1. The population is increasing for 0 < P < 4100. The answer in interval notation is (0, 4100).
2. The population is decreasing for P > 4100. The answer in interval notation is (4100, ∞).
3. The equilibrium solutions are P = 0 and P = 4100.
To determine when the population is increasing or decreasing, we need to examine the sign of the derivative dP/dt.
1. For what values of P is the population increasing?
The population is increasing when dP/dt > 0.
In this case, we have dP/dt = 1.1P(1 - P/4100).
To find the values of P for which the population is increasing, we need to solve the inequality 1.1P(1 - P/4100) > 0.
To do this, we can consider the sign of each factor:
1.1 is positive.
P is the variable.
(1 - P/4100) is positive when P < 4100 and negative when P > 4100.
From this, we can determine the intervals where the population is increasing:
When P < 0 (since P cannot be negative in a population context), the term 1.1P is negative, so the entire expression is negative. The population is not increasing in this interval.
When 0 < P < 4100, both 1.1P and (1 - P/4100) are positive, so the entire expression is positive. The population is increasing in this interval.
When P > 4100, 1.1P is positive, but (1 - P/4100) is negative. The entire expression is negative. The population is not increasing in this interval.
Therefore, the population is increasing for 0 < P < 4100. The answer in interval notation is (0, 4100).
2. For what values of P is the population decreasing?
The population is decreasing when dP/dt < 0.
In this case, we have dP/dt = 1.1P(1 - P/4100).
To find the values of P for which the population is decreasing, we need to solve the inequality 1.1P(1 - P/4100) < 0.
Using the same analysis as in the previous part, we can determine the intervals where the population is decreasing:
When P < 0, the population is not decreasing.
When 0 < P < 4100, the population is not decreasing.
When P > 4100, the population is decreasing.
Therefore, the population is decreasing for P > 4100. The answer in interval notation is (4100, ∞).
3. What are the equilibrium solutions?
Equilibrium solutions occur when the population remains constant, meaning dP/dt = 0.
In this case, we have dP/dt = 1.1P(1 - P/4100)
= 0.
To find the equilibrium solutions, we solve the equation 1.1P(1 - P/4100) = 0.
This equation is satisfied when either 1.1P = 0 or (1 - P/4100) = 0.
From 1.1P = 0, we have P = 0.
From (1 - P/4100) = 0, we have P = 4100.
Therefore, the equilibrium solutions are P = 0 and P = 4100.
Learn more about Equilibrium Solutions at
brainly.com/question/32080559
#SPJ4
Calculate the volume of 2.45 moles of hydrogen gas at STP.
One Mol of any gas occupies a volume of 22.4
We have been given the number of miles of hydrogen and the volume of any gas at STP
Therefore you can use that and substitute it into the formula to calculate the Volume of hydrogen for that number of moles
Let X be a Markov chain with transition probability matrix 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1 The Markov chain starts at time zero in state Xo = 0. Let T = min{n > 0: X = 2} be the first time that the process reaches state 2. Eventually, the process will reach and be absorbed into state 2. If in some experiment we observed such a process and noted that absorption had not yet taken place, we might be interested in the conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Determine P(X3 = 0 T > 3).
The conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Therefore, P(X3 = 0, T > 3) = 0.075.
The Markov chain starts at time zero in state Xo = 0 and the transition probability matrix of Markov Chain is as follows: 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1
P(X3 = 0, T > 3).We know that the probability of moving from state i to state j in two steps is given by P2(i, j).
Thus, the probability of moving from state i to state j in three steps is given by P3(i, j). We have P2(i, j) = P(i, ·)P(j, ·) = Σk P(i, k)P(k, j).
For a 3-step transition probability, we use the equation P3 = P2P = P2 (P2) and so on. Therefore,P3(1, 2) = P2(1, 1)P(1, 2) + P2(1, 2)P(2, 2) + P2(1, 3)P(3, 2) = (0.7)(0.3) + (0.2)(0.5) + (0.1)(0) = 0.235
Similarly,P3(1, 0) = P2(1, 0)P(0, 0) + P2(1, 1)P(1, 0) + P2(1, 2)P(2, 0) = (0)(0.7) + (0.3)(0) + (0.235)(0.2) = 0.047
Since we are interested in finding P(X3 = 0, T > 3), we need to find the probability that absorption had not yet taken place at time 3 and that the process is in state 0 at time 3, which can be expressed as:
P(X3 = 0, T > 3) = P(X3 = 0, X4 ≠ 2) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1)
We know that T is the first time that the process reaches state 2 and the process will reach and be absorbed into state 2.
Thus, T is the absorption time for state 2 and it has a geometric distribution with parameter P2(2, 2) = 1, which implies that P(T = t) = (1 – 1)P2(2, 2) = 0 for all t < 1.
Therefore, P(X3 = 0, X4 = 0) = P(X3 = 0, X4 = 0, T > 3)
= P(X3 = 0, T > 3)P(X4 = 0 | X3 = 0, T > 3) = P(X3 = 0, T > 3)P(0, 0) / P(X3 = 0, T > 3)P(0, 0) + P(X3 = 1, T > 3)
P(1, 0) + P(X3 = 2, T > 3)
P(2, 0) = (0.047)(1) / [(0.047)(1) + (0.235)(0.7) + (0)(0.2)]
= 0.067
Therefore, P(X3 = 0, T > 3) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1) = (0.067)(0.7) + (0.235)(0.2) = 0.075.
Therefore, P(X3 = 0, T > 3) = 0.075.
Learn more about Markov chain rule:https://brainly.com/question/30465344
#SPJ11
A recent survey of the alumni of a university indicated that the average salary of 10,000 of its 200,000 graduates was $130,000. The $130,000 would be considered a: a. Population. b. Parameter. c. Sample. d. Statistic.
The $130,000 would be considered as Statistic.
In statistics, a population refers to the entire group of individuals or items of interest, while a sample is a subset of the population. A parameter is a numerical value that describes a characteristic of a population.
In this scenario, the survey results are based on a sample of 10,000 graduates out of a total population of 200,000 graduates. The average salary of $130,000 is calculated from the data collected within this sample. Since it is derived from the sample, it is considered a statistic.
A parameter would be used to describe the average salary of the entire population of 200,000 graduates if data were collected from all of them. However, in this case, the given information only pertains to the subset of the sample.
LEARN MORE ABOUT Statistic here: brainly.com/question/31538429
#SPJ11
Suppose that the mean score for a critical reading test is 580 with a population standard deviation of 115 points. What is the probability that a random sample of 500 students will have a mean score of more than 590? Less than 575? Solve using Excel.
the mean score for a critical reading test, using Excel, the probability that a random sample of 500 students will have a mean score of more than 590 can be calculated to be approximately 0.408.
To calculate the probabilities using Excel, we can utilize the standard normal distribution. First, we need to convert the sample means to z-scores by using the formula: z = (sample mean - population mean) / (population standard deviation / sqrt(sample size)). For the sample mean of more than 590, we can calculate the probability of z being greater than the corresponding z-score using the formula "=1-NORM.S.DIST(z-score,TRUE)". In this case, the z-score is (590 - 580) / (115 / sqrt(500)), which gives approximately 0.408.
Similarly, for the sample mean of less than 575, we calculate the probability of z being less than the corresponding z-score using the formula "=NORM.S.DIST(z-score,TRUE)". The z-score is (575 - 580) / (115 / sqrt(500)), which gives approximately 0.084.
Therefore, the probability that a random sample of 500 students will have a mean score of more than 590 is approximately 0.408, and the probability that the sample mean is less than 575 is approximately 0.084.
Learn more about mean here:
https://brainly.com/question/31101410
#SPJ11
Point S is on line segment RT. Given ST=3 and RT=20, determine the length RS.
Answer:
RS = 17
Step-by-step explanation:
Given S lies on RT , then
RS + ST = RT , that is
RS + 3 = 20 ( subtract 3 from both sides )
RS = 17
A water supply system is to be installed at a distance of 54 meters using 6 meters long PVC pipe with a diameter of 100mm. determine the number of length of PVC pipe to be used? a. 7 b. 8 c. 9 d. 10
To determine the number of lengths of PVC pipe to be used, we need to divide the total distance to be covered (54 meters) by the length of each PVC pipe (6 meters) and round up to the nearest whole number.
Number of lengths of PVC pipe = Total distance / Length of each PVC pipe
Number of lengths of PVC pipe = 54 meters / 6 meters
Number of lengths of PVC pipe = 9
Therefore, the number of lengths of PVC pipe to be used is 9.
So, the answer is option c. 9.
The moment of inertia depends on the distribution of masses relative to the axis of rotation. It is a measure of an object's resistance to rotational motion. The formula for the moment of inertia varies depending on the specific shape and distribution of masses.
If you can provide more details about the arrangement of masses and the axis of rotation, I can help you derive the expression for the moment of inertia in terms of m and l.
To know more about lengths of PVC pipe:- https://brainly.com/question/29887360
#SPJ11
A pentagon has angles that measure
80°, 90°, 130°, 120°, and t. What is t?
t=
Answer:
120°
Step-by-step explanation:
To find the measure of angle t in the pentagon, we can use the property that the sum of the interior angles of a pentagon is equal to 540 degrees.
So, we can set up an equation:
80° + 90° + 130° + 120° + t = 540°
Simplifying the equation:
420° + t = 540°
Next, we can isolate the variable t by subtracting 420° from both sides of the equation:
t = 540° - 420°
t = 120°
Therefore, the measure of angle t in the pentagon is 120 degrees.
Hope this helps!
PLZZZ HELP MEE BIG BRAIN PLP PLZZ
Represent each linear situation with an equation in slope-intercept form.
A family bucket meal at Chicken Deluxe costs twenty-six dollars plus $1.50 for every extra piece of chicken added to the bucket.
Answer:
y = 1.5x + 26Step-by-step explanation:
Total cost is $26 add x-pieces each $1.50:
y = 1.5x + 26help pleaseeeeee ♀️
Answer:
9
my bad I realized mistake
Answer:
perimeter of square =4×side
36 inches =4×side
36 inches/4 =side
9 inches =side
Show that sin(π/2 + x) = cos x using the compound angle formulas.
Given the trigonometric identity: sin (π/2 + x) = cos x. We are to show that it can be derived from compound angle formulas for sine and cosine functions.
In order to prove the identity using the compound angle formulas, we have to start by recalling the formulas for sin(A + B) and cos(A + B).The compound angle formulas are given as:$$\begin{aligned}\sin (A+B)&=\sin A\cos B+\cos A\sin B\\\cos(A+B)&=\cos A\cos B-\sin A\sin B\end{aligned}$$
Let us set A = π/2 and B = x. Hence, A + B = π/2 + x. Then, we can write:$$\begin{aligned}\sin \left(\frac{\pi}{2}+x\right)&=\sin\frac{\pi}{2}\cos x+\cos\frac{\pi}{2}\sin x \\&= \cos x\end{aligned}$$And, $$\begin{aligned}\cos\left(\frac{\pi}{2}+x\right)&=\cos\frac{\pi}{2}\cos x - \sin\frac{\pi}{2}\sin x \\&= -\sin x\end{aligned}$$
Therefore, sin(π/2 + x) = cos x, as desired.
Know more about trigonometric:
https://brainly.com/question/29156330
#SPJ11
When you find volume of a 3-D shape, you are finding the measurement of the outside of the shape.
Group of answer choices
True
False
B. use the figur on the right side to classify the pair of Angeles.
________1.<5 and <1
________2.<6 and <3
________3.<7 and <2
The highest value on the domain of the function is called its ____ ____. (2 words)
starts with ab
Answer:
It is called the maximum value.
Step-by-step explanation:
The highest value on the domain of the function is called its maximum value.
What is domain and range of the function?The domain and range are defined for a relation and they are the sets of all the x-coordinates and all the y-coordinates of ordered pairs respectively. For example, if the relation is, R = {(1, 2), (2, 2), (3, 3), (4, 3)}, then:
Domain = the set of all x-coordinates = {1, 2, 3, 4}
Range = the set of all y-coordinates = {2, 3}
The maximum value of a function is the place where a function reaches its highest point, or vertex, on a graph. There is no point above the maximum value of the function. Thus the highest point on the graph is known as the maximum value of the domain of the function.
The maximum value is one of the extreme values of the domain of the function. The other extreme value is known as the minimum value. It is on one side of the graph and the maximum value is on the other side of the graph.
Therefore, the highest value on the domain of the function is called its maximum value.
To learn more about the domain and range visit:
brainly.com/question/28135761.
#SPJ6
Find the flux of the field F=axi-3yj F outward across the ellipse: x-cost, y - 4 sint ostaan Use Green's thm. to find the area enclosed by the ellipse; x = a cose, yabsine.
Using Green's theorem the flux of the field F across the ellipse is 0, indicating that there is no net flow across the ellipse.
To find the flux of the field F = a × i - 3y × j outward across the ellipse, we can use Green's theorem. Green's theorem relates the flux of a vector field across a closed curve to the circulation of the field around the curve.
Let's denote the ellipse as C, which is parameterized by x = a cos(t) and y = b sin(t), where a and b are the semi-major and semi-minor axes of the ellipse, and t varies from 0 to 2π.
Calculate the curl of the vector field F:
∇ × F = (∂Fₓ/∂y - ∂Fᵧ/∂x) k
= (-3)k
Determine the area enclosed by the ellipse using Green's theorem:
The flux of F across the ellipse is equal to the circulation of F around the ellipse:
∮C F · dr = ∬R (∇ × F) · dA
Since the curl of F is -3k, the flux simplifies to:
∮C F · dr = ∬R (-3k) · dA
= -3 ∬R dA
= -3A
Therefore, the flux of F across the ellipse is -3 times the area enclosed by the ellipse.
Find the area enclosed by the ellipse:
The equation of the ellipse is given as x = a cos(t) and y = b sin(t).
To find the limits of integration, we note that t varies from 0 to 2π, which represents one complete revolution around the ellipse.
∬R dA = ∫₀²π ∫₀²π (a cos(t))(b) dt
= ab ∫₀²π cos(t) dt
= ab [sin(t)]₀²π
= ab (sin(2π) - sin(0))
= 0
Therefore, the area enclosed by the ellipse is 0.
Learn more about Green's theorem at
https://brainly.com/question/30763441
#SPJ4
help me please it’s math
Answer:
1) 345 2) 62 3) 36 4) 51 5) 96 6) 142
Step-by-step explanation:
I hope you and your 3rd or 4th grader self enjoy the answers i gave you.
If a car can travel 78 miles using 3 gallons of gas, what is the unit rate?
Answer:
26
Step-by-step explanation:
Well since a car can travel 78 miles Using 3 gallons
78/3=26
26 miles for each gallon
Answer:
26
Step-by-step explanation:
The unit rate refers to the distance the car can travel using 1 gallon of gas. One way to solve this is to create an algebraic equation like [tex]78=3x[/tex]. Then, to find the unit rate isolate x by dividing both sides by 3. This means that the unit rate is 26.
Please please help please
Yikes i dont know this but thanks for the coins C:
which of the following key characteristics is not true of a quadratic function
Here are the characteristics of Quadratic Functions
Axis of symmetry
x and y-intercepts
Zeroes
vertex
Point symmetric to y-intercept
Determine the range of the function [tex]\displaystyle f(x)=a\sqrt[3]{bx-c}+d[/tex].
Hi there!
[tex]\large\boxed{(-\infty, \infty)}}[/tex]
We are given the function:
[tex]f(x) = a\sqrt[3]{bx-c}+d[/tex]
This is the transformation form of a cubic root function. Recall these properties of cubic root functions:
Domain: -∞ < x < ∞ (all real numbers)
Range: -∞ < x < ∞ (all real numbers)
Therefore, the range of the given function is all real numbers, or on (-∞, ∞).
Which of the following is not determined in the case of multiple linear regression?
Question 4 options:
1)Potential confounding effects from "lurking" variables included in the analysis
2)Observed relationship between predictor and outcome variables adjusting for confounding relationships
3)Causal relationships between predictor and outcome variables adjusting for confounding relationships
4) Potential observed interaction effects between variables Save
It does not determine the observed interaction effects between variables.
In the case of multiple linear regression, the option that is not determined is:
Variable interactions that have been observed to have effects.
Option 1: Multiple linear regression can estimate the observed relationship between predictor and outcome variables after adjusting for confounding relationships and option 2: it can determine the potential confounding effects from lurking variables. Furthermore, numerous direct relapse can evaluate the causal connections among indicator and result factors adapting to jumbling connections (choice 3).
However, the potential observed interaction effects between variables cannot be determined by multiple linear regression on its own. Association impacts happen when the connection between two indicators and the result variable isn't added substance however relies upon the mix of the indicator values. In order to evaluate the effects of interactions, additional analysis methods like including interaction terms and carrying out specific interaction tests are required.
To know more about linear regression refer to
https://brainly.com/question/32505018
#SPJ11
A cylindrical canister has a radius of 7 cm and a height of 16 cm. Find the volume of
the canister.
pls show your work thx pls help me will give brainly.
Answer:
7/16 = 0.4375
Step-by-step explanation:
sry
According to the Chronicles of Contrived Statistics (March, 2016), the probability that Elon Musk will offer you a free trip into space in the next month is 41%. The probability that scientists from Area 51 will start selling pet aliens next month is 27%. Finally, the probability that either Elon Musk will offer you a free trip to space or that scientists will begin sales of pet aliens next month is 24%. What is the probability that both Elon Musk will offer you a free trip into space and scientists from Area 51 will start selling pet aliens in the next month?
Answer:
[tex]P(A\ and\ B) = 44\%[/tex]
Step-by-step explanation:
Given
Represents the event as follows:
A = [tex]Elon\ Musk[/tex] will offer you a [tex]free\ trip[/tex] into space in the next month\
B = Scientists from [tex]Area\ 51[/tex] will [tex]start\ selling[/tex] pet aliens next month
So, we have:
[tex]P(A) = 41\%[/tex]
[tex]P(B) = 27\%[/tex]
[tex]P(A\ or\ B) = 24\%[/tex]
Required
Determine [tex]P(A\ and\ B)[/tex]
This is calculated as:
[tex]P(A\ and\ B) = P(A) + P(B) - P(A\ or\ B)[/tex]
So, we have:
[tex]P(A\ and\ B) = 41\% + 27\% - 24\%[/tex]
[tex]P(A\ and\ B) = 44\%[/tex]
find the amount to be paid at the end of 3years in each case. a) principal =1,200 at12% p.a
Answer:
$1632
Step-by-step explanation:
Given data
Principal= $1200
Rate= 12%
Time= 3 years
The simple interest expression is given as
A= P(1+rt)
substitute
A=1200(1+0.12*3)
A=1200(1+0.36)
A=1200*1.36
A=$1632
Hence the amount is $1632
Expert Answer Please Find The Area Of A Circle With R=13.8 Explain Answer!
Answer:
if r= to 13.8 so the answer is 598.28
PLEASE HELP ASAP!!!!!!!!!!!!!!!
Answer:
the answer is D
Step-by-step explanation:
y=[tex]x^{2}[/tex]+1
use the chain rule to find dz/dt. z = xy7 − x2y, x = t2 1, y = t2 − 1
If z = xy⁷ − x²y, x = t² + 1, y = t² − 1, then using chain rule the value of dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
To find dz/dt using the chain rule, we need to differentiate z with respect to t while considering the relationship between z, x, y, and t.
Given:
z = xy⁷ − x²y
x = t² + 1
y = t² − 1
We can start by finding dz/dx and dz/dy separately and then use the chain rule to combine them to find dz/dt.
First, let's find dz/dx:
Differentiating z with respect to x, keeping y constant:
dz/dx = (d/dx)(xy⁷) - (d/dx)(x²y)
Using the product rule for differentiation:
dz/dx = y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)
Next, let's find dz/dy:
Differentiating z with respect to y, keeping x constant:
dz/dy = (d/dy)(xy⁷) - (d/dy)(x²y)
Using the product rule again:
dz/dy = x(y⁷ + 7y⁶(dy/dy)) - x²
Since dy/dy = 1, we can simplify dz/dy as:
dz/dy = xy⁷ + 7xy⁶ - x²
Now, let's use the chain rule to find dz/dt:
dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)
Substituting the expressions we found earlier for dz/dx and dz/dy, and using dx/dt = 2t and dy/dt = 2t:
dz/dt = (y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx))(2t) + (xy⁷ + 7xy⁶ - x²)(2t)
Simplifying this expression gives us dz/dt in terms of t, x, and y:
dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
This is the complete calculation for finding dz/dt using the chain rule.
To learn more about chain rule click on,
https://brainly.com/question/32367259
#SPJ4
Complete question is:
Use the chain rule to find dz/dt. z = xy⁷ − x²y, x = t² + 1, y = t² − 1
Which measure is equivalent to 65 kilograms?
659
6509
65000
65,0000
Answer:
65,000 grams
Step-by-step explanation:
Because 1000 grams is in 1 kilogram, you would multiply the 65 kilograms by 1,000 to convert it to grams.
So to convert 65 kilograms to grams you would do:
65 * 1,000 = 65,000 grams
Find the length of side x in simplest radical form with a rational denominator
Answer:
x = 8/√3
Step-by-step explanation:
We solve that above question using the trigonometric function of sin
Sin theta = Opposite/Hypotenuse
Theta = 60°
Opposite = 4
Hypotenuse = x
Hence,
Sin 60 = 4/x
Cross Multiply
sin 60 × x = 4
x= 4/sin 60
in rational form
sin 60 = √3/2
Hence, x = 4/ √3/2
= 4 ÷ √3/2
= 4 × 2/√3
= 8/√3
find an equation of the tangent plane to the given parametric surface at the specified point. x = u + v, y = 5u², z = u − v; (2, 5, 0)
The equation of the tangent plane to the parametric surface at the point (2, 5, 0) is x + 20y + z - 102 = 0.
To find the equation of the tangent plane to the given parametric surface at the point (2, 5, 0), we need to compute the partial derivatives and evaluate them at the given point.
The parametric surface is defined by the equations:
x = u + v
y = 5u^2
z = u - v
First, we find the partial derivatives with respect to u and v:
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 10u
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
Next, we evaluate the partial derivatives at the given point (2, 5, 0):
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 10u = 10(2) = 20
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
At the point (2, 5, 0), the partial derivatives are:
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 20
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
The equation of the tangent plane can be written as:
(x - x₀) (∂x/∂u) + (y - y₀) (∂y/∂u) + (z - z₀) (∂z/∂u) = 0,
where (x₀, y₀, z₀) is the given point.
Substituting the values, we have:
(x - 2)(1) + (y - 5)(20) + (z - 0)(1) = 0.
Simplifying further, we get:
x - 2 + 20(y - 5) + z = 0.
Expanding and rearranging the terms, the equation of the tangent plane to the parametric surface at the point (2, 5, 0) is:
x + 20y + z - 102 = 0.
Learn more about tangent plane here
https://brainly.com/question/30619505
#SPJ11
A researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. According to the Empirical rule, approximately 95% of women had their first child between the ages of a. 11.1 years and 43.5 years b. 16.5 years and 38.1 years C. 21.9 years and 32.7 years d. 25.0 years and 29.6 years
According to the Empirical rule, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. The correct option is b. 16.5 years and 38.1 years.
The given question states that a researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. We are to find the range of age at which approximately 95% of women had their first child, according to the empirical rule.
There are three ranges for the empirical rule as follows:
Approximately 68% of the observations fall within the first standard deviation from the mean.
Approximately 95% of the observations fall within the first two standard deviations from the mean.
Approximately 99.7% of the observations fall within the first three standard deviations from the mean.
Now, we will apply the empirical rule to find the age range at which approximately 95% of women had their first child. The mean age is 27.3 years and the standard deviation is 5.4 years, hence:
First, find the age at which 2.5% of women had their first child:
µ - 2σ = 27.3 - (2 × 5.4) = 16.5
Then, find the age at which 97.5% of women had their first child:
µ + 2σ = 27.3 + (2 × 5.4) = 38.1
Therefore, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. Hence, the correct answer is option b. 16.5 years and 38.1 years.
To know more about Empirical rule, refer to the link below:
https://brainly.com/question/30404590#
#SPJ11