The answer is:
You shouldn't expect to get exactly 5000 heads, because you cannot predict precisely how many heads will occur.
The outcome in tossing a fair coin is based on chance.
However, according to the law of large numbers, the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances.
For example, in the case of a fair coin, where both head and tail have equal probability of occurrence, as the number of tosses becomes sufficiently large (say 1 million tosses), the ratio heads to tails in the outcome will be extremely close to 1:1.
So according to the law, we should expect to approach a point where half of the outcomes are heads and the other half are tails, as the number of tosses become very large.
The answer is (a). You should not expect to get exactly 5000 heads, because you cannot predict precisely how many heads will occur.
The law of large numbers states that as the number of independent trials of a random experiment increases, the observed frequency of each outcome approaches the expected frequency. In other words, the more times you toss a fair coin, the closer the proportion of heads will get to 50%.
However, the law of large numbers does not guarantee that you will get exactly 5000 heads even if you toss a fair coin 10,000 times. It is still possible to get more or fewer than 5000 heads, even though it is unlikely.
For example, if you toss a fair coin 100 times, you might get 55 heads and 45 tails. This is within the normal range of variation, even though it is not exactly 50 heads and 50 tails.
As the number of tosses increases, the probability of getting exactly 50/50 heads and tails decreases. However, the probability of getting close to 50/50 heads and tails increases.
In conclusion, you should not expect to get exactly 5000 heads even if you toss a fair coin 10,000 times. However, you can expect the proportion of heads to be close to 50%.
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Find the expression for the electric field, E [infinity] , of the ring as the point P becomes very far from the ring ( x ≫ R ) in terms of the radius R , the distance x , the total charge on the ring q , and the constant k = 1 / ( 4 π ϵ 0 ).
Answer:
The expression of the field E as the point P becomes very far from the ring is:
[tex]\vec{E}(x)=\displaystyle\frac{q}{4\pi\epsilon_0} \frac{sgn(x)}{x^2}\vec{x} \\\left \{ {{\vec{E}(x)=\frac{q}{4\pi\epsilon_0} \frac{1}{x^2}\vec{x} \mapsto x>0} \atop {\vec{E}(x)=\frac{q}{4\pi\epsilon_0} \frac{-1}{x^2}\vec{x} \mapsto x< 0 }} \right.[/tex]
Step-by-step explanation:
The Electric field expression is:
[tex]\vec{E}(x)=\displaystyle\frac{q}{4\pi\epsilon_0} \frac{x}{(R^2+x^2)^{\frac{3}{2}}}\vec{x}[/tex]
To determine the asked expression we use limits. If we consider that x≫R, this is the same as considering the radius insignificant respect the x distance. Therefore we can considerate than from this distance X, the radius R tends to zero:
[tex]\displaystyle\lim_{R \to{}0}{\vec{E}(x)}=\lim_{R \to{}0}{\frac{q}{4\pi\epsilon_0} \frac{x}{(R^2+x^2)^{\frac{3}{2}}}\vec{x}}\rightarrow\frac{q}{4\pi\epsilon_0} \frac{x}{(0^2+x^2)^{\frac{3}{2}}}\vec{x}=\frac{q}{4\pi\epsilon_0} \frac{x}{(x^2)^{\frac{3}{2}}}\vec{x}=\frac{q}{4\pi\epsilon_0} \frac{\cancel{x}}{|x|^{\cancel{3}}}\vec{x}=\displaystyle\frac{q}{4\pi\epsilon_0} \frac{sgn(x)}{x^2}\vec{x}[/tex]
The expression for the electric field of the ring as the point P becomes very far from the ring is [tex]E_{z} = \frac{k\cdot Q}{x^{2}}[/tex].
How to estimate an electric field for a ring with an uniform chargeLet suppose that the ring has an uniform linear electric density ([tex]\lambda[/tex]). A formula for the electric field at point P ([tex]E[/tex]) in rectangular coordinates is shown below:
[tex]\vec E = (E_{x}, E_{y}, E_{z})[/tex] (1)
Where:
[tex]E_{x}[/tex] - Electric field in the x-direction.[tex]E_{y}[/tex] - Electric field in the y-direction.[tex]E_{z}[/tex] - Electric field in the z-direction.Each component of the electric field are defined by the following integral formulae:
[tex]E_{x} = \int\limits^{2\pi}_{0} {\sin \theta \cdot \cos \phi} \, dE[/tex] (2)
[tex]E_{y} = \int\limits^{2\pi}_{0} {\sin \theta\cdot \sin\phi} \, dE[/tex] (3)
[tex]E_{z} = \int\limits^{2\pi}_{0} {\cos \theta} \, dE[/tex] (4)
Where:
[tex]\theta[/tex] - Axial angle, in radians.[tex]\phi[/tex] - Radial angle, in radians.By Coulomb's law and trigonometric and geometric relationships, we expand and solve each integral as following:
[tex]E_{x} = \frac{R}{\sqrt{x^{2}+R^{2}}}\int\limits^{2\pi}_{0} {\cos \phi} \, dE = \frac{k\cdot \lambda\cdot R^{2}}{(x^{2}+R^{2})^{3/2}}\int\limits^{2\pi}_{0} {\cos \phi} \, d\phi = 0[/tex]
[tex]E_{y} = \frac{R}{\sqrt{x^{2}+R^{2}}}\int\limits^{2\pi}_{0} {\sin \phi} \, dE = \frac{k\cdot \lambda\cdot R^{2}}{(x^{2}+R^{2})^{3/2}}\int\limits^{2\pi}_{0} {\sin \phi} \, d\phi = 0[/tex]
[tex]E_{z} = \frac{k\cdot \lambda\cdot x \cdot R}{(x^{2}+R^{2})^{3/2}} \int\limits^{2\pi}_{0}\, d\phi = \frac{x\cdot k \cdot (2\pi\cdot \lambda\cdot R)}{(x^{2}+R^{2})^{3/2}} = \frac{x\cdot k\cdot Q}{(x^{2}+R^{2})^{3/2}}[/tex] (5)
Where [tex]k[/tex] is the electrostatic constant.
If [tex]x >> R[/tex], (5) is simplified into the following expression:
[tex]E_{z} = \frac{k\cdot Q}{x^{2}}[/tex] (6)
Where [tex]Q[/tex] is the electric charge of the entire ring.
Please notice that (6) tends to be zero when [tex]x \to \infty[/tex]. The expression for the electric field of the ring as the point P becomes very far from the ring is [tex]E_{z} = \frac{k\cdot Q}{x^{2}}[/tex]. [tex]\blacksquare[/tex]
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Find the sample standard deviation s for the following sample data. Round your answer to the nearest hundredth. 23 20 14 35 28
Answer:
The standard deviation for given sample is 7.97
Step-by-step explanation:
We are given the following data set:
23, 20, 14, 35, 28
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{120}{5} = 24[/tex]
Sum of squares of differences = 1 + 16 + 100 + 121 + 16 = 254
[tex]S.D = \sqrt{\dfrac{254}{4}} = 7.97[/tex]
The standard deviation for given sample is 7.97
To find the sample standard deviation for the data sets 23, 20, 14, 35, 28, we calculate the mean, subtract the mean from each data point and square the result, sum these squares, divide by one less than the sample size to find the variance, and finally take the square root to find the standard deviation which is approximately 7.97.
To calculate the sample standard deviation (s), follow these steps:
Find the mean (average) of the sample data.
Subtract the mean from each data point and square the result.
Sum all the squared values.
Divide this sum by the sample size minus one (n-1) to get the sample variance.
Take the square root of the sample variance to find the sample standard deviation.
Let's apply these steps to the given data: 23, 20, 14, 35, 28.
Mean = (23 + 20 + 14 + 35 + 28) / 5 = 120 / 5 = 24.
Subtract the mean and square: (23 - 24)² = 1, (20 - 24)² = 16, (14 - 24)² = 100, (35 - 24)² = 121, (28 - 24)² = 16.
Sum of squares = 1 + 16 + 100 + 121 + 16 = 254.
Variance = 254 / (5 - 1) = 254 / 4 = 63.5.
Standard Deviation = √63.5 ≈ 7.97 (rounded to the nearest hundredth).
The sample standard deviation s is approximately 7.97.
Which equation can be used to find the total number of toothpicks?
Answer:B
Step-by-step explanation:
Evaluate the integral by changing the order of integration in an appropriate way. Integral from 0 to 1 Integral from 0 to 1 Integral from x squared to 1 3 x font size decreased by 7 z e Superscript z font size decreased by 5 y squared Baseline dy dx dz
Answer:
The question is not clear, but here is a similar question with the same approach.
Integral from 0 to 1 Integral from 1 to 2 Integral from 2 to 3, { (x+y+z) } dx dy dz by changing the order of integration in an appropriate way.
Step-by-step explanation:
The approach is that of multiple integral where changing the order of integration is done appropriately
The step by step with detailed workings are shown in the attachment below.
Suppose that 96% of bolts and 91% of nails meet specifications. One bolt and one nail are chosen independently. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part.
What is the probability that at least one of them meets specifications? (Round the final answer to four decimal places.)
The probability that at least one of them meets specifications is_______
Answer:
0.9964 is the probability that at least one of them meets specifications.
Step-by-step explanation:
We are given the following in the question:
B: Bolts meet the specification
N: Nails meet the specification
P(B) = 96% = 0.96
P(N) = 91% = 0.91
One bolt and one nail are chosen independently.
Thus, we can write
[tex]P(B\cap N) = P(B) \times P(N) = 0.96\times 0.91 = 0.8736[/tex]
We have to find the probability that at least one of them meets specifications.
[tex]P(B\cup N) = P(B) + P(N) -P(B\cap N)\\P(B\cup N) =0.96 + 0.91-0.8736\\P(B\cup N) =0.9964[/tex]
0.9964 is the probability that at least one of them meets specifications.
The most appropriate study design depends, among other things, on the distribution of:______
Option:
A) The risk factor in the population of interest
B) The participants
C) The outcome in the population of interest
D) A & C
Answer:
D) A & C
Step-by-step explanation:
Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area: where V = volume (mm3 ), t = time (min), k = the evaporation rate (mm/min), and A = surface area (mm2 ). Use Euler’s method to compute the volume of the droplet from t = 0 to 10 min using a step size of 0.25 min. Assume that k = 0.08 mm/min and that the droplet initially has a radius of 2.5 mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that it is consistent with the evaporation rate.
Answer:
V = 20.2969 mm^3 @ t = 10
r = 1.692 mm @ t = 10
Step-by-step explanation:
The solution to the first order ordinary differential equation:
[tex]\frac{dV}{dt} = -kA[/tex]
Using Euler's method
[tex]\frac{dVi}{dt} = -k *4pi*r^2_{i} = -k *4pi*(\frac {3 V_{i} }{4pi})^(2/3)\\ V_{i+1} = V'_{i} *h + V_{i} \\[/tex]
Where initial droplet volume is:
[tex]V(0) = \frac{4pi}{3} * r(0)^3 = \frac{4pi}{3} * 2.5^3 = 65.45 mm^3[/tex]
Hence, the iterative solution will be as next:
i = 1, ti = 0, Vi = 65.45[tex]V'_{i} = -k *4pi*(\frac{3*65.45}{4pi})^(2/3) = -6.283\\V_{i+1} = 65.45-6.283*0.25 = 63.88[/tex]
i = 2, ti = 0.5, Vi = 63.88[tex]V'_{i} = -k *4pi*(\frac{3*63.88}{4pi})^(2/3) = -6.182\\V_{i+1} = 63.88-6.182*0.25 = 62.33[/tex]
i = 3, ti = 1, Vi = 62.33[tex]V'_{i} = -k *4pi*(\frac{3*62.33}{4pi})^(2/3) = -6.082\\V_{i+1} = 62.33-6.082*0.25 = 60.813[/tex]
We compute the next iterations in MATLAB (see attachment)
Volume @ t = 10 is = 20.2969
The droplet radius at t=10 mins
[tex]r(10) = (\frac{3*20.2969}{4pi})^(2/3) = 1.692 mm\\[/tex]
The average change of droplet radius with time is:
Δr/Δt = [tex]\frac{r(10) - r(0)}{10-0} = \frac{1.692 - 2.5}{10} = -0.0808 mm/min[/tex]
The value of the evaporation rate is close the value of k = 0.08 mm/min
Hence, the results are accurate and consistent!
Using Euler's method, we approximated the volume and surface area of the object over a period of 10 minutes. Starting with an initial radius of 2.5 mm, and given a decay constant of 0.08 mm/min, we computed the final volume to be approximately 26.19 mm³ and the final surface area to be about 31.70 mm². This resulted in a final radius of approximately 2.06 mm.
To solve this problem using Euler's method, we'll use the following formulas:
dV/dt = -kA
A = 4πr²
V = (4/3)πr³
Given that the initial radius r₀ = 2.5 mm, we can compute the initial volume V₀ and surface area A₀. Then, we'll iterate through time steps using Euler's method:
Vₙ₊₁ = Vₙ - kA * Δt
Aₙ₊₁ = 4π(rₙ - (k/3) * Δt)²
Using k = 0.08 mm/min and a step size of Δt = 0.25 min, we perform the calculations:
V₀ = (4/3)π(2.5)³ ≈ 65.45 mm³
A₀ = 4π(2.5)² ≈ 98.17 mm²
Iterating:
t = 0.25 min:
V₁ ≈ 65.45 - 0.08 * 98.17 * 0.25 ≈ 61.44 mm³
A₁ ≈ 4π(2.44)² ≈ 59.17 mm²
t = 0.5 min:
V₂ ≈ 61.44 - 0.08 * 59.17 * 0.25 ≈ 59.58 mm³
A₂ ≈ 4π(2.42)² ≈ 58.01 mm²
Continuing this process until t = 10 min, we obtain:
V₄₀ ≈ 26.19 mm³
A₄₀ ≈ 31.70 mm²
Finally, calculating the radius r₄₀ corresponding to V₄₀:
r₄₀ = ((3V₄₀)/(4π))^(1/3) ≈ 2.06 mm
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An article reported the following data on oxygen consumption (mL/kg/min) for a sample of ten firefighters performing a fire-suppression simulation: 28.6 49.4 30.3 28.2 28.9 26.4 33.8 29.9 23.5 30.2Compute the following. (Round your answers to four decimal places.) a. The sample range mL/kg/minb. The sample variance s2 from the definition (i.e., by first computing deviations, then squaring them, etc.) mL2/kg2/min2c. The sample standard deviation mL/kg/mind. s2 using the shortcut method mL2/kg2/min
Answer:
a) The sample range 25.9 [tex]ml\slash kg\slash \min[/tex]
b) The sample variance is 49.344 [tex]ml^2 \slash kg^2 \slash min^2[/tex]
c) The sample standard deviation 7.0245 [tex]ml\slash kg\slash \min[/tex]
Step-by-step explanation:
We are given the following data on oxygen consumption (mL/kg/min):
28.6, 49.4, 30.3, 28.2, 28.9, 26.4, 33.8, 29.9, 23.5, 30.2
a) The sample range
Range = Maximum - Minimum
[tex]\text{Range} = 49.4 - 23.5 = 25.9[/tex]
The sample range 25.9 [tex]ml\slash kg\slash \min[/tex]
b) The sample variance
[tex]\text{Variance} = \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{309.2}{10} = 30.92[/tex]
Sum of squares of differences =
5.3824 + 341.5104 + 0.3844 + 7.3984 + 4.0804 + 20.4304 + 8.2944 + 1.0404 + 55.0564 + 0.5184 = 444.096
[tex]s^2 = \dfrac{444.096}{9} = 49.344[/tex]
The sample variance is 49.344 [tex]ml^2 \slash kg^2 \slash min^2[/tex]
c) The sample standard deviation
It is the square root of sample variance.
[tex]s = \sqrt{s^2} = \sqrt{49.344} = 7.0245[/tex]
The sample standard deviation 7.0245 [tex]ml\slash kg\slash \min[/tex]
Which equation represent the relationship between the total number of pages N that Ronalds can read in M minutes?
Answer:option A is the correct answer.
Step-by-step explanation:
Ronald can read at a constant rate of p pages per minute.
If Roland can read a total number of N pages in minutes, then the equation representing the relationship between the number of pages, N and the time, m minutes would be
p pages = 1 minute
N pages = m minutes
Crossmultiplying, it becomes
p × m = N × 1
N = pm
4. Using the geometric sum formulas, evaluate each of the following sums and express your answer in Cartesian form.
Answer:
[tex]\sum_{n=0}^9cos(\frac{\pi n}{2})=1[/tex]
[tex] \sum_{k=0}^{N-1}e^{\frac{i2\pi kk}{2}}=0[/tex]
[tex] \sum_{n=0}^\infty (\frac{1}{2})^n cos(\frac{\pi n}{2})=\frac{1}{2}[/tex]
Step-by-step explanation:
[tex] \sum_{n=0}^9cos(\frac{\pi n}{2})=\frac{1}{2}(\sum_{n=0}^9 (e^{\frac{i\pi n}{2}}+ e^{\frac{i\pi n}{2}}))[/tex]
[tex]=\frac{1}{2}(\frac{1-e^{\frac{10i\pi}{2}}}{1-e^{\frac{i\pi}{2}}}+\frac{1-e^{-\frac{10i\pi}{2}}}{1-e^{-\frac{i\pi}{2}}})[/tex]
[tex]=\frac{1}{2}(\frac{1+1}{1-i}+\frac{1+1}{1+i})=1[/tex]
2nd
[tex]\sum_{k=0}^{N-1}e^{\frac{i2\pi kk}{2}}=\frac{1-e^{\frac{i2\pi N}{N}}}{1-e^{\frac{i2\pi}{N}}}[/tex]
[tex]=\frac{1-1}{1-e^{\frac{i2\pi}{N}}}=0[/tex]
3th
[tex] \sum_{n=0}^\infty (\frac{1}{2})^n cos(\frac{\pi n}{2})==\frac{1}{2}(\sum_{n=0}^\infty ((\frac{e^{\frac{i\pi n}{2}}}{2})^n+ (\frac{e^{-\frac{i\pi n}{2}}}{2})^n))[/tex]
[tex]=\frac{1}{2}(\frac{1-0}{1-i}+\frac{1-0}{1+i})=\frac{1}{2}[/tex]
What we use?
We use that
[tex] e^{i\pi n}=cos(\pi n)+i sin(\pi n)[/tex]
and
[tex]\sum_{n=0}^k r^k=\frac{1-r^{k+1}}{1-r}[/tex]
Geometric sum formulas are used to evaluate sums of a geometric series, with the result expressed in Cartesian form (a + bi) where a is the real part and bi is the imaginary part. The sum of a geometric series is calculated with the formula: Sum = a * (1 - r^n) / (1 - r), where a is the first term and r is the ratio. Please provide the specific sums for a detailed step-by-step calculation.
Explanation:The problem at hand revolves around the usage of geometric sum formulas to evaluate sums and to express the result in Cartesian form. The critical point to remember is that a geometric series is a series with a constant ratio between successive terms. The sum of the first 'n' terms of a geometric sequence can be calculated using the formula:
Sum = a * (1 - rⁿ) / (1 - r)
Assuming 'a' represents the first term in the series and 'r' is the ratio.To convert a complex number into Cartesian form, you simply map the real and imaginary parts of the number 'a + bi', where 'a' is the real part, and 'bi' is the imaginary part.Unfortunately, without the specifics of the sums you're looking to evaluate, it's impossible to give a concrete step-by-step calculation. However, understanding the formulas and how they're applied should provide you with a good start.
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The average age in a sample of 190 students at City College is 22. As a result of this sample, it can be concluded that the average age of all the students at City College
Answer:
The true mean [tex]\mu[/tex] it probably could be larger, smaller, or equal to 22
Step-by-step explanation:
False.
By definition the sample mean is defined as:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
For this case the value for the sample size is n =190 and the calculated sample mean is [tex] \bar X=22[/tex]. This value represent the sample and for this case we can't assume that this value represent at all the population as the population mean [tex]\mu[/tex] since we probably have variability from the data of the students at City College.
So we can conclude that the true mean [tex]\mu[/tex] it probably could be larger, smaller, or equal to 22
The true mean it probably could be larger, smaller, or equal to 22.
Given that:
Total student, n = 190.
Average age of 190 students, [tex]\bar X = 22\\[/tex].
By definition the sample mean is defined as:
[tex]\bar X =\dfrac{{\sum_{i=1}^nX_i}}{n}[/tex]
For this case, the sample size n =190 and the calculated sample mean is [tex]\bar X = 22\\[/tex]. This value represent the sample and for this case can't assume that this value represent at all the population as the population mean since probably have variability from the data of the students at city college.
Hence, conclude that the true mean it probably could be larger, smaller, or equal to 22
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Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.5 millimeters (mm) and a standard deviation of 1.7 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)
(a) the thickness is less than 3.0 mm.
(b) the thickness is more than 7.0 mm.
(c) the thickness is between 3.0 mm and 7.0 mm.
Answer:
a) [tex]P(X<3.0)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-4.5}{1.7})=P(z<-0.882)[/tex]
[tex]P(z<-0.882)=0.189[/tex]
b) [tex]P(X>7.0)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z<\frac{7-4.5}{1.7})=P(z>1.47)[/tex]
[tex]P(z>1.47)=1-P(z<1.47) = 1-0.929=0.071[/tex]
c) [tex]P(3<X<7)=P(\frac{3-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{7-\mu}{\sigma})=P(\frac{3-4.5}{1.7}<Z<\frac{7-4.5}{1.7})=P(-0.882<z<1.47)[/tex]
[tex]P(-0.882<z<1.47)=P(z<1.47)-P(z<-0.882)=0.929-0.189=0.740[/tex]
Step-by-step explanation:
Previous concepts
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".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the thickness of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(4.5,1.7)[/tex]
Where [tex]\mu=4.5[/tex] and [tex]\sigma=1.7[/tex]
We are interested on this probability
[tex]P(X<3.0)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X<3.0)=P(\frac{X-\mu}{\sigma}<\frac{3-\mu}{\sigma})=P(Z<\frac{3-4.5}{1.7})=P(z<-0.882)[/tex]
And we can find this probability using excel or the normal standard table:
[tex]P(z<-0.882)=0.189[/tex]
Part b
We are interested on this probability
[tex]P(X>3.0)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X>7.0)=P(\frac{X-\mu}{\sigma}>\frac{7-\mu}{\sigma})=P(Z<\frac{7-4.5}{1.7})=P(z>1.47)[/tex]
And we can find this probability using excel or the normal standard table:
[tex]P(z>1.47)=1-P(z<1.47) = 1-0.929=0.071[/tex]
Part c
[tex]P(3<X<7)=P(\frac{3-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{7-\mu}{\sigma})=P(\frac{3-4.5}{1.7}<Z<\frac{7-4.5}{1.7})=P(-0.882<z<1.47)[/tex]
And we can find this probability using excel or the normal standard table liek this:
[tex]P(-0.882<z<1.47)=P(z<1.47)-P(z<-0.882)=0.929-0.189=0.740[/tex]
in a class there are
8 students who play football and cricket
4 students who do not play football or cricket
14 students who play football
20 students who play cricket
find the probability that a student chosen at random plays football or cricket or both
dont necessarily need an explanation but if you have a simple one i can understand then please do tell me x
Answer:
the probability that a student chosen at random plays football or cricket or both = [tex]\frac{1}{5} + \frac{2}{5} + \frac{4}{15} = \frac{13}{15}[/tex]
Step-by-step explanation:
i) 8 students play football and cricket
ii) 4 students do not play football or cricket
iii) total of 14 students play football.
iv) therefore the number of students who play only football is = 14 - 8 = 6
v) total of 20 students play cricket.
vi) therefore the number of students who play only cricket is = 20 - 8 = 12
vii) therefore the total number of students = 8 + 4 + 6 + 12 = 30
viii) the probability a student chosen at random plays football = [tex]\frac{6}{30} = \frac{1}{5}[/tex]
ix) the probability a student chosen at random plays cricket = [tex]\frac{12}{30} = \frac{2}{5}[/tex]
x) the probability a student chosen at random plays both football and cricket = [tex]\frac{8}{30} = \frac{4}{15}[/tex]
xi) therefore the probability that a student chosen at random plays football or cricket or both = [tex]\frac{1}{5} + \frac{2}{5} + \frac{4}{15} = \frac{13}{15}[/tex]
The probability that a student chosen at random plays football or cricket or both is [tex]\frac{13}{15}[/tex].
We have
Number of students play football and cricket = 8
Number of students do not play football or cricket = 4
Total Number of students play football = 14
Therefore, the number of students who play only football
= 14 - 8
= 6
Total Number of students play cricket = 20
Therefore, the number of students who play only cricket
= 20 - 8
= 12
So, the total number of students
= 8 + 4 + 6 + 12
= 30
Now, the probability that a student chosen at random plays football
[tex]=\frac{6}{30} \\=\frac{1}{5}[/tex]
The probability that a student chosen at random plays cricket
[tex]=\frac{12}{30} \\=\frac{2}{5}[/tex]
The probability a student chosen at random plays both football and cricket [tex]=\frac{8}{30} \\=\frac{4}{15}[/tex]
Therefore, the probability that a student chosen at random plays football or cricket or both
[tex]=\frac{1}{5} +\frac{2}{5}+\frac{4}{15}\\=\frac{3}{15} +\frac{6}{15}+\frac{4}{15}\\=\frac{13}{15}[/tex]
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If we collect a large sample of blood platelet counts and if our sample includes a single outlier, how will that outlier appear in a histogram?
A. The outlier will appear as a tall bar near one side of the distribution.
B. Since a histogram shows frequencies, not individual data values, the outlier will not appear. Instead, the outlier increases the frequency for its class by 1
C. The outlier will appear as the tallest bar near the center of the distribution
D. The outlier will appear as a bar far from all of the other bars with a height that corresponds to a frequency of 1.
Answer:
D. The outlier will appear as a bar far from all of the other bars with a height that corresponds to a frequency of 1.
Step-by-step explanation:
An histogram measures how many times each value appears in the set we are studying. That is, it is a frequency measure.
Suppose we have the following set:
S = {1,1,1,1,1,1, 2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,100}
1 appears 6 times. That means that when the X axis is 1, the y axis is 6.
2 appears 8 times. The means that when the X axis is 2, the y axis is 8.
...
100 appears 1 time. This means that when the X axis is 100, the y axis is 1. The X is the outlier, and it is quite far from the other values.
So the correct answer is:
D. The outlier will appear as a bar far from all of the other bars with a height that corresponds to a frequency of 1.
simplify -1/64........
Answer:
- 2⁻⁶
Step-by-step explanation:
To simplify this we have to know the following rules.
(i) (xᵃ)ᵇ = xᵃᵇ
(ii) 1/xᵃ = x⁻ᵃ
Given: [tex]$ \frac{-1}{64} $[/tex]
= [tex]$ \frac{-1}{4^3} $[/tex]
= [tex]$ \frac{-1}{(2^2)^3} $[/tex]
= [tex]$ \frac{-1}{2^6} \hspace{10mm} $[/tex] [using (i)]
= [tex]$ 2^{-6} $[/tex] [using (ii)]
Hence, the simplified form would be: [tex]$ 2^{-6} $[/tex]
An engineer in charge of water rationing for the U.S. Army wants to determine if the average male soldier spends less time in the shower than the average female soldier. Let μm represent the average time in the shower of male soldiers and μf represent the average time in the shower of female soldiers.
a) What are the appropriate hypotheses for the engineer?
H0: μm = μf versus Ha: μm > μf
H0: σm = σf versus Ha: σm > σf
H0: μm = μf versus Ha: μm ≠ μf
H0: μm = μf versus Ha: μm < μf
b) Among a sample of 66 male soldiers the average shower time was found to be 2.68 minutes and the standard deviation was found to be 0.65 minutes. Among a sample of 69 female soldiers the average shower time was found to be 2.7 minutes and the standard deviation was found to be 0.5 minutes. What is the test statistic? Give your answer to three decimal places.
c) What is the P-value for the test? Give your answer to four decimal places.
d) Using a 0.1 level of significance, what is the appropriate conclusion?
Reject the claim that the average shower times are different for male and female soldiers because the P-value is greater than 0.1.
Conclude that the average shower time for males is less than the average shower time for females because the P-value is less than 0.1.
Fail to reject the claim that the average shower times are the same for male and female soldiers because the P-value is greater than 0.1.
Conclude that the average shower time for males is equal to the average shower time for females because the P-value is less than 0.1.
Answer:
a) H0: μm = μf versus Ha: μm < μf
b) [tex]t=\frac{(2.68-2.7)-0}{\sqrt{\frac{0.65^2}{66}+\frac{0.5^2}{69}}}}=-0.200[/tex]
c) [tex]p_v =P(t_{133}<-0.200)=0.421[/tex]
d) Fail to reject the claim that the average shower times are the same for male and female soldiers because the P-value is greater than 0.1.
Step-by-step explanation:
Data given and notation
[tex]\bar X_{m}=2.68[/tex] represent the mean for the sample male
[tex]\bar X_{f}=2.7[/tex] represent the mean for the sample female
[tex]s_{m}=0.65[/tex] represent the sample standard deviation for the males
[tex]s_{f}=0.5[/tex] represent the sample standard deviation for the females
[tex]n_{m}=66[/tex] sample size for the group male
[tex]n_{f}=69[/tex] sample size for the group female
t would represent the statistic (variable of interest)
Part a
Concepts and formulas to use
We need to conduct a hypothesis in order to check if the average male soldier spends less time in the shower than the average female soldier, the system of hypothesis would be:
Null hypothesis:[tex]\mu_{m}-\mu_{f}\geq 0[/tex]
Alternative hypothesis:[tex]\mu_{m} - \mu_{f}< 0[/tex]
Or equivalently:
Null hypothesis:[tex]\mu_{m}-\mu_{f}= 0[/tex]
Alternative hypothesis:[tex]\mu_{m} - \mu_{f}< 0[/tex]
And the best option is:
H0: μm = μf versus Ha: μm < μf
Part b
We don't have the population standard deviation, so for this case is better apply a t test to compare means, and the statistic is given by:
[tex]t=\frac{(\bar X_{m}-\bar X_{f})-\Delta}{\sqrt{\frac{s^2_{m}}{n_{m}}+\frac{s^2_{f}}{n_{f}}}}[/tex] (1)
And the degrees of freedom are given by [tex]df=n_m +n_f -2=66+69-2=133[/tex]
t-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.
What is the test statistic?
With the info given we can replace in formula (1) like this:
[tex]t=\frac{(2.68-2.7)-0}{\sqrt{\frac{0.65^2}{66}+\frac{0.5^2}{69}}}}=-0.200[/tex]
Part c What is the p-value?
Since is a left tailed test the p value would be:
[tex]p_v =P(t_{133}<-0.200)=0.421[/tex]
Part d
The significance level given is [tex] \alpha =0.1[/tex] since the p value is higher than the significance level we can conclude:
Fail to reject the claim that the average shower times are the same for male and female soldiers because the P-value is greater than 0.1.
The question relates to hypothesis testing in statistics, insights into average shower times for male and female soldiers. After formulating the hypotheses, we calculate the test statistic using the provided sample data, and then find the corresponding P-value. If the P-value is less than our significance level, we reject the null hypothesis and side with the alternative hypothesis.
Explanation:The subject of this question falls under Mathematics, specifically it deals with hypothesis testing statistics.
a) The appropriate hypotheses for the engineer to consider would be: H0: μm = μf versus Ha: μm > μf
b) To calculate the test statistic, we use the formula for the test statistic in a independent two-sample t-test which incorporates the sample sizes, means, and standard deviations from the two groups. The formula is (avg(male soldiers)-avg(female soldiers))/sqrt(((sd(male soldiers))^2/number(male soldiers))+((sd(female soldiers))^2/number(female soldiers))). Plug in given values, we can obtain the test statistic.
c) The P-value can be obtained by looking up the test statistic in the T distribution table.
d) If the P-value is greater than the level of significance (0.1), we would fail to reject the null hypothesis. If the P-value is less than the level of significance, we would reject the null hypothesis. The conclusion is based on the specific P-value we computed.
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Marian went shopping and bought clothes for $76.17 and books for $44.98. She then had a meal at the mall for $19.15. Which is the best estimate of the total cost of her shopping trip?
A. $130
B. $120
C. $150
D. $140
Answer:
D. $140.
Step-by-step explanation:
Given:
Cost of clothes = $76.17
Cost of books = $44.98
Cost of meal = $19.15
We need to find the best estimate of total cost of her shopping trip.
Solution:
First we will find the total cost of her shopping trip.
total cost of her shopping trip is equal to sum of Cost of clothes, Cost of books and Cost of meal.
framing in equation form we get;
total cost of her shopping trip = [tex]76.17+44.98+19.15 = \$140.3[/tex]
Now we can say that;
140.3 is close to 140
Hence Best estimate of total cost of shopping trip is $140.
Compute each of the following complex numbers, giving your answers in both rectangular and exponential forms. Sketch each complex number, on individual pairs of axes, and indicate on each plot the real part, imaginary part, magnitude, and phase in radians.(a) q = [(e - jπ)/(π - je)]^(2/9)(b) r = abcdf, wherea = √3(1 + j) + (1- j) d = 1 + j√3b = √3 + j f = jc = 1+ j
Answer:
The complex numbers computed are:
A) [tex]q=0.8752+j0.4838=1e^{-j0.5049}[/tex]
B) [tex]r=-8-j8\sqrt{3} =16e^{j\pi \frac{4}{3}}[/tex]
The sketches are attached to this answer
Step-by-step explanation:
To compute these complex numbers you have to remember these rules:
[tex]Z=a+jb=(a^2+b^2)^{\frac{1}{2}}e^{jtan^{-1}(b/a)}[/tex] (a)
[tex]Z=|z|e^{j\alpha}=|z|cos(\alpha)+j|z|sin(\alpha)[/tex] (b)
Also for multiplication, division, and powers, if W and U are complex numbers and k is a real number:
[tex]{W}\cdot{U}={|W|e^{j\alpha}}{|U|e^{j\beta}}={|W|}{|U|}e^{j(\alpha+\beta)}[/tex] (1)
[tex]\frac{W}{U}=\frac{|W|e^{j\alpha}}{|U|e^{j\beta}}=\frac{|W|}{|U|}e^{j(\alpha-\beta)}[/tex] (2)
[tex]W^{k}=|w|^{k}e^{j(\alpha\cdot k)}[/tex] (3)
With these rules we will do the followings steps:
for A:
1) We solve first the divition, writing the 2 complex numbers exponential form (equation (a)).
2) With the rule (2) we solve the division.
3) with rule (3) we solve the power.
For B:
1)We write the numbers a, b, c, d, and f in exponential form (equation (a)).
2) We use the rule (1) for the product.
While conducting experiments, a marine biologist selects water depths from a uniformly distributed collection that vary between 2.00 m and 7.00 m. What is the expected value of the water depth
Answer: The expected value of the water depth is 4.5 m.
Step-by-step explanation:
Let x be a random variable which is uniformly distributed in interval [a,b] .
Then the mean of the distribution is ghiven by :-
[tex]E(x)=\dfrac{a+b}{2}[/tex]
Given : While conducting experiments, a marine biologist selects water depths from a uniformly distributed collection that vary between 2.00 m and 7.00 m.
Then, the expected value of the water depth = [tex]\dfrac{2+7}{2}=\dfrac{9}{2}=4.5[/tex]
Hence, the expected value of the water depth is 4.5 m.
Help me plsss I need it by tonight
Answer:
[tex]y=2x+2[/tex]
Step-by-step explanation:
we know that
The equation of a linear function has no exponents higher than 1, and the graph of a linear function is a straight line.
Verify each case
case 1) we have
[tex]y=2x+2[/tex]
Is the equation of a line in slope intercept form
so
Is a straight line and has no exponents higher than 1
therefore
Is a linear equation
case 2) we have
[tex]y=2x^{2}+2[/tex]
Is a quadratic equation
Is a curved line and has at least one exponent higher than 1,
therefore
Is a non-linear equation
case 3) we have
[tex]y=2x^{3}+2[/tex]
Is a cubic equation
Is a curved line and has at least one exponent higher than 1,
therefore
Is a non-linear equation
case 4) we have
[tex]y=2x^{4}+2[/tex]
Is a quartic equation
Is a curved line and has at least one exponent higher than 1,
therefore
Is a non-linear equation
Ryan is a record executive for a hip hop label in Atlanta, Georgia. He has a new album coming out soon, and wants to know the best way to promote it, so he is considering many variables that may have an effect. He is considering three different album covers that may be used, four different television commercials that may be used, and two different album posters that may be used. Determine the number of different combinations he needs in order to test each album cover, television commercial, and album poster.
Answer: 24
Step-by-step explanation:
Given : Choices for album covers = 3
Choices for television commercials = 4
Choices for album posters = 2
Now , the number of different combinations he needs in order to test each album cover, television commercial, and album poster = ( Choices for album covers ) x (Choices for television commercials) x (Choices for album posters)
= 3 x 4 x 2 = 24
Hence, the number of different combinations he needs in order to test each album cover, television commercial, and album poster is 24.
An aircraft seam requires 22 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. (Round your answers to four decimal places.)
(a) If 19% of all seams need reworking, what is the probability that a rivet is defective?
(b) How small should the probability of a defective rivet be to ensure that only 9% of all seams need reworking?
Answer:
Part A:
[tex]p=0.0095[/tex]
Part B:
[tex]p=0.0043[/tex]
Step-by-step explanation:
Part A:
The number of rivets=22 rivets
Probability that no rivet is defective= (1-p)^22
The probability that at least one rivet is defective=1-(1-p)^22
For 19% of all seams need reworking, probability that a rivet is defective is given by:
[tex]1-(1-p)^{22}=0.19[/tex]
[tex](1-p)^{22}=0.81\\p=1-\sqrt[22]{0.81} \\p=0.0095[/tex]
Part B:
For 9% of all seams need reworking, probability of a defective rivet is:
[tex]1-(1-p)^{22}=0.09\\p=1-\sqrt[22]{0.91} \\p=0.0043[/tex]
To find the probability of a defective rivet in a seam and the smallest probability of a defective rivet to ensure a certain reworking percentage, we use the concept of independent events and probability calculations.
Explanation:(a) To find the probability that a rivet is defective:
Let p be the probability of a defective rivet.
Since 19% of seams need reworking, 19% of the seams have at least one defective rivet.
Therefore, 19% of all seams equals the probability that at least one rivet is defective:
P(at least one defective rivet) = 1 - P(no defective rivets) = 0.19
P(no defective rivets) = 1 - P(at least one defective rivet) = 1 - 0.19
P(no defective rivets) = 0.81
Since each rivet is defective independently of one another, the probability that a rivet is defective is:
p = 1 - P(no defective rivet)
p = 1 - 0.81
p = 0.19
Therefore, the probability that a rivet is defective is 0.19 or 19%.
(b) To find the smallest probability of a defective rivet:
Let p be the probability of a defective rivet that ensures only 9% of seams need reworking.
We need to find the value of p such that P(at least one defective rivet) = 0.09.
From part (a), we know that P(at least one defective rivet) = 1 - P(no defective rivets) = 0.19.
Therefore, we can set up the equation:
0.19 = 1 - (1 - p)22
Solving this equation will give us the smallest value of p that satisfies the condition.
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Determine whether the value given is a parameter or statistic. Two thirds of all the students in this class are womena. Parameterb. Statistic
Answer: a. Parameter
Step-by-step explanation:
Parameter can be defined as a fact/characteristic about a whole population.
For example:
i. 20% of my class are boys
ii. 40% of Canadian senators are women.
Parameter usually deal with a small measurable population.
While statistic is a characteristic/fact about the sample.
Therefore the case above is a parameter because the students in the class is the population and the stated (Two thirds of all the students in this class are women) is a fact about the population.
Answer:
Parameter
Step-by-step explanation:
The given value is Parameter because it contains the measurement of population. In short, the given value 2/3 is the estimated value of population.
The population consists of all the students in the class. If the ratio of women from all the students in the class is calculated, then it is a measure of population. Thus, the given value is parameter.
The California State University (CSU) system consists of 23 campuses, from San Diego State in the south to Humboldt State near the Oregon border. A CSU administrator wishes to make an inference about the average distance between the hometowns of students and their campuses. Describe and discuss several different sampling methods that might be employed. (Select all that apply.)
a. One could take a simple random sample of students from all students in the California State University system and ask each student in the sample to report the distance from their hometown to campus.
b. There are no potential problems with self reporting of distances.
c. Certain problems arise with self reporting of distances, such as recording error or poor recall.
d. Instead of taking a random sample, every student should be included in the study.
e. The sample could be generated by taking a stratified random sample by taking a simple random sample from each of the 23 campuses and again asking each student in the sample to report the distance from their hometown to campus.
A Simple Random Sample or a Stratified Random Sample of students across the CSU campuses would allow for reliable data collection on the average distance between student hometowns and their campus, taking into account that self-reporting may introduce errors.
Explanation:When considering methods to sample the average distance between the hometowns of students and their California State University (CSU) campuses, there are several sampling techniques that can be considered:
(a) Simple Random Sample - This involves randomly selecting students from the entire CSU system, which could help ensure that each student has an equal chance of being included in the sample.(c) Self-reporting issues – When students report distances, errors can occur due to recording mistakes or poor recall. This is an important consideration that can affect data accuracy.(e) Stratified Random Sample - This method involves taking a simple random sample from each of the 23 campuses to avoid overrepresentation or underrepresentation of any single campus and can provide a more accurate reflection of the entire system.Option (b) is incorrect as there are potential problems with self-reporting of distances, and option (d) is impractical for such a large population and not necessary for making inferences. Therefore, options (a), (c), and (e) are relevant to the question.
For each of the following questions, select a research technique that is likely to yield a useful answer. For instance, if the question is "Which companies within a 20-mile radius of our company headquarters sell recycled paper?" a search of the web is likely to provide a useful answer.
a. Does the Honda CR-V include traction control as a standard feature?
b. How much money has our company's philanthropic foundation donated to colleges and universities in each of the last three years?
c. How does a 3D printer work?
d. Could our Building 3 support a rooftop green space?
e. How can we determine whether we would save more money by switching to LED lighting in our corporate offices?
Answer:
Web searching, specialists consultations and comparisons.
Step-by-step explanation:
a. Does the Honda CR-V iclude traction control as a standard feature?
Research about the Honda CR-V on the internet, or reading an article about it.
b. How much money has our company´s philanthropic foundation donated to colleges and universities in each of the last three year?
Look over the company´s administrative records.
c.How does a 3D printer work?
Search on the web about the 3D printer function.
d. Could our Building 3 support a rooftop green space?
Consultation with an architect.
e. How can we determine whether we would save more money by switching to LED lighting in our corporate offices?
Search on the web about the LED lighting use of electricty and the use of electricty of the type of lighting that the company is already using and compare for the best one.
Which equation shows this relationship?
Answer: y=x+2
Step-by-step explanation:
a shirt is on sale for 40% off, and you have an additional 20% off coupon. true or false: the shirt will ultimately be 60% off the original price
Answer:
Google it
Step-by-step explanation:
google is very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very very useful
Answer:it is false.
Step-by-step explanation:
Let us assume that the regular price of the shirt is $x.
The shirt is on sale for 40% off the regular price. The amount that is taken off the shirt would be
40/100 × x = 0.4 × x = 0.4x
The new price of the shirt would be x - 0.4x = $0.6x
you have an additional 20% off coupon. The value of the coupon would be
20/100 × 0.6x = 0.2 × 0.6x = 0.12x
The cost of the shirt if the coupon is applied would be
0.6x - 0.12x = 0.48x
If you assumed that the shirt will ultimately be 60% off the original price, the cost of the shirt would be
x - 60/100 × x = x - 0.6x = 0.4x
Therefore, they are not equal and si, it is false.
What is next in this sequence of numbers: 1, 11, 21, 1211, 111221, 312211, ...?
Answer:
13112221
Step-by-step explanation:
Each sequence of numbers is a verbal representation of the sequence before it. Thus, starting with 1, the next sequence would be "one one," or "11." That sequence is followed by "two one," or "21," and so on and so forth.
This may also be a good explanation:
The first number is just ONE (amount) "1" (0-9 numeral). So if you say there's ONE "1" (seriously just say it aloud) the next number would be an 11. Then there are TWO "1's", creating 21. Then ONE "2" and ONE "1" which creates 1,211. Then ONE "1", ONE "2", and TWO "1's" creating 111,221 ... and so on.
The first number 1 is read as one one, so the second number is written as 11, this is read as two ones, so the next number is written as 21 ( two ones)
This continues throughout the sequence.
The last number written is 312211 which is read as one three, one one, two twos, two ones
This gets written as 13112221
A consulting company must hire 20 new associates per year to replace those who have left the company for other positions or have retired. The company employs 117 associates overall. How long is the average associate employed at the consulting company?
Answer: 5.85 years
Therefore, an average associate is employed for 5.85 years.
Step-by-step explanation:
Given:
Rate of employment yearly = 20 associates per year
Total number of associates = 117 associates
Since the total number of associates remain constant the rate at which they employ new associates is equal to the rate at which associates leave = 20 per year
If 20 new associates are employed in a particular year it would take aan average of :
Average employment year A = total number of associates divided by the rate at which associates leave
A = 117/20
A = 5.85 years
Therefore, an average associate is employed for 5.85 years.
The average length of employment for an associate at the consulting company is calculated by dividing the total number of associates (117) by the annual turnover (20), yielding an average of approximately 5.85 years.
Explanation:To calculate the average length of employment for associates at the consulting company, we can use the concept of employee turnover rate, which is the rate at which employees leave an organization and are replaced. Since the company must hire 20 new associates each year to replace those who have departed and the total number of associates is 117, we can use the formula for the average employment duration: Total Number of Associates / Annual Turnover = Average Length of Employment.
Using the numbers provided: 117 associates / 20 associates per year = 5.85 years.
This result signifies that the average associate is employed at the consulting company for approximately 5.85 years before they leave the company, although this is a simplification assuming a constant replacement and turnover rate.
A company had 110 employees whose salaries are summarized in the frequency distribution below. Find the mean salary.Salary ($) Employees5,001-10,000 2210,001-15,000 2015,001-20,000 2120,001-25,000 2325,001-30,000 24
Answer:
Mean salary=$17818.68
Step-by-step explanation:
Salary($) Employees(f)
5001-10,000 22
10,001-15,000 20
15,001-20,000 21
20,001-25,000 23
25,001-30,000 24
We know that company had 110 employees so ∑f should be equal to 110.
∑f=22+20+21+23+24=110
The mean salary can be computed as
[tex]xbar=\frac{sum(fx)}{sum(f)}[/tex]
The x be the midpoint can be calculated by taking the average of upper and lower class limit.
Class Interval Frequency(f) x fx
5001-10,000 22 7500.5 165011
10,001-15,000 20 12500.5 250010
15,001-20,000 21 17500.5 367510.5
20,001-25000 23 22500.5 517511.5
25,001-30,000 24 27500.5 660012
fx can be computed by multiplying each x value with frequency in the respective class.
∑fx=165011+250010+367510.5+517511.5+660012=1960055
[tex]xbar=\frac{1960055}{110}=17818.68[/tex]
Thus, the mean salary is $17818.68.
The mean salary is approximately $17,818.18.
To find the mean salary, we need to calculate the average of all the salaries. Here’s the step-by-step process:
Determine the midpoint of each salary range, which is the average of the lower and upper bounds of that range.Multiply the midpoint of each range by the number of employees in that range to find the total contribution of each range to the sum of all salaries.Add up the contributions from all ranges to get the total sum of salaries.Divide the total sum of salaries by the total number of employees (110).Here's the detailed calculation:
[tex]Midpoint \ for \ $5,001-$10,000=(5001 + 10000) / 2 = 7,500[/tex][tex]Midpoint \ for\ $10,001-$15,000: (10001 + 15000) / 2 = 12,500\\Midpoint \ for \ $15,001-$20,000: (15001 + 20000) / 2 = 17,500\\Midpoint \ for \ $20,001-$25,000: (20001 + 25000) / 2 = 22,500\\Midpoint\ for\ $25,001-$30,000: (25001 + 30000) / 2 = 27,500[/tex]Now, multiply each midpoint by the number of employees in that range:
22 * 7,500 = 165,00020 * 12,500 = 250,00021 * 17,500 = 367,50023 * 22,500 = 517,50024 * 27,500 = 660,000Add these values together to get the total sum:
165,000 + 250,000 + 367,500 + 517,500 + 660,000 = 1,960,000
Now, divide by the total number of employees:
1,960,000 / 110 ≈ 17,818.18
Therefore, the mean salary is approximately $17,818.18.