Answer:
X is a discrete random variable.
X can take values from 0 to 12:
[tex]X\in[0,1,2,3,4,5,6,7,8,9,10,11,12][/tex]
Step-by-step explanation:
(a) X = the number of unbroken eggs in a randomly chosen standard egg carton
X is a discrete random variable.
The minimum amount of eggs broken is 0 and the maximum amount of eggs broken is 12 (assuming a dozen egg carton).
Then, X can take values from 0 to 12:
[tex]X\in[0,1,2,3,4,5,6,7,8,9,10,11,12][/tex]
About the probability ditribution nothing can be said, because there is no information about it (it can be a binomial, uniform or non-standard distribution).
The set of possible values for the random variable X, representing the number of unbroken eggs in an egg carton, is {0, 1, 2, 3, ..., 12}. X is a discrete random variable because it takes on distinct, countable values with no gaps in between.
The random variable X represents the number of unbroken eggs in a randomly chosen standard egg carton. To describe the set of possible values for this variable, we can consider the number of eggs in a standard carton, which is typically 12. Therefore, the set of possible values for X would be {0, 1, 2, 3, ..., 12}, as it encompasses all possible outcomes, ranging from having no unbroken eggs (0) to having all 12 unbroken eggs.
Now, let's determine whether the variable X is discrete or not. A discrete random variable is one that can take on a countable number of distinct values with gaps in between.
In this case, X is indeed a discrete random variable because it can only take on integer values from 0 to 12, and there are no intermediate values between these whole numbers. Each value represents a distinct and countable outcome based on the number of unbroken eggs, making it a discrete random variable.
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Dandelions are studied for their effects on crop production and lawn growth. In one region, the mean number of dandelions per square meter was found to be 2. We are interested in the number of dandelions in this region. (a) Find the probability of no dandelions in a randomly selected area of 1 square meter in this region. (Round your answer to four decimal places.) (b) Find the probability of at least one dandelion in a randomly selected area of 1 square meter in this region. (Round your answer to four decimal places.)
Answer:
(a) The probability that there are no dandelions in a randomly selected area of 1 square meter in this region is 0.1353.
(b) The probability that there are at least one dandelion in a randomly selected area of 1 square meter in this region is 0.8647.
Step-by-step explanation:
Let X = number of dandelions per square meter.
The average number of dandelions per square meter is, λ = 2.
The random variable X follows a Poisson distribution with parameter λ = 2.
The probability mass function of X is:
[tex]P(X=x)=\frac{e^{-2}2^{x}}{x!};\ x=0,1,2,3...[/tex]
(a)
Compute the value of P (X = 0) as follows:
[tex]P(X=0)=\frac{e^{-2}2^{0}}{0!}=\frac{0.13534\times 1}{1}=0.1353[/tex]
Thus, the probability that there are no dandelions in a randomly selected area of 1 square meter in this region is 0.1353.
(b)
Compute the value of P (X ≥ 1) as follows:
P (X ≥ 1) = 1 - P (X < 1)
= 1 - P (X = 0)
[tex]=1-0.1353\\=0.8647[/tex]
Thus, the probability that there are at least one dandelion in a randomly selected area of 1 square meter in this region is 0.8647.
To find the probability of no dandelions, use the Poisson distribution formula with the mean value of 2. To find the probability of at least one dandelion, use the complement rule.
Explanation:To find the probability of no dandelions in a randomly selected area of 1 square meter in this region, we need to use the Poisson distribution. The mean number of dandelions per square meter is 2. The probability of no dandelions is given by the formula: P(X=0) = e^(-2) * (2^0 / 0!). Calculating this gives us approximately 0.1353.
To find the probability of at least one dandelion, we can use the complement rule. The probability of at least one dandelion is equal to 1 minus the probability of no dandelions. So, P(X >= 1) = 1 - P(X=0). Calculating this gives us approximately 0.8647.
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Please help..........
bro I hope someone can help you like an expert or something...
Solve for x: 2 over x - 2 + 7 over x^2 - 4 = 5 over x
A. x = -4/3 and x = −5
B. x = -4/3 and x = 5
C. x = 4/3 and x = −5
D. x = 4/3 and x = 5
Answer:
B. x = -4/3 and x = 5
Step-by-step explanation:
2 / (x − 2) + 7 / (x² − 4) = 5 / x
2 / (x − 2) + 7 / ((x + 2) (x − 2)) = 5 / x
Multiply both sides by x − 2:
2 + 7 / (x + 2) = 5 (x − 2) / x
Multiply both sides by x + 2:
2 (x + 2) + 7 = 5 (x − 2) (x + 2) / x
Multiply both sides by x:
2x (x + 2) + 7x = 5 (x − 2) (x + 2)
Simplify:
2x² + 4x + 7x = 5 (x² − 4)
2x² + 4x + 7x = 5x² − 20
0 = 3x² − 11x − 20
Factor:
0 = (3x + 4) (x − 5)
x = -4/3 or x = 5
Prove the superposition principle for nonhomogeneous equations. Suppose that y1 is a solution to Ly1 = f(x) and y2 is a solution to Ly2 = g(x) (same linear operator L). Show that y = y1 + y2 solves Ly = f(x) + g(x). Differential Equation.
Answer:
Step-by-step explanation:
Given Data
Suppose That [tex]y_{1}[/tex] is a solution of L[tex]y_{1}[/tex] = F(x)
and [tex]y_{2}[/tex] is a solution of [tex]Ly_{2}[/tex] = g(x)
L is liner operator
∴ [tex]L(y_{1}+y_{2} )[/tex] = L[tex]y_{1}[/tex] +[tex]Ly_{2}[/tex]
L(y) = F(x) + g(x)
[tex]y_{1}+y_{2}[/tex] is the solution to L(y) = F(x) + g(x)
E.g the liner operator be L = [tex]\frac{d}{dx}[/tex]
[tex]\frac{d}{dx}[/tex] [tex]y_{1}[/tex] = f(x)
[tex]\frac{d}{dx}[/tex] [tex]y_{2}[/tex] = g(x)
[tex]\frac{d}{dx}[/tex] [tex](y_{1}+y_{2} )[/tex] = [tex]\frac{d}{dx}[/tex] [tex]y_{1}[/tex] + [tex]\frac{d}{dx}[/tex] [tex]y_{2}[/tex] = f(x) + g(x)
Final Answer:
y = y1 + y2 solves the equation Ly = f(x) + g(x).
Explanation:
Certainly! To prove the superposition principle for nonhomogeneous linear differential equations, we will utilize the properties of linear operators and the given solutions for the differential equations.
Let's define L as a linear differential operator. Being linear implies that for any functions u(x) and v(x), and any constants a and b, the operator satisfies the following properties:
1. L(u + v) = L(u) + L(v) (additivity)
2. L(au) = aL(u) (homogeneity)
Given there are two functions y1(x) and y2(x) that are solutions to the nonhomogeneous linear differential equations:
Ly1 = f(x)
Ly2 = g(x)
We need to prove that if you take a linear combination of y1 and y2, the result y = y1 + y2 will also be a solution to the combined nonhomogeneous equation Ly = f(x) + g(x).
Here's how we do it:
Consider a linear combination of y1 and y2, denoted as y = y1 + y2. Apply the linear operator L to both sides of this equation:
L(y) = L(y1 + y2)
Since L is a linear operator, we can apply the additivity property:
L(y) = L(y1) + L(y2)
Now, we know that y1 and y2 are solutions to their respective nonhomogeneous equations, so we can substitute f(x) for L(y1) and g(x) for L(y2):
L(y) = f(x) + g(x)
Thus, we have shown that y = y1 + y2 solves the equation Ly = f(x) + g(x).
This is the proof of the superposition principle for nonhomogeneous linear differential equations. It shows that solutions to such equations can be added together to obtain a new solution corresponding to the sum of the nonhomogeneous parts.
Based on the provided information about the characteristic roots and the right hand side function g(t), determine the appropriate form of a particular solution to be used with the undetermined coefficient method.
(a) r1=-2i; r2=2i g(t)=2sin(2t) + 3cos(2t)
(b) r1=r2=0; r3=1 g(t)= t^2 +2t + 3
Answer:
Yp = t[Asin(2t) + Acos(2t)]
Yp = t²[At² + Bt + C]
Step-by-step explanation:
The term "multiplicity" means when a given equation has a root at a given point is the multiplicity of that root.
(a) r1=-2i; r2=2i g(t)=2sin(2t) + 3cos(2t)
As you can notice the multiplicity of this equation is 1 since the roots r1 = 2i and r2 = 2i appear for only once.
The form of a particular solution will be
Yp = t[Asin(2t) + Acos(2t)]
where t is for multiplicity 1
(b) r1=r2=0; r3=1 g(t)= t² +2t + 3
As you can notice the multiplicity of this equation is 2 since the roots r1 = r2 = 0 appears 2 times.
The form of a particular solution will be
Yp = t²[At² + Bt + C]
where t² is for multiplicity 2
Suppose Gabe, an elementary school student, has just finished dinner with his mother, Judy. Eyeing the nearby cookie jar, Gabe asks his mother if he can have a cookie for dessert. She tells Gabe that she needs to check his backpack to make sure that he finished his homework. Gabe cannot remember where he left his backpack, but he knows for sure that he did not complete his homework and will not be allowed to eat a cookie. Gabe believes his only option is to quickly steal a cookie while his mother is out of the room. Judy then leaves the room to look for Gabe's backpack. Assume that Judy could return at any time in the next 90 seconds with equal probability. For the first 30 seconds, Gabe sheepishly wonders if he will get caught trying to grab a nearby cookie. After waiting and not seeing his mother, Gabe decides that he needs a cookie and begins to take one from the jar. Assuming it takes Gabe 15 seconds to grab a cookie from the jar and devour it without a trace, what is the probability that his mother returns in time to catch Gabe stealing a cookie? Please round your answer to the nearest two decimal places.
Answer:
the probability that his mother would return on time to catch Gabe stealing is given by:[tex]\frac{15}{60}=0.25[/tex].
Step-by-step explanation:
probability is given by; the required outcome over the number of possible outcome.
Step1; we have to subtract the first initial 30 secs that Gabe spent wondering if he will be caught. i.e 90 sec-30 secs= 60 Sec.
Step2: we divide the number of seconds that he can finish a cookie without being caught over the number of seconds left before his mother returns. that is why we have [tex]\frac{15}{60} = 0.25[/tex]
The probability that his mother returns in time to catch Gabe stealing a cookie is 0.25.
Calculation of the probability:Judy could return at any time in the next 90 seconds with equal probability. For the first 30 seconds, Gabe sheepishly wonders if he will get caught trying to grab a nearby cookie. And, it takes Gabe 15 seconds to grab a cookie from the jar.
Now the probability is
[tex]= 15 \div (90 - 30)\\\\= 15 \div 60[/tex]
= 0.25
Hence, we can conclude that The probability that his mother returns in time to catch Gabe stealing a cookie is 0.25.
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The number of cars sold weekly by a new automobile dealership grows according to a linear growth model. The first week the dealership sold six cars ( P 0 = 6 ). The second week the dealership sold eight cars ( P 1 = 8 ). Write the recursive formula for the number of cars sold, P n , in the ( n + 1 )th week. P n = P n − 1 + Write the explicit formula for the number of cars sold, P n , in the ( n + 1 )th week. P n = If this trend continues, how many cars will be sold in the fourth week?
(a) Recursive formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week: [tex]P_n = P_{n-1} + 2[/tex]
(b) Explicit formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week:
[tex]P_n = 6 + 2n[/tex]
(c) In the fourth week, the dealership will sell 12 cars.
To find the recursive formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week, we can observe the pattern from the given information.
Given data:
[tex]P_0 = 6[/tex] (number of cars sold in the first week)
[tex]P_1 = 8[/tex] (number of cars sold in the second week)
We can see that each week, the number of cars sold increases by 2. So, the recursive formula can be expressed as:
[tex]P_n = P_{n-1} + 2[/tex]
This formula states that the number of cars sold in the nth week [tex](P_n)[/tex] is equal to the number of cars sold in the previous week [tex](P_{n-1})[/tex] plus 2.
Next, let's find the explicit formula for the number of cars sold, [tex]P_n[/tex], in the (n + 1)th week.
To do this, we need to identify the initial value ([tex]P_0[/tex]) and the common difference (d) in the arithmetic sequence. In this case, the initial value is 6 (P₀ = 6), and the common difference is 2 (the number of cars sold increases by 2 each week).
The explicit formula for an arithmetic sequence is given by:
[tex]P_n = P_0 + n * d[/tex]
Substitute the given values:
[tex]P_n = 6 + n * 2[/tex]
Therefore, the explicit formula for the number of cars sold, P_n, in the (n + 1)th week is [tex]P_n = 6 + 2n[/tex].
Now, let's find how many cars will be sold in the fourth week (n = 3):
[tex]P_3 = 6 + 2 * 3[/tex]
[tex]P_3 = 12[/tex]
In the fourth week, the dealership will sell 12 cars.
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The recursive formula for the number of cars sold weekly is Pn = Pn - 1 + 2, where Pn represents the number of cars in the nth week. The explicit formula is Pn = 6 + 2n. Based on this model, the dealership would sell 14 cars in the fourth week.
Explanation:In your problem, the number of cars sold each week is increasing by a constant amount, following a linear growth model. The first week, 6 cars were sold and the second week, 8 cars were sold. This indicates an increase of 2 cars from week 1 to week 2.
To write the recursive formula for the number of cars sold, Pn, in the (n + 1)th week, we determine the growth by subtraction: P1 - P0 = 8 - 6 = 2. So, every week, the number of cars sold increases by 2. The recursive formula would therefore be Pn = Pn - 1 + 2.
For the explicit formula which gives us the number of cars sold in the n-th week directly, we observe that it started with 6 cars (base) and increments by 2 each week. Therefore, the explicit formula would be Pn = 6 + 2n.
By the fourth week, the number of cars sold would be P4 = 6 + 2 * 4 = 6 + 8 = 14 cars sold.
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The function f(x,y)=2x + 2y has an absolute maximum value and absolute minimum value subject to the constraint 9x^2 - 9xy + 9y^2 =25. Use Lagrange multipliers to find these values.
Answer:
the minimum is located in x = -5/3 , y= -5/3
Step-by-step explanation:
for the function
f(x,y)=2x + 2y
we define the function g(x)=9x² - 9xy + 9y² - 25 ( for g(x)=0 we get the constrain)
then using Lagrange multipliers f(x) is maximum when
fx-λgx(x)=0 → 2 - λ (9*2x - 9*y)=0 →
fy-λgy(x)=0 → 2 - λ (9*2y - 9*x)=0
g(x) =0 → 9x² - 9xy + 9y² - 25 = 0
subtracting the second equation to the first we get:
2 - λ (9*2y - 9*x) - (2 - λ (9*2x - 9*y))=0
- 18*y + 9*x + 18*x - 9*y = 0
27*y = 27 x → x=y
thus
9x² - 9xy + 9y² - 25 = 0
9x² - 9x² + 9x² - 25 = 0
9x² = 25
x = ±5/3
thus
y = ±5/3
for x=5/3 and y=5/3 → f(x)= 20/3 (maximum) , while for x = -5/3 , y= -5/3 → f(x)= -20/3 (minimum)
finally evaluating the function in the boundary , we know because of the symmetry of f and g with respect to x and y that the maximum and minimum are located in x=y
thus the minimum is located in x = -5/3 , y= -5/3
A sampling method is ________ dependent quantitative independent qualitative when an individual selected for one sample does not dictate which individual is to be in the second sample.
Answer:
A sampling method is independent qualitative when an individual selected for one sample does not dictate which individual is to be in the second sample.
Step-by-step explanation:
individuals selected for one sample do not dictate which individuals are to be in the second sample, while dependent quantitative involves individuals selected to be in one sample are used to determine the individuals in the second sample.
Final answer:
An independent sampling method allows for selection of individuals for one sample without influencing the selection for another, maintaining the integrity of quantitative research.
Explanation:
A sampling method is considered independent when an individual selected for one sample does not dictate which individual is to be in the second sample. This type of sampling strategy ensures that the selection of participants for one sample does not influence the selection for another, which is crucial for maintaining the validity and reliability of research findings, particularly in quantitative research.
In research, particularly in studies involving statistical analysis, it is essential to use appropriate sampling methods to ensure that the results are representative of the wider population. Probability sampling methods are often employed in quantitative research as they provide a means to make generalizations from the sample to the population. Types of probability samples include simple random samples, stratified samples, and cluster samples.
Suppose there is currently a tax of $50 per ticket on airline tickets. Sellers of airline tickets are required to pay the tax to the government. If the tax is reduced from $50 per ticket to $30 per ticket, then thea. demand curve will shift upward by $20, and the price paid by buyers will decrease by less than $20.b. demand curve will shift upward by $20, and the price paid by buyers will decrease by $20.c. supply curve will shift downward by $20, and the effective price received by sellers will increase by less than $20.d. supply curve will shift downward by $20, and the effective price received by sellers will increase by $20.
Answer:
the demand curve will shift upward by $20 and the price paid by buyers will decrease by $20.
Step-by-step explanation:
the reduction in the fixed amount of tax from $50 t0 $30 will bring about reduction of $20 in the price of ticket. the reduction in the price of the ticket, other factors held constant, will brings about change in the demand curve.
The correct answer is D, as supply curve will shift downward by $20, and the effective price received by sellers will increase by $20.
This is so because by reducing the tax, the value that each airline will effectively increase. In this way, as they obtain greater income for each ticket, the airlines will in turn increase the offer of tickets.
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Multiply 2 1/2 x 3 2/3
Answer: 9.16666666667
Step-by-step explanation:
PLEASE HELP 69 POINTS, BRAINLIEST, 5 STARS, AND THANKS.
Answer:
a) 125 scoops b) 31 scoops
Step-by-step explanation:
The sink = 4000/3 pi
a) The volume of a Cone is 1/3 (pi)(h)(r)^2, meaning...
(1/3)(pi)(8)(2^2)=32/3 pi, and...
(4000pi/3)/(32pi/3) =
4000/32 =
125
b) The volume of a Cone is 1/3 (pi)(h)(r)^2, meaning...
(1/3)(pi)(8)(4^2) = 128pi/3, and...
(4000pi/3)/(128pi/3) =
4000/128 =
31.25, which is approx. 31
Answer:
a) 125
b) 31
Step-by-step explanation:
a) radius = 2
Volume of cup = ⅓×pi×r²×h
= 32pi/3
No. of scoops = 4000pi/3 ÷ 32pi/3
= 125
b) radius = 4
Volume of cup = ⅓×pi×r²×h
= 128pi/3
No. of scoops = 4000pi/3 ÷ 128pi/3
= 31.25
Nearest whole number: 31
In a sample of 100 steel canisters, the mean wall thickness was 8.1 mm with a standard deviation of 0.5 mm. Someone says that the mean thickness is less than 8.2 mm. With what level of confidence can this statement be made? (Express the final answer as a percent and round to two decimal places.)
Answer:
This statement can be made with a level of confidence of 97.72%.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 8.1 mm
Standard Deviation, σ = 0.5 mm
Sample size, n = 100
We are given that the distribution of thickness is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
Standard error due to sampling:
[tex]=\dfrac{\sigma}{\sqrt{n}} = \dfrac{0.5}{\sqrt{100}} = 0.05[/tex]
P(mean thickness is less than 8.2 mm)
P(x < 8.2)
[tex]P( x < 8.2)\\\\ = P( z < \displaystyle\frac{8.2 - 8.1}{0.05})\\\\ = P(z < 2)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 8.2) =0.9772 = 97.72\%[/tex]
This statement can be made with a level of confidence of 97.72%.
The z-score calculation reveals that the mean thickness is less than 8.2 mm with approximately 97.50% confidence.
Explanation:To determine the level of confidence that the mean thickness is less than 8.2 mm, we can calculate the z-score of the sample mean and then find the corresponding percentile in the standard normal distribution. The z-score is calculated by the formula:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.
Using the given information, x = 8.1 mm, μ = 8.2 mm (the mean to test against), σ = 0.5 mm, and n = 100, we get:
z = (8.1 mm - 8.2 mm) / (0.5 mm / √100) = -0.1 / 0.05 = -2
The z-score corresponds to a percentile in the standard normal distribution, which can be found using a z-table or statistical software. A z-score of -2 corresponds to approximately the 2.5th percentile, which means that the sample mean is less than 8.2 mm with about 97.5% confidence.
Therefore, the statement that the mean thickness is less than 8.2 mm can be made with approximately 97.50% confidence.
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The number of accidents per week at a hazardous intersection varies with mean 2.2 and standard deviation 1.1. This distribution takes only whole-number values, so it is certainly not Normal. (a) Let x be the mean number of accidents per week at the intersection during a year (52 weeks). What is the approximate distribution of x according to the central limit theorem
Answer:
Step-by-step explanation:
Hello!
The definition of the Central Limi Theorem states that:
Be a population with probability function f(X;μ,δ²) from which a random sample of size n is selected. Then the distribution of the sample mean tends to the normal distribution with mean μ and variance δ²/n when the sample size tends to infinity.
As a rule, a sample of size greater than or equal to 30 is considered sufficient to apply the theorem and use the approximation.
X[bar]≈N(μ;σ²/n)
If the variable of interest is X: the number of accidents per week at a hazardous intersection.
There is no information about the distribution of this variable, but a sample of n= 52 weeks was taken, and since the sample is large enough you can approximate the distribution of the sample mean to normal. With population mean μ= 2.2 and standard deviation σ/√n= 1.1/√52= 0.15
I hope it helps!
Determine which of the following numbers could not be probabilities, and why? Select all that apply. A. The number 175% could not be a probability because it is larger than 100%. B. The number 5.91 could not be a probability because it is larger than 1. C. The number 0.66 could not be a probability because it is smaller than 1. D. The number 0.002 could not be a probability because it is extremely small. E. The number negative 110% could not be a probability because it is negative.
Answer:
A Not a probability B.Probability C.not a probability d.Probability E propability F.not a propability
Step-by-step explanation:
A propability of any event is a number between zero and one
A)175% converted 1.75 so lies outside zero and one not a propability
B)5.91 lies outside the zero and one rage not a propability
C)0.66 lies in the zero and one rage therefore a propability
D)0.002 Lies within the zero and one range therefore a propability
E) -110% converted equals -1.1 so does not lies in this range not a propability
The following numbers could not be probabilities:
A. The number 175% could not be a probability because it is larger than 100%.
B. The number 5.91 could not be a probability because it is larger than 1.
E. The number negative 110% could not be a probability because it is negative.
Probabilities must be between 0 and 1, inclusive. This is because a probability represents the likelihood of an event occurring, and there is no event that can be more likely than certain (1) or less likely than impossible (0).
The numbers 0.66 and 0.002 are valid probabilities.
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What length is the shortest path from A to Cin the graph below?
Answer: It's C.5
Step-by-step explanation:
plsss help i don't knwo thai
Question 4: The value of x is 4.5.
Question 5: The value of x is 15.6.
Solution:
Question 4:
The given triangle is a right triangle.
Using trigonometric formulas for a right triangle.
[tex]$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}[/tex]
[tex]$\sin C=\frac{AB}{BC}[/tex]
[tex]$\sin 45^\circ=\frac{x}{6.4}[/tex]
[tex]$\frac{1}{\sqrt{2} } =\frac{x}{6.4}[/tex]
Multiply 6.4 on both sides, we get
[tex]$\frac{6.4}{\sqrt{2} }=x[/tex]
x = 4.5
The value of x is 4.5.
Question 5:
The given triangle is a right triangle.
Using trigonometric formulas for a right triangle.
[tex]$\sin \theta=\frac{\text { opposite }}{\text { hypotenuse }}[/tex]
[tex]$\sin E=\frac{EF}{DF}[/tex]
[tex]$\sin 60^\circ=\frac{x}{18}[/tex]
[tex]$\frac{\sqrt3}{2} =\frac{x}{18}[/tex]
Multiply 18 on both sides, we get
x = 15.6
The value of x is 15.6.
Gabe Amodeo, a nuclear physicist, needs 60 liters of a 60% acid solution. He currently has a 40% solution and a 70% solution. How many liters of each does he need
to make the needed 60 liters of 60% acid solution?
Answer:
20 liters of the 40% solution
60-20=40 liters of the 70% solution
Step-by-step explanation:
40%x+70%(60-x)=60%(60)
0.4x+0.7(60-x)=0.6(60)
0.4x+42-0.7x=36
-0.3x+42=36
-0.3x=-42+36
-0.3x=-6
0.3x=6
3x=60
x=60/3
x=20 liters of the 40% solution
60-20=40 liters of the 70% solution
check: 0.4*20=8
0.7*40=28
8+28=36 and 0.6*60=36 also
a medical director found mean blood pressure x = 126.07 for an SRS of 72 executives. The standard deviation of the blood pressures of all executives is σ = 15. Give a 90% confidence interval for the mean blood pressure μ of all executives.
Answer:
[tex]126.07-1.64\frac{15}{\sqrt{72}}=123.17[/tex]
[tex]126.07+1.64\frac{15}{\sqrt{72}}=128.97[/tex]
So on this case the 90% confidence interval would be given by (123.17;128.97)
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
Solution to the problem
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]
Now we have everything in order to replace into formula (1):
[tex]126.07-1.64\frac{15}{\sqrt{72}}=123.17[/tex]
[tex]126.07+1.64\frac{15}{\sqrt{72}}=128.97[/tex]
So on this case the 90% confidence interval would be given by (123.17;128.97)
In Quebec, 90 percent of the population subscribes to the Roman Catholic religion. In a random sample of eight Quebecois, find the probability that the sample contains at least five Roman Catholics.
Answer:
Probability that the sample contains at least five Roman Catholics = 0.995 .
Step-by-step explanation:
We are given that In Quebec, 90 percent of the population subscribes to the Roman Catholic religion.
The Binomial distribution probability is given by;
P(X = r) = [tex]\binom{n}{r}p^{r}(1-p)^{n-r}[/tex] for x = 0,1,2,3,.......
Here, n = number of trials which is 8 in our case
r = no. of success which is at least 5 in our case
p = probability of success which is probability of Roman Catholic of
0.90 in our case
So, P(X >= 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
= [tex]\binom{8}{5}0.9^{5}(1-0.9)^{8-5} + \binom{8}{6}0.9^{6}(1-0.9)^{8-6} + \binom{8}{7}0.9^{7}(1-0.9)^{8-7} + \binom{8}{8}0.9^{8}(1-0.9)^{8-8}[/tex]
= 56 * [tex]0.9^{5} * (0.1)^{3}[/tex] + 28 * [tex]0.9^{6} * (0.1)^{2}[/tex] + 8 * [tex]0.9^{7} * (0.1)^{1}[/tex] + 1 * [tex]0.9^{8}[/tex]
= 0.995
Therefore, probability that the sample contains at least five Roman Catholics is 0.995.
The probability that a random sample of eight Quebecois contains at least five Roman Catholics is approximately 0.9950 or 99.50%.
Given:
- The probability of success (subscribing to the Roman Catholic religion) p = 0.9 .
- The number of trials (sample size) n = 8 .
- We want to find the probability of getting at least 5 successes.
The binomial probability formula is:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Where:
- [tex]\( \binom{n}{k} \)[/tex] is the binomial coefficient [tex]\(\frac{n!}{k!(n-k)!} \)[/tex],
- k is the number of successes,
- p is the probability of success,
- (1-p) is the probability of failure.
We need to find [tex]\( P(X \geq 5) \)[/tex], which is the sum of the probabilities of getting exactly 5, 6, 7, and 8 successes.
[tex]\[ P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) \][/tex]
Calculating each term individually:
1.
[tex]\( P(X = 5) \):\[ P(X = 5) = \binom{8}{5} (0.9)^5 (0.1)^3 \]\[ \binom{8}{5} = \frac{8!}{5!3!} = 56 \]\[ P(X = 5) = 56 \times (0.9)^5 \times (0.1)^3 \]\[ P(X = 5) = 56 \times 0.59049 \times 0.001 \]\[ P(X = 5) = 56 \times 0.00059049 \]\[ P(X = 5) \approx 0.0331 \][/tex]
2. P(X = 6) :
[tex]\[ P(X = 6) = \binom{8}{6} (0.9)^6 (0.1)^2 \]\[ \binom{8}{6} = \frac{8!}{6!2!} = 28 \]\[ P(X = 6) = 28 \times (0.9)^6 \times (0.1)^2 \]\[ P(X = 6) = 28 \times 0.531441 \times 0.01 \]\[ P(X = 6) = 28 \times 0.00531441 \]\[ P(X = 6) \approx 0.1488 \][/tex]
3. P(X = 7) :
[tex]\[ P(X = 7) = \binom{8}{7} (0.9)^7 (0.1)^1 \]\[ \binom{8}{7} = \frac{8!}{7!1!} = 8 \]\[ P(X = 7) = 8 \times (0.9)^7 \times 0.1 \]\[ P(X = 7) = 8 \times 0.4782969 \times 0.1 \]\[ P(X = 7) = 8 \times 0.04782969 \]\[ P(X = 7) \approx 0.3826 \][/tex]
4. P(X = 8) :
[tex]\[ P(X = 8) = \binom{8}{8} (0.9)^8 (0.1)^0 \]\[ \binom{8}{8} = 1 \]\[ P(X = 8) = 1 \times (0.9)^8 \times 1 \]\[ P(X = 8) = (0.9)^8 \]\[ P(X = 8) = 0.43046721 \]\[ P(X = 8) \approx 0.4305 \][/tex]
Now sum these probabilities:
[tex]\[ P(X \geq 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) \]\[ P(X \geq 5) \approx 0.0331 + 0.1488 + 0.3826 + 0.4305 \]\[ P(X \geq 5) \approx 0.9950 \][/tex]
So, the probability that the sample contains at least five Roman Catholics is approximately 0.9950 or 99.50%.
Consider a sample with a mean of 500 and a standard deviation of 100. What are the z-scores for the following data values: 560, 650, 500, 450, and 300? z-score for 560 z-score for 650 z-score for 500 z-score for 450 z-score for 300
The z-scores for the given data values are approximately:
560: 0.6
650: 1.5
500: 0
450: -0.5
300: -2
Given the mean (μ) of 500 and the standard deviation (σ) of 100, we can calculate the z-scores for the provided data values:
The z-score (also known as the standard score) measures how many standard deviations a data point is away from the mean. It is calculated using the formula:
z = (x - μ) / σ
Where:
x is the data value
μ is the mean of the sample
σ is the standard deviation of the sample
For x = 560:
z = (560 - 500) / 100 = 0.6
For x = 650:
z = (650 - 500) / 100 = 1.5
For x = 500:
z = (500 - 500) / 100 = 0
For x = 450:
z = (450 - 500) / 100 = -0.5
For x = 300:
z = (300 - 500) / 100 = -2
So, the z-scores for the given data values are approximately:
560: 0.6
650: 1.5
500: 0
450: -0.5
300: -2
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The z-scores for the data values 560, 650, 500, 450, and 300 in a sample with a mean of 500 and a standard deviation of 100 are 0.6, 1.5, 0, -0.5, and -2 respectively. They are computed using the formula z = (X - μ) / σ.
Explanation:This question refers to the concept of z-scores in statistics, which is a part of Mathematics. A z-score indicates how many standard deviations a given data point is from the mean. The formula to calculate the z-score is: z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation.
The given sample has a mean (μ) of 500 and a standard deviation (σ) of 100. Let's calculate the z-scores:
For X = 560: z = (560 - 500) / 100 = 0.6 For X = 650: z = (650 - 500) / 100 = 1.5 For X = 500: z = (500 - 500) / 100 = 0 For X = 450: z = (450 - 500) / 100 = -0.5 For X = 300: z = (300 - 500) / 100 = -2
So the z-scores for the data values 560, 650, 500, 450, and 300 are 0.6, 1.5, 0, -0.5, and -2 respectively.
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In an experiment, A,B, C, andD are events with probabilitiesP[A UB] = 5/8,P[A] =3/8,
P[C ∩D] = 1/3, andP[C] =1/2. Furthermore, Aand B are disjoint, whileC and D areindependent.
(a) Find P[A∩ B],P[B],P[A ∩Bc], andP[A UBc].
(b) Are A andB independent?
(c) FindP[D],P[C ∩Dc],P[Cc∩ Dc],andP[C|D].
(d) Find P[CU D] andP[C UDc].
(e) Are C andDcindependent?
Answer:
Step-by-step explanation:
Hello!
You have 4 events A, B, C and D
With probabilities:
P(A∪B)= 5/8
P(A)= 3/8
P(C∩D)= 1/3
P(C)= 1/2
A and B are disjoint events, this means that there are no shared elements between then and their intersection is void, symbolically A∩B= ∅, in consequence, these events are mutually exclusive.
C and D are independent events, this means that the occurrence of one of them does not affect the probability of occurrence of the other one in two consecutive repetitions.
a.
i. P(A∩B)= 0
⇒ Since A and B are disjoint events, the probability of their intersection is zero.
ii. A and B are mutually exclusive events, this means that P(A∪B)= P(A)+P(B)
⇒ From this expression, you can clear the probability of b as P(B)= P(A∪B)-P(A)= 5/8-3/8= 1/4
iii. If Bc is the complementary event of B, its probability would be P(Bc)= 1 - P(B)= 1 - 1/4= 3/4. If the events A and B are mutually exclusive and disjoint, it is logical to believe that so will be the events A and Bc, so their intersection will also be void:
P(A∩Bc)= 0
vi.P(A∪Bc)= P(A) + P(Bc)= 3/8+3/4= 9/8
b.
If A and B are independent then the probability of A is equal to the probability of A given B, symbolically:
P(A)= P(A/B)
[tex]P(A/B)= \frac{P(AnB)}{P(B)}= \frac{0}{1/4}= 0[/tex]
P(A)= 3/8
P(A) ≠ P(A/B) ⇒ A and B are not independent.
c.
i. P(D) ⇒ Considering C and D are two independent events, then we know that P(C∩D)= P(C)*P(D)
Then you can clear the probability of D as:
P(D)= P(C∩D)/P(C)= (1/3)/(1/2)= 2/3
ii. If Dc is the complementary event of D, then its probability is P(Dc)= 1 - P(D) = 1 - 2/3= 1/3
P(C∩Dc)= P(C)*P(Dc)= (1/2)*(1/3)= 1/6
iii. Now Cc is the complementary event of C, its probability is P(Cc)= 1 - P(C)= 1 - 1/2= 1/2
P(Cc∩Dc)= P(Cc)*P(Dc)= (1/2)*(1/3)= 1/6
vi. and e.
[tex]P(C/D)= \frac{P(CnD)}{P(D)} = \frac{1/3}{2/3} = 1/2[/tex]
P(C)=1/2
As you can see the P(C)=P(C/D) ⇒ This fact proves that the events C and D are independent.
d.
i. P(C∪D)= P(C) + P(D) - P(C∩D)= 1/2 + 2/3 - 1/3= 5/6
ii. P(C∪Dc)= P(C) + P(Dc) - P(C∩Dc)= 1/2 + 1/3 - 1/6= 2/3
I hope it helps!
A number between 1 and 15
is chosen at random. What
is the probability that the
number chosen will be a
multiple of 5? please help just learning this will mark brainest got people answering it maki g up junk to get the answer then needing help
The probability that the chosen will be a multiple of 5 between 1 and 15 is [tex]\frac{2}{13}[/tex].
Solution:
Given data:
Number between 1 and 15 is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14.
Total numbers between 1 and 15 = 13
N(S) = 13
Multiple of 5 between 1 and 15 = 5, 10
Number of multiples of 5 between 1 and 15 = 2
N(A) = 2
Probability of multiple of 5 between 1 and 15:
[tex]$P(A)=\frac{N(A)}{N(S)}[/tex]
[tex]$P(A)=\frac{2}{13}[/tex]
The probability that the chosen will be a multiple of 5 between 1 and 15 is [tex]\frac{2}{13}[/tex].
Final answer:
To determine the probability of selecting a multiple of 5 between 1 and 15, count the multiples (5 and 10) and divide by the total number (15), resulting in a probability of 2/15.
Explanation:
The question is related to the concept of probability in mathematics. To find the probability that a randomly chosen number between 1 and 15 is a multiple of 5, you simply count how many multiples of 5 are in that range and divide by the total number of possibilities (15). The multiples of 5 between 1 and 15 are 5 and 10, so there are 2 favorable outcomes. Hence, the probability is 2 divided by 15, which equals 2/15.
The length of time needed to complete a certain test is normally distributed with mean 35 minutes and standard deviation 15 minutes. Find the probability that it will take between 31 and 40 minutes to complete the test.
Answer:
Probability = 0.23572 .
Step-by-step explanation:
We are given that the length of time needed to complete a certain test is normally distributed with mean 35 minutes and standard deviation 15 minutes.
Let X = length of time needed to complete a certain test
Since, X ~ N([tex]\mu,\sigma^{2}[/tex])
The z probability is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1) where, [tex]\mu[/tex] = 35 and [tex]\sigma[/tex] = 15
So, P(31 < X < 40) = P(X < 40) - P(X <= 31)
P(X < 40) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{40-35}{15}[/tex] ) = P(Z < 0.33) = 0.62930
P(X <= 31) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{31-35}{15}[/tex] ) = P(Z < -0.27) = 1 - P(Z <= 0.27)
= 1 - 0.60642 = 0.39358
Therefore, P(31 < X < 40) = 0.62930 - 0.39358 = 0.23572 .
To find the probability that it takes between 31 and 40 minutes to complete the test when the test completion times are normally distributed with a mean (μ) of 35 minutes and a standard deviation (σ) of 15 minutes, we can use the properties of the normal distribution.
Firstly, we need to standardize our times (31 and 40 minutes) to find the corresponding z-scores. The z-score is a measure of how many standard deviations an element is from the mean. We use the following formula to calculate the z-score:
\[ z = \fraction{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value for which we are finding the z-score
- \( \mu \) is the mean
- \( σ \) is the standard deviation
We will calculate the z-scores for both 31 minutes and 40 minutes.
For \( X = 31 \):
\[ z_{31} = \fraction{(31 - 35)}{15} = \fraction{-4}{15} \approx -0.267 \]
For \( X = 40 \):
\[ z_{40} = \fraction{(40 - 35)}{15} = \fraction{5}{15} \approx 0.333 \]
Next, we look up these z-scores in the standard normal distribution (z-distribution) table, which will give us the area to the left of each z-score.
If we don't have the z-distribution table available, we might use statistical software or a calculator with a normal distribution function. But let's proceed as if we are using the z-table.
Let's say our z-table gives us the following areas:
Area to the left of \( z_{31} \approx -0.267 \): Approximately 0.3944 (please note that actual values will depend on the specific z-table or calculator you are using).
Area to the left of \( z_{40} \approx 0.333 \): Approximately 0.6304.
Now we want the area between these two z-scores, which represents the probability that the test completion time is between 31 and 40 minutes. To find this, we can subtract the area for \( z_{31} \) from the area for \( z_{40} \):
Probability \( P(31 < X < 40) = P(z_{40}) - P(z_{31}) \)
Substituting the values,
\[ P(31 < X < 40) = 0.6304 - 0.3944 = 0.2360 \]
So the probability that it will take between 31 and 40 minutes to complete the test is approximately 0.2360, or 23.60%.
Consider the population of four juvenile condors. Their weights in pounds are : 4, 5, 7, 12 (a) Let x be the weight of a juvenile condor. Write the possible unique values for x: (NOTE: Separate each value in the list with a comma.) . (b) Find the mean of the population: (c) Let x¯ be the average weight from a sample of two juvenile condors. List all possible outcomes for x¯. (If a value occurs twice, make sure to list it twice.) This is the sampling distribution for samples of size 2: (NOTE: Separate each value in the list with a comma.) . (d) Find the mean of the sampling distribution: Note: You can earn partial credit on this problem.
Answer:
a) 4, 5, 7, 12
b) 7
c) 4.5, 6.5, 8, 6, 8.5, 9.5
d) 7.167
Step-by-step explanation:
We are given the following in the question:
4, 5, 7, 12
a) unique values for x
4, 5, 7, 12
b) mean of the population
[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]
[tex]\mu =\displaystyle\frac{28}{4} = 7[/tex]
c) sampling distribution for samples of size 2
Sample size, n = 2
Possible samples of size 2 are (4,5),(4,7),(4,12),(5,7),(5,12),(7,12)
Sample means are:
[tex]\bar{x_1} = \dfrac{4+5}{2} = 4.5\\\\\bar{x_2} = \dfrac{4+7}{2} = 6.5\\\\\bar{x_3} = \dfrac{4+12}{2} = 8\\\\\bar{x_4} = \dfrac{5+7}{2} = 6\\\\\bar{x_5} = \dfrac{5+12}{2} = 8.5\\\\\bar{x_6} = \dfrac{7+12}{2} = 9.5[/tex]
Thus, the list is 4.5, 6.5, 8, 6, 8.5, 9.5
d) mean of the sampling distribution
[tex]\bar{x} = \dfrac{4.5 + 6.5 + 8+ 6 + 8.5+ 9.5}{6} = \dfrac{43}{6} = 7.167[/tex]
The unique values for x are 4, 5, 7, and 12 pounds. The population mean is 7 pounds, and the sampling distribution for the average weight from a sample of two juvenile condors has possible outcomes of 4.5, 5.5, 6, 8, 8.5, and 9.5 pounds, with a mean of 7 pounds.
Explanation:To answer the student's questions regarding juvenile condor weights:
Possible unique values for x: The unique weights of the juvenile condors are 4, 5, 7, 12 pounds.Mean of the population: The mean (μ) is the average of all the values. To calculate the mean, add all the weights and divide by the number of condors: (4 + 5 + 7 + 12) / 4 = 28 / 4 = 7 pounds.Possible outcomes for ¯x (the average weight from a sample of two juvenile condors): To list all possible outcomes for ¯x, calculate the mean of all possible pairs of condor weights:Since some pairs can be chosen in two ways (e.g., condor 1 and 2, or condor 2 and 1), record the mean for both occurrences. However, in this example, no average is repeated because all condor weights are unique.Mean of the sampling distribution: We can find this by averaging the possible outcomes for ¯x: (4.5 + 5.5 + 8 + 6 + 8.5 + 9.5) / 6 = 42 / 6 = 7 pounds, which is identical to the population mean.
A manufacturing process outputs parts having a normal distribution with a mean of 30 cm and standard deviation of 2 cm. From a production sample of 80 parts, what proportion of the sample can be expected to fall between 28 and 32 cm
Answer:
68% of the sample can be expected to fall between 28 and 32 cm
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 30
Standard deviation = 2
What proportion of the sample can be expected to fall between 28 and 32 cm
28 = 30 - 2
28 is one standard deviation below the mean
32 = 30 + 2
32 is one standard deviation above the mean.
By the Empirical Rule, 68% of the sample can be expected to fall between 28 and 32 cm
A fishing company operates a search plane to find schools of fish in the ocean. Schools of fish are randomly located in the ocean. On average, there is one school of fish per 100,000 square miles of ocean. During any one day, the plane can search 10,000 square miles.
COMPLETE QUESTION:
What is the expected number of schools of fish found in one day of searching?
Answer: 0.1
Step-by-step explanation:
One school of fish per 100,000 square miles of ocean
Plane can search 10,000 square miles
Therefore,
Expected number of schools of fish found in one day of searching
= 10,000 / 100,000
= 0.1
A normally distributed population has mean 57,800 and standard deviation 750. Find the probability that a single randomly selected element X of the population is between 57,000 and 58,000. Find the mean and standard deviation of X - for samples of size 100. Find the probability that the mean of a sample of size 100 drawn from this population is between 57,000 and 58,000.
Answer:
(a) Probability that a single randomly selected element X of the population is between 57,000 and 58,000 = 0.46411
(b) Probability that the mean of a sample of size 100 drawn from this population is between 57,000 and 58,000 = 0.99621
Step-by-step explanation:
We are given that a normally distributed population has mean 57,800 and standard deviation 75, i.e.; [tex]\mu[/tex] = 57,800 and [tex]\sigma[/tex] = 750.
Let X = randomly selected element of the population
The z probability is given by;
Z = [tex]\frac{X-\mu}{\sigma}[/tex] ~ N(0,1)
(a) So, P(57,000 <= X <= 58,000) = P(X <= 58,000) - P(X < 57,000)
P(X <= 58,000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] <= [tex]\frac{58000-57800}{750}[/tex] ) = P(Z <= 0.27) = 0.60642
P(X < 57000) = P( [tex]\frac{X-\mu}{\sigma}[/tex] < [tex]\frac{57000-57800}{750}[/tex] ) = P(Z < -1.07) = 1 - P(Z <= 1.07)
= 1 - 0.85769 = 0.14231
Therefore, P(31 < X < 40) = 0.60642 - 0.14231 = 0.46411 .
(b) Now, we are given sample of size, n = 100
So, Mean of X, X bar = 57,800 same as before
But standard deviation of X, s = [tex]\frac{\sigma}{\sqrt{n} }[/tex] = [tex]\frac{750}{\sqrt{100} }[/tex] = 75
The z probability is given by;
Z = [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
Now, probability that the mean of a sample of size 100 drawn from this population is between 57,000 and 58,000 = P(57,000 < X bar < 58,000)
P(57,000 <= X bar <= 58,000) = P(X bar <= 58,000) - P(X bar < 57,000)
P(X bar <= 58,000) = P( [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] <= [tex]\frac{58000-57800}{\frac{750}{\sqrt{100} } }[/tex] ) = P(Z <= 2.67) = 0.99621
P(X < 57000) = P( [tex]\frac{Xbar-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{57000-57800}{\frac{750}{\sqrt{100} } }[/tex] ) = P(Z < -10.67) = P(Z > 10.67)
This probability is that much small that it is very close to 0
Therefore, P(57,000 < X bar < 58,000) = 0.99621 - 0 = 0.99621 .
Using the normal distribution and the central limit theorem, it is found that:
There is a 0.4641 = 46.41% probability that a single randomly selected element X of the population is between 57,000 and 58,000.For samples of size 100, the mean is of 57800 and the standard deviation is 75.There is a 0.9965 = 99.65% probability that the mean of a sample of size 100 drawn from this population is between 57,000 and 58,000.In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.By the Central Limit Theorem, for sampling distributions of samples of size n, the mean is [tex]\mu[/tex] and the standard deviation is [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].In this problem:
Mean of 57800, thus [tex]\mu = 57800[/tex].Standard deviation of 750, thus [tex]\sigma = 750[/tex].The probability that a single randomly selected element X of the population is between 57,000 and 58,000 is the p-value of Z when X = 58000 subtracted by the p-value of Z when X = 57000, thus:
X = 58000:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{58000 - 57800}{750}[/tex]
[tex]Z = 0.27[/tex]
[tex]Z = 0.27[/tex] has a p-value of 0.6064.
X = 57000:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{57000 - 57800}{750}[/tex]
[tex]Z = -1.07[/tex]
[tex]Z = -1.07[/tex] has a p-value of 0.1423.
0.6064 - 0.1423 = 0.4641.
0.4641 = 46.41% probability that a single randomly selected element X of the population is between 57,000 and 58,000.
For samples of size 100, [tex]n = 100[/tex], and then:
[tex]s = \frac{750}{\sqrt{100}} = 75[/tex]
For samples of size 100, the mean is of 57800 and the standard deviation is 75.
Then, the probability is:
X = 58000:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{58000 - 57800}{75}[/tex]
[tex]Z = 2.7[/tex]
[tex]Z = 2.7[/tex] has a p-value of 0.9965.
X = 57000:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{57000 - 57800}{75}[/tex]
[tex]Z = -10.7[/tex]
[tex]Z = -10.7[/tex] has a p-value of 0.
0.9965 - 0 = 0.9965.
0.9965 = 99.65% probability that the mean of a sample of size 100 drawn from this population is between 57,000 and 58,000.
A similar problem is given at https://brainly.com/question/24663213
Based on outcomes since 1967, the probability a team from the National League wins the baseball World Series in any given year is 0.44. Based on outcomes since 1967, the probability a team from the National Football conference wins the Superbowl in any given year is 0.53. Assume whether a team from the National League wins the World Series in a year is independent of whether a team from the National Football Conference wins the Superbowl in the same year. What is the probability a team from the National League wins the World Series in a year if a team from the National Football Conference wins the Superbowl in the same year
Answer:
Probability a team from national football conference wins the same year will be equal to 0.53
Step-by-step explanation:
Probability that a National League team wins baseball world series = 0.44
Probability that a National Football conference team wins the Superbowl = 0.53
The two events are independent, since the sports played are different (baseball and football). This is a logical assumption to make.
Since these events are independent, the occurrence of one event will not change the probability of the other event. This means that it does not matter whether the national league team won or lost the world series. The probability of the football team winning the Superbowl will remain the same, which is 0.53.
A mechanical assembly consists of a rod with a bearing on each end. The three parts are manufactured independently, and all vary a bit from part to part. The length of the rod has mean 23 centimeters (cm) and standard deviation 0.18 millimeters (mm). The length of a bearing has mean 2 cm and standard deviation 0.03 mm. What are the mean and standard deviation of the total length of the assembly? (Round your standard deviation answer to four decimal places.)
Answer:
Total mean= 27cm
Total standard deviation = 0.1849mm
Step-by-step explanation:
The mechanical assembly has three parts consisting of the length of the rod (the this be called x), the length of the bearing (be called y). You might be wondering where the third part is but the assembly usually has two bearings. Then the second length of the bearing (be called z).
From the problem,
μx = 23, μy =2, μz = 2
σx = 0.18, σy = 0.03, σz = 0.03
We can find the total Mean by adding all the means,
μxyz = 23 + 2 + 2 = 27cm
Since the length of the assembly are independent,
To find the total standard deviation, we must first find the total variance(square of standard deviation)
σ² = (0.18)² + (0.03)² + (0.03)² = 0.0342
Now, we find the standard deviation (square root of the variance)
σ = √0.0342 = 0.18493242
σ ≈ 0.1849mm (you can change to cm by multiplying by 10)
The mean total length of the mechanical assembly is 27 cm and the standard deviation is approximately 0.0183 cm when rounded to four decimal places.
Explanation:The question is asking for the mean and standard deviation of the total length of a mechanical assembly with a rod and two bearings. To find the mean total length, we simply add the mean lengths of the rod and two bearings.
The mean length of the rod is 23 cm and of each bearing is 2 cm, which sums up to a mean total length of 27 cm for the assembly since there are two bearings (23 + 2 + 2).
For standard deviation, since the parts are manufactured independently, we apply the rule of variances: The variance of the sum of independent variables is the sum of their variances. The standard deviation of the rod is 0.18 mm (or 0.018 cm), and of each bearing is 0.03 mm (or 0.003 cm).
Thus, the total variance for the assembly is the sum of the variances: (0.0182 + 0.0032 + 0.0032). After calculating, we take the square root to find the total standard deviation, which is approximately 0.0183 cm, rounded to four decimal places.