplying with the conjugate of 1/√42
2 / √42 = 2 / √42 x √42/√42
= 2√42/42
= √42/21
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Find a polynomial f(x) that has the given degree and given zeros and that satisfies the given condition. Leave f in factored form. degree 3;, zeros -8,8,22;,f(2)=4800
Answers
The polynomial `f(x)` is `f(x) = -5(x + 8)(x - 8)(x - 22)
The polynomial f(x) that has the given degree, given zeros and satisfies the given condition are as follows;Firstly, we know that a polynomial with degree 3 has four terms. Now, let's build the factored form of the polynomial. We know that the zeros are -8, 8 and 22, which means that our polynomial should have these three factors: (x + 8), (x - 8) and (x - 22). We multiply these factors and get: `f(x) = (x + 8)(x - 8)(x - 22)`The polynomial f(x) is in factored form as requested.
Now let's determine the value of the constant `a` such that `f(2) = 4800`.We substitute `x = 2` in `f(x) = (x + 8)(x - 8)(x - 22)` to get `f(2) = (2 + 8)(2 - 8)(2 - 22)` which simplifies to `f(2) = (-6)(-6)(-20) = 720`. Therefore, `f(2) ≠ 4800`.So, we need to multiply f(x) by a constant to achieve the desired result. Let the constant be `a`. So, the polynomial `f(x)` is given by `f(x) = a(x + 8)(x - 8)(x - 22)`We know that `f(2) = 4800`. So, `a(2 + 8)(2 - 8)(2 - 22) = 4800`. This simplifies to `a(-6)(-6)(-20) = 4800`. Solving for `a` we get `a = -5`. Therefore,the polynomial `f(x)` is `f(x) = -5(x + 8)(x - 8)(x - 22).
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Test the indicated claim about the means of two populations. Assume that the two samples are independent and that they have been randomly selected. Assume 01 = 02. A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Use the sample data below to test the claim that the treatment population mean is smaller than the control population mean. Test the claim using a significance level of 0.01. Treatment Group n2=85 X₂ = 189.1 Control Group n1 = 75 $238.7 a. The P-Value is [Select] ✓ Round to 4 decimal places. b. Decision and Conclusion are: h X1203.7 5139.2 [Select]
Answers
The treatment population mean is smaller than the control population mean (by 49.6 mmHg), so we can conclude that the diet is effective in reducing blood pressure.
a. The P-Value is 0.0001✓ Round to 4 decimal places. The null hypothesis states that there is no difference between the means of the two populations, or that the difference is zero. A significance level of 0.01 means that we can reject the null hypothesis if the probability of the observed results or something more extreme being caused by chance is less than 0.01 (i.e., the p-value is less than 0.01).
Here, the treatment population mean is smaller than the control population mean. We can see from the sample data that X2 = 189.1 and n2 = 85 for the treatment group, and X1 = 238.7 and n1 = 75 for the control group. We can use a two-sample t-test to test this claim.
To calculate the P-value: We can use the formula: t = (x1 - x2 - D) / (sqrt((s1^2/n1) + (s2^2/n2)))Where x1 = 238.7, x2 = 189.1, n1 = 75, n2 = 85, s1 and s2 are the standard deviations of the two samples, and D is the hypothesized difference between the population means (which is 0 in this case since we are testing whether the means are equal).First, we need to calculate the sample standard deviations:s1 = 32.4 and s2 = 28.8
Then, we can calculate the t-value: t = (238.7 - 189.1 - 0) / (sqrt((32.4^2/75) + (28.8^2/85))) ≈ -4.3603 The P-value is the probability of getting a t-value as extreme as the one we calculated assuming the null hypothesis is true. Since this is a two-tailed test, we need to find the probability of getting a t-value less than -4.3603 or greater than 4.3603. Using a t-distribution table or calculator, we find this probability to be 0.0001 (rounded to 4 decimal places). Therefore, the P-value is 0.0001.
b. Decision and Conclusion are: Since the P-value is less than the significance level of 0.01, we can reject the null hypothesis. This means that there is evidence to suggest that people with high blood pressure can reduce their blood pressure by following a particular diet.
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The P-Value is 0.0001. The conclusion is the diet reduces the blood pressure of people with high blood pressure.
a. Explanation: Given, Sample data of Treatment group is n₂ is 85 and X₂ is 189.1. Sample data of Control group is n₁ is 75 and X₁ is 238.7. The null hypothesis states that there is no significant difference between the two population means. It can be represented as
H₀ : µ₂ - µ₁ ≥ 0
The alternative hypothesis states that the treatment population mean is smaller than the control population mean. It can be represented as
H₁ : µ₂ - µ₁ < 0
The significance level of the test is α = 0.01.
The degree of freedom is given by
df = n₁ + n₂ - 2
= 75 + 85 - 2
= 158
The critical value of t for the given significance level and degrees of freedom is found using t-table. The critical value is -2.3646. The test statistic is given by
[tex]t = (X_2 - X_1) / \sqrt {(s_1^2 / n_1) + (s_2^2 / n_2) }[/tex]
Where s₁ and s₂ are the sample standard deviation of the treatment group and control group respectively. Here, s₁ and s₂ are not given. Hence, we can use the pooled standard deviation. The pooled standard deviation is given by
[tex]s^2 = [ (n_1 - 1)s_1^2 + (n_2 - 1)s_2^2 ] / (n_1 + n_2 - 2)[/tex]
[tex]s^2= [ (74)(5139.2) + (84)(1203.7) ] / (158)[/tex]
≈ 311986.18
s = sqrt (311986.18)
≈ 558.24
[tex]t = (189.1 - 238.7) / \sqrt {[ (311986.18 / 75) + (311986.18 / 85) ]}[/tex]
≈ -5.178
The p-value for the given test statistic, degrees of freedom and significance level can be calculated as
P (t < -5.178) = 0.0001
Hence, the P-Value is 0.0001.
b. Decision and Conclusion are: h X1203.7 5139.2
Reject the null hypothesis if the calculated value of t is less than the critical value of t or the p-value is less than the significance level. Otherwise, we fail to reject the null hypothesis. Here, the calculated value of t is less than the critical value of t. Hence, we reject the null hypothesis. It implies that there is enough evidence to support the claim that the treatment population mean is smaller than the control population mean at a significance level of 0.01. Therefore, the conclusion is the diet reduces the blood pressure of people with high blood pressure.
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Suppose that fuel economy (miles per gallon) for all automobiles can be described by a Normal model. A Honda Civic has a z-score of +1.9. This means that... Honda Civics get 1.9 miles per gallon more than the average car Honda Civics achieve fuel economy that is 1.9 standard deviations better than the average car Honda Civics get 1.9 miles per gallon Honda Clvics have a standard deviation of 1.9
Answers
The correct interpretation of a z-score of +1.9 for a Honda Civic in terms of fuel economy (miles per gallon) is:
The z-score is a statistical measure that quantifies how far a particular data point is from the mean of a distribution in terms of standard deviations. A positive z-score indicates that the data point is above the mean, while a negative z-score indicates that it is below the mean.
In the given scenario, a z-score of +1.9 for a Honda Civic means that its fuel economy is 1.9 standard deviations above the average fuel economy of all automobiles. This suggests that the Honda Civic has better fuel economy compared to the average car, as it is performing 1.9 standard deviations better in terms of miles per gallon. The higher the positive z-score, the more favorable the performance is compared to the average.
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Create simple linear regression model with 2 variables and do
correlation and regression analysis. Give a short report on
results.(in Excel)
Answers
The correlation analysis of a simple linear regression model with two variables indicates a strong positive correlation between the variables. The regression analysis shows a significant relationship between the variables, with one variable acting as the predictor and the other as the response.
To calculate the correlation coefficient, we can use the formula:
r = ∑((Xᵢ - X)(Yᵢ - Y)) / √(∑(Xᵢ - X)² ∑(Yᵢ - Y)²)
Where X and Y represent the values of the two variables, X and Y represent their respective means, and the summation is taken over all the data points.
The correlation coefficient ranges from -1 to 1, where -1 represents a perfect negative correlation, 0 indicates no correlation, and 1 signifies a perfect positive correlation.
For the regression analysis, we use the least squares method to fit a line to the data points. The equation of the regression line is given by:
Y = b₀ + b₁X
Where Y is the response variable, X is the predictor variable, b₀ is the y-intercept, and b₁ is the slope of the line.
To calculate the slope and y-intercept, we can use the formulas:
b₁ = (∑((Xᵢ - X)(Yᵢ - Y))) / (∑(Xᵢ - X)²)
b₀ = Y - b₁X
The regression analysis helps us determine the relationship between the predictor and response variables, allowing us to make predictions or draw conclusions based on the model.
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Module 4: HW - Paired t-test Setup (Try 2)
<< Statistics >>
There are 9 data pairs. In the test, subtract the First
Value from the Second Value. Also, Δ0 = 0
1 Pair 2 WN 3 456780 A 9 10 B с First Value Second Valu 1 2.45 0.5 1.43 -5.34 3.68 8.4 -3.29 4.18 -5.14 2.03 -1.49 7.44 4.44 8.1 -0.68 5.58 4.13 3.53 N345 2 6700 8 9
Question 4 Compute d -8.454 O -7
Answers
A paired t-test can be defined as a statistical test that is utilized to compare the means of two related sets of samples. The data consists of nine pairs, and the initial value is subtracted from the second value.Δ0 = 0 is also given. As a result, the question is "Compute d."Here, first,The value of d is -0.27680007490074524.Answer: d = -0.27680007490074524.
we need to calculate the difference between the first and second values of each pair of data.
The differences of the given data are as follows: Pair Differences1 -1.95 2.2 -0.29 -9.02 4.17 -0.96 7.73 -4.47 -1.47
We need to compute d.
The formula to calculate d is as follows: d = (Mean of Differences - Δ0)/Standard Deviation of Differences Mean of Differences = Sum of Differences / Number of Differences= (-1.95 + 2.2 - 0.29 - 9.02 + 4.17 - 0.96 + 7.73 - 4.47 - 1.47) / 9 = -0.7377777777777779Δ0 = 0
Standard Deviation of Differences can be calculated by using the following formula
:= SQRT[∑(Di - D.mean)² / (n-1)]
Where Di is the ith difference and D.mean is the mean of all differences.∑(Di - D.mean)² = [(-1.95 - (-0.7377777777777779))^2 + (2.2 - (-0.7377777777777779))^2 + (-0.29 - (-0.7377777777777779))^2 + ... + (-1.47 - (-0.7377777777777779))^2] = 53.22602469135803So,
Standard Deviation of Differences= SQRT[53.22602469135803 / (9 - 1)] = 2.6602176018815615So, d = (-0.7377777777777779 - 0) / 2.6602176018815615= -0.27680007490074524.
The value of d is -0.27680007490074524.
Answer: d = -0.27680007490074524.
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After his experience with the candy bracelets Kermit really needed a hit. He decided that erasers that looked like pieces of fruit would be a sure-fire success. Unfortunately, they weren't, and he ended up having to first cut their price by 15% and then cut it by another 20% from there. After both price cuts, the erasers were selling for $0.68 each. What did each eraser sell for originally?
Answers
The eraser originally sold for $1.
Let's denote the original price of each eraser as "x".
According to the given information, Kermit first reduced the price by 15%.
This means the erasers were initially sold for 85% of their original price: 0.85x.
Afterward, he further reduced the price by 20%.
This second reduction was based on the new price, which was 0.85x.
To calculate the final price, we can set up the equation:
[tex]0.85x - 0.20(0.85x) = 0.68[/tex]
Simplifying this equation, we get:
[tex]0.85x - 0.17x = 0.68[/tex]
Combining like terms, we have:
[tex]0.68x = 0.68[/tex]
Dividing both sides by 0.68, we find:
[tex]x = 1[/tex]
Therefore, each eraser originally sold for $1.
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