Final answer:
To find the probability that the mean weight of a sample of 107 babies differs from the population mean by less than 40 grams, we can use the Central Limit Theorem. The probability is approximately 0.0441.
Explanation:
To find the probability that the mean weight of a sample of 107 babies differs from the population mean by less than 40 grams, we can use the Central Limit Theorem. According to the Central Limit Theorem, the distribution of sample means will be approximately normal as long as the sample size is large enough. In this case, our sample size is 107, which is considered large enough.
The formula to calculate the standard deviation of the sampling distribution of the sample mean is sigma/sqrt(n), where sigma is the population standard deviation and n is the sample size. Plugging in the values, we have 446/sqrt(107) = 42.96 grams.
To find the probability, we need to calculate the z-score for a difference of 40 grams from the population mean. The z-score formula is (x - mu) / (sigma / sqrt(n)), where x is the desired difference, mu is the population mean, sigma is the population standard deviation, and n is the sample size.
Plugging in the values, we have (40 - 3242) / (446/sqrt(107)) = -1.7. We want to find the probability that the z-score is less than -1.7. Using a standard normal distribution table or calculator, we find that the probability is approximately 0.0441.
To solve this question, we can use the Central Limit Theorem (CLT), which implies that the sampling distribution of the sample means will be normally distributed if the sample size is large enough (usually n > 30 is considered sufficient). Since we are dealing with a sample of 107 babies, which is quite large, we can safely use the CLT.
According to the CLT, the mean of the sample means will be equal to the population mean (μ), and the standard deviation of the sample means (often called the standard error, SE) will be equal to the population standard deviation (σ) divided by the square root of the sample size (n).
Given:
- The population mean (μ) = 3242 grams
- The population standard deviation (σ) = 446 grams
- The sample size (n) = 107 babies
First, we must calculate the standard error (SE):
SE = σ / √n
SE = 446 grams / √107
SE ≈ 446 grams / 10.344
SE ≈ 43.13 grams
Now, we want to find the probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams. This means we are looking at weights from (μ - 40) grams to (μ + 40) grams.
So we will find the z-scores corresponding to (μ - 40) grams and (μ + 40) grams:
Z = (X - μ) / SE
For the lower limit (μ - 40 grams = 3202 grams):
Z Lower = (3202 - 3242) / 43.13
Z Lower ≈ -40 / 43.13
Z Lower ≈ -0.927
For the upper limit (μ + 40 grams = 3282 grams):
Z Upper = (3282 - 3242) / 43.13
Z Upper ≈ 40 / 43.13
Z Upper ≈ 0.927
Using the standard normal distribution (Z-distribution), we can find the probability that a Z-score falls between -0.927 and 0.927. These values correspond to the area under the standard normal curve between these two Z-scores.
If we look these values up in a Z-table, or use a calculator:
- The probability of Z being less than 0.927 is approximately 0.8238.
- The probability of Z being less than -0.927 is approximately 0.1762.
So, the probability of the mean weight being between μ - 40 grams and μ + 40 grams is the area between these two Z-scores, which can be found by subtracting the lower probability from the upper probability:
P(-0.927 < Z < 0.927) = P(Z < 0.927) - P(Z < -0.927)
P(-0.927 < Z < 0.927) = 0.8238 - 0.1762
P(-0.927 < Z < 0.927) = 0.6476
Rounding to four decimal places:
P(-0.927 < Z < 0.927) ≈ 0.6476
Therefore, the probability that the mean weight of the sample babies would differ from the population mean by less than 40 grams is approximately 0.6476, or 64.76%.
Find an explicit solution to the Bernoulli equation. y'-1/3 y = 1/3 xe^xln(x)y^-2
[tex]y'-\dfrac13y=\dfrac13xe^x\ln x\,y^{-2}[/tex]
Divide both sides by [tex]\dfrac13y^{-2}(x)[/tex]:
[tex]3y^2y'-y^3=xe^x\ln x[/tex]
Substitute [tex]v(x)=y(x)^3[/tex], so that [tex]v'(x)=3y(x)^2y'(x)[/tex].
[tex]v'-v=xe^x\ln x[/tex]
Multiply both sides by [tex]e^{-x}[/tex]:
[tex]e^{-x}v'-e^{-x}v=x\ln x[/tex]
The left side can be condensed into the derivative of a product.
[tex](e^{-x}v)'=x\ln x[/tex]
Integrate both sides to get
[tex]e^{-x}v=\dfrac12x^2\ln x-\dfrac14x^2+C[/tex]
Solve for [tex]v(x)[/tex]:
[tex]v=\dfrac12x^2e^x\ln x-\dfrac14x^2e^x+Ce^x[/tex]
Solve for [tex]y(x)[/tex]:
[tex]y^3=\dfrac12x^2e^x\ln x-\dfrac14x^2e^x+Ce^x[/tex]
[tex]\implies\boxed{y(x)=\sqrt[3]{\dfrac14x^2e^x(2\ln x-1)+Ce^x}}[/tex]
A sample of 20 printed labels is selected from a process that is 20% nonconforming. What is the probability of 3 nonconforming labels in the sample? Use the Poisson distribution. Show your work.
Answer:
15%
Step-by-step explanation:
Lauren agrees to pay back $3,400 in one year at an interest rate of 15%. The bank subtracts the 15% discount from the $3,400, and gives the rest to Lauren. Find the amount of the discount and the proceeds to Lauren.
Discount = $
Proceeds = $
Answer:
hence proceeds is $2513.05 and discount is $443.47
Step-by-step explanation:
Lauren pays = $3,400
interest = 15%
time = 1 yr
[tex]P(1+\dfrac{r}{100} )^n = 3400\\P(1+\dfrac{15}{100} )^1 = 3400\\P=2956.52[/tex]
the amount to be discounted is
[tex]\dfrac{15}{100} \times 2956.52 = 443.47[/tex]
discount = $443.47
proceeds = $2956.52 -$443.47
proceeds = $2513.05
hence proceeds is $2513.05 and discount is $443.47
The discount on the loan is $510, calculated as 15% of $3,400. Lauren receives proceeds of $2,890 after the discount is subtracted from the principal amount.
Explanation:To calculate the discount and proceeds given to Lauren by the bank, we use the interest rate and the principal amount. Since the bank subtracts the interest upfront (known as discounting), we need to find 15% of $3,400.
First, find the discount:
Discount = Principal x Interest Rate
Discount = $3,400 x 0.15 = $510
Next, to calculate the proceeds, subtract the discount calculated above from the principal:
Proceeds = Principal - Discount
Proceeds = $3,400 - $510 = $2,890
Therefore, the bank subtracts the discount of $510 from the original $3,400 and gives Lauren the remaining proceeds of $2,890.
Learn more about Loan Discount Calculation here:https://brainly.com/question/33110454
#SPJ12
A bookmark has a perimeter of 46 centimeters and an area of 102 square centimeters. What are the dimensions of the bookmark?
Answer:
6 cm by 17 cm
Step-by-step explanation:
The area is the product of the dimensions; the perimeter is double the sum of the dimensions.
So, we want to find two numbers whose product is 102 and whose sum is 23.
102 = 1·102 = 2·51 = 3·34 = 6·17
The last of these factor pairs has a sum of 23.
The dimensions are 6 cm by 17 cm.
Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.) 5 sin(2θ) − 2 sin(θ) = 0
Answer:
x = {kπ, arccos(1/5) +2kπ, 2kπ -arccos(1/5)}
Step-by-step explanation:
The double-angle trig identity for sine is useful:
5(2sin(θ)cos(θ)) -2sin(θ) = 0
2sin(θ)(5cos(θ) -1) = 0
This has solutions that make the factors zero:
θ = arcsin(0) = kπ
and ...
cos(θ) = 1/5
θ = arccos(1/5) +2kπ . . . . or . . . . 2kπ -arccos(1/5)
_____
Some numerical values are shown on the graph attached. values for multiples of pi are ...
{..., -12.566, -9.425, -6.283, -3.142, 0, 3.142, 6.283, 9.425, 12.566, ...}
a. Find dy/dx if y^2 + x^2 = 16 b. Find the equation of the tangent line that contains the point (2, 2 squareroot 3).
Answer:
[tex]x+\sqrt{3}y=8[/tex]
Step-by-step explanation:
Given equation of curve,
[tex]y^2+x^2=16[/tex]
[tex]\implies y^2=16-x^2[/tex]
Differentiating with respect to x,
[tex]2y\frac{dy}{dx}=-2x[/tex]
[tex]\implies \frac{dy}{dx}=-\frac{x}{y}[/tex]
Since, the tangent line of the curve contains the point (2, 2√3),
Thus, the slope of the tangent line,
[tex]m=\left [ \frac{dy}{dx} \right ]_{(2, 2\sqrt{3})}=-\frac{1}{\sqrt{3}}[/tex]
Hence, the equation of tangent line would be,
[tex]y-2\sqrt{3}=-\frac{1}{\sqrt{3}}(x-2)[/tex]
[tex]\sqrt{3}y-6=-x+2[/tex]
[tex]\implies x+\sqrt{3}y=8[/tex]
PLEASE HELP
Identify the radius and center
x^2 +y^2 -6x -2y + 1 = 0
Answer:
The center is the point (3,1) and the radius is 3 units
Step-by-step explanation:
we know that
The equation of a circle in standard form is equal to
[tex](x-h)^{2}+(y-k)^{2}=r^{2}[/tex]
we have
[tex]x^{2}+y^{2}-6x-2y+1=0[/tex]
Convert to standard form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex](x^{2}-6x)+(y^{2}-2y)=-1[/tex]
Complete the square twice. Remember to balance the equation by adding the same constants to each side.
[tex](x^{2}-6x+9)+(y^{2}-2y+1)=-1+9+1[/tex]
[tex](x^{2}-6x+9)+(y^{2}-2y+1)=9[/tex]
Rewrite as perfect squares
[tex](x-3)^{2}+(y-1)^{2}=9[/tex]
The center is the point (3,1) and the radius is 3 units
PLEASE HELP I ONLY NEED THIS ONE TO FINISH THE SECTION
The functions r and s are defined as follows.
r(x) = -2x + 1
s(x) = -x^2 + 2
Find the value of .
r(s(3))
Answer:
The value of r(s(3)) = -21
Step-by-step explanation:
It is given that,
r(x) = -2x + 1
s(x) = -x^2 + 2
To find the value of r(s(3))
s(x) = -x^2 + 2
s(3) = (-3)^2 + 2 [Substitute 3 instead of x]
= 9 + 2
= 11
Therefore s(3) = 11
r(x) = -2x + 1
r(s(3)) = r(11) [Substitute 11 instead of x]
= -2(11) + 1
= -22 + 1
= -21
Therefore the value of r(s(3)) = -21
The answer is:
[tex]r(s(3))=15[/tex]
Why?To solve the problem, first, we need to compose the functions, and then evaluate the obtained function. Composing function means evaluating a function into another function.
We have that:
[tex]f(g(x))=f(x)\circ g(x)[/tex]
From the statement we know the functions:
[tex]r(x)=-2x+1\\s(x)=-x^{2}+2[/tex]
We need to evaluate the function "s" into the function "r", so:
[tex]r(s(x))=-2(-x^2+2)+1\\\\r(s(x))=2x^{2}-4+1=2x^{2}-3[/tex]
Now, evaluating the function, we have:
[tex]r(s(3))=2(3)^{2}-3=2*9-2=18-3=15[/tex]
Have a nice day!
A town recently dismissed 9 employees in order to meet their new budget reductions. The town had 7 employees over 50 years of age and 15 under 50. If the dismissed employees were selected at random, what is the probability that exactly 1 employee was over 50? Express your answer as a fraction or a decimal number rounded to four decimal places.
Using the hypergeometric distribution, the required probability is 0.0129.
Given that, the town had 7 employees over 50 years of age and 15 under 50.
Let, n₁=7 and n₂=15
Total employees in the town = 7+15 = 22
We have to find the probability that exactly 1 dismissed employee was over 50.
That means the remaining 8 dismissed employees are under 50.
Let the number of dismissed employees under 50 is denoted by x and above 50 denoted by y.
Then, using the hypergeometric distribution, the required probability can be calculated as:
[tex]P(x=1, y=8) = \dfrac{\binom{7}{1}+\binom{15}{8}}{\binom{22}{9}}[/tex]
[tex]P(x=1, y=8) = \dfrac{7+6435}{ 497420}[/tex]
[tex]P(x=1, y=8) = \dfrac{6442}{ 497420}[/tex]
[tex]P(x=1, y=8) = 0.0129[/tex]
Hence, the probability that exactly 1 employee was over 50 is 0.0129.
Learn more about hypergeometric distribution here:
https://brainly.com/question/30911049
#SPJ4
What is a regular tessellation? How many regular tessellations are possible? Why aren’t there infinitely many regular tessellations?
Answer:
Step-by-step explanation:
What is a regular tessellation?
A regular tessellation is a pattern made by repeating a regular polygon. In simpler words regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex.
How many regular tessellation are possible?
There are only 3 regular tessellation.
1. Triangle
2. Square
3. Hexagon
Why aren't there infinitely many regular tessellations?
Not more than 3 regular tessellations are possible because the sums of the interior angles are either greater than or less than 360 degrees....
Final answer:
A regular tessellation is a repeating pattern of a regular polygon that fills a plane without gaps or overlaps, with only three possible: equilateral triangles, squares, and regular hexagons. There aren't infinitely many because a tessellation requires the angles at a vertex to sum to 360 degrees, a condition only satisfied by these three shapes.
Explanation:
A regular tessellation is a pattern made by repeating a regular polygon to fill a plane without any gaps or overlaps. There are exactly three regular tessellations possible, which are constituted by:
Equilateral trianglesSquaresRegular hexagonsThere aren't infinitely many regular tessellations because for a regular polygon to tessellate, the sum of the angles at a vertex where polygons meet must be exactly 360 degrees.
This criterion is satisfied only by triangles (each angle is 60 degrees), squares (each angle is 90 degrees), and hexagons (each angle is 120 degrees).
Polygons with more sides have larger interior angles, such that the sum exceeds 360 degrees, preventing them from tessellating regularly.
rain gutter is 36 feet long, 8 inches in height,3 inches across base, 12 inches across top..how many gallons of water will it hold when full?
Answer:
112.77 gallons of water
Step-by-step explanation:
We will calculate the area by Trapezoid formula :
[tex]A=\frac{a+b}{2}\times h\times l[/tex]
Given Base a = 12 inches = 1 feet
Base b = 3 inches = 0.25 feet
height = 8 inches = 0.67 feet
length = 36 feet = 36 feet
[tex]Area=\frac{1+2.5}{2}\times 0.67\times 36[/tex]
= 0.625 × 0.67 × 36
= 15.075 cubic feet.
As we know 1 cubic feet = 7.48052 per liquid gallon
Therefore, 15.075 cubic feet = 15.075 × 7.48052
= 112.768839 ≈ 112.77 liquid gallon
When full it will hold 112.77 gallons of water
Write y = 2x^2 + 8x + 3 in vertex form.
y = 2(x – 2)^2 – 5
y = (x – 4)^2 + 3
y = 2(x + 2)^2 – 5
y = (x + 4)^2 + 3
Answer:
y = 2(x + 2)^2 – 5
Step-by-step explanation:
When y = ax^2 +bx +c is written in vertex form, it becomes ...
y = a(x +b/(2a))^2 +(c -b^2/(4a))
The constant term in the squared binomial is b/(2a) = 8/(2(2)) = +2. Only one answer choice matches:
y = 2(x +2)^2 -5
The volume of a cylinder is 4pi x(3) cubic units and its height is x units. Which expression represents the radius of the cylinder in units? 2x, 4x, 2 pi x^, 4 pi x^
Answer:
2x
Step-by-step explanation:
Volume of a cylinder is:
V = πr²h
If V = 4πx³ and h = x, then:
4πx³ = πr²x
4x² = r²
r = 2x
For this case we have that by definition, the volume of a cylinder is given by:
[tex]V = \pi * r ^ 2 * h[/tex]
Where:
r: It's the radio
h: It's the height
We have as data that the volume of the cylinder is:
[tex]V = 4 \pi * x ^ 3[/tex]
They also tell us that the height is "x", then:
[tex]4 \pi * x ^ 3 = \pi * r ^ 2 * x[/tex]
We have similar terms on both sides of the equation:
[tex]4x ^ 2 = r ^ 2[/tex]
We apply root to both sides of the equation:
[tex]r = \pm \sqrt {4x ^ 2}\\r = \pm2x[/tex]
We choose the positive value. Now the radio is 2x
Answer:
[tex]2x[/tex]
A car is travelling at exactly 6 miles per hour and accelerates at a constant rate to exactly 65 miles per hour.
This acceleration takes exactly 14.8 seconds.
What is the rate of acceleration, in miles per hour2?
Answer:
[tex]14878.04878miles/hours^2[/tex]
Step-by-step explanation:
Let's find a solution by understanding the following:
The acceleration rate is defined as the change of velocity within a time interval, which can be written as:
[tex]A=(Vf-Vi)/T[/tex] where:
A=acceleration rate
Vf=final velocity
Vi=initial velocity
T=time required for passing from Vi to Vf.
Using the problem's data we have:
Vf=65miles/hour
Vi=6miles/hour
T=14.8seconds
Using the acceleration rate equation we have:
[tex]A=(65miles/hour - 6miles/hour)/14.8seconds[/tex], but look that velocities use 'hours' unit while 'T' uses 'seconds'.
So we need to transform 14.8seconds into Xhours, as follows:
[tex]X=(14.8seconds)*(1hours/60minutes)*(1minute/60seconds)[/tex]
[tex]X=0.0041hours[/tex]
Using X=0.0041hours in the previous equation instead of 14.8seconds we have:
[tex]A=(65miles/hour - 6miles/hour)/0.0041hours[/tex]
[tex]A=(61miles/hour)/0.0041hours[/tex]
[tex]A=(61miles)/(hour*0.0041hours)[/tex]
[tex]A=61miles/0.0041hours^2[/tex]
[tex]A=14878.04878miles/hours^2[/tex]
In conclusion, the acceleration rate is [tex]14878.04878miles/hours^2[/tex]
Scores on the Critical Reading part of the SAT exam in a recent year were roughly Normal with mean 495495 and standard deviation 118118 . You choose an SRS of 100100 students and average their SAT Critical Reading scores. If you do this many times, the mean of the average scores you get will be close to 495118√=45.57495118=45.57 . 495495 . 495100√=49.5495100=49.5 . 495100=4.95495100=4.95 .
Answer: [tex] 495[/tex]
Step-by-step explanation:
We know that if the population is normally distributed with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], then the sampling distribution of the sample mean , [tex]\overline{x}[/tex] is also normally distribution with :-
Mean : [tex]\mu_{\overlien{x}}=\mu[/tex]
Given : The scores on the Critical Reading part of the SAT exam in a recent year were roughly Normal with mean [tex]\mu= 495[/tex]
Then , the mean of the average scores you get will be close to [tex] 495[/tex]
During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 1010:00 AM there were 4 comma 0004,000 bacteria present in the culture. At noon, the number of bacteria grew to 4 comma 6004,600. How many bacteria will there be at midnight?
Answer:
68600 will there be at midnight ( approx )
Step-by-step explanation:
Let P shows the population of the bacteria,
Since, the number of bacteria in a culture grew at a rate proportional to its size,
[tex]\implies \frac{dP}{dt}\propto P[/tex]
[tex]\frac{dP}{dt}=kP[/tex]
Where, k is the constant of proportionality,
[tex]\frac{dP}{P}=kdt[/tex]
[tex]\int \frac{dP}{P}=\int kdt[/tex]
[tex]ln P=kt + C_1[/tex]
[tex]P=e^{kt+C_1}[/tex]
[tex]P=e^{kt}.e^{C_1}=C e^{kt}[/tex]
Now, let the population of bacteria is estimated from 10:00 AM,
So, at t = 0, P = 4,000 ( given )
[tex]4000 = Ce^{0}[/tex]
[tex]\implies C=4000[/tex]
Now, at noon there are 4,600 bacterias,
That is, at t = 2, P = 4600
[tex]4600=Ce^{2k}[/tex]
[tex]4600 = 4000 e^{2k}[/tex]
[tex]\implies e^{2k}=\frac{4600}{4000}=1.15[/tex]
[tex]2k=ln(1.5)\implies k=\frac{ln(1.5)}{2}=0.202732554054\approx 0.203[/tex]
Hence, the equation that represents the population of bacteria after t hours,
[tex]P=4000 e^{0.203t} [/tex]
Therefore, the population of the bacteria at midnight ( after 14 hours ),
[tex]P=4000 e^{0.203\times 14}=4000 e^{2.842}= 68600.1252903\approx 68600[/tex]
2. Which of the following are terms of the series with nth term T-3n +17? a) 80 b) 170 c)217 d) 312 e) 278 f) 3566
Answer:
The correct options are a,b,e and f.
Step-by-step explanation:
It is given that the nth terms of the series is defined as
[tex]T_n=3n+17[/tex]
Subtract 17 from both the sides.
[tex]T_n-17=3n[/tex]
Divide both sides by 3.
[tex]\frac{T_n-17}{3}=n[/tex]
The term Tₙ is a term of given series if n is a positive integer.
(a) The given term is 80.
[tex]n=\frac{80-17}{3}=21[/tex]
Since n is a positive integer, therefore 80 is a term of given series.
(b) The given term is 170.
[tex]n=\frac{170-17}{3}=51[/tex]
Since n is a positive integer, therefore 170 is a term of given series.
(c) The given term is 217.
[tex]n=\frac{217-17}{3}=66.67[/tex]
Since n is not a positive integer, therefore 217 is a term of given series.
(d) The given term is 312.
[tex]n=\frac{312-17}{3}=98.33[/tex]
Since n is not a positive integer, therefore 312 is a term of given series.
(e) The given term is 278.
[tex]n=\frac{278-17}{3}=87[/tex]
Since n is a positive integer, therefore 278 is a term of given series.
(f) The given term is 3566.
[tex]n=\frac{3566-17}{3}=1183[/tex]
Since n is a positive integer, therefore 3566 is a term of given series.
Thus, the correct options are a, b, e and f.
Which of the following actors starred in Dr. Strangelove? a. Peter Fonda b. Paul Newman c. Rod Stieger d. None of the above.
Answer:
D. None of the Above
Step-by-step explanation:
At Information and Communications University: 31 female seniors were on the dean’s list 62 women were on the dean’s list who were not seniors 45 male seniors were on the dean’s list 87 female seniors were not on the dean’s list 96 male seniors were not on the dean’s list 275 women were not senior and were not on the dean’s list 88 men were on the dean’s list who were not seniors 227 men were not seniors and were not on the dean’s list (a) How many were seniors? (b) How many were women? (c) How many were on the dean’s list? (d) How many were seniors on the dean’s list? (e) How many were female seniors? (f) How many were women on the dean’s lists? (g) How many were students at the college?
Step-by-step explanation:
There's a lot of information here, so first things first, let's get organized.
Let's start by assigning variable names to each group. We want the variables to be short but easy to understand.
For example, let's say the number of female seniors on the dean's list is FSD (F for female, S for senior, and D for dean's list).
FSD = 31
Sticking to this naming scheme:
FD = 62
MSD = 45
FS = 87
MS = 96
F = 275
MD = 88
M = 227
Now we can begin.
a) We want to know how many seniors there are. So all we have to do is add up all the variables with an S in them.
FSD + MSD + FS + MS
31 + 45 + 87 + 96
259
b) We want to know how many women there are. So add up all the variables with an F in them.
FSD + FD + FS + F
31 + 62 + 87 + 275
455
c) We want to know how many are on the dean's list. So add up all the variables with a D in them.
FSD + FD + MSD + MD
31 + 62 + 45 + 88
226
d) Now we want to know how many are seniors AND on the dean's list. So add up all the variables that have both an S and a D.
FSD + MSD
31 + 45
76
e) We want to know how many female seniors there are, so add up all the variables with both an F and an S.
FSD + FS
31 + 87
118
f) We want to know how many women were on the dean's list, so add up all the variables with both an F and a D.
FSD + FD
31 + 62
93
g) Finally, we want to know how many students there are total. So add up all the variables.
FSD + FD + MSD + FS + MS + F + MD + M
62 + 45 + 87 + 96 + 275 + 88 + 227
880
Using arithmetic operations based on the provided data, we deduced the total number of seniors, women, students on the dean's list, and overall students at the college, among other specific categorizations.
Explanation:To solve these problems, we will first interpret the given data, then perform simple arithmetic operations based on the information provided to answer each part of the question.
(a) To find how many were seniors, we sum the number of female and male seniors on the dean's list and those not on the dean's list: 31 (female seniors on the dean's list) + 87 (female seniors not on the dean's list) + 45 (male seniors on the dean's list) + 96 (male seniors not on the dean's list) = 259.(b) To calculate how many were women, we add the women on the dean's list and those not, including seniors and non-seniors: 31 (female seniors on the dean's list) + 62 (women on the dean's list who were not seniors) + 87 (female seniors not on the dean's list) + 275 (women not senior and not on the dean's list) = 455.(c) How many were on the dean's list? Summing the numbers for all students on the dean's list gives us: 31 (female seniors) + 62 (women not seniors) + 45 (male seniors) + 88 (men not seniors) = 226.(d) How many were seniors on the dean's list? Adding together the number of female and male seniors on the dean's list: 31 (female seniors) + 45 (male seniors) = 76.(e) How many were female seniors? Adding the number of female seniors on the dean's list and those not gives: 31 (on the dean's list) + 87 (not on the dean's list) = 118.(f) How many were women on the dean's list? This is the total number of women, both seniors and not, on the dean's list: 31 (female seniors) + 62 (women not seniors) = 93.(g) Summing students from all categories to find how many were students at the college: (259 seniors, both genders) + (275 women not senior/not on dean's list) + (88 men on the dean's list who were not seniors) + (227 men who were not seniors and not on the dean's list) = 849.To win at LOTTO in one state, one must correctly select 7 numbers from a collection of 48 numbers (1 through 48). The order in which the selection matter. How many different selections are possible? made does not There are different LOTTO selections.
The number of different possible 7-number selections from a pool of 48, where order does not matter, can be calculated using the mathematical concept of combinations. The formula to calculate the total number of combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of options and k is the number of options selected.
Explanation:In the LOTTO game you described, you must select 7 numbers from a pool of 48, and the order of the numbers does not matter. This is a problem of combinations in mathematics. The formula for combinations is C(n, k) = n! / [k!(n-k)!], where n is the total number of options, k is the number of options selected, and '!' denotes a factorial. In this case, n=48 (the numbers 1 through 48) and k=7 (the seven numbers you select).
By plugging these values into the formula, we can calculate the total number of different selections possible: C(48, 7) = 48! / [7!(48-7)!]. This calculation would give us the total number of combinations of 7 numbers that can be selected from a pool of 48, which represents all the different possible LOTTO selections. It should be remembered that factorials such as 48! or 7! represent a product of an integer and all the integers below it, down to 1.
Learn more about Combinations here:https://brainly.com/question/39347572
#SPJ12
There are 85,900,584 different selections possible in the LOTTO game.
Explanation:To calculate the number of different selections possible in the LOTTO game, we need to focus on the concept of permutations.
With 48 numbers to choose from and 7 numbers to be selected, the number of permutations can be calculated using the formula P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items to be selected.
In this case, n = 48 and r = 7.
Using the formula, we have P(48, 7) = 48! / (48-7)! = (48 * 47 * 46 * 45 * 44 * 43 * 42) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
= 85,900,584 different selections possible.
Learn more about Permutations here:https://brainly.com/question/23283166
#SPJ2
The concept of determining which reactant is limiting and which is in excess is akin to determining the number of sandwiches that can be made from a set number of ingredients. Assuming that a cheese sandwich consists of 2 slices of bread and 3 slices of cheese, determine the number of whole cheese sandwiches that can be prepared from 44 slices of bread and 69 slices of cheese.
_____ sandwiches
Which of the two ingredients limits the number of sandwiches that can be made?
slices of cheese
slices of bread
What quantity of the ingredient in excess remains?
With 22 packed cheese sandwiches made, bread is the 'limiting' ingredient, and we have three slices of cheese in excess.
Explanation:The number of sandwiches that can be made depends on the availability of the critical ingredients needed to complete a sandwich: 2 slices of bread and three slices of cheese. To calculate the number of sandwiches, we can determine how many sandwiches each ingredient batch could provide, then find the minimum of those two, as the 'limiting' factor will be the ingredient that finishes first.
For bread, we have 44 pieces/ 2 pieces per sandwich = 22 sandwiches. With cheese, we have 69 pieces/ 3 pieces per sandwich = 23 sandwiches. Therefore, we can fully make a maximum of 22 sandwiches because we would run out of bread first. This means the bread is the 'limiting' ingredient, while cheese is in excess. With 22 sandwiches made, we would use three slices of cheese per sandwich for 66 slices used, leaving three slices of cheese remaining. The quantity in excess is three slices of cheese.
Learn more about Limiting and Excess Quantity here:https://brainly.com/question/33576857
#SPJ3
There are 0 slices of bread in excess, and there are 3 slices of cheese in excess.
Let's start by calculating the maximum number of sandwiches that can be made using 44 slices of bread:
Each sandwich requires 2 slices of bread.
So, the maximum number of sandwiches that can be made using 44 slices of bread is 44/2 = 22 sandwiches.
Now, let's calculate the maximum number of sandwiches that can be made using 69 slices of cheese:
Each sandwich requires 3 slices of cheese.
So, the maximum number of sandwiches that can be made using 69 slices of cheese is 69/3 = 23 sandwiches.
Now, we compare the results:
With 44 slices of bread, we can make 22 sandwiches.
With 69 slices of cheese, we can make 23 sandwiches.
Since we can only make 22 sandwiches due to the limitation of the bread, the bread is the limiting ingredient.
The quantity of the ingredient in excess can be found by subtracting the number of sandwiches that can be made using the limiting ingredient from the total number of slices of that ingredient.
For the bread:
Quantity of bread in excess = Total slices of bread - (Number of sandwiches × Slices of bread per sandwich)
Quantity of bread in excess = 44 - (22 *2)
= 44 - 44
= 0
For the cheese:
Quantity of cheese in excess = Total slices of cheese - (Number of sandwiches × Slices of cheese per sandwich)
Quantity of cheese in excess = 69 - (22 *s 3)
= 69 - 66
= 3
So, there are 0 slices of bread in excess, and there are 3 slices of cheese in excess.
A supply curve slopes upward because
a.
â as more is produced, total cost of production falls.
b.
â many firms will experience increases in their costs of production as their output rises so they need a higher price in order to induce them to produce more output.
c.
â the higher the price per unit, the greater the profitability generated by supplying more of that good.
d.
â both (b) and (c).
Answer:
D.) Both (B) and (C)
Step-by-step explanation:
The slope upward depicts higher sales option (B) and (C) are causes for higher production
HOPE IT HELPS....
The supply curve slopes upward due to two primary reasons. Firstly, as a firm increases output, the cost of production typically increases, necessitating a higher price to cover the increasing costs. Also, at higher prices, it's more profitable for firms to supply more of a good. Both these factors cause an upward slope in the supply curve.
Explanation:The supply curve in Economics slopes upward because it represents the relationship between the cost of production and the quantity supplied. The correct answer is option d. As a firm produces more goods (output), its cost of production often rises, typically due to diminishing returns in the short run. So, they require a higher price to cover their increasing costs, hence option (b). Further, when the price per unit of a good in a market is higher, it's more profitable for firms to supply more of that good, conforming with option (c). Both these factors together explain the upward slope of the supply curve.
Learn more about Supply Curve here:https://brainly.com/question/31849606
#SPJ6
In the following probability distribution, the random variable x represents the number of activities a parent of a 6 th dash to 8 th -grade student is involved in. Complete parts (a) through (f) below.
x 0 1 2 3 4
P(x) 0.216 0.072 0.322 0.289 0.101
Compute and interpret the mean of the random variable x.
The mean is ___ activities.
(Type an integer or a decimal.)
(d) Compute the standard deviation of the random variable x.
The standard deviation is _____ activities.
(Round to one decimal place as needed.)
(e) What is the probability that a randomly selected student has a parent involved in three activities?
The probability is _____ (Type an integer or a decimal.)
(f) What is the probability that a randomly selected student has a parent involved in three or four activities?
The probability is_____ (Type an integer or a decimal.)
The mean/expected value is
[tex]E[X]=\displaystyle\sum_x x\,P(X=x)=0P(X=0)+1P(X=1)+2P(X=2)+3P(X=3)+4P(X=4)[/tex]
[tex]\implies E[X]=\boxed{1.987}[/tex]
d. The standard deviation is the square root of the variance, which itself is
[tex]V[X]=E[(X-E[X])^2]=E[X^2]-E[X]^2[/tex]
We have
[tex]E[X^2]=\displaystyle\sum_xx^2\,P(X=x)=0^2P(X=0)+1^2P(X=1)+2^2P(X=2)+3^2P(X=3)+4^2P(X=4)[/tex]
[tex]\implies E[X^2]=5.577[/tex]
Then the variance is
[tex]V[X]=5.577-1.987^2\approx1.629[/tex]
and so the standard deviation is
[tex]\sqrt{V[X]}\approx\boxed{1.276}[/tex]
e. We know this immediately from the table:
[tex]P(X=3)=\boxed{0.289}[/tex]
f. A parent can be involved in either 3 or 4 activities, but not simultaneously 3 and 4 activities (i.e. these events are disjoint), so
[tex]P(X=3\text{ or }X=4)=P(X=3)+P(X=4)=\boxed{0.39}[/tex]
Applying statistical concepts, we have that:
c) The mean is of 1.987 activities.
d) The standard deviation is of 1.67 activities.
e) The probability is 0.289.
f) The probability is 0.39.
The probability distribution is given by:
[tex]P(X = 0) = 0.216[/tex]
[tex]P(X = 1) = 0.072[/tex]
[tex]P(X = 2) = 0.322[/tex]
[tex]P(X = 3) = 0.289[/tex]
[tex]P(X = 4) = 0.101[/tex]
Item c:
The mean is given by the sum of the multiplications of each outcome by it's probability, thus:
[tex]E(X) = 0.216(0) + 0.072(1) + 0.322(2) + 0.289(3) + 0.101(4) = 1.987[/tex]
The mean is of 1.987 activities.
Item d:
The standard deviation is given by the square root of the sum of the squares of each outcome subtracted by the mean, multiplied by it's probability, thus:
[tex]\sqrt{V(X)} = \sqrt{0.216(0-1.987)^2 + 0.072(1-1.987)^2) + 0.322(2-1.987)^2) + ...}[/tex]
[tex]\sqrt{V(X)} = 1.67[/tex]
The standard deviation is of 1.67 activities.
Item e:
This probability is P(X = 3), thus 0.289.
Item f:
This probability is:
[tex]p = P(X = 3) + P(X = 4) = 0.289 + 0.101 = 0.39[/tex]
A similar problem is given at https://brainly.com/question/14395358
The centers for disease control and prevention reported that 25% of baby boys 6-8 months old in the united states weigh more than 20 pounds. A sample of 16 babies is studied. What is the probability that fewer than 3 weigh more than 20 pounds?
Answer:
0.1971 ( approx )
Step-by-step explanation:
Let X represents the event of weighing more than 20 pounds,
Since, the binomial distribution formula is,
[tex]P(x)=^nC_r p^r q^{n-r}[/tex]
Where, [tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Given,
The probability of weighing more than 20 pounds, p = 25% = 0.25,
⇒ The probability of not weighing more than 20 pounds, q = 1-p = 0.75
Total number of samples, n = 16,
Hence, the probability that fewer than 3 weigh more than 20 pounds,
[tex]P(X<3) = P(X=0)+P(X=1)+P(X=2)[/tex]
[tex]=^{16}C_0 (0.25)^0 (0.75)^{16-0}+^{16}C_1 (0.25)^1 (0.75)^{16-1}+^{16}C_2 (0.25)^2 (0.75)^{16-2}[/tex]
[tex]=(0.75)^{16}+16(0.25)(0.75)^{15}+120(0.25)^2(0.75)^{14}[/tex]
[tex]=0.1971110499[/tex]
[tex]\approx 0.1971[/tex]
calculate simple interest and maturity value
Principal is. 6,000. interest rate 5%. the time is 15. what is the solution
Answer:
Simple interest=4500
maturity value= 10500
Step-by-step explanation:
Given: Principal P = 6000
Rate % R = 5%
time T = 15 years
we know that simple interest [tex]SI=\frac{PRT}{100}[/tex]
⇒[tex]SI=\frac{6000\times5\times15}{100}[/tex]
on calculating we get SI = 4500
and maturity value = Principal amount + Simple interest
maturity value = 6000+4500 = 10500
hence the final answers are
Simple interest= 4500
and maturity value= 10500
hope this helps!!
Let an = –3an-1 + 10an-2 with initial conditions a1 = 29 and a2 = –47. a) Write the first 5 terms of the recurrence relation. b) Solve this recurrence relation. Show your reasoning. c) Using the explicit formula you found in part b, evaluate a5. You must show that you are using the equation from part b.
We can express the recurrence,
[tex]\begin{cases}a_1=29\\a_2=-47\\a_n=-3a_{n-1}+10a_{n-2}7\text{for }n\ge3\end{cases}[/tex]
in matrix form as
[tex]\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}\begin{bmatrix}a_{n-1}\\a_{n-2}\end{bmatrix}[/tex]
By substitution,
[tex]\begin{bmatrix}a_{n-1}\\a_{n-2}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}\begin{bmatrix}a_{n-2}\\a_{n-3}\end{bmatrix}\implies\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}^2\begin{bmatrix}a_{n-2}\\a_{n-3}\end{bmatrix}[/tex]
and continuing in this way we would find that
[tex]\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}\begin{bmatrix}a_2\\a_1\end{bmatrix}[/tex]
Diagonalizing the coefficient matrix gives us
[tex]\begin{bmatrix}-3&10\\1&0\end{bmatrix}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}-5&0\\0&2\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}[/tex]
which makes taking the [tex](n-2)[/tex]-th power trivial:
[tex]\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}-5&0\\0&2\end{bmatrix}^{n-2}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}[/tex]
[tex]\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}(-5)^{n-2}&0\\0&2^{n-2}\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}[/tex]
So we have
[tex]\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}(-5)^{n-2}&0\\0&2^{n-2}\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}\begin{bmatrix}a_2\\a_1\end{bmatrix}[/tex]
and in particular,
[tex]a_n=\dfrac{29\left(2(-5)^{n-1}+5\cdot2^{n-1}\right)-47\left(-(-5)^{n-1}+2^{n-1}\right)}7[/tex]
[tex]a_n=\dfrac{105(-5)^{n-1}+98\cdot2^{n-1}}7[/tex]
[tex]a_n=15(-5)^{n-1}+14\cdot2^{n-1}[/tex]
[tex]\boxed{a_n=-3(-5)^n+7\cdot2^n}[/tex]
Biologists stocked a lake with 160 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 4,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 2000 (half of the carrying capacity)?
Answer:
The equation to find the number of fish after t years where y is the number of fish is:
y(t) = 160×3^(t) ( t <= 2.93 assuming that the maximum number of fish is 4000 )
Therefore when y(t) = 2000 2000 = 160×3^(t)
3^(t) = 25/2 log 3 (3^(t)) = log 3 (25/2) t = 2.29 years.
Using the logistic equation, we have that:
a)
The equation is:
[tex]P(t) = \frac{4000}{1 + 24e^{-1.1856t}}[/tex]
b)
It will take 2.68 years for the population to increase to 2000.
The logistic equation is:
[tex]P(t) = \frac{K}{1 + Ae^{-kt}}[/tex]
With:
[tex]A = \frac{K - P(0)}{P(0)}[/tex]
The parameters are:
The carrying capacity K.The decay rate k.The initial population P(0).In this problem:
Initial population of 160, thus [tex]P(0) = 160[/tex].Carrying capacity of 4,000, thus [tex]K = 4000[/tex].Then:
[tex]A = \frac{4000 - 160}{160} = 24[/tex]
Thus:
[tex]P(t) = \frac{4000}{1 + 24e^{-kt}}[/tex]
Item a:
Tripled during the first year, thus [tex]P(1) = 3P(0) = 3(160) = 480[/tex].
This is used to find k.
[tex]480 = \frac{4000}{1 + 24e^{-k}}[/tex]
[tex]480 + 11520e^{-k} = 4000[/tex]
[tex]e^{-k} = \frac{3520}{11520}[/tex]
[tex]\ln{e^{-k}} = \ln{\frac{3520}{11520}}[/tex]
[tex]k = -\ln{\frac{3520}{11520}}[/tex]
[tex]k = 1.1856[/tex]
Thus, the equation is:
[tex]P(t) = \frac{4000}{1 + 24e^{-1.1856t}}[/tex]
Item b:
This is t for which P(t) = 2000, thus:
[tex]P(t) = \frac{4000}{1 + 24e^{-1.1856t}}[/tex]
[tex]2000 = \frac{4000}{1 + 24e^{-1.1856t}}[/tex]
[tex]\frac{1}{1 + 24e^{-1.1856t}} = 0.5[/tex]
[tex]0.5 + 12e^{-1.1856t} = 1[/tex]
[tex]e^{-1.1856t} = \frac{1}{24}[/tex]
[tex]\ln{e^{-1.1856t}} = \ln{\frac{1}{24}}[/tex]
[tex]t = -\frac{\ln{\frac{1}{24}}}{1.1856t}[/tex]
[tex]t = 2.68[/tex]
It will take 2.68 years for the population to increase to 2000.
A similar problem is given at https://brainly.com/question/24215157
A new game is being introduced at the Hard Rock Cafe. A ball is spun around a wheel until it comes to rest in one of many spots. Whatever is listed in that spot will be the player's winnings. If the wheel has 9 spots labeled $1, 18 spots labeled $2, and 1 spots labeled $10, how much should a player expect to win on average? Round to the nearest cent. Your Answer: Question 5 options: Answer
Answer:
$ 1.96
Step-by-step explanation:
Number of spots with outcome of $1 = 9
Number of spots with outcome of $2 = 18
Number of spots with outcome of $10 = 1
Total number of spots = 28
Probability that ball will land on $1 = [tex]\frac{9}{28}[/tex]
Probability that ball will land on $2 = [tex]\frac{18}{28}[/tex]
Probability that ball will land on $10 = [tex]\frac{1}{28}[/tex]
The amount that player should expect to win on average in equal to expected value of the game. Expected value is calculated as the summation of product of probabilities with their respective outcomes.
i.e. for this case:
Expected Value will be:
[tex](1 \times \frac{9}{28})+(2 \times \frac{18}{28})+(10 \times \frac{1}{28})\\\\ =1.96[/tex]
This means, on average the player should expect to win $ 1.96
Probability is the ratio of the favorable event to the total number of events. The average amount is $1.96.
What is probability?Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1. Its basic meaning is something is likely to happen.
A new game is being introduced at the Hard Rock Cafe.
A ball is spun around a wheel until it comes to rest in one of many spots. Whatever is listed in that spot will be the player's winnings.
If the wheel following spots
Number of spots labeled $1 = 9
Number of spots labeled $2 = 18
Number of spots labeled $10 = 1
The total number of spots will be
Total spots = 28
The probability of the ball landing on $1 will be
[tex]\rm P(\$1) = \dfrac{9}{28}[/tex]
The probability of the ball landing on $2 will be
[tex]\rm P(\$2) = \dfrac{18}{28}[/tex]
The probability of the ball landing on $10 will be
[tex]\rm P(\$10) = \dfrac{1}{28}[/tex]
The amount that player should expect to win on average is equal to the expected value of the game will be
[tex]\rm Average\ amount = 1*\dfrac{9}{28} + 2*\dfrac{18}{28} + 10*\dfrac{1}{28}\\\\\\Average \ amount = 1.96[/tex]
More about the probability link is given below.
https://brainly.com/question/795909
With a short time remaining in the day, a delivery driver has time to make deliveries at 7 locations among the 9 locations remaining. How many different routes are possible?
There are 9,072 different routes possible for the delivery driver.
Explanation:To find the number of different routes possible, we can use the concept of permutations. Since the driver has to deliver to 7 out of the 9 remaining locations, we can calculate the number of ways to choose 7 out of 9 and then multiply it by the number of ways to arrange those 7 locations. The formula for permutations is P(n, r) = n! / (n - r)!. In this case, n = 9 and r = 7.
P(9, 7) = 9! / (9 - 7)! = 9! / 2! = 9 × 8 × 7 × 6 × 5 × 4 × 3 = 9,072
Therefore, there are 9,072 different routes possible for the delivery driver.
Learn more about Permutations here:https://brainly.com/question/23283166
#SPJ12
To find the number of different routes possible, we can use the combination formula: C(n, k) = n! / (k!(n-k)!). Plugging in the values, we get C(9, 7) = 36. Therefore, there are 36 different routes possible.
Explanation:To find the number of different routes possible, we can use the combination formula:
C(n, k) = n! / (k!(n-k)!)
Where n is the total number of locations remaining (9) and k is the number of locations the driver can make deliveries to (7).
Plugging in the values, we get:
C(9, 7) = 9! / (7!(9-7)!)
= 9! / (7!2!)
= (9 * 8 * 7!)/(7! * 2!)
= (9 * 8)/(2 * 1)
= 36
Therefore, there are 36 different routes possible.
Suppose that on the average, 7 students enrolled in a small liberal arts college have their automobiles stolen during the semester. What is the probability that more than 1 student will have his automobile stolen during the current semester? Round your answer to four decimal places.
To find the probability that more than 1 student will have their automobile stolen during the current semester, we can use a Poisson distribution. The average number of automobile thefts per semester is 7. By calculating 1 minus the probability of 0 or 1 thefts, we find that the probability of more than 1 theft is approximately 0.9991.
Explanation:To find the probability that more than 1 student will have their automobile stolen during the current semester, we can use a Poisson distribution. The Poisson distribution models the number of events that occur in a fixed interval of time or space. In this case, the average number of automobile thefts per semester is given as 7. We want to find the probability that more than 1 student will have their automobile stolen, so we need to calculate 1 minus the probability that 0 or 1 student will have their automobile stolen.
The probability mass function of a Poisson distribution is given by P(X=k) = e^{-λ} * (λ^k) / k! where λ is the average number of events. In this case, λ = 7.
So, the probability that more than 1 student will have their automobile stolen is P(X>1) = 1 - (P(X=0) + P(X=1)).
To calculate P(X=0) and P(X=1), we can substitute k=0 and k=1 into the probability mass function and sum them up.
P(X=0) = e^{-7} * (7^0) / 0! = e^{-7}
P(X=1) = e^{-7} * (7^1) / 1! = 7e^{-7}
Therefore, P(X>1) = 1 - (e^{-7} + 7e^{-7}).
Rounding this answer to four decimal places, we get P(X>1) ≈ 0.9991.
Learn more about Probability here:https://brainly.com/question/32117953
#SPJ3