Using the formula of combination, a committee of six can be chosen in 336 ways from a group of eleven if Barry and Harry refuse to serve together on the same committee
1. First, find the total number of ways to select six people from eleven without considering the restriction using the combination formula:
C(n, r) = n! / [r!(n-r)!],
where n is the total number of people, and r is the number of people to be selected.
In this case, n = 11 and r = 6. So the total number of ways without considering the restriction is:
C(11, 6) = 11! / [6!(11-6)!] = 11! / [6! * 5!] = 462 ways.
2. Now, consider Barry and Harry as a single unit, and we have 10 units in total (9 remaining people + 1 unit of Barry and Harry). We need to choose 4 more people from the remaining 9. So the number of ways is:
C(9, 4) = 9! / [4!(9-4)!] = 9! / [4! * 5!] = 126 ways.
3. Finally, apply the subtraction principle to exclude the cases where Barry and Harry are together. The total number of ways to form a committee of six, considering the restriction, is:
Total ways - Ways with Barry and Harry together = 462 - 126 = 336 ways.
So, in 336 ways, a committee of six can be chosen from a group of eleven if Barry and Harry refuse to serve together on the same committee.
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What is the answer after you have factored it?
9^2+34-20=2
The factored version of the equation would be (b + 9) (b - 7) = 0.
How to find the factored version ?We are given the equation:
b ^ 2 + 2b - 72 + 9 = 0
b ^ 2 + 2b - 63 = 0
We can rewrite this equation, based on two numbers that multiply to - 63 and add up to 2. These numbers would be 9 and 7 so the equation becomes :
(b + 9) (b - 7) = 0
The answers after the simplification would be:
b + 9 = 0
b = -9
b - 7 = 0
b = 7
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Carol purchased one basket of fruit consisting of 4 apples and 2 oranges and another basket of fruit consisting of 3 apples and 5 oranges. Carol is to select one piece of fruit at random from each of the two baskets. What is the probability that one of the two pieces of fruit selected will be an apple and the other will be an orange
Answer:13/24
Step-by-step explanation 1. the desired probability is the sum of the probabilities of two disjoint events. In the first event, an apple is selected from the first basket and an orange is selected from the second basket; the probability of this event is (4/6)(5/8)=20/48. 2. In the second event, an orange is selected from the first basket and an apple is selected from the second basket; the probability of this event is (2/6)(3/8)=6/48. Therefore, the desired probability is 20/48+6/48=26/48=13/24.
6. In a study with a 4 x 2 x 3 factorial design, a. There are _______ independent variables. b. There are _______ possible main effects. c. There are ______ different conditions. d. There are _______ possible interactions.
a. There are three independent variables in a 4 x 2 x 3 factorial design.
b. There are four possible main effects in a 4 x 2 x 3 factorial design, one for each independent variable.
c. There are 24 different conditions in a 4 x 2 x 3 factorial design, which is the product of the levels of each independent variable (4 x 2 x 3 = 24).
d. There are several possible interactions in a 4 x 2 x 3 factorial design, including two-way and three-way interactions between the independent variables.
a) How many independent variables are there in a 4 x 2 x 3 factorial design ?a). In a factorial design, the independent variables are systematically manipulated in all possible combinations. In a 4 x 2 x 3 factorial design, there are three independent variables, each with a different number of levels.
The first independent variable has four levels, the second independent variable has two levels, and the third independent variable has three levels.
Therefore, there are three independent variables in this design.
b) What is a main effect in a factorial design ?b). A main effect is the effect of a single independent variable on the dependent variable, averaged across all levels of the other independent variables.
In a 4 x 2 x 3 factorial design, there are three independent variables, each of which can have a main effect. Therefore, there are three possible main effects in this design.
Additionally, there can be two-way and three-way interactions between the independent variables, which can also affect the dependent variable. However, a main effect is distinct from an interaction effect and is not dependent on any other independent variables.
c) How is the total number of different conditions calculated in a 4 x 2 x 3 factorial design ?c).The number of different conditions in a 4 x 2 x 3 factorial design is calculated by multiplying the number of levels for each independent variable.
In this case, the first independent variable has four levels, the second independent variable has two levels, and the third independent variable has three levels.
Therefore, the total number of different conditions is obtained by multiplying these three numbers (4 x 2 x 3 = 24). Each condition represents a unique combination of the levels of the three independent variables.
d) How do interactions occur in factorial designs ?d). Interactions in factorial designs occur when the effect of one independent variable on the dependent variable depends on the level of another independent variable. In a 4 x 2 x 3 factorial design, there are several possible interactions between the independent variables.
For example, there may be a two-way interaction between the first and second independent variables, meaning that the effect of the first independent variable on the dependent variable depends on the level of the second independent variable.
Similarly, there may be a two-way interaction between the second and third independent variables, or between the first and third independent variables.
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Find the probability of getting any triple-digit number where all the digits are the same in a lottery game that consists of selecting a three-digit number.
Thus, the probability is 1/100 or 0.01, which means there is a 1% chance of getting a triple-digit number with all the same digits in this lottery game.
To calculate the probability of getting a triple-digit number where all digits are the same in a lottery game that consists of selecting a three-digit number, you should first determine the number of favorable outcomes and the total possible outcomes.
There are 9 favorable outcomes, as the triple-digit numbers with the same digits are: 111, 222, 333, 444, 555, 666, 777, 888, and 999. Note that we don't include 000 since it's not a triple-digit number.
Now, let's find the total possible outcomes. In a three-digit number, there are 10 possible digits (0-9) for each of the three positions. However, the first position cannot be 0, as that would not be a triple-digit number.
So, there are 9 possible digits for the first position and 10 for the other two. The total possible outcomes are 9 x 10 x 10, which equals 900.
Finally, to find the probability of getting a triple-digit number with all the same digits, divide the number of favorable outcomes by the total possible outcomes:
Probability = Favorable Outcomes / Total Possible Outcomes
Probability = 9 / 900
The probability is 1/100 or 0.01, which means there is a 1% chance of getting a triple-digit number with all the same digits in this lottery game.
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A two-inch cube ($2\times2\times2$) of silver weighs 3 pounds and is worth $\$200$. How much is a three-inch cube of silver worth
The three-inch cube would be worth [tex]9\text{ lb}\times $20/\text{lb} = $180$.[/tex]
Since the cube is made of silver, we know that the ratio of volume to weight is constant, meaning that the density of silver is the same throughout the cube. Therefore, if a two-inch cube of silver weighs 3 pounds, then its volume is [tex]$(2\text{ in})^3=8\text{ in}^3$[/tex] and its density is [tex]3\text{ lb}/8\text{ in}^3 = 0.375\text{ lb/in}^3$.[/tex]
Now let's consider a three-inch cube. Its volume is [tex]$(3\text{ in})^3=27\text{ in}^3$[/tex], which is 3 times the volume of the two-inch cube. Since the density is the same, the weight of the three-inch cube will be 3 times the weight of the two-inch cube, or [tex]3\times 3\text{ lb}=9\text{ lb}$.[/tex]
To find the value of the three-inch cube, we need to know the price of silver per pound. Let's assume it's [tex]$$20$[/tex] per pound (this is just an example). Then the three-inch cube would be worth [tex]9\text{ lb}\times $20/\text{lb} = $180$.[/tex]
In general, the value of the three-inch cube would be proportional to the weight and the price of silver per pound.
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The results of the survey showed that 41% of participants chose Science, 38% chose humanities, and 11% chose both. What percentage of all students like science but not humanities
To find the percentage of all students who like science but not humanities, we need to subtract the percentage of students who chose both science and humanities from the percentage of students who chose science only. 30% of all students like Science but not Humanities.
So, the percentage of students who like science but not humanities is:
41% (students who chose science) - 11% (students who chose both science and humanities) = 30%
Therefore, 30% of all students like science but not humanities.
To find the percentage of students who like Science but not Humanities, we need to subtract the percentage of students who chose both from the percentage who chose Science. Here's the step-by-step explanation:
1. The percentage of students who chose Science: 41%
2. The percentage of students who chose both Science and Humanities: 11%
3. Subtract the percentage of students who chose both from the percentage who chose Science: 41% - 11% = 30%
So, 30% of all students like Science but not Humanities.
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Suppose N(t) is a Poisson process with rate 3. Let Tn denote the time of the nth arrival. Find (a) E(Ti2), (6) E(Tiz/N(2) = 5), (c) E(N(S) N(2) = 5).
(a) To find E(Ti2), we can use the fact that the interarrival times of a Poisson process are exponentially distributed. Since T2 is the time of the second arrival,
We can write T2 = T1 + X, where X is the time between the first and second arrivals. Thus, we have:
E(T2) = E(T1 + X) = E(T1) + E(X)
Since the Poisson process has rate 3, the interarrival times are exponentially distributed with parameter λ = 3. Therefore, we have E(X) = 1/λ = 1/3. Also, the time of the first arrival is distributed as an exponential random variable with parameter λ, so we have E(T1) = 1/λ = 1/3. Putting it all together, we get:
E(T2) = E(T1) + E(X) = 1/3 + 1/3 = 2/3
Therefore, E(Ti2) = 2/3.
(b) To find E(Tiz/N(2) = 5), we need to condition on the value of N(2). We have:
E(Ti2/N(2) = 5) = ∑k≥2 E(Ti2/N(2) = 5, N(2) = k) P(N(2) = k)
Since the Poisson process has independent and stationary increments, we know that the distribution of N(2) is Poisson with parameter 6. Therefore, we have:
P(N(2) = k) = e^(-6) 6^k / k!
For a fixed value of N(2) = k, we can think of the process up to time T2 as a Poisson process with rate 3, and condition on the times of the first k arrivals. The time of the ith arrival, given the times of the first i-1 arrivals, is distributed as an exponential random variable with parameter λ = 3. Therefore, we have:
E(Ti2/N(2) = 5, N(2) = k) = E(Ti2 | T1, T2, ..., Tk)
Using the memoryless property of the exponential distribution, we can write:
E(Ti2 | T1, T2, ..., Ti-1) = Ti + E(T2 | T1, T2, ..., Ti-1) = Ti + 2/3
Therefore, we have:
E(Ti2/N(2) = 5, N(2) = k) = Ti + 2/3
Putting it all together, we get:
E(Tiz/N(2) = 5) = ∑k≥2 ∑i≥1 (Ti + 2/3) e^(-6) 6^k / k!
Using the fact that the interarrival times are exponentially distributed, we can compute the sum over i as:
∑i≥1 (Ti + 2/3) = E(T2) + 2/3 = 8/3
Therefore, we have:
E(Tiz/N(2) = 5) = (8/3) ∑k≥2 e^(-6) 6^k / k! = (8/3) (1 - e^(-12))
Thus, E(Tiz/N(2) = 5) ≈ 1.81.
(c) To find E(N(S) N(2) = 5), we can use the fact that the number of arrivals in a Poisson process of rate λ in an interval of length t is a Poisson random variable with parameter λt. Therefore, we have:
E(N(S) N(2) = 5) = E(N(5) N(2) = 5) = E(N(5)^2 | N(2) = 5) P(N(2) = 5)
For a fixed value of N(2) = 5, we can think of the process up to time 5 as a Poisson process with rate 3, and condition on the times of the first 5 arrivals. Therefore, we have:
E(N(5)^2 | N(2) = 5) = E((N(5) - 5)^2 | N(2) = 5) + E(10 N(5) - 25 | N(2) = 5) + 25
Using the fact that the number of arrivals in an interval of length t is Poisson with parameter λt, we have:
E((N(5) - 5)^2 | N(2) = 5) = Var(N(3)) = 3
Also, we have:
E(10 N(5) - 25 | N(2) = 5) = 10 E(N(5) | N(2) = 5) - 25 = 10 (5 + 2) - 25 = 15
Putting it all together, we get:
E(N(S) N(2) = 5) = (3 + 15 + 25) P(N(2) = 5) = 43 e^(-6) 6^5 / 5!
Thus, E(N(S) N(2) = 5) ≈ 1.94.
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Consider the following. x = e^t, y = e^(−3t)
(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
To indicate the direction in which the curve is traced as the parameter increases, we can draw an arrow that points from lower left to upper right, since the curve moves from the point (1, 1) at t = 0 to the right and upward as t increases.
To eliminate the parameter, we can use the fact that x = e^t and y = e^(-3t) to solve for t in terms of x and y. Taking the natural logarithm of both sides of x = e^t, we get ln(x) = t. Similarly, taking the natural logarithm of both sides of y = e^(-3t), we get ln(y) = -3t, or t = (-1/3)ln(y). Substituting this expression for t into the equation we found for ln(x), we get ln(x) = (-1/3)ln(y), which simplifies to ln(x^3) = ln(y^(-1)), or x^3 = 1/y. This is the Cartesian equation of the curve.
To sketch the curve, we can start by noting that both x and y are positive for all values of t, since e^t and e^(-3t) are always positive. As t increases, x and y both increase, but y increases much more slowly than x since e^(-3t) decreases rapidly as t increases. This means that the curve starts out very steep (since the slope of the tangent line at t = 0 is dx/dt = 1 and dy/dt = -3), but becomes flatter and flatter as t increases. The curve approaches the x-axis as t approaches infinity, but never touches it.
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A bus comes to a station once every 10 minutes and waits at the station for 3 minutes. Assume that you arrive at the station at a random time. Express the probability as a decimal. Find the probability that you will have to wait more than 6 minutes to board a bus.
The time between buses is 10 minutes and the bus waits at the station for 3 minutes, so the time between the departure of one bus and the departure of the next bus is 10 + 3 = 13 minutes.
Since you arrive at a random time, the time you have to wait for the next bus can be any number from 0 to 13 minutes, with each value between 0 and 13 being equally likely.
The probability that you will have to wait more than 6 minutes to board a bus is the probability that you arrive at the station between 0 and 7 minutes after a bus has left.
This is because if you arrive at the station more than 7 minutes after a bus has left, you will have to wait less than 6 minutes for the next bus to arrival time.
The probability that you arrive at the station between 0 and 7 minutes after a bus has left is 7/13, or approximately 0.538.
Therefore, the probability that you will have to wait more than 6 minutes to board a bus is approximately 0.462 (1 - 0.538).
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television retails for $4,500. What is the purchase price in dollars of the television if it has been marked up 25% of the retail price
The purchase price in dollars of the television that has been marked up 25% of the retail price of $4,500 would be $5,625.
To find the purchase price of the television in dollars, follow these steps:
1. Determine the markup amount: 25% of the retail price ($4,500).
2. Subtract the markup amount from the retail price to find the purchase price.
Step 1: Calculate the markup amount:
25% of $4,500 = 0.25 × 4,500 = $1,125
Step 2: Subtract the markup amount from the retail price:
$4,500 (retail price) - $1,125 (markup) = $3,375
The purchase price of the television in dollars is $3,375.
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4. Researchers measured the sugared-beverage consumption and obesity of the same children for twoyears. They found that when these children added one sugared drink a day to their diet, their risk ofobesity increased 60%. Which quasi-experimental design did they use
The researchers monitored the same children's consumption of sugar-sweetened beverages and obesity over the course of two years using a longitudinal quasi-experimental methodology.
The researchers used a longitudinal quasi-experimental design in this study, as they measured the same children's sugared-beverage consumption and obesity for two years.
This design allows the researchers to observe changes over time and make causal inferences about the relationship between the variables.
By comparing the same group of children's obesity rates before and after the addition of one sugared drink per day, the researchers were able to establish a causal link between the consumption of sugared drinks and increased risk of obesity.
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what is 8x + 7 = 6x + 15.
Answer:
To solve for x in the equation 8x + 7 = 6x + 15, you need to isolate the variable (x) on one side of the equation.
First, you can start by subtracting 6x from both sides of the equation to get:
8x + 7 - 6x = 15
Simplifying this gives: 2x + 7 = 15
Next, you can subtract 7 from both sides of the equation to get: 2x = 8
Finally, divide both sides of the equation by 2 to solve for x: x = 4
Therefore, the solution for x in the equation 8x + 7 = 6x + 15 is x = 4
Answer:
x = 4
Step-by-step explanation:
To solve the equation 8x + 7 = 6x + 15, you can start by isolating the variable on one side of the equation. To do this, you can subtract 6x from both sides of the equation to get 2x + 7 = 15. Then, you can subtract 7 from both sides of the equation to get 2x = 8. Finally, you can divide both sides of the equation by 2 to get x = 4 1.
I hope that helps!
How many $3$-digit positive integers are there whose middle digit is equal to the sum of the first and last digits
The number of 3-digit positive integers whose middle digit is equal to the sum of the first and last digits is 55 such numbers
Let's first consider the possible values for the middle digit, which is the sum of the first and last digits:
If the first digit is 1, then the middle digit can only be 2, and the last digit can be any digit from 0 to 8.
There are 9 possible numbers in this case.
If the first digit is 2, then the middle digit can be 2 or 4, and the last digit can be any digit from 0 to 6.
There are 14 possible numbers in this case.
If the first digit is 3, then the middle digit can be 2, 4, or 6, and the last digit can be any digit from 0 to 4.
There are 15 possible numbers in this case.
If the first digit is 4, then the middle digit can be 2, 4, 6, or 8, and the last digit can be any digit from 0 to 2.
There are 13 possible numbers in this case.
If the first digit is 5, then the middle digit can be 4, 6, or 8, and the last digit can only be 0.
There are 3 possible numbers in this case.
If the first digit is 6, then the middle digit can only be 6, and the last digit can only be 0.
There is only 1 possible number in this case.
In total, the number of 3-digit positive integers whose middle digit is equal to the sum of the first and last digits is:
9 + 14 + 15 + 13 + 3 + 1 = 55
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Complete the statement. 6 – 6 = 6 + , or
6 - 6 = 6 - 6 is the correct statement.
To complete the statement 6 - 6 = 6 + x, we need to find the value of x that makes the statement true.
Simplifying the left-hand side of the equation, we have:
6 - 6 = 0
On the right-hand side, we have:
6 + x
To make the statement true, we need to find the value of x that satisfies:
0 = 6 + x
We can solve for x by subtracting 6 from both sides:
0 - 6 = 6 + x - 6
-6 = x
Therefore, the completed statement is:
6 - 6 = 6 - 6
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After finding a null result in her study, a researcher decides to conduct a follow-up study where she limits who the findings will generalize to. Which cause of within-group variance is she trying to reduce
The researcher is trying to reduce the cause of within-group variance related to individual differences or heterogeneity within the sample.
In statistics, within-group variance, also known as within-group variation or error variance, refers to the variability of data within a specific group or sample. It represents the differences among individuals within the same group or sample.
There are several causes of within-group variance, such as individual differences, measurement error, and uncontrolled extraneous variables.
In this scenario, the researcher is trying to limit who the findings will generalize to, which means that she wants to control or reduce the effect of individual differences or heterogeneity within the sample.
By limiting the sample to a specific population or subgroup, the researcher can reduce the within-group variance related to individual differences and increase the likelihood of finding significant results.
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40% is nearly half of the days in the workweek! Why was Dilbert's boss wrong to use this number to think that the employees were cheating
Dilbert's boss was wrong to use the number 40% to assume that the employees were cheating because it only represents a percentage and not the actual number of days that the employees were absent.
In reality, if the workweek is 5 days, then 40% would only be 2 days. Therefore, assuming that the employees were cheating based on this percentage alone is not a fair or accurate assessment. It's important to consider the actual number of days missed by each employee and investigate the reasons for their absence before jumping to conclusions.
While 40% might seem like nearly half of the days in a workweek, it doesn't necessarily imply that employees are cheating. To make a fair assessment, one should consider other factors such as workload, productivity, and employee performance, rather than solely focusing on a percentage.
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Consumer Reports rated 76 breakfast cereals from 0 to 100, the higher the number, the healthier the cereal. The following scatter plots show this rating paired with 5 ingredients from the nutritional label of each cereal: fat, sugar, protein, fiber grams and sodium in mgs. Which ingredients are positively correlated with nutritional rating
It's important to consider the overall nutrient profile of a cereal, rather than just focusing on a few specific ingredients.
To determine which ingredients are positively correlated with the nutritional rating, we can examine the scatter plots and look for a general trend between each ingredient and the rating.
In general, the higher the nutritional rating, the lower the values for fat, sugar, and sodium, and the higher the values for protein and fiber.
Based on this observation, we can conclude that protein and fiber are positively correlated with the nutritional rating, while fat, sugar, and sodium are negatively correlated with the nutritional rating.
It's worth noting that correlation does not imply causation, and there may be other factors that contribute to the nutritional value of a cereal beyond these five ingredients.
Therefore, it's important to consider the overall nutrient profile of a cereal, rather than just focusing on a few specific ingredients.
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what are the zeros of the function? f(x)=x^2+15x+36
x1=_ x2=_
The figure represents a water trough in the shape of a rectangular prism. The dimensions of the water trough are given in feet.
What is the volume of water in the trough in cubic feet when the trough is full?
Answer: 240 cubic feet
Please do these 4 questions. It needs to be in standard position. Please please help!!!!
A graph of the given angle measures is shown in the image attached below.
What is a rotation?In Mathematics and Geometry, a rotation refers to a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.
In this scenario, we would use an online graphing to calculator to plot or draw each of the angle measures that are provided as shown in the graph attached below.
For instance, we would convert -π/2 to degrees;
θ = -π/2 × 180/π
θ = -180/2
θ = -90 degrees.
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If $1.00 U.S. bought $1.40 Canadian dollars in 2006 and in 2010 it bought $1.00 Canadian dollar, then;
The statements are inconsistent and cannot both be true.
How to compare the exchange rates between USD and CAD in 2006 and 2010?To answer this question, we need to compare the exchange rate between the US dollar (USD) and the Canadian dollar (CAD) in 2006 and 2010.
In 2006, 1.00 USD bought 1.40 CAD. This can be expressed as:
1 USD = 1.40 CAD
In 2010, 1.00 USD bought 1.00 CAD. This can be expressed as:
1 USD = 1.00 CAD
To compare the two exchange rates, we can set them equal to each other and solve for CAD:
1 USD = 1.40 CAD
1 USD = 1.00 CAD
Setting the two equations equal to each other, we get:
1.40 CAD = 1.00 CAD
Subtracting 1.00 CAD from both sides, we get:
0.40 CAD = 0
This is a contradiction, which means that there is no consistent exchange rate that can explain both statements.
Therefore, the statements are inconsistent and cannot both be true.
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Problem
There are 888 employees on The Game Shop's sales team. Last month, they sold a total of ggg games. One of the sales team members, Chris, sold 171717 fewer games than what the team averaged per employee.
How many games did Chris sell?
Write your answer as an expression.
There are 888 employees on The Game Shop's sales team. Last month, they sold a total of ggg games. Chris sold 171717 fewer games than the team averaged per employee, so the number of games he sold can be expressed as: (ggg/888) - 171717.
To find the average number of games sold per employee, we divide the total number of games sold by the total number of employees.
The problem gives us the total number of employees, which is 888, and the total number of games sold, which is ggg. So, the average number of games sold per employee is:
ggg games ÷ 888 employees = ggg/888 games per employee
Next, we're told that Chris sold 171717 fewer games than the team averaged per employee.
This means that the number of games he sold is equal to the average number of games sold per employee minus 171717.
Thus, the expression for the number of games Chris sold is: (ggg/888) - 171717.
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Moses is twice as old as Methuselah was when Methuselah was one-third as old as Moses will be when Moses is as old as Methuselah is now. If the difference of their ages is 666, how old is Methuselah
Moses is currently 3996 years old.
Let's denote Methuselah's current age as "M" and Moses's current age as "M2".
"Moses is twice as old as Methuselah was when Methuselah was one-third as old as Moses will be when Moses is as old as Methuselah is now." This can be written as:
M2 = 2 × (M - (1/3) × M2)
We can simplify this equation by multiplying both sides by 3:
3 × M2 = 6 × (M - (1/3) × M2)
3 × M2 = 6M - 2 × M2
5 × M2 = 6M
"If the difference of their ages is 666" can be written as:
M2 - M = 666
We can use equation (5) to substitute for M2 in equation (6):
5 × M2 = 6M
5 × (M + 666) = 6M
5M + 3330 = 6M
M = 3330
Therefore, Methuselah is currently 3330 years old. We can use equation (6) to find Moses's current age:
M2 - M = 666
M2 - 3330 = 666
M2 = 3996
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): The SAT mathematics scores across the population of high school seniors is a normal random variable with mean 500 and standard deviation 100. Assume that the SAT mathematics test scores of different students are independent. (a) Find the probability that five randomly chosen seniors all score below 600. (b) What score must a student obtain to be in the 99th percentile
Normal random variable with mean 500 and standard deviation 100 (a) the probability that five randomly chosen seniors all score below 600 is approximately 0.4437. (b) a student must obtain a score of 766.5 or higher to be in the 99th percentile.
(a) To find the probability that five randomly chosen seniors all score below 600, we need to use the normal distribution formula. We know that the mean (μ) is 500 and the standard deviation (σ) is 100. To find the probability of a score below 600, we need to find the z-score for 600.
z = (x - μ) / σ
z = (600 - 500) / 100 = 1
Using a standard normal distribution table or calculator, we can find that the probability of a score below 600 is 0.8413. To find the probability of five randomly chosen seniors all scoring below 600, we need to multiply this probability by itself five times.
P(X < 600)^5 = 0.8413^5 = 0.4437
Therefore, the probability that five randomly chosen seniors all score below 600 is approximately 0.4437.
(b) To be in the 99th percentile, a student must score higher than 99% of the population. We can use the normal distribution table or calculator to find the z-score for the 99th percentile.
z = 2.33
Using the same formula as before, we can solve for x (the score needed to be in the 99th percentile).
2.33 = (x - 500) / 100
x = 500 + 2.33(100)
x = 766.5
Therefore, a student must obtain a score of 766.5 or higher to be in the 99th percentile.
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if a circle has a diameter of 46 centimeter what is the area of the circle
US Crime Commission wants to estimate the proportion of crimes in which firearms are used to within 3% with 95% confidence. In case no guess about a preliminary estimate of a sample proportion is available, what is minimum required sample size they have to take
The US Crime Commission needs to take a sample of at least 1068 crimes to estimate the proportion of crimes in which firearms are used with a margin of error of 3% and a 95% confidence level, assuming no preliminary estimate is available.
To calculate the minimum required sample size, we can use the formula:
[tex]n = [(Z^2) \times p \times q] / E^2[/tex]
where:
n is the sample size
Z is the Z-score corresponding to the desired confidence level (95% in this case), which is 1.96
p is the estimated proportion of crimes in which firearms are used (since no preliminary estimate is available, we can use 0.5 as a conservative estimate to get the maximum sample size)
q is the complement of p, which is 1 - p
E is the margin of error, which is 3% or 0.03
Substituting the values, we get:
[tex]n = [(1.96^2) \times 0.5 \times 0.5] / 0.03^2[/tex]
n = 1067.11
Rounding up to the nearest integer, the minimum required sample size is 1068. Therefore, the US Crime Commission needs to take a sample of at least 1068 crimes to estimate the proportion of crimes in which firearms are used with a margin of error of 3% and a 95% confidence level, assuming no preliminary estimate is available.
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Suppose that in a class of 10 stat majors and 10 engineers, 5 students are randomly chosen to present work at the board. What is the probability that exactly 4 of the students selected to present are stat majors
The probability that exactly 4 of the students selected to present are stat majors is 0.219, or about 22%.
The total number of ways to choose 5 students from a class of 20 is:
C(20,5) = (20!)/[(5!)(15!)] = 15504
To find the probability that exactly 4 of the students selected are stat majors, we need to count the number of ways to choose 4 stat majors and 1 engineer, and divide by the total number of ways to choose 5 students:
[C(10,4) * C(10,1)] / C(20,5) = [(10!)/[(4!)(6!)]] * [(10!)/[(1!)(9!)]] / [(20!)/[(5!)(15!)]] = 0.219
Therefore, the probability that exactly 4 of the students selected to present are stat majors is 0.219, or about 22%.
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Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12. Enter your answer as a list of ordered pairs separated by commas.
The foci of the ellipse are (4, 0) and (9, 0).
To find the foci of the ellipse, we first need to determine its standard form equation and identify the values of the major and minor axes. Given the endpoints of the major axis are (0,0) and (13,0), we can determine that the length of the major axis (2a) is 13 units. Thus, a = 6.5 units.
The minor axis has a length of 12 units, so the length of the minor axis (2b) is 12 units. Therefore, b = 6 units.
Next, we will find the value of c, which is the distance from the center of the ellipse to each focus. Using the relationship [tex]c^2 = a^2 - b^2[/tex], we get:
[tex]c^2 = (6.5)^2 - (6)^2[/tex]
[tex]c^2 = 42.25 - 36[/tex]
[tex]c^2 = 6.25[/tex]
c = √6.25
c ≈ 2.5 units
Now, we have all the necessary information to find the foci of the ellipse. Since the major axis is along the x-axis, the foci will be located at a distance of c units to the left and right of the center (which is the midpoint of the major axis). The center of the ellipse is (6.5, 0), so the foci will be at (6.5 - 2.5, 0) and (6.5 + 2.5, 0), which are (4, 0) and (9, 0).
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4. Find the critical number(s) of the function F(x) = x-1/x^2-x+2
5. Find the critical number(s) of the function F(x) = x^3/4 – 2x^1/4
6. Find the critical number(s) of the function F(x) = x^4/5(x-4)^2
The critical number is also x = 4.To find the critical number(s) of a function, we need to first take the derivative of the function and then find where the derivative is equal to zero or undefined.
4. F(x) = x-1/x^2-x+2
To find the derivative, we can use the quotient rule:
F'(x) = [(x^2-x+2)(1) - (x-1)(2x-1)] / (x^2-x+2)^2
Next, we need to find where F'(x) is equal to zero or undefined.
Setting the numerator equal to zero gives:
(x^2-x+2) - (2x^2-3x+1) = 0
-x^2 + 4x - 1 = 0
Using the quadratic formula, we can solve for x:
x = (4 ± sqrt(16-4(-1)(-1))) / (-2)
x = (4 ± sqrt(20)) / (-2)
x = 1 ± sqrt(5)
So the critical numbers are 1 + sqrt(5) and 1 - sqrt(5).
5. F(x) = x^3/4 – 2x^1/4
To find the derivative, we can use the power rule:
F'(x) = (3/4)x^-1/4 - (1/2)x^-3/4
Next, we need to find where F'(x) is equal to zero or undefined.
Setting the numerator equal to zero gives:
3x^-1/4 - 2x^-3/4 = 0
Multiplying both sides by x^3/4 gives:
3 - 2x = 0
Solving for x gives:
x = 3/2
So the critical number is 3/2.
6. F(x) = x^4/5(x-4)^2
To find the derivative, we can use the quotient rule:
F'(x) = [(x-4)^2(4x^3/5) - x^4/5(2(x-4)(1))] / (x-4)^4
Simplifying gives:
F'(x) = (2x^3 + 16x^2 - 32x) / 5(x-4)^3
Next, we need to find where F'(x) is equal to zero or undefined.
Setting the numerator equal to zero gives:
2x(x^2 + 8x - 16) = 0
Using the quadratic formula, we can solve for x:
x = (-8 ± sqrt(64 + 8(16))) / 2
x = (-8 ± sqrt(192)) / 2
x = -4 ± 2sqrt(6)
So the critical numbers are -4 + 2sqrt(6) and -4 - 2sqrt(6). However, we also need to check if the derivative is undefined at x = 4.
Plugging in x = 4 gives:
F'(4) = (2(4)^3 + 16(4)^2 - 32(4)) / 5(4-4)^3
F'(4) = undefined
So the critical number is also x = 4.
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Two archers shoot at a target. The distance of each shot from the center of the target is uniformly distributed from 0 to 1, independent of the other shot. What is the PDF of the distance of the losing shot from the center
The distance of the losing shot from the center is also uniformly distributed from 0 to 1. To find the PDF of the distance of the losing shot from the center, we need to first determine the probability of one shot being closer to the center than the other.
Let X be the distance of the first shot from the center, and Y be the distance of the second shot from the center. Then, the probability that X is closer to the center than Y is given by the area of the region where X < Y, which is a triangular region with base 1 and height 1/2 (since the probability of X being closer to the center than Y is the same as the probability of Y being closer to the center than X). Therefore, the probability of X being closer to the center than Y is 1/4.
Now, let Z be the distance of the losing shot from the center. We know that Z is equal to the distance of the second shot from the center if the second shot is closer to the center than the first shot, and it is equal to the distance of the first shot from the center if the first shot is closer to the center than the second shot. Therefore, the PDF of Z is given by:
fZ(z) = (1/4)fY(z) + (3/4)fX(z)
where fX(x) and fY(y) are the PDFs of X and Y, respectively. Since X and Y are uniformly distributed from 0 to 1, their PDFs are both equal to 1 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. Therefore, we have:
fZ(z) = (1/4) + (3/4) = 1
for 0 ≤ z ≤ 1. This means that the distance of the losing shot from the center is also uniformly distributed from 0 to 1.
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