Answer:
0.1724 or 17.24%
Step-by-step explanation:
There are
10+13+7 = 30 people
so there are C(30,5) combinations of 30 elements taken 5 at a time ways to select groups of 5 people.
[tex]C(30,5)=\displaystyle\binom{30}{5}=\frac{30!}{5!25!}=142,506[/tex]
There are C(10,2) combinations of 10 elements taken 2 at a time ways to select the 2 Democrats
[tex]C(10,2)=\displaystyle\binom{10}{2}=\frac{10!}{2!8!}=45[/tex]
There are C(13,2) combinations of 13 elements taken 2 at a time ways to select the 2 Republicans
[tex]C(13,2)=\displaystyle\binom{13}{2}=\frac{13!}{2!11!}=78[/tex]
There are C(7,1) combinations of 7 elements taken 1 at a time ways to select the Independent
[tex]C(7,1)=\displaystyle\binom{7}{1}=\frac{7!}{1!6!}=7[/tex]
By the Fundamental Principle of Counting, there are
45*78*7 = 24,570
ways of making the group of 5 people with 2 Democrats, 2 Republicans and 1 independent, and the probability of forming the group is
[tex]\displaystyle\frac{24,570}{142,506}=0.1724[/tex] or 17.24%
Suppose that blood chloride concentration (mmol/L) has a normal distribution with mean 104 and standard deviation 5
a. What is the probability that chloride concentration equals 105? b. What is the probability that chloride concentration is less than 105?
b. What is the probability that chloride concentration differs from the mean by more than 1 standard deviation?
Answer:
a) 0
b) 0.579 is the probability that chloride concentration is less than 105.
c) 0.32 is the probability that chloride concentration differs from the mean by more than 1 standard deviation.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 104
Standard Deviation, σ = 5
We are given that the distribution of blood chloride concentration is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) Since normal distribution s a continuous distribution, the probability for a particular value is zero. Therefore,
[tex]P(x =105) = 0[/tex]
b) P(chloride concentration is less than 105)
[tex]P(x < 105) = P(z < \displaystyle\frac{105-104}{5}) = P(z < 0.2)[/tex]
Calculating the value from the standard normal table we have,
[tex]P(z<0.2) = 0.579 = 57.9\%\\P( x < 105) = 57.9\%[/tex]
c) P(chloride concentration differs from the mean by more than 1 standard deviation)
Since it is a normal distribution, the Empirical rule shows that 68% falls within the first standard deviation, 95% within the first two standard deviations, and 99.7% within the first three standard deviations.
= 1 - P(chloride concentration within the mean by 1 standard deviation)
= 1 - 0.68 = 0.32
Final answer:
The probability of a single value is 0 in a normal distribution. The probability that the chloride concentration is less than 105 mmol/L can be found using the z-score and CDF of the normal distribution. The probability of the concentration being more than one standard deviation from the mean involves calculating the z-scores for one standard deviation away from the mean and using standard normal distribution probabilities.
Explanation:
The question involves the application of probability theory and statistics, specifically the properties of the normal distribution. Here are the answers to the student's questions:
(a) In a normal distribution, the probability of a continuous random variable taking on any single value is technically zero because there are an infinite number of potential values it could take. Therefore, the probability that the chloride concentration equals exactly 105 mmol/L is 0.
(b) To find the probability that the chloride concentration is less than 105 mmol/L, we use the cumulative distribution function (CDF) for a normal distribution. The z-score is calculated by subtracting the mean from the observed value and dividing by the standard deviation: z = (105 - 104) / 5 = 0.20. Using standard normal distribution tables or a calculator, we find the probability corresponding to a z-score of 0.20.
(b) The probability that the chloride concentration differs from the mean by more than 1 standard deviation involves finding the probability of being outside the interval [99, 109], since one standard deviation from the mean is 104 ± 5. We calculate the z-scores for 99 and 109, which are -1 and +1, respectively. Using the normal distribution, we find the probabilities associated with these z-scores and then subtract them from 1 to get the probability of being more than one standard deviation away from the mean.
In an agricultural study, the average amount of corn yield is normally distributed with a mean of 185.2 bushels of corn per acre, with a standard deviation of 23.5 bushels of corn. If a study included 1200 acres, about how many would be expected to yield more than 190 bushels of corn per acre?
Out of 1200 acres, approximately 504 of them would be expected to yield more than 190 bushels of corn per acre, assuming a normal distribution of yields with a mean of 185.2 bushels and a standard deviation of 23.5 bushels.
Explanation:Your question pertains to the field of statistics and normal distribution. When given a mean of 185.2 bushels and a standard deviation of 23.5 bushels for corn yield, you're interested in knowing how many acres out of the total surveyed, in this case 1200 acres, would yield more than 190 bushels per acre.
We first need to calculate the z-score of the 190 bushels. The z-score is the number of standard deviations from the mean a data point is. The formula for a z-score is:
Z = (X - μ) / σ
In this instance, X is the value we are comparing to the mean (190), μ represents the mean (185.2), and σ is the standard deviation (23.5). Plugging these values in gives us a z-score of approximately 0.204.
Now, we consult a standard z-table, which gives us the area to the left of the z-score under a standard normal distribution curve. If we look up the z-score of 0.204, we'd find that the proportion of data less than this score is approximately 0.5803 (i.e., 58.03%). So, the proportion that is higher (expected to yield more than 190 bushels) is about 41.97%.
If we apply this percentage to the number of acres in our study (1200), we find that approximately 504 acres are expected to yield more than 190 bushels of corn per acre.
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According to a survey, 67% of murders committed last year were cleared by arrest or exceptional means. Fifty murders committed last year are randomly selected, and the number cleared by arrest or exceptional means is recorded.
a) Find the probability that exactly 41 murders were cleared.
b)The probability that between 36 and 38 of the murders, inclusive were cleared is?
c) Would it be unusual if fewer than 20 of the murders were cleared? why or why not?
Answer:
a) P(X=41)= 0.0086 = 0.86%
b) P(36≤X≤38)== 0.1215 = 12.15%
c) would be unusual that fewer that 20 murders were cleared because P(X≤20) = 0.000081 =0.0081%
Step-by-step explanation:
if the random variable X= number of cleared murders , then X follows a binomial distribution:
P(X=x)= n!/[(n-x)!*x!]*p^x*(1-p)^(n-x)
where P(X=x)= probability of x cleared murders, n= number of murders selected=50 , p= probability for a murder to be cleared (67%)
therefore
a) P(X=41)=50!/[(50-41)!*41!]*0.67^41*0.33^(50-41)= 0.0086
b) P(36≤X≤38)= P(X≤38) - P(X≤36) = 0.9371- 0.8156 = 0.1215
c) P(X≤20) = 0.000081
therefore would be unusual that fewer that 20 murders were cleared
Consider a triangle ABC for which ∠A=100∘,a=34,b=13. If such a triangle can not exist, then write NONE in each answer box. If there could be more than one such triangle, then enter dimensions for the one with the smallest value for side c. Finally, if there is a unique triangle ABC, then enter its dimensions.
B is _______ degrees;
∠C is_________ degrees;
c= _________.
Answer:
B is 22.12 degrees; ∠C is 57.88°; c=29.24
Step-by-step explanation:
So, first, it's important to draw a diagram of the triangle the problem is talking about (see attached picture).
Once the triangle has been drawn, we can visualize it better and determine what to do. So first, we are going to find what the value of angle B is by using law of sines:
[tex]\frac{sin B}{b}=\frac{sin A}{a}[/tex]
which can be solved for angle B:
[tex]sin B=b\frac{sin A}{a}[/tex]
[tex] B= sin^{-1}(b\frac{sin A}{a})[/tex]
and substitute the values we already know:
[tex] B= sin^{-1}(13\frac{sin 100 ^{o}}{34})[/tex]
which yields:
B=22.12°
Once we know what the angle of B is, we can now find the value of angle C by using the fact that the sum of the angles of any triangle is equal to 180°. So:
A+B+C=180°
When solving for C we get:
C=180°-A-B
C=180°-22.12°-|00°=57.88°
So once we know what angle C is, we can go ahead and find the length of side c by using the law of sines again:
[tex]\frac{c}{sin C}=\frac{a}{sin A}[/tex]
and solve for c:
[tex]c=sin C \frac{a}{sin A}[/tex]
so we can now substitute for the values we already know:
[tex]c=sin(57.88^{o})\frac{34}{sin(100^{o})}[/tex]
which yields:
c=29.24
A surveying team set up some equipment 104104 ft from the base of a tree in order to sight the top of the tree. From ground level, they measure the angle of elevation to be 27.5∘27.5∘. If the calculated height needs to be accurate to within 1.51.5%, what is the allowed error in the angle measurement? (Assume the 104104 ft measurement is 100% accurate.)
In order to estimate the allowed angle error when measuring the height of a tree, you should first use the given distance and the angle of elevation to calculate the tree's height. Then calculate the potential height range considering the 1.5% error, and figure out the corresponding angles with these heights. The angle error can be estimated by the difference between these two angles.
Explanation:To solve this problem, one can use the principles of trigonometry. Specifically, the tangent function in trigonometry (tan θ = height / distance) can be used to estimate the height of the tree. Here, the distance from the tree (104 feet) and the angle of elevation (27.5°) are given.
We must first calculate the tree height using tan θ = height / distance. After that, we find that the tree is approximately 58.4 ft tall.
In this case, a 1.5% error in the height measurement would be 0.015 * 58.4 ft = 0.876 feet. Using the equation θ = arctan (height / distance), we can figure out the angle error. Enter the heights 58.4 + 0.876 ft and 58.4 - 0.876 into the equation and use the distance 104104 ft to calculate the corresponding angles. The allowed error for the angle follows as the difference of these two angles.
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To ensure the tree height is within 1.5% accuracy, we calculated that the allowable error in the angle measurement is ±0.22 degrees. This was determined by converting the tolerable height range back to angles using inverse tangent. The primary method used is trigonometry.
To determine the allowed error in the angle measurement for a tree with a height accurate to within 1.5%, we follow these steps:
Let's denote the height of the tree as h, the measured distance from the tree as d = 104 ft, and the angle of elevation as θ = 27.5∘.Using the tangent function: tan(θ) = h / d, we find the height h using trigonometry: h = d × tan(θ)Given that d = 104 ft and tan(27.5°) ≈ 0.5197, we can calculate: h ≈ 104 × 0.5197 ≈ 54.05 ft
To find the acceptable range of h within 1.5% accuracy: h ± 0.015h:So, 54.05 ft ± 0.015 × 54.05 ft ≈ 54.05 ft ± 0.81 ft
Tree Height thus ranges from approximately 53.24 ft to 54.86 ft.
Next, we need to determine the allowed error in the angle θ:We convert the bounds of height back to angles using inverse tangent:
For minimum height: tan⁻¹(53.24/104) ≈ 27.28°For maximum height: tan⁻¹(54.86/104) ≈ 27.72°The allowed error in the angle measurement is the difference from the original angle:
± |27.5° - 27.28°| ≈ ±0.22° and ± |27.72° - 27.5°| ≈ ±0.22°
Therefore, the allowable error in the angle measurement to ensure a height accuracy within 1.5% is ±0.22 degrees.
It is known that x=7 is a root of the equation ax^2+bx+2=0, where a<0. Solve the inequality ax^4+bx^2+2>0.
Answer:
|x| < √7
Step-by-step explanation:
The product of the roots of the given quadratic equation is 2/a, so the other root (the one not given) is 2/(7a). It will be negative, since "a" is negative.
The roots of ax^2 +bx +2 = 0 are the values of x^2 that satisfy ax^4 +bx^2 +2 = 0. That is, roots of the latter equation will be the square root of the roots of the former equation.
We know that two of the zeros of the quartic are ±√7, and the other two are complex, as they are the square roots of a negative number. So, the graph of the quartic opens downward (because a < 0), and has real zeros at x=±√7.
The solution to the inequality must be ...
-√7 < x < √7
_____
The graph shows an example of the quadratic (green) and quartic (black). The ripple in the quartic changes amplitude with different values of "a", but the locations of the zeros do not change.
A researcher at a major hospital wishes to estimate the proportion of the adult population of the United States that has high blood pressure. How large must the sample be in order to be 90% confident that the sample proportion will not differ from the true proportion by more than 5%?
The researcher would need a sample size of 271 individuals to be 90% confident that the sample proportion will not differ from the true proportion of adults
How to find the sample size ?
To determine the sample size needed to estimate the proportion of the adult population of the United States with high blood pressure with a 90% confidence level and a margin of error (precision) of 5%, you can use the formula for sample size for estimating proportions. The formula is:
[tex]\[n = \frac{Z^2 \cdot p(1-p)}{E^2}\][/tex]
E is the margin of error as a proportion (in this case, 5% or 0.05).
Plug in the values:
[tex]\(n = \frac{(1.645)^2 \cdot 0.5(1-0.5)}{(0.05)^2}\)[/tex]
[tex]\(n = \frac{2.706025 \cdot 0.25}{0.0025}\)[/tex]
[tex]\(n = \frac{0.67650625}{0.0025}\)[/tex]
n = 270.6025
= 271 people
An electric scale gives a reading equal to the true weight plus a random error that
is normally distributed with mean 0 and standard deviation s = .1 mg. Suppose
that the results of five successive weighings of the same object are as follows: 3.142,
3.163, 3.155, 3.150, 3.141.
(a) Determine a 95 percent confidence interval estimate of the true weight.
(b) Determine a 99 percent confidence interval estimate of the true weight.
The true weight, according to a 95% confidence interval, is likely between 3.062 mg and 3.238 mg. According to a 99% confidence interval, it's likely between 3.035 mg and 3.265 mg.
Explanation:This question is about finding the confidence interval of the true weight based on a series of measurements and given the standard deviation of the instrument's error is 0.1 mg.
The error is normally distributed, thus the average of the measurements is likely to be the true value, and can be calculated by adding the measurements up and divide by the number of measurements, in this case, 5. The result is 3.1502.
(a) To determine a 95 percent confidence interval, we need to use a standard Z-score, which for 95% is ±1.96. The standard error is the standard deviation divided by the square root of the number of measurements, in this case, 0.1/√5 = 0.04472. The 95% confidence interval can be found by multiplying the standard error by the Z-score and subtracting/adding this from/to the mean. In this case, the interval is [3.1502 -1.96(0.04472), 3.1502 + 1.96(0.04472)] = [3.062, 3.238] mg.
(b) For a99 percent confidence interval, the Z-score is 2.576. The calculations are the same as for part (a), resulting in the interval [3.1502 - 2.576(0.04472), 3.1502 + 2.576(0.04472)] = [3.035, 3.265] mg.
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The 95% and 99% confidence intervals for the true weight based on the five readings and the given standard deviation can be calculated using the average weight and the concept of standard error in combination with the respective z-scores for 95% and 99%.
First, calculate the average weight ([tex]\bar{x}[/tex]) from the 5 readings. That is (3.142 + 3.163 + 3.155 + 3.150 + 3.141) / 5 = 3.1502 mg. Our goal is to find the confidence interval around this average.
(a) For a 95 percent confidence interval, we use a z-score of 1.96 (which corresponds to 95% in a standard normal distribution). The standard error (SE) is the standard deviation (s) divided by the square root of the sample size (n), i.e., SE = s / sqrt(n) = 0.1 / sqrt(5). Hence, the 95 percent confidence interval is [tex]\bar{x}[/tex] ± z * SE = 3.1502 ± 1.96 * SE.
(b) For a 99 percent confidence interval, we use a z-score of 2.576. So, the 99 percent confidence interval is [tex]\bar{x}[/tex] ± z * SE = 3.1502 ± 2.576 * SE.
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The number of calories consumed by customers at the Chinese buffet is normally distributed with mean 2689 and standard deviation 560. One randomly selected customer is observed to see how many calories X that customer consumes. Round all answers to 4 decimal places where possible. What is the distribution of X?
Answer: [tex]X\sim N(2689,\560)[/tex]
Step-by-step explanation:
Let X be a random variable that represents the number of calories consumed.
Given : The number of calories consumed by customers at the Chinese buffet is normally distributed with mean 2689 and standard deviation 560.
i.e. Population mean : [tex]\mu=2689[/tex]
Population standard deviation: [tex]\sigma=560[/tex].
We can say X follows normal distribution [tex]\mu=2689[/tex] and [tex]\sigma=560[/tex]
So , the distribution of X : [tex]X\sim N(\mu,\ \sigma)[/tex]
i.e. [tex]X\sim N(2689,\560)[/tex]
The consumer demand equation for tissues is given by q = (108 − p)2, where p is the price per case of tissues and q is the demand in weekly sales.
(a) Determine the price elasticity of demand E when the price is set at $32. (Round your answer to three decimal places.) E = Interpret your answer. The demand is going by % per 1% increase in price at that price level.
(b) At what price should tissues be sold to maximize the revenue? (Round your answer to the nearest cent.) $
(c) Approximately how many cases of tissues would be demanded at that price? (Round your answer to the nearest whole number.) cases per week
Answer:
a) -0.842
b) $0
c) 11,664 cases
Step-by-step explanation:
a)
The price elasticity of demand E at a given point [tex]\large (p_1,q_1)[/tex] is defined as
[tex]\large E=\frac{p_1}{q_1}.\frac{\text{d}q}{\text{d}p}(p_1)[/tex]
and in this case, it would measure the possible response of tissues demand due to small changes in its price when the price is at [tex]\large p_1[/tex]
When the price is set at $32 the demand is
[tex]\large q=(108-32)^2=5,776 [/tex]
cases of tissues, so
[tex]\large (p_1,q_1)=(32,5776)\Rightarrow \frac{p_1}{q_1}=\frac{32}{5776}=0.00554[/tex]
Also, we have
[tex]\large \frac{\text{d}q}{\text{d}p}=-2(108-p)\Rightarrow \frac{\text{d}q}{\text{d}p}(32)=-2(108-32)=-152[/tex]
hence
[tex]\large E=\frac{p_1}{q_1}.\frac{\text{d}q}{\text{d}p}(p_1)=0.00554(-152)=-0.842[/tex]
That would mean the demand is going down about 0.842% per 1% increase in price at that price level.
b)
When the price is $108 the demand is 0, so the price should always be less than $108.
On the other hand, the parabola
[tex]\large q=(108-32)^2=5,776 [/tex] is strictly decreasing between 0 and 108, that means the maximum demand would be when the price is 0.
c)
When the price is 0 the demand is
[tex]\large (108)^2=11,664[/tex] cases
The elasticity for p = 32 is E = -0.84.
The maximum revenue is obtained when p = 32, and the demand for that price is q = 5,184.
How to determine the price elasticity?
It is given by:
[tex]E = \frac{p}{q} *\frac{dq}{dp} (p)[/tex]
if p = $32, then:
q(32) = (108 − 32)^2 = 5,776
and:
dq/dp = -2*(108 - p)
Evaluating that in p = 32 we get:
-2*(108 - 32) = -152
Then the elasticity is:
E = (32/5,776)*-152 = -0.84.
How to maximize the revenue?The revenue is equal to the demand times the price, so we get:
R = p*q = p*(108 - p)^2 = p*(p^2 - 216p + 11,664)
To maximize this we need to find the zeros of the derivation, we have:
R' = 3p^2 - 2*216*p + 11,664
The zeros of that equation are given by:
[tex]p = \frac{432 \pm \sqrt{(-432)^2 - 4*3*11,664} }{2*3} \\\\p = (432 \pm 216)/6[/tex]
Notice that we need to use the smaller of these values, so the demand never becomes zero, then we use:
p = (432 - 216)/6 = 36
(the other root gives p = 108, so that is a minimum).
c) For this price, the number of cases demanded is:
q = (108 - 36)^2 = 5,184
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When you sample the first individuals you can find, you call it a _____; it's cheap and easy to do, but statistically not a very strong method.
A.
cluster
B.
stratified random sample
C.
convenience sample
D.
cluster sample
E.
simple random sample
Answer:
E.
simple random sample
Step-by-step explanation:
Answer:
C.
convenience sample
Step-by-step explanation:
The amount of cola in a 355 ml bottle from a certain company is a random variable with a mean of 355 ml and a standard deviation of 2 ml. For a sample of size 32, perform the following calculations. a. Find an approximate probability that the sample mean is less than 354.8 ml. b. Suppose the amount of cola is distributed as N(355, 4). Find an approximate probability that 10 of the bottles in the sample contain less than 354.8 ml of cola.
The probability of the sample mean being less than 354.8 ml can be calculated using the z-score and the standard normal distribution. For the second part, if the standard deviation is indeed 4, it's necessary to calculate the probability for a single bottle first, then apply the binomial distribution to find the probability for 10 out of 32 bottles.
Explanation:The calculation of the probability that the sample mean is less than 354.8 ml when the population mean is 355 ml and the standard deviation is 2 ml for a sample size of 32 uses the standard normal distribution. Applying the Central Limit Theorem, we can find the z-score and use it to calculate the probability.
For part b, assuming the normal distribution N(355, 4), to find the probability that 10 out of 32 bottles contain less than 354.8 ml, we would use the binomial distribution with the probability calculated in part a. However, the second part of question b appears to have a typo with the standard deviation. If the distribution is N(355, 4), we would need to first find the probability of a single bottle having less than 354.8 ml, then use the binomial formula to find the probability of 10 bottles having less than that amount.
Suppose a sociologist is interested in testing if the average age at puberty of girls living is Thailand is different from the average age of puberty of girls living in Japan. To test her theory, she selects a random sample of 50 young Thai women and 50 young Japanese women and determines that the average age at puberty of the women in Japan is 1.7 years younger than the average age at puberty of the Thai women. Which of the following would change if instead the average age at puberty for Japanese women is 0.8 years younger?a) The p-value would increase.b) The p-value would decrease.c) There would be no difference because Japanese women still went through puberty at a younger age.d) The sample standard deviation would be larger if the difference were less.e) Exactly 2 of the above are correct.
Answer:
Option b is right
Step-by-step explanation:
Given that a sociologist is interested in testing if the average age at puberty of girls living is Thailand is different from the average age of puberty of girls living in Japan. To test her theory, she selects a random sample of 50 young Thai women and 50 young Japanese women and determines that the average age at puberty of the women in Japan is 1.7 years younger than the average age at puberty of the Thai women
If average of Japanese women is decreased by 0.8 years, we would get mean difference = [tex]1.7+0.8=2.5[/tex]
Since mean difference increases test statistic would increase thus reducing p value, i.e. chances of rejecting H0 higher.
The sample std devition would not change because of this.
Evaluate f(-12) given f(x) = -21x - 12
Answer:
The answer for the given function is f( -12 ) = 240.
Step-by-step explanation:
Given:
[tex]f(x)=-21x-12[/tex]
To Find :
F ( -12 ) = ?
Solution:
Function:
A function is like a machine that gives an output for a given input. A function has an independent variable which is called the input of the function. The output for a given input is called the dependent variable.For Example
f ( x ) = x + 3
x = independent variable
f ( x ) = Output for a given input
Here we have
[tex]f(x)=-21x-12\\Put\ x=-21\\\therefore f(-12) = -21\times -12 - 12\\\therefore f(-12) = 252 - 12\\\therefore f(-12) = 240[/tex]
Therefore f(-12) = 240
Answer for f(-12) = 240 for function f(x) = -21x -12
Step-by-step explanation:
Theory: f(x) is the" Function " and defined as a relation between the set of input values and a respective set of output values. set of input values is term as " Domain " and a respective set of output values is term as " Range "
In given question, f(x) = -21x -12
Note: Question didn't mention the domain so that, we can take it as Real numbers.
To find value of f( -12 )
Replacing value of x with (-12) in f(x)
we get,
f(-12) = -21(-12) -12
f(-12) = 240.
A tank with a capacity of 1600 L is full of a mixture of water and chlorine with a concentration of 0.0125 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 16 L/s. The mixture is kept stirred and is pumped out at a rate of 40 L/s. Find the amount of chlorine in the tank as a function of time. (Let y be the amount of chlorine in grams and t be the time in seconds.)
Answer:
y(t) = 20 [1600^(-5/3)] x (1600-24t)^ (5/3)
Step-by-step explanation:
1) Identify the problem
This is a differential equation problem
On this case the amount of liquid in the tank at time t is 1600−24t. (When the process begin, t=0 ) The reason of this is because the liquid is entering at 16 litres per second and leaving at 40 litres per second.
2) Define notation
y = amount of chlorine in the tank at time t,
Based on this definition, the concentration of chlorine at time t is y/(1600−24t) g/ L.
Since liquid is leaving the tank at 40L/s, the rate at which chlorine is leaving at time t is 40y/(1600−24t) (g/s).
For this we can find the differential equation
dy/dt = - (40 y)/ (1600 -24 t)
The equation above is a separable Differential equation. For this case the initial condition is y(0)=(1600L )(0.0125 gr/L) = 20 gr
3) Solve the differential equation
We can rewrite the differential equation like this:
dy/40y = - (dt)/ (1600-24t)
And integrating on both sides we have:
(1/40) ln |y| = (1/24) ln (|1600-24t|) + C
Multiplying both sides by 40
ln |y| = (40/24) ln (|1600 -24t|) + C
And simplifying
ln |y| = (5/3) ln (|1600 -24t|) + C
Then exponentiating both sides:
e^ [ln |y|]= e^ [(5/3) ln (|1600-24t|) + C]
with e^c = C , we have this:
y(t) = C (1600-24t)^ (5/3)
4) Use the initial condition to find C
Since y(0) = 20 gr
20 = C (1600 -24x0)^ (5/3)
Solving for C we got
C = 20 / [1600^(5/3)] = 20 [1600^(-5/3)]
Finally the amount of chlorine in the tank as a function of time, would be given by this formula:
y(t) = 20 [1600^(-5/3)] x (1600-24t)^ (5/3)
An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 3 km east of the refinery. The cost of laying pipe is $400,000 per km over land to a point P on the north bank and $800,000 per km under the river to the tanks. To minimize the cost of the pipeline, how far from the refinery should P be located? (Round your answer to two decimal places.)
Answer:
P is exactly 3km east from the oil refinery.
Step-by-step explanation:
Let's d be the distance in km from the oil refinery to point P. So the horizontal distance from P to the storage is 3 - d and the vertical distance is 2. Hence the diagonal distance is:
[tex]\sqrt{(3 - d)^2 + 2^2} = \sqrt{(3 - d)^2 + 4}[/tex]
So the cost of laying pipe under water with this distance is
[tex]800000\sqrt{(3 - d)^2 + 4}[/tex]
And the cost of laying pipe over land from the refinery to point P is 400000d. Hence the total cost:
[tex]800000\sqrt{(3 - d)^2 + 4} + 400000d[/tex]
We can find the minimum value of this by taking the 1st derivative and set it to 0
[tex]800000\frac{2*0.5*(3-d)(-1)}{\sqrt{(3 - d)^2 + 4}} + 400000 = 0[/tex]
We can move the first term over to the right hand side and divide both sides by 400000
[tex]1 = 2\frac{3 - d}{\sqrt{(3 - d)^2 + 4}}[/tex]
[tex]\sqrt{(3 - d)^2 + 4} = 6 - 2d[/tex]
From here we can square up both sides
[tex](3 - d)^2 + 4 = (6 - 2d)^2[/tex]
[tex]9 - 6d + d^2 + 4 = 36 - 24d + 4d^2[/tex]
[tex]3d^2-18d+27 = 0[/tex]
[tex]d^2 - 6d + 9 = 0[/tex]
[tex](d - 3)^2 = 0[/tex]
[tex]d -3 = 0[/tex]
d = 3
So the cost of pipeline is minimum when P is exactly 3km east from the oil refinery.
A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed that 15% percent of all teenagers smoke. A Gallup survey interviewed a nationally representative sample of 785 teenagers aged 13 to 17. Seventy-one teenagers in the survey acknowledged having smoked at least once in the past week. Does the study provide adequate evidence to conclude that the percentage of teenagers who smoke is now different than 15%
1. Are the requirements for both a confidence intervale and a hypothesis test met?
2. Determine the appropriate 95% confidence intervale for the true proportion of teenage smokers. Input your answers for the margin of error, lower bound and upper bound.
Final answer:
The requirements for constructing a 95% confidence interval and performing a hypothesis test are met. The 95% confidence interval for the true proportion of teenage smokers is approximately 6.99% to 11.1%, indicating that the proportion may have changed from the previous 15%.
Explanation:
To ascertain if there is sufficient evidence to conclude that the percentage of teenagers who smoke has changed from 15%, we should conduct both a confidence interval analysis and a hypothesis test.
Requirements for Confidence Interval and Hypothesis Testing
Both require a random sample and a sufficiently large sample size to warrant using a normal approximation for the distribution of sample proportions. For a 95% confidence interval and hypothesis testing regarding proportions, the sample should satisfy the conditions np≥15 and n(1-p)≥15 where n is the sample size and p is the hypothesized proportion.
In this case, with a sample size of 785 and the previous proportion of 0.15, the conditions 785(0.15)≥ 15 and 785(0.85)≥ 15 are both met, satisfying the criteria to proceed with the confidence interval and hypothesis test.
95% Confidence Interval for True Proportion
To calculate the 95% confidence interval for the proportion of teenage smokers, use the formula:
Proportion = x/n
Margin of Error (ME) = z*(√[p(1-p)/n])
Lower Bound = Proportion - ME
Upper Bound = Proportion + ME
Where:
x is the number of successes in the sample (71 teenagers acknowledged smoking)
n is the sample size (785)
z is the z-score corresponding to the desired confidence level (approximately 1.96 for 95%)
Substituting in the values we get:
Proportion = 71/785 = 0.0904
ME = 1.96*√[0.0904(1-0.0904)/785] ≈ 0.0205
Lower Bound = 0.0904 - 0.0205 ≈ 0.0699
Upper Bound = 0.0904 + 0.0205 ≈ 0.111
Based on this confidence interval, we have evidence that the proportion of smokers may differ from the previous proportion of 15% as it does not include 0.15 in the interval.
Julie has $48 to spend at a carnival. The carnival charges $8 for admission and $5 per ride. What is the maximum number of rides Julie can go on?
Answer:
8
Step-by-step explanation:48 is the total amount of money Julie has it's 8 dollars for admission and 5 dollars per ride subtract the 8 from 48 which will leave you 40 dollars left now divide that from g and the total of ride juile will be able to ride is 8
Suppose the force acting on a column that helps to support a building is a normally distributed random variable X with mean value 15.0 (kips) and standard deviation 1.25 (kips). Calculate the following probabilities: (1) (2 points) P(X ≥ 17.5). (2) (2 points) P(14 ≤ X ≤ 18).
Answer:
(1) 0.4207
(2) 0.7799
Step-by-step explanation:
Given,
Mean value,
[tex]\mu = 15.0[/tex]
Standard deviation,
[tex]\sigma = 1.25[/tex]
(1) P(X ≥ 17.5) = 1 - P( X ≤ 17.5)
[tex]= 1- P(\frac{x-\mu}{\sigma} \leq \frac{17.5-\mu}{\sigma})[/tex]
[tex]=1-P(z\leq \frac{17.5 - 15}{1.25})[/tex]
[tex]=1-P(z\leq \frac{2.5}{1.25})[/tex]
[tex]=1-P(z\leq 2)[/tex]
[tex]=1- 0.5793[/tex] ( By using z-score table )
= 0.4207
(2) P(14 ≤ X ≤ 18) = P(X ≤ 18) - P(X ≤ 14)
[tex]=P(z\leq \frac{18 - 15}{1.25}) - P(z\leq \frac{14 - 15}{1.25})[/tex]
[tex]=P(z\leq \frac{3}{1.25}) - P(z\leq -\frac{1}{1.25})[/tex]
[tex]=P(z\leq 2.4) - P(z\leq -0.8)[/tex]
= 0.9918 - 0.2119
= 0.7799
The probabiliity that the force is greater than 17.5 kips is 2.28% while the probabiliity that the force is between 14 and 18 kips is 20.37%
What is z score?
Z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:
z = (raw score - mean) / standard deviation
Mean value 15.0 (kips) and standard deviation 1.25 (kips).
1) For 17.5:
z = (17.5 - 15) / 1.25 = 2
P(x > 17.4) = 1 - P(z < 2) = 1 - 0.9772 = 0.0228
2) For 14:
z = (14 - 15)/1.25 = 0.8
For 18:
z = (18 - 15)/1.25 = 2.4
P(14 < x < 18) = P(z < 2.4) - P(z < 0.8) = 0.9918 - 0.7881 = 0.2037
The probabiliity that the force is greater than 17.5 kips is 2.28% while the probabiliity that the force is between 14 and 18 kips is 20.37%
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Ashok arrives at Starbucks at a random time in between 9:00 am and 9:20 am and Melina arrives at Starbucks at a random time in between 9:10 am and 9:30 am. Both stay for exactly 15 minutes. What is the probability that the two of them are in the Starbucks at the exact same time?
Answer:
0.5
Step-by-step explanation:
Since Ashok stays for exactly 15 minutes after his arrival, they will meet if he arrives in the 10 minutes between 9:10 and 9:20 independently of the time Melina arrives.
Since there are 20 minutes from 9:00 and 9:20 and Ashok arrives at a random time, the distribution of his arrival is uniform, so the probability that he arrives between 9:10 and 9:20 is
10/20 = 0.5
Match each item in the lef column with the correct item in the right column. p-value = 0.02 a. do not reject H0 at α = 0.1 p-value = 0.07 b. reject H0 at α = 0.1 but not at 0.05 p-value = 0.3 c. reject H0 at α = 0.05 but not at 0.01 p-value = 0.006 d. reject H0 at α = 0.01
Answer:
Step-by-step explanation:
Hello!
You have some p-values and different decisions that can be made using these p-values.
Remember the decision rule for the p-value method.
If p-value > α, you don't reject the null hypothesis.
If p-value ≤ α, you reject the null hypothesis.
With this in mind, I've arranged the decisions with the given p-values.
a. do not reject H₀ at 10% ⇒ p-value: 0.3
b. do reject H₀ at 10% but not at 5% p-value: ⇒ 0.07
c. do reject H₀ at 5% but not at 1% ⇒ p-value: 0.02
d. do reject H₀ at 1% ⇒ p-value: 0.006
I hope it helps!
Which function is not linear?
Answer: y=x^2+2
Step-by-step explanation:
y=x^2+2 (answer choice 3) is parabolic and not linear
that is the answer
A publisher reports that 49% of their readers own a personal computer. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 200 found that 42% of the readers owned a personal computer. Determine the P-value of the test statistic.
Answer:
0.0239
Step-by-step explanation:
A publisher reports that 49% of their readers own a personal computer.
Claim : . A marketing executive wants to test the claim that the percentage is actually different from the reported percentage.
[tex]H_0:p = 0.49\\H_a:p\neq 0.49[/tex]
A random sample of 200 found that 42% of the readers owned a personal computer.
No. of people owned a personal computer = [tex]42\% \times 200[/tex]
= [tex]\frac{42}{100} \times 200[/tex]
= [tex]84[/tex]
[tex]x = 84\\n = 200[/tex]
We will use one sample proportion test
[tex]\widehat{p}=\frac{x}{n}[/tex]
[tex]\widehat{p}=\frac{84}{200}[/tex]
[tex]\widehat{p}=0.42[/tex]
Formula of test statistic =[tex]\frac{\widehat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}[/tex]
=[tex]-1.98[/tex]
Now refer the p value from the z table
p value =0.0239
Hence The p value of test statistic is 0.0239
A national newspaper reported that the state with the longest mean life span is Hawaii, where the population mean life span is 76 years. A random sample of 20 obituary notices in the Honolulu Advertizer gave the following information about life span (in years) of Honolulu residents.72 68 81 93 56 19 78 94 83 8477 69 85 97 75 71 86 47 66 27(i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)x-bar = ______ yrs = ______ yr
Answer:
Mean = 71.4 years
Standard Deviation = 20.64 years
Step-by-step explanation:
We are given the following data set:
72, 68, 81, 93, 56, 19, 78, 94, 83, 84, 77, 69, 85, 97, 75, 71, 86, 47, 66, 27
Formula:
[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]
where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]Mean =\displaystyle\frac{1428}{20} = 71.4[/tex]
Sum of squares of differences =
0.36 + 11.56 + 92.16 + 466.56 + 237.16 + 2745.76 + 43.56 + 510.76 + 134.56 + 158.76 + 31.36 + 5.76, 184.96 + 655.36 + 12.96 + 0.16 + 213.16 + 595.36 + 29.16 + 1971.36 = 8100.8
[tex]S.D = \sqrt{\displaystyle\frac{8100.8}{19}} = 20.64[/tex]
Mean = 71.4 years
Standard Deviation = 20.64 years
A study is conducted regarding shatterproof glass used in automobiles. Twenty-six glass panes are coated with an anti-shattering film. Then a 5-pound metal ball is fired at 70mph at each pane. Five of the panes shatter. We wish to determine whether, in the population of all such panes, the probability the glass shatters under these conditions is different from π = 0.2
Compute the power of the test if the true π was in fact 0.3
Answer:
0.2773
Step-by-step explanation:
Please see attachment.
A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2902 occupants not wearing seat belts, 30 were killed. Among 7866 occupants wearing seat belts, 20 were killed. Use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities. Complete parts (a) through (c) below. a. Test the claim using a hypothesis test. Consider the first sample to be the sample of occupants not wearing seat belts and the second sample to be the sample of occupants wearing seat belts. What are the null and alternative hypotheses for the hypothesis? test?
Answer:
Null hypothesis:[tex]p_{SB} \geq p_{NSB}[/tex]
Alternative hypothesis:[tex]p_{SB} < \mu_{NSB}[/tex]
The p value is a very low value and using the significance given [tex]\alpha[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of people death using seat belts is significant lower than the proportion of deaths no using seat belts .
Step-by-step explanation:
1) Data given and notation
[tex]X_{NSB}=30[/tex] represent the number of people killed not using seat belts
[tex]X_{SB}=20[/tex] represent the number of people killed using seat belts
[tex]n_{NSB}=2902[/tex] sample of people not wearing seat belts
[tex]n_{SB}=7866[/tex] sample of people wearing seat belts
[tex]p_{NSB}=\frac{30}{2902}=0.0103[/tex] represent the proportion of people killed not using seat belts
[tex]p_{SB}=\frac{20}{7866}=0.00254[/tex] represent the proportion of people killed using seat belts
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
2) Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that seat belts are effective in reducing fatalities (If using seat belts reduce the proportion of deaths we need to see that the proportion of death using seat belts is lower than not using seat belts) , the system of hypothesis would be:
Null hypothesis:[tex]p_{SB} \geq p_{NSB}[/tex]
Alternative hypothesis:[tex]p_{SB} < \mu_{NSB}[/tex]
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{SB}-p_{NSB}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{SB}}+\frac{1}{n_{NSB}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{SB}+X_{NSB}}{n_{SB}+n_{NSB}}=\frac{20+30}{7866+2902}=0.00464[/tex]
3) Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.00254-0.0103}{\sqrt{0.00464(1-0.00464)(\frac{1}{7866}+\frac{1}{2902})}}=-5.26[/tex]
4) Statistical decision
Using the significance level provided [tex]\alpha=0.05[/tex], the next step would be calculate the p value for this test.
Since is a one side lower test the p value would be:
[tex]p_v =P(Z<-5.26)=7.2x10^{-8}[/tex]
So the p value is a very low value and using the significance given [tex]\alpha[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of people death using seat belts is significant lower than the proportion of deaths no using seat belts .
g My Notes Water flows from the bottom of a storage tank at a rate of r(t) = 200 − 4t liters per minute, where 0 ≤ t ≤ 50. Find the amount of water that flows from the tank during the first 30 minutes. 4400 Incorrect: Your answer is incorrect. liters
Answer:
4,200 liters
Step-by-step explanation:
The flow rate is given by:
[tex]r(t) = 200 - 4t[/tex]
Integrating the flow rate expression from t=0 to t=30 minutes, yields the total volume that flows out of the tank during that period:
[tex]\int\limits^{30}_0 {r(t)} = \int\limits^{30}_0 {(200 - 4t} )\, dt \\V=(200t - 2t^2)|_0^{30}\\V= (200*30 -2*30^2)-(200*0 -2*0^2)\\V=4,200\ liters[/tex]
4,200 liters of water flow from the tank during the first 30 minutes.
Consider the following functions.G(x) = 4x2; f(x) = 8x(a)
a. Verify that G is an antiderivative of f.G(x) is an antiderivative of f(x) because f '(x) = G(x) for all x.
A. G(x) is an antiderivative of f(x) because G(x) = f(x) for all x.
B. G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.
C. G(x) is an antiderivative of f(x) because G(x) = f(x) + C for all x.
D. G(x) is an antiderivative of f(x) because f(x) = G(x) + C for all x.
b. Find all antiderivatives of f. (Use C for the constant of integration.)
Answer:
(a) B. G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.
(b) Every function of the form [tex]4x^2+C[/tex] is an antiderivative of 8x
Step-by-step explanation:
A function F is an antiderivative of the function f if
[tex]F'(x)=f(x)[/tex]
for all x in the domain of f.
(a) If [tex]f(x) = 8x[/tex], then [tex]G(x)=4x^2[/tex] is an antiderivative of f because
[tex]G'(x)=8x=f(x)[/tex]
Therefore, G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.
Let F be an antiderivative of f. Then, for each constant C, the function F(x) + C is also an antiderivative of f.
(b) Because
[tex]\frac{d}{dx}(4x^2)=8x[/tex]
then [tex]G(x)=4x^2[/tex] is an antiderivative of [tex]f(x) = 8x[/tex]. Therefore, every antiderivative of 8x is of the form [tex]4x^2+C[/tex] for some constant C, and every function of the form [tex]4x^2+C[/tex] is an antiderivative of 8x.
Final answer:
G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x. The general antiderivative of f(x) = 8x is F(x) = 4x^2 + C.
Explanation:
To determine if G(x) is an antiderivative of f(x), we must consider the derivative of G(x), which is G'(x). If G'(x) is the same function as f(x), then by definition G is an antiderivative of f. Starting with G(x) = 4x2, the derivative is G'(x) = 8x, which is exactly the function f(x). Thus, the correct choice is B: G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.
Now, to find all antiderivatives of the function f(x) = 8x, we integrate f(x) to get the general antiderivative, which includes a constant of integration, C. The antiderivative of 8x is F(x) = 4x2 + C, representing all antiderivatives of f(x).
For students who first enrolled in two year public institutions in a recent semester, the proportion who earned a bachelor's degree within six years was 0.398 . The president of a certain college believes that the proportion of students who enroll in her institution have a lower completion rate.
(a) Determine the null and alternative hypotheses.
(b) Explain what it would mean to make a Type I error.
(c) Explain what it would mean to make a Type II error.
Answer:
a) Null hypothesis: [tex]p \geq 0.368[/tex]
Alternative hypothesis: [tex]p<0.368[/tex]
b) A type of error I for this case would be reject the null hypothesis that the population proportion is greater or equal than 0.368 when actually is not true.
c) A type of error II for this case would be FAIL to reject the null hypothesis that the population proportion is greater or equal than 0.368 when actually the alternative hypothesis is the true.
Step-by-step explanation:
A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".
The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".
The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".
Type I error, also known as a “false positive” is the error of rejecting a null hypothesis when it is actually true. Can be interpreted as the error of no reject an alternative hypothesis when the results can be attributed not to the reality.
Type II error, also known as a "false negative" is the error of not rejecting a null hypothesis when the alternative hypothesis is the true. Can be interpreted as the error of failing to accept an alternative hypothesis when we don't have enough statistical power.
Part a
On this case we want to test if the proportion of students who enroll in her institution have a lower completion rate (0.398), so the system of hypothesis would be:
Null hypothesis: [tex]p \geq 0.368[/tex]
Alternative hypothesis: [tex]p<0.368[/tex]
Part b
A type of error I for this case would be reject the null hypothesis that the population proportion is greater or equal than 0.368 when actually is not true.
Part c
A type of error II for this case would be FAIL to reject the null hypothesis that the population proportion is greater or equal than 0.368 when actually the alternative hypothesis is the true.
The null hypothesis would state that the proportion of students who earn a bachelor's degree within six years at the college is the same as the national proportion. Making a Type I error would mean wrongly rejecting the null hypothesis, while making a Type II error would mean failing to reject the null hypothesis when it is actually false.
Explanation:(a) The null hypothesis would state that the proportion of students who earn a bachelor's degree within six years at the college is the same as the national proportion of 0.398. The alternative hypothesis would state that the proportion of students who earn a bachelor's degree within six years at the college is lower than 0.398.
(b) Making a Type I error in this context would mean rejecting the null hypothesis when it is actually true. This would imply that the college has a lower completion rate when in fact it does not.
(c) Making a Type II error in this context would mean failing to reject the null hypothesis when it is actually false. This would imply that the college has the same completion rate as the national proportion when it actually has a lower completion rate.
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Toys for boys and for girls are donated to a benefit event by two groups. The Lakeville Do-Gooders donated 5 toys for boys and 7 for girls. The Southside Champions Club donated 6 toys for boys and 5 for girls. The master of ceremonies pulls the first toy out of a bin and it’s a toy for a boy. Find the probability that it was donated by the Do-Gooders.
Answer:
0.4545 or 45%
Step-by-step explanation:
Do-Gooders toys for boys = 5
Southside toys for boys = 6
Given that the first toy pulled out of the bin is a toy for boys, to find the probability that this toy was donated by the Do-Gooders, the number of toys for girls donate shouldn't be considered since there's a zero-chance that the toy in question is for girls. Therefore:
[tex]Total\ sample \ space = 5+6 =11\\P(Do-Gooders)=\frac{5}{11} =0.4545[/tex]
There's a 0.4545 or 45% probability that it was donated by the Do-Gooders.