Do you tailgate the car in front of you? About 35% of all drivers will tailgate before passing, thinking they can make the car in front of them go faster. Suppose that you are driving a considerable distance on a two-lane highway and are passed by 12 vehicles.
(a) Let r be the number of vehicles that tailgate before passing. Make a histogram showing the probability distribution of r for r = 0 through r = 12.
(b) Compute the expected number of vehicles out of 12 that will tailgate. (Round your answer to two decimal places.)
vehicles
(c) Compute the standard deviation of this distribution. (Round your answer to two decimal places.)
vehicles
Answers
Answer:
(a) The histogram is shown below.
(b) E (X) = 4.2
(c) SD (X) = 2.73
Step-by-step explanation:
Let X = r = a driver will tailgate the car in front of him before passing.
The probability that a driver will tailgate the car in front of him before passing is, P (X) = p = 0.35.
The sample selected is of size n = 12.
The random variable X follows a Binomial distribution with parameters n = 12 and p = 0.35.
The probability function of a binomial random variable is:
[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x}[/tex]
(a)
For X = 0 the probability is:
[tex]P(X=0)={12\choose 0}(0.35)^{0}(1-0.35)^{12-0}=0.006[/tex]
For X = 1 the probability is:
[tex]P(X=1)={12\choose 1}(0.35)^{1}(1-0.35)^{12-1}=0.037[/tex]
For X = 2 the probability is:
[tex]P(X=2)={12\choose 2}(0.35)^{2}(1-0.35)^{12-2}=0.109[/tex]
Similarly the remaining probabilities will be computed.
The probability distribution table is shown below.
The histogram is also shown below.
(b)
The expected value of a Binomial distribution is:
[tex]E(X)=np[/tex]
The expected number of vehicles out of 12 that will tailgate is:
[tex]E(X)=np=12\times0.35=4.2[/tex]
Thus, the expected number of vehicles out of 12 that will tailgate is 4.2.
(c)
The standard deviation of a Binomial distribution is:
[tex]SD(X)=np(1-p)[/tex]
The standard deviation of vehicles out of 12 that will tailgate is:
[tex]SD(X)=np(1-p)=12\times0.35\times(1-0.35)=2.73\\[/tex]
Thus, the standard deviation of vehicles out of 12 that will tailgate is 2.73.
To determine the probability distribution, create a histogram showing the possible values of r and their probabilities. The expected number of vehicles that will tailgate can be calculated by multiplying each value of r by its probability and summing up the results. The standard deviation can be found by calculating the variance and taking the square root of it.
Explanation:In order to determine the probability distribution of the number of vehicles that tailgate before passing, we need to consider the given information. We know that about 35% of all drivers tailgate. Since we are passed by 12 vehicles, the number of vehicles that tailgate can range from 0 to 12. To create a histogram showing the probability distribution, we need to calculate the probability of each possible value of r and represent them in a bar graph.
(b) To compute the expected number of vehicles that will tailgate, we need to multiply each possible value of r by its corresponding probability and sum up the results. This will give us the average number of vehicles that tailgate out of the 12 vehicles that passed us.
(c) The standard deviation of this distribution can be calculated by determining the variance and taking the square root of it. Variance is calculated by summing up the squared differences between each value of r and the expected value, multiplying each squared difference by its corresponding probability, and summing up the results.
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Related Questions
A random number generator on a computer selects two integers from 1 through 40. What is the probability that (a) both numbers are even, (b) one number is even and one number is odd, (c) both numbers are less than 30, and (d) the same number is selected twice?
Answers
Answer:
(a) 0.25
(b) 0.5
(c) 0.5256
(d) 0.025
Step-by-step explanation:
(a) There are 20 even numbers out of 40 the probability that both numbers are even is:
[tex]P=\frac{20}{40} *\frac{20}{40} =\frac{1}{4}=0.25[/tex]
(b) The events for which one number is even and one number is odd are:
- First is odd, second is even
- First is even, second is odd.
The probability is:
[tex]P = \frac{20}{40}*\frac{20}{40}+\frac{20}{40}*\frac{20}{40}=\frac{1}{2}=0.5[/tex]
(c) There are 29 numbers that are less than 30, the probability that both numbers are less than 30 is:
[tex]P=\frac{29}{40}*\frac{29}{40}=\frac{841}{1600}=0.5256[/tex]
(d) If any number from 1 to 40 is selected in the first pick, the probability that the same number is selected again is:
[tex]P=\frac{1}{40} =0.025[/tex]
Drilling down beneath a lake in Alaska yields chemical evidence of past changes in climate. Biological silicon, left by the skeletons of single-celled creatures called diatoms, measures the abundance of life in the lake. A rather complex variable based on the ratio of certain isotopes relative to ocean water gives an indirect measure of moisture, mostly from snow. As we drill down, we look farther into the past. Here are data from 2300 to 12,000 years ago:Isotope(%) Silicon(mg/g) Isotope(%) Silicon(mg/g) Isotope(%) Silicon(mg/g)?19.90 95 ?20.71 152 ?21.63 226?19.84 106 ?20.80 263 ?21.63 233?19.46 114 ?20.86 265 ?21.19 186?20.20 139 ?21.28 298 ?19.37 337
(b) Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)What is the correlation without this point? (Round your answer to two decimal places.)(c) Is the outlier also strongly influential for the regression line? Calculate the regression line with the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ? x ( )Calculate the regression line without the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ?( ) x
Answers
Answer:
The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.
What is the correlation with this point? (Round your answer to two decimal places.)
> cor(x,y)
[1] 0.34
What is the correlation without this point? (Round your answer to two decimal places.)
> cor(x1,y1)
[1] 0.79
Calculate the regression line with the outlier.
y = 35.2 x-523.9
Calculate the regression line without the outlier.
y = 77.9 x -1422.3
Step-by-step explanation:
We have the following data:
Isotope %: 19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37 (representing X)
Silicon : 85,152,226,106,263,233,114,265,186,139,298,337 (representing Y)
Find the single outlier in the data. This point strongly influences the correlation. What is the correlation with this point? (Round your answer to two decimal places.)
We can use the scatter plot in order to see any potential outlier. With the following R code:
> x<-c(19.90, 20.71, 21.63, 19.84, 20.80, 21.63, 19.46, 20.86, 21.19,20.20, 21.28, 19.37)
> y<-c(85,152,226,106,263,233,114,265,186,139,298,337)
> plot(x,y, main="Scatter plot Silicon vs Isotope")
And we can see the plot on the figure attached.
The outlier for this case would be (19.37 , 337) since this point is far away from the others and this point probably strongly influences the correlation value.
What is the correlation with this point? (Round your answer to two decimal places.)
> cor(x,y)
[1] 0.34
What is the correlation without this point? (Round your answer to two decimal places.)
> x1<-x[-12]
> x1
[1] 19.90 20.71 21.63 19.84 20.80 21.63 19.46 20.86 21.19 20.20 21.28
> y1<-y[-12]
> y1
[1] 85 152 226 106 263 233 114 265 186 139 298
> cor(x1,y1)
[1] 0.79
As we can see the correlation changes significantly without the outlier.
c) Is the outlier also strongly influential for the regression line? Calculate the regression line with the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ? x ( )
We can calculate the regression line with the following R code
> linearmod1<-lm(y~ x)
> linearmod1
Call:
lm(formula = y ~ x)
Coefficients:
(Intercept) x
-523.9 35.2
So our equation would be: y = 35.2 x-523.9
Calculate the regression line without the outlier. (Round your slope to two decimal places, round your y-intercept to one decimal place.)y = ( ) ?( ) x
> linearmod2<-lm(y1~x1)
> linearmod2
Call:
lm(formula = y1 ~ x1)
Coefficients:
(Intercept) x1
-1422.26 77.85
The new equation would be y = 77.9 x -1422.3
So as we can see the outlier also changes significantly the estimation for the slope and the intercept of the linear model
To answer this question one would need to identify the outlier in the data then calculate both the correlation and regression lines with and without this outlier. Changes in these calculations can demonstrate the influence of the outlier.
Explanation:This question involves finding statistical outliers and calculating correlation and regression lines in a dataset. The outlier in a dataset is a data point that is remarkably distinct from the rest of the data. As your question doesn't provide a clear dataset, it's impossible to identify the outlier clearly. However, once identified, to determine if the point strongly influences the correlation you would find the correlation with the outlier and without the outlier and compare these two values.
If the
correlation
drastically changes when the outlier is removed, it can confirm that the point is strong influencing the correlation. The
regression line
could be calculate using the standard formulas for slope and intercept, both with and without the outlier in the data. If the regression line meaningfully shifts after removing the outlier, it indicates that outlier is impactful to the regression line as well. This analytical work typically requires skills in interpreting scatterplots and using statistical programs or calculators.
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Similarly, approaching along the y-axis yields a limit equal to 0. Since these two limits are the same, we will examine another approach path. Approach (0, 0) along the curve y = x2. When x is positive, we have lim (x, y) → (0, 0) xy x2 + y2 =
Answers
Answer:
This approach to (0,0) also gives the value 0
Step-by-step explanation:
Probably, you are trying to decide whether this limit exists or not. If you approach through the parabola y=x², you get
[tex]\lim_{(x,y)\rightarrow (0,0)}\frac{xy}{\sqrt{x^2+y^2}}=\lim_{(x,x^2)\rightarrow (0,0)}\frac{xx^2}{\sqrt{x^2+(x^2)^2}}=\lim_{x\rightarrow 0}\frac{x^3}{|x|\sqrt{1+x^2}}=0[/tex]
It does not matter if x>0 or x<0, the |x| on the denominator will cancel out with an x on the numerator, and you will get the term x²/(√(1+x²) which tends to 0.
If you want to prove that the limit doesn't exist, you have to approach through another curve and get a value different from zero.
However, in this case, the limit exists and its equal to zero. One way of doing this is to change to polar coordinates and doing a calculation similar to this one. Polar coordinates x=rcosФ, y=rsinФ work because the limit will only depend on r, no matter the approach curve.
We have N cars on a circular one-way road; they have the same make, same model, same year and the same fuel economy. The total amount of gas in all cars is sufficient to make the full circle.
Prove by induction that it is always possible to find a car that can make the full circle, taking gas from other cars as it passes them.
Answers
Answer: Satisfied for n=1, n=k and n=k+1
Step-by-step explanation:
The induction procedure involves two steps
First is
Basic Step
Here we consider that for the value n=1, there is one car and it will always make the full circle.
Induction Step
Since basic step is satisfied for n=1
Now we do it for n=k+1
Now according to the statement a car makes full circle by taking gas from other cars as it passes them. This means there are cars that are there to provide fuel to the car. So we have a car that can be eliminated i.e. it gives it fuels to other car to make full circle so it is always there.
Now ,go through the statement again that the original car gets past the other car and take the gas from it to eliminate it. So now cars remain k instead of k+1 as it's fuel has been taken. Now the car that has taken the fuel can make the full circle. The gas is enough to make a circle now.
So by induction we can find a car that satisfies k+1 induction so for k number of cars, we can also find a car that makes a full circle.
In an area of the Midwest, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Which is the best predicted value for y given x = 7.3? Assume that the variables x and y have a significant correlation. 36.5 36.0 36.7 36.2
Answers
Answer:
The correct answer is 36.2
Step-by-step explanation:
Hello!
Given the data:
Rainfall(inch) x: 10.5, 8.8, 13.4, 12.5, 18.8, 10.3, 7.0, 15.6, 16.0
Yield (bushels per acre) y: 50.5, 46.2, 58.8, 59.0, 82.4, 49.2, 31.9, 76.0, 78.8
The response variable is
Y: Yield of wheat (bushels per acre)
Explanatory variable:
X: Rainfall in a certain period (inch)
I've calculated the equation of regression using a statistics software, the estimated equation is:
^Y= 4.27 + 4.38X
To calculate the value that will take the response variable for a given value of X:
^Y= 4.27 + 4.38*7.3= 36.24 bushels per acre.
I hope it helps!
Without the necessary data such as a regression equation or the correlation coefficient, we are unable to predict the yield of wheat when rainfall is 7.3 inches. A regression equation is a statistical tool that helps understand the relationship between predictor and response variables, in this case, rainfall and yield of wheat.
Explanation:In order to predict the value of yield of wheat (y) given the rainfall (x = 7.3), we would need to understand the relationship between x and y, typically via a regression equation. A regression equation is a statistical tool used to understand the relationship between predictor variables (in this case, rainfall) and response variables (in this case, yield of wheat).
However, from the details provided, it seems we do not have the necessary data to conduct a regression analysis or connect x and y values in a meaningful way. Therefore, we're unable to directly predict the specific yield of wheat when rainfall is 7.3 inches over this data gap.
An accurate value can only be predicted if we are provided with more data or the specific correlation coefficient between the two variables and the regression equation (y = ax + b).
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Solar-heat installations successfully reduce the utility bill 60% of the time. What is the probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill?
Answers
Answer:
4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.
Step-by-step explanation:
For each installation, there are only two possible outcomes. Either it reduces the utility bill, or it does not. The probabilities for each installation reducing the utility bill are independent. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
Solar-heat installations successfully reduce the utility bill 60% of the time, which means that [tex]p = 0.6[/tex]
What is the probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill?
This is [tex]P(X \geq 9)[/tex] when [tex]n = 10[/tex]. So
[tex]P(X \geq 9) = P(X = 9) + P(X = 10)[/tex]
In which
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 9) = C_{10,9}.(0.6)^{9}.(0.4)^{1} = 0.0363[/tex]
[tex]P(X = 10) = C_{10,10}.(0.6)^{10}.(0.4)^{0} = 0.0060[/tex]
So
[tex]P(X \geq 9) = P(X = 9) + P(X = 10) = 0.0363 + 0.0060 = 0.0423[/tex]
4.23% probability that at least 9/10 solar-heat installations are successful and will reduce the utility bill.
The probability of at least 90% success in solar-heat installations is found by using the binomial probability formula to calculate and add together the probabilities of exactly 9 and 10 successful installations out of 10.
Explanation:The problem in question is a classic scenario of binomial probability. Here, each solar-heat installation attempt is independent and each attempt is a success (reduces the utility bill) 60% of the time. We are interested in the probability of having 90% or more success in ten attempts.
In a binomial distribution, the formula for calculating the probability of k successes in n attempts is:
P(X=k) = C(n, k) * (p^k) * (1-p)^(n-k)
where C(n, k) is the binomial coefficient ('n choose k'), p is the probability of success on an individual trial, n is the number of trials, and k is the number of successes.
To calculate the probability that at least 9 out of 10 solar-heat installations are successful, we need to calculate P(X=9) and P(X=10) and add these probabilities together.
Calculations like these help inform decisions in a range of fields - from individual choices about energy saving at homes to policy and planning decisions at the level of energy utilization for entire nations.
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The employees of a company work in six departments: 31 are in sales, 54 are in research, 42 are in marketing, 20 are in engineering, 47 are in finance, and 58 are in production. The payroll department loses one employee's paycheck. What is the probability that the employee works in the research department?
Answers
Answer:
[tex]\frac{3}{14}[/tex]
Step-by-step explanation:
There are 252 (=31+54+42+20+47+58) employees in total. 54 of those are in research. So the chances that one check that gets lost belongs to a research employee can be calculated as follows:
[tex]P=\frac{54}{252}= \frac{3}{14}[/tex]
The amount of juice in a grapefruit is directly proportional to the cube of its diameter. Let j represent the amount of juice (in fluid ounces) in a grapefruit, let d represent the diameter (in inches) of the grapefruit, and let k be the constant of proportionality. Write an equation that relates j to d.
Answers
Answer:
The equation relating j to d can be written as:
If J = the amount of juice in fluid ounces and D = its diameter, then:
J=kD³ where k = the constant of proportionality.
The equation that represents the relationship between the amount of juice in a grapefruit (j) and its diameter (d) with constant of proportionality (k), given that they are directly proportional, is j = kd^3.
Explanation:The question given is a classic example of direct variation, more specifically, cubic direct variation since the amount of juice in a grapefruit varies with the cube of its diameter. When two quantities are directly proportional or vary directly, they form a relationship that can be described by an equation of the form y = kx^n, where k is the constant of proportionality, x is one quantity and y is the other. Here, the amount of juice in a grapefruit (j) is directly proportional to the cube of the diameter (d^3), with k being the constant of proportionality. Therefore, the equation that represents this relationship is j = kd^3.
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ln(x+1)3=3
ln
(
x
+
1
)
=
3
Question 7 options:
Answers
Answer:
x=e-1 0r 1.71828
Step-by-step explanation:
Step-by-step explanation:
[tex]In(x + 3)^{3} = 3 \\ \therefore \: 3In(x + 3) = 3 \:\\ \therefore \: In(x + 3) = \frac{3}{3} \: \\ \therefore \: In(x + 3) = 1\\ \therefore \: In(x + 3) = In \: 10\\..( \because \: In \: 10 = 1) \\ \therefore \:x + 3 = 10 \\ \therefore \:x = 10 - 3 \\ \: \: \: \: \: \huge \red{ \boxed{\therefore \:x = 7}}[/tex]
Determine whether the results below appear to have statistical significance, and also determine whether the results have practical significance.
In a study of a weight loss program, 5 subjects lost an average of 50 lbs. It is found that there is about 28% chance of getting such results with a diet that has no effect.
Part A. Does the weight loss program have statistical significance?
a. No, the program is not statistically significant because the results are likely to occur by chance.
b. Yes, the program is statistically significant because the results are unlikely to occur by chance.
c. No, the program is not statistically significant because the results are unlikely to occur by chance.
d. Yes, the program is statistically significant because the results are likely to occur by chance.
PART B. Does the weight loss program have practical significance?
a. Yes, the program is practically significant because the results are too unlikely to occur by chance.
b. No, the program is not practically significant because the amount of weight lost is trivial.
c. No, the program is not practically significant because the results are likely to occur even if the weight loss program has no effect.
d. Yes, the program is practically significant because the amount of lost weight is large enough to be considered practically significant.
Answers
Answer:
A: d. Yes, the program is statistically significant because the results are likely to occur by chance.
B: d. Yes, the program is practically significant because the amount of lost weight is large enough to be considered practically significant.
Step-by-step explanation:
A likely event is the one which is likely to occur or show the same results.
An unlikely event is one in which there's a chance of not getting the desired result or may not show the same results.
In the given scenario the program is statistically significant because even if the 28 % results occur likely by chance the rest 72 % will show some desired results.
In part B the program is statistically practical as the remaining 72 % represent a significant stats.
A bacteria population starts with 400 bacteria and grows at a rate of r(t) = (450.263)e1.12567t bacteria per hour. How many bacteria will there be after three hours?
Answers
Answer:
11,713 bacteria
Step-by-step explanation:
Integrating the growth rate function gives us the population of bacteria at any given moment 't', in hours:
[tex]r(t) = (450.263)e^{1.12567t} \\\int\ {r(t)} \, dt =P(t) = \frac{450.263}{1.12567}*e^{1.12567t} +C[/tex]
Since at t=0, P(t) = 400, the value of C is:
[tex]P(0) = \frac{450.263}{1.12567}*e^{1.12567*0} +C\\400 = 400*1+C\\C=0[/tex]
The number of bacteria after 3 hours is:
[tex]P(3) = 400*e^{1.12567*3}\\P(3) =11,713[/tex]
Consider a room that is 20 ft long, 15 ft wide, and 8 ft high. For standard sea level conditions, calculate the mass of air in the room in slugs. Calculate the weight in pounds.
Answers
Answer:
5.70456 slug
Step-by-step explanation:
Data provided in the question:
Dimensions of the room = 20 ft long, 15 ft wide, and 8 ft high
Now,
Volume of the room = 20 × 15 × 8
or
Volume of the room = 2400 ft³
we know,
Density of air = 0.0023769 slug/ft³
Therefore,
Mass of air in the room = Volume × Density
= 0.0023769 × 2400
= 5.70456 slug
The mass of air in the room is approximately 1268 slugs and the weight is approximately 40825.6 pounds.
Explanation:To calculate the mass of air in a room, we first need to find the volume of the room. The volume of a rectangular room can be calculated by multiplying its length, width, and height. So, the volume of the room is 20 ft x 15 ft x 8 ft = 2400 ft³.
Next, we need to convert the volume from cubic feet to cubic meters. Since 1 ft³ is approximately equal to 0.0283 m³, we can multiply the volume in cubic feet by 0.0283 to get the volume in cubic meters. Therefore, the volume of the room is 2400 ft³ x 0.0283 m³/ft³ = 67.92 m³.
Lastly, we need to find the mass of air in the room. The average molar weight of air is approximately 28.8 g/mol. Since the mass of one cubic meter of air is 1.28 kg, the mass of air in the room is 67.92 m³ x 1.28 kg/m³ = 86.86 kg. To convert the mass from kg to slugs, we divide it by the conversion factor of 0.0685218 slugs/kg. Therefore, the mass of air in the room is 86.86 kg / 0.0685218 slugs/kg ≈ 1268 slugs.
To calculate the weight in pounds, we multiply the mass in slugs by the acceleration due to gravity. The acceleration due to gravity is approximately 32.2 ft/s². Therefore, the weight of the air in the room is 1268 slugs x 32.2 ft/s² = 40825.6 lb.
A university's housing and residence office wants to know how much students pay per month for rent in off-campus housing. The university does not have enough on-campus housing for students, and this information will be used in a brochure about student housing. They obtain a list of the 12,304 students who live in off-campus housing and have not yet graduated and mail a questionnaire to 200 students selected at random. Only 78 questionairres are returned 8.1 (a) What is the population in this study? Be careful: about what group do they want information? (b) What is the sample? Be careful: from what group do they actually obtain information? can redefine the population about which infor- The important message in this problem is that the sample mation is obtained
Answers
Answer:
a) For this case the population of study is " the 12,304 students who live in off-campus housing and have not yet graduated " since the objective of the study is know how much students pay per month for rent in off-campus housing. So thn th epopulation size is N = 12304.
b) They select 200 students who live in off-campus housing and have not yet graduated selected using random sampling from the total population of 12304. So the sample size is n=200
As we can see in the info they got response just for 78 questionairres, so then the response rate is (78/200)*100 =39% and the non response rate is 100-39= 61.
So at the end the final sample consist on 39% of the original sample with 78 responses available for the analysis.
For this case we can redefine the sample as "The k students selected at random who live in off-campus housing and have not yet graduated and answer the mail questionairries ". And as we can see we have a better definition fo the real sample for this situation.
Step-by-step explanation:
Previous concepts
The term population represent to the total set of observations in a sample space S defined, and with an specified characteristics.
The term sample represent a set of individuals or objects who are a subsample of the population who are "collected or selected from a statistical population by a defined procedure".
Part a
For this case the population of study is " the 12,304 students who live in off-campus housing and have not yet graduated " since the objective of the study is know how much students pay per month for rent in off-campus housing. So thn th epopulation size is N = 12304.
Part b
They select 200 students who live in off-campus housing and have not yet graduated selected using random sampling from the total population of 12304. So the sample size is n=200
As we can see in the info they got response just for 78 questionairres, so then the response rate is (78/200)*100 =39% and the non response rate is 100-39= 61.
So at the end the final sample consist on 39% of the original sample with 78 responses available for the analysis.
For this case we can redefine the sample as "The k students selected at random who live in off-campus housing and have not yet graduated and answer the mail questionairries ". And as we can see we have a better definition fo the real sample for this situation.
The population in this study is the 12,304 students who live in off-campus housing and have not yet graduated. The sample is the 200 students selected at random who received the questionnaire. The sample provides information about the population of students living in off-campus housing.
Explanation:(a) Population: The population in this study refers to all the 12,304 students who live in off-campus housing and have not yet graduated. This is the group about which the university's housing and residence office wants information.
(b) Sample: The sample in this study refers to the 200 students selected at random who received the questionnaire. This is the group from which the office obtains information. The sample can be seen as a representation of the population.
The important message in this problem is that the sample can help provide information about the population of students living in off-campus housing.
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A small grocery store had 10 cartons of milk, 2 of which were sour. If you are going to buy the 6th carton of milk sold that day at random, find the probability of selecting a carton of sour milk.
Answers
Answer:
The probability that the sixth customer buys sour milk is [tex]\frac{1}{5}[/tex].
Step-by-step explanation:
The grocery store has a total of 10 cartons of milk.
The number of cartons of milk that are sour is, 2.
If none of the sour cartons of milk were bought by the first 5 buyers, then the probability of this event is:P (Both the sour cartons are available to be sold to the sixth customer)
= [tex]P(2\ sour\ cartons)=\frac{2}{5}[/tex]
If only one sour carton of milk is sold to the first 5 buyers then the probability is:P (Only one sour cartons is available to be sold to the sixth customer)
= [tex]P(1\ sour\ cartons)=\frac{1}{5}[/tex]
If both the sour carton of milk is sold to the first 5 buyers then the probability is:P (None of the sour cartons is available to be sold to the sixth customer)
= [tex]P(0\ sour\ cartons)=\frac{0}{5}[/tex]
Compute the probability that the sixth customer buys sour milk:
= P (Both sour milk is available for the 6th customer) +
P (Only one sour milk is available for the 6th customer) +
P (None of the sour milk is available for the 6th customer)
[tex]=\frac{{8\choose 5}{2\choose 0}}{{10\choose 5}} \times\frac{2}{5} +\frac{{8\choose 4}{2\choose 1}}{{10\choose 5}} \times\frac{1}{5} +\frac{{8\choose 3}{2\choose 2}}{{10\choose 5}} \times\frac{0}{5} \\=\frac{56\times2}{252\times5} +\frac{140\times1}{252\times5} +0\\=\frac{1}{5}[/tex]
Thus, the probability that the sixth customer buys sour milk is [tex]\frac{1}{5}[/tex].
A total of 584 tickets were sold for the school play. They were adult tickets or student tickets. The number of student tickets sold was three times the number of adult tickets sold. How many adult tickets were sold
Answers
Answer:
The answer to your question is 146 adult tickets
Step-by-step explanation:
Data
Total number of tickets = 584
student ticket = s
adult ticket = a
Condition
3a = s
Equation
s + a = 584
Substitution
3a + a = 584
Simplification
4a = 584
Solve for a
a = 584/4
a = 146
Find s
s = 3(146)
s = 438
Answer:146 adult tickets were sold.
Step-by-step explanation:
Let x represent the number of adult tickets that were sold.
Let y represent the number of student tickets that were sold.
A total of 584 tickets were sold for the school play. This means that
x + y = 584 - - - - - - - - - - - - - - 1
The number of student tickets sold was three times the number of adult tickets sold. This means that
y = 3x
Substituting y = 3x into equation 1, it becomes
x + 3x = 584
4x = 584
x = 584/4 = 146
y = 3x = 3 × 146
y = 438
A manufacturer can produce digital recorders at a cost of 50 dollars each. It is estimated that if the recorders are sold for p dollars apiece, consumers will buy q=120p recorders each month. a) Express the manufacturer's profit P as a function of q.
b)What is the average rate of profit obtained as the level of production increases from q=0 to q=15? (at dollars per unit)
c) At what rate is profit changing when q=15 recorders are produced? (at dollars per unit)
Answers
The manufacturer's profit is P(p) = 120p^2 - 6000p. The average profit for producing 0 to 15 recorders is 15 * 120. At 15 units, the rate of profit change is -3000.
Explanation:The function for the manufacturer's profit P is given by P(p) = pq - 50q where q=120p. Thus, P(p) = 120p * p - 50 * 120p, simplifying to P(p) = 120p^2 - 6000p.
To express the average rate of profit from q = 0 to q=15, we need to calculate P(15)-P(0) and divide it by 15 - 0. Thus, ((120 * 15^2 - 6000 * 15) - (120 * 0^2 - 6000 * 0)) / (15 - 0) = 15*120.
The rate at which profit is changing when q = 15 is the derivative of the profit function evaluated at q = 15. We obtain P'(q) = 240p - 6000, and by inserting 15 for p, we obtain -3000 (at dollars per unit).
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a) The manufacturer's profit can be expressed as P = (p - 50)q. b) The average rate of profit obtained as the level of production increases from q=0 to q=15 is p - 50 dollars per unit. c) When q=15 recorders are produced, the rate at which profit is changing is p - 50 dollars per unit.
Explanation:a) To express the manufacturer's profit P as a function of q, we can use the equation: P = (p - 50)q. This equation represents the total revenue minus the total cost, where the total revenue is the selling price p multiplied by the quantity q, and the total cost is the cost of production per unit (50 dollars) multiplied by the quantity q.
b) The average rate of profit obtained as the level of production increases from q=0 to q=15 can be found by calculating the change in profit divided by the change in quantity. In this case, the change in profit is (p - 50)(15) - (p - 50)(0) = 15(p - 50) dollars, and the change in quantity is 15 units. Therefore, the average rate of profit is (15(p - 50))/15 = p - 50 dollars per unit.
c) To find the rate at which profit is changing when q=15 recorders are produced, we can calculate the derivative of the profit function with respect to q. Taking the derivative of P = (p - 50)q with respect to q gives us dP/dq = p - 50 dollars per unit. So, when q=15, the rate at which profit is changing is p - 50 dollars per unit.
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Trying to find length and area from this triangle.
Answers
Answer:
24, 204
Step-by-step explanation:
to find the height
we would use the heron formula when all the sides are given
S which is equal to half of the perimeter of the triangle which is (a+b+c)/2.
S = (17+ 25+ 26)/2 = 68/2 = 34
Area = √S(S-A)(S-B)(S-C)
Area= (1/2)bh
we equate it back to the formula Area = √S(S-A)(S-B)(S-C)
it becomes
(1/2)bh = √S(S-A)(S-B)(S-C)
A = IABI= 25
B= IBCIM = 26
C= IACI = 17
b = base = IACI = 17
S = 34
(1/2)bh = √S(S-A)(S-B)(S-C)
(1/2)17h = √34(34-25)(34-26)(34-17)
(17/2)h = √34(9)(8)(17) = √34 x 9 x 8 x 17
(17/2)h = √41616 = 204
17h/2 = 204
17h = 204 x 2 = 408
h = 408/17 = 24 inch
height = h = IBDI = 24 in
Area = (1/2)bh
= (1/2) x 17 x 24
= 12 x 17 = 204 or we use the heron formula just like the above which we get 204 before multiplication by 2.
A local fraternity is conducting a raffle where 55 tickets are to be sold—one per customer. There are three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what are the following probabilities? (Round your answers to five decimal places.) (a) What is the probability that the four organizers win all of the prizes? (b) What is the probability that the four organizers win exactly two of the prizes? (c) What is the probability that the four organizers win exactly one of the prizes? (d) What is the probability that the four organizers win none of the prizes?
Answers
Answer:
(a) 0.0152%
(b) 1.1663%
(c) 19.4297%
(d) 79.3787%
Step-by-step explanation:
Tickets bought by organizers = 4
Number of tickets = 55
Prizes = 3
(a) The probability that the four organizers win all of the prizes is:
[tex]P = \frac{4}{55}*\frac{3}{54}*\frac{2}{53}\\P=0.0152\%[/tex]
(b) The probability that the four organizers win exactly two of the prizes is:
[tex]P = \frac{4}{55}*\frac{3}{54}*\frac{51}{53}+\frac{4}{55}*\frac{51}{54}*\frac{3}{53}+\frac{51}{55}*\frac{4}{54}*\frac{3}{53}\\P=1.1663\%[/tex]
(c) The probability that the four organizers win exactly one of the prizes is:
[tex]P = \frac{4}{55}*\frac{51}{54}*\frac{50}{53}+\frac{51}{55}*\frac{4}{54}*\frac{50}{53}+\frac{51}{55}*\frac{50}{54}*\frac{4}{53}\\P=19.4397\%[/tex]
(d) The probability that the four organizers win none of the prizes is:
[tex]P = \frac{51}{55}*\frac{50}{54}*\frac{49}{53}}\\P=79.3787\%[/tex]
Giorgio offers the person who purchases an 8-ounce bottle of Allure two free gifts, chosen from the following: an umbrella, a 1-ounce bottle of Midnight, a feminine shaving kit, a raincoat, or a pair of rain boots. If you purchased Allure, what is the probability you randomly selected an umbrella and a shaving kit in that order?a. 0.20 b. 0.00c. 0.05d. 1.00
Answers
Answer:
P(T) = 1/20 = 0.05
The probability of randomly selecting an umbrella and a shaving kit in that order is 0.05
Step-by-step explanation:
The probability of randomly selecting an umbrella and a shaving kit in that order.
P(T) = Probability of selecting umbrella first P(U) × probability of selecting shaving kit second P(S)
P(U) = 1/5 (1 umbrella out of five possible gifts)
P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)
P(T) = 1/5 × 1/4
P(T) = 1/20 = 0.05
The probability that a person will select an umbrella and a shaving kit in that order is 1/20.
The following can be deduced from the information given:
P(U) = 1/5 (1 umbrella out of five possible gifts)
P(S) = 1/4 (1 shaving kit out of four remaining possible gifts)
Therefore, the probability that a person will select an umbrella and a shaving kit in that order will be:
P(T) = 1/5 × 1/4
P(T) = 1/20 = 0.05
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Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (Enter your answer using interval notation.) (16 − t2)y' + 9ty = 5t2, y(−5) = 1
Answers
Answer: t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )
Step-by-step explanation:
The standard form of O.D.E is written as :
[tex]y^{1}[/tex] + [tex]p(t) = g(t)[/tex]
Equation given :
[tex](16-t^{2} )y^{1}[/tex] + [tex]9ty[/tex] = [tex]5t^{2}[/tex] , [tex]y(-5) = 1[/tex]
The first thing to do is to write the O.D.E in standard form , that is we will divide through by [tex]16 - t^{2}[/tex] , so we have
[tex]y^{1} + \frac{9ty}{16-t^{2}}=\frac{5t^{2}}{16-t^{2}}[/tex]
With this , we can see that [tex]p(t)[/tex] and [tex]g(t)[/tex] are both continuous in the same domain. Therefore , the intervals are :
[tex]I_{1}[/tex] = ( -∞ , -4 )
[tex]I_{2}[/tex] = ( - 4 , 4 )
[tex]I_{3}[/tex] = ( 4 , -∞ )
recall that y(−5) = 1 , then t = -5
This means that :
t = - 5 ∈ [tex]I_{1}[/tex] = ( -∞ , -4 )
1. A class of 16 students contains five Math majors, eight Engineering majors and three Physics majors. A group of four students from the class is to be selected to form a team for an academic competition.How many teams can be formed that have a representative from each major?
Answers
Answer:
495
Step-by-step explanation:
Using combination without repetition, the formular is given by n!/(r!(n-1)!
= (12 x11 ×10 ×9) /4 !
=11880/24
=495
From the group of 16 students with 5 math majors, 3 physics major and 8 engineering majors a team of 4 students where there has been a student from each majors total no. of teams that can be formed is 111.
What is combination ?Combination is selecting some things in which order does not matter.
According to the given question
A class of 16 students contains 5 Math majors, 8 Engineering majors and 3 Physics majors. A group of four students from the class is to be selected to form a team for an academic competition.
Here three cases will be formed as group is formed by 4 students and the no. of majors is 3.
First case when 2 math majors, 1 physics major and 1 engineering major is chosen from 16 students.
= ⁵C₂ + ³C₁ + ⁸C₁
= 8 + 5 + 3
= 16.
Second case when 1 math major, 2 physics major and 1 engineering major is chosen from 16 students.
= ⁵C₁ + ³C₂ + ⁸C₁
= 8 + 20 + 3
= 31.
In the third case when 1 math major, 1 physics major and 2 engineering major is chosen from 16 students.
= ⁵C₁ + ³C₁ + ⁸C₂
= 56 + 5 + 3
= 64.
∴total no of teams that can be formed is
= 16 + 31 + 64
= 111.
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class is made up of 40% women and has 20 women in it. What is the total number of students in the class?
Answers
The total number of students in the class is 50, which is calculated based on the given that 20 women make up 40% of the class.
Explanation:The question deals with the concept of percentages. In this case, the number of women in the class represents 40% of the total number of students. We have been told that there are 20 women in the class.
Here's how we can solve it step by step:
Given that 20 women represent 40% of all students in the class.So, if we want to find 100% (the total number of students), we'll divide 20 by 40 to find the value that represents 1%, which equals 0.5.Then we multiply 0.5 by 100 to get the total number of students which equals 50.Therefore, the total number of students in the class is 50.
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An electronics firm sells four models of stereo receivers, three CD decks, and six speaker brands. When the three types of components are sold together, they form a "system." How many different systems can the electronics firm offer?
A. 169
B. 72
C. 13
D. 36
Answers
Answer:
B. 72
Step-by-step explanation:
The total number of different systems that can be bundled together is the product of the possible number of ways to select 1 out of 4 stereo receivers, by 1 out of 3 CD decks and by 1 out of 6 speakers. Assuming that the order at which products are picked does not matter, the number of different systems is:
[tex]n=\frac{4!}{(4-1)!1!}*\frac{3!}{(3-1)!1!}*\frac{6!}{(6-1)!1!}\\n=4*3*6\\n=72\ systems[/tex]
The electronics firm can offer 72 different systems.
The electronics firm can offer 72 different systems by combining 4 models of stereo receivers, 3 CD decks, and 6 speaker brands together.
To find the number of different systems the electronics firm can offer, we use the combinations formula. Since there are 4 models of stereo receivers, 3 CD decks, and 6 speaker brands, the total number of different systems can be calculated as:
4 models * 3 CD decks * 6 speaker brands = 72 different systems
Therefore, the answer to the question is 72 different systems (option B).
In a Caesar Cipher system, if encryption is done by the rule c = E(p+k) = p + 9, then the letter ‘q’ is encrypted as what other letter?
Answers
Answer:
i
Step-by-step explanation:
There has to be a mod 26 operation for c, since the alphabet only has 26 letters.
But otherwise
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
p is the initial position of the letter. So for example, a = 0, b = 1, c = 2, q = 16, z = 25...
9 is by how much the encrypted character will change relative to the original.
We want to cipher the letter q.
So p = 25.
[tex]c = E(p+k) = p + 9 \mod{26}[/tex]
[tex]c = 25 + 9 \mod{26}[/tex]
[tex]c = 34 \mod{26}[/tex]
The remainder of 34 divided by 26 is 8.
So
[tex]c = 8[/tex]
Which means that q is encrypted as 'i'.
In a Caesar Cipher system, when the encryption rule is shifting by 9 places, the letter 'q' is encrypted as 'c'.
Explanation:In a Caesar Cipher system, the letter 'q' is encrypted using the formula:
c = E(p + k)
Where:
c is the ciphertext (the encrypted letter).
p is the plaintext (the original letter).
k is the encryption key.
You mentioned that the encryption is done with the rule c = p + 9. In this case, k is 9.
To encrypt the letter 'q', you would substitute p with 'q' in the formula:
c = 'q' + 9
Adding 9 to 'q' in the alphabet results in:
'q' + 9 = 'z'
So, the letter 'q' is encrypted as the letter 'z' using this Caesar Cipher system.
Therefore, the encrypted letter for 'q' in a Caesar Cipher system with a shift of 9 is 'c'.
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Describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities."x2 + y2 + 4, z = y
Answers
Answer:
Step-by-step explanation:
This is a circle with radius 2 and z = y
All points on or within the circle x2 + y2 +4 and in the plane z = y
Two cards are drawn without replacement from a standard deck of 52 playing cards. What is the probability of choosing a king for the second card drawn, if the first card, drawn without replacement, was a king? Express your answer as a fraction or a decimal number rounded to four decimal places.
Answers
Answer:
the probability of choosing a king for the second card drawn is 3/51 , if the first card, drawn without replacement, was a king
Step-by-step explanation:
defining the variable F= choosing a king in the first drawn , then the probability is
P(F)= 4/52
then using the theorem of Bayes for conditional probability and denoting the event S= choosing a king in the second drawn , then
P(S/F)= P(S∩F)/P(F)
where
P(S∩F) = probability of choosing a king in the first drawn and second drawn = 4/52* 3/51
P(S∩F) =probability of choosing a king in the second drawn given that a king was chosen in the first drawn
then
P(S/F)= P(S∩F)/P(F) = 4/52* 3/51 / 4/52 = 3/51
The probability of choosing a king for the second card drawn is 1/17 or approximately 0.0588.
Explanation:To find the probability of choosing a king for the second card drawn, given that the first card was a king, we need to consider the total number of cards and the number of kings remaining after the first card was drawn.
After the first king is drawn, there are 51 cards left in the deck, with 3 remaining kings. Therefore, the probability of choosing a king for the second card drawn is 3/51, which can be simplified to 1/17 or approximately 0.0588.
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An urn contains 7 black and 9 green balls. Six balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 6 balls drawn from the urn are green? Round your answer to three decimal places.
Answers
Step-by-step explanation:
There are 16 balls total. Since the balls are replaced after selection, the probability that the ball is green is 9/16 each time. The probability that a green ball is selected 6 times is:
P = (9/16)^6
P = 0.032
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
Determine the sample space for the experiment.
Answers
Answer:
The sample space for the experiment is {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
Step-by-step explanation:
Consider the provided information.
Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan.
The sample space for the experiment is:
{(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}
Hence, the possible outcomes are {(A,B), (A,C), (A,D), (A,E), (B,C), (B,D), (B,E), (C,D), (C,E), (D,E)}.
A regular square pyramid has a height of 6 and a base with sides of length 12. What is the: Lateral area of the pyramid?
Answers
Answer: Lateral area of the pyramid is 144
Step-by-step explanation:
The formula for determining the lateral surface area of a regular pyramid is expressed as
Lateral area = (perimeter of base x slant height of pyramid) /2
From the information given,
Length of base = 12
Since the base is a square, the perimeter of the base is
4L = 4 × 12 = 48
Height of pyramid = 6
Therefore, lateral surface area of the pyramid is
(48 × 6)/2 = 288/2 = 144
Lateral area of square pyramid with height 6 and base side 12 is 144 square units.
To find the lateral area of a square pyramid, you need to calculate the area of each triangular face and then sum them up.
1. Find the slant height (l):
In a regular square pyramid, the slant height (l) can be found using the Pythagorean theorem. Each triangular face is an isosceles right triangle. So,
[tex]\[ l = \sqrt{(s/2)^2 + h^2} \][/tex]
Where:
[tex]- \( s \) is the length of one side of the square base (given as 12 in this case).- \( h \) is the height of the pyramid (given as 6).[/tex]
[tex]\[ l = \sqrt{(12/2)^2 + 6^2} \]\[ l = \sqrt{(6)^2 + 6^2} \]\[ l = \sqrt{36 + 36} \]\[ l = \sqrt{72} \]\[ l = 6\sqrt{2} \][/tex]
2. **Calculate the lateral area (A):**
Each triangular face has a base of length 12 and a height of 6 (same as the height of the pyramid). So, the area of each triangular face (A) can be calculated as:
Since there are four triangular faces in a square pyramid, the total lateral area (LA) is:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]\[ A = \frac{1}{2} \times 12 \times 6 \]\[ A = 36 \][/tex] [tex]\[ LA = 4 \times A \]\[ LA = 4 \times 36 \]\[ LA = 144 \][/tex]
So, the lateral area of the square pyramid is 144 square units.
Ralph wants to estimate the percentage of coworkers that use the company's healthcare. He asks a randomly selected group of 200 coworkers whether or not they use the company's healthcare. What is the parameter?
a.the percentage of surveyed coworkers that use the company's healthcare
b.the 200 coworkers surveyed
c.specific "yes" or "no" responses to the survey
d.all coworkers that use the company's healthcare
e.the percentage of all coworkers that use the company's healthcare
Answers
Answer:
e.
Step-by-step explanation:
The parameter is used to measure the population or in other words the measurement taken from a population is known as parameter. Here, the population will be all the co-workers that use company's healthcare. The percentage calculated for all co-workers that use company's healthcare will be the measurement calculated from population. Thus, the parameter will be the percentage of all the co-workers that use the company's healthcare.
Final answer:
The parameter Ralph is trying to estimate is 'e. the percentage of all coworkers that use the company's healthcare,' which reflects the actual percentage for the entire population of coworkers, not just the surveyed sample.
Explanation:
The question posed by the student is concerned with identifying the parameter involved in a statistical study. In the scenario provided, Ralph wants to estimate the percentage of coworkers that use the company's healthcare by surveying a random sample of 200 coworkers. The parameter, in this case, is the actual percentage of all coworkers at the company who use the company's healthcare, because it describes the entire population that is being studied, not just the sample that was surveyed. Therefore, the correct answer is e. the percentage of all coworkers that use the company's healthcare.
A parameter is a summary measure that describes an entire population, whereas a statistic is a summary measure that describes a sample taken from the population. An example that illustrates this distinction is an insurance company determining the proportion of all medical doctors who have been involved in malpractice lawsuits by surveying a random sample of 500 doctors. The true proportion of all doctors who have faced lawsuits is a parameter, while the proportion calculated from the sample of 500 doctors is a statistic.