Answer:
Option A.
Step-by-step explanation:
Let M be the milk per gallon.
C be the cookies per dozen.
B be the bundle (one gallon of milk and a dozen cookies ).
Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen and bundle is sold for a profit of $3.00 per bundle.
[tex]Profit= 1.5M + 2.5C + 3B[/tex]
We need to maximize the profits. So, our objective function is
Therefore, the correct option is A.
According to the empirical rule, if the data form a bell-shaped normal distribution, __________ of the observations will fall within two standard deviations around the mean.
Answer:
95%
Step-by-step explanation:
The empirical rule states that if data follows normal distribution then the percentage of observations falls within one, two and three standard deviation around the mean are
i) 68% falls within one standard deviation
ii) 95% falls within two standard deviation
iii) 99.7% falls within three standard deviation.
Hence 95% of the observations will fall within two standard deviations around the mean if the data follows normal distribution.
According to the empirical rule, approximately 95 percent of observations in a bell-shaped normal distribution fall within two standard deviations of the mean.
According to the empirical rule, if the data form a bell-shaped normal distribution, approximately 95 percent of the observations will fall within two standard deviations around the mean. This statistical concept is an important part of descriptive statistics and is crucial when studying the Normal or Gaussian probability distribution.
In a normal distribution, the data is symmetric around the mean, and as dictated by the empirical rule, about 68% of the data lies within one standard deviation, while approximately 95% lies within two standard deviations, and over 99% within three standard deviations of the mean.
The water content of soil from a borrow site is normally distributed with a mean of 14% and a standard deviation of 2.5%. What is the probability that a sample taken from the site will have a water content above 16% or below 12%?
The probability that a sample taken from the site will have water content above 16% or below 12% is 0.3862.
Given that,
Mean = µ
= 14%
= 0.14
Standard deviation = σ
= 2.5%
= 0.025
[Using standard normal table]
16% or 12% = 0.16 or 0.12
P(X < 0.16 or X > 0.12)
Using standard normal table,
To see the z value -0.9 in the row and 0.06 in the column of the standard normal table the cumulative probability of z = -0.96 is = 0.1685
To see the z value 0.7 in the row and 0.08 in the column of the standard normal table the cumulative probability of z = 0.78 is = 0.7823
[tex]P(X < 0.6\ or \ x > 0.12)=1-[P(0.16 < X < 0.12)][/tex]
[tex]=1-{P[\frac{0.16-0.142}{0.023} < \frac{x-\mu}{\sigma} < \frac{0.12-0.142}{0.023}]}[/tex]
[tex]= 1-[P(0.78 < z < -0.96)][/tex]
[tex]= [P(z < -0.96)-P(z < 0.78)][/tex]
[tex]=1-(0.1685-0.7823)[/tex]
[tex]= 1-0.6138[/tex]
[tex]P(X < 0.16 \ or \ X > 0.12)=0.3862[/tex]
Therefore, the probability that a sample taken from the site will have water content above 16% or below 12% is 0.3862.
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The probability that a random sample’s water content will be higher than 16% or lower than 12% is calculated by converting the values to Z-scores and then using the standard normal distribution's cumulative probability. The values from a Z-table or calculator are used to determine this probability.
Explanation:The subject of this question is Probability and Statistics, specifically related to the normal distribution. In this scenario, the water content of soil from a borrow site is normally distributed with a mean (μ) of 14% and a standard deviation (σ) of 2.5%.
To find the probability of the soil having a water content above 16% or below 12%, we first need to calculate the Z-scores for these values. The Z-score represents how many standard deviations an element is from the mean. It is calculated as (X - μ)/σ.
For X = 16%, Z = (16 - 14) / 2.5 = 0.8
For X = 12%, Z = (12 - 14) / 2.5 = -0.8
Using a Z-table or a calculator function that outputs standard normal probabilities, we would find that P(Z > 0.8) or P(Z < -0.8). By convention, Z-tables only give values as P(Z < z), for z > 0, so to find P(Z > 0.8), we can do 1 - P(Z < 0.8). Same for P(Z < -0.8).
As such, the desired probability is the sum of these two probabilities.
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You randomly select 500 students and observe that 85 of them smoke. What is your estimate of the probability that a randomly selected student smokes?
Answer:
17% probability that a randomly selected student smokes.
Step-by-step explanation:
A probability is the number of desired outcomes divided by the number of total outcomes.
What is your estimate of the probability that a randomly selected student smokes?
This estiamte is the number of smoking students divided by the total number of students.
We have that:
There are 500 students.
85 smoke
So there is an 85/500 = 0.17 = 17% probability that a randomly selected student smokes.
If events A and B are mutually exclusive, P(A or B) = 0.5, and P(B) = 0.3; then what is P(A)?
If events A and B are mutually exclusive, the probability of A or B occurring is equal to the sum of the probabilities of A and B. Given that P(A or B) = 0.5 and P(B) = 0.3, P(A) can be calculated as 0.2.
Explanation:If events A and B are mutually exclusive, the probability of A or B occurring is equal to the sum of the probabilities of A and B. So, we have P(A or B) = P(A) + P(B). Given that P(A or B) = 0.5 and P(B) = 0.3, we can substitute these values into the formula to solve for P(A).
0.5 = P(A) + 0.3.
Now, subtract 0.3 from both sides to isolate P(A):
0.5 - 0.3 = P(A).
P(A) = 0.2.
Therefore, the probability of event A occurring is 0.2.
Find all values of x in the interval [0, 2π] that satisfy the equation.
6sin²(x) = 3
Answer:
The solutions are π/4, 3π/4,5π/4,7π/4
Step-by-step explanation:
The given equation is
6sin²(x) = 3
Divide by 6 to get:
[tex] { \sin}^{2} (x) = \frac{1}{2} [/tex]
This implies that;
[tex] \sin(x) = \pm \frac{ \sqrt{2} }{2} [/tex]
If
[tex]\sin(x) = \frac{ \sqrt{2} }{2}[/tex]
[tex]x = \frac{\pi}{4} [/tex]
in the first quadrant
[tex]x = \frac{3\pi}{4} [/tex]
in the second quadrant.
If
[tex]\sin(x) = - \frac{ \sqrt{2} }{2}[/tex]
[tex]x = \frac{5\pi}{4} [/tex]
in the third quadrant
[tex]x = \frac{7\pi}{4} [/tex]
It takes you 47 seconds to walk from the first (ground) floor of a building to the third floor.
How long will it take you to walk from the first floor to the sixth floor (at the same pace, assuming all floors have the same height)?
Answer: 117.5 seconds
Step-by-step explanation:
Given : It takes you 47 seconds to walk from the first (ground) floor of a building to the third floor.
Since from first floor to third , a person need to pass two levels of staircase.
We assume that the person walk at the same pace and the height of all floors are same.
The time taken to pass each level = (Time taken to pass 2 levels) ÷ 2
= (47 seconds) ÷ (2) = 23.5 seconds
Also, the from first floor to sixth floor , a person need to pass 5 levels of staircase.
Then, the time taken to pass each level = 5 x (Time taken to pass one level)
= 5 x 23.5 =117.5 seconds
Hence, it will take 117.5 seconds to walk from the first floor to the sixth floor.
The curve given by x=sin(t),y=sin(t+sin(t)) has two tangent lines at the point (x,y)=(0,0). List both of them in order of increasing slope. Your answers should be in the form of y=f(x) without t′s.
Answer:
y = 0
y =2x
Step-by-step explanation:
Given parametric equations:
x (t) = sin (t)
y (t) = sin (t + sin (t))
The slope of the curve at any given point is given by dy / dx we will use chain rule to find dy / dx
(dy / dx) * (dx / dt) = (dy / dt)
(dy / dx) = (dy / dt) / (dx / dt)
Evaluate dx / dt and dy / dt
dx / dt = cos (t)
dy / dt = cos (t + sin (t)) * (1+cos (t))
Hence,
dy / dx = (1+cos(t))*cos(t + sin (t))) / cos (t)
@Given point (x,y) = 0 we evaluate t
0 = sin (t)
t = 0 , pi
Input two values of t and compute dy / dx
@ t = 0
dy / dx = (1 + cos (0))*cos (0 + sin (0))) / cos (0)
dy / dx = (1+1)*(1) / (1) = 2 @ t = 0
@t = pi
dy / dx = ( 1 + cos (pi))* cos (pi + sin (pi)) / cos (pi)
dy / dx = (1-1) * (-1) / (-1) = 0 @ t = pi
The corresponding gradients are 0 and 2 in increasing order and their respective equations are:
y = 2x
y = 0
The equation of the two tangent lines at the point (x,y) = (0,0) in order of increasing slope are; y = 0 and y = 2x
We are given the parametric equations of the curve as;
x = sin(t)
y = sin(t + sin(t))
Now, since we want to find slope, the we need to find dy/dx from;
dy/dx = (dy/dt) ÷ (dx/dt)
Thus;
dx/dt = cos(t)
Using chain rule;
dy/dt = cos (t + sin(t)) × (1 + cos(t))
Thus;
dy/dx = [cos (t + sin(t)) × (1 + cos(t))]/(cos(t))
At (0, 0), we have;
0 = sin(t) ---(1)
0 = sin(t + sin(t)) ---(2)
From eq (1), values of t that makes the function 0 are;
0 and π
Thus;
At t = 0;
dy/dx = [cos (0 + sin(0)) × (1 + cos(0))]/(cos(0))
dy/dx = (1 + 1)/1
dy/dx = 2
At t = π;
dy/dx = [cos (π + sin(π)) × (1 + cos(π))]/(cos(π))
dy/dx = 0
Using the point slope form; y - y₁ = m(x - x₁)
At m = 2, we have;
y - 0 = 2(x - 0)
y = 2x
At m = 0, we have;
y - 0 = 0(x - 0)
y = 0
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The only swimming pool at the El Cheapo Motel is outdoors. It is 5.0 m wide and 12.0 m long. If the weekly evaporation is 2.35 in., how many gallons of water must be added to the pool if it does not rain?
To determine the volume of water evaporated from the pool at the El Cheapo Motel, we converted all measurements to a common unit and calculated the volume of water evaporated. The motel has to add approximately 946.13 gallons of water weekly, considering there is no rain.
Explanation:To answer this question, we first need to convert the measurements to a common unit. Given that the pool is 5.0 m wide and 12.0 m long (a total area of 60.0 m2) and the weekly evaporation is 2.35 inches, we first convert the inches to meters. Since 1 inch is equal to 0.0254 meters, 2.35 inches equals 0.05969 meters.
Then, we calculate the volume of water evaporated in a week, which is calculated by multiplying the surface area of the pool by the depth of the water evaporated. Hence, it's 60.0 m2 * 0.05969 m = 3.58 m3. As 1 m3 is approximately 264.17 gallons, 3.58 m3 equals 946.1296 gallons approximately.
In conclusion, the El Cheapo Motel needs to add around 946.13 gallons of water to their pool on a weekly basis, if there is no rain.
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(a) State the Cauchy-Schwarz inequality in an equation, in words, and in a planar sketch. (b) Use part (a) to give all solutions to the following. Let x E Rn: arg max ļaTa 2 subject to |c11-1
Complete question
The complete question is shown on the first uploaded image
Answer:
The solution and the explanation is on the second third and fourth uploaded image
what is the slope intercept equation of the line that goes through the points (1,-5) and (-3,7)?
Answer:
y=-3x-2
Step-by-step explanation:
The slope intercept equation is y=mx+b. y2-y1/x2-x1 is the equation for the slope. 7-(-5)/-3-1 is -3. Now you have y=-3x+b, you plug in one set of cordinates so you would have 7=-3(-3)+b, 7=9+b, b=-2
y=-3x-2
Hope that helps :)
Answer:
Step-by-step explanation:
The equation of a straight line can be represented in the slope-intercept form, y = mx + c
Where c = intercept
Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis
change in the value of y = y2 - y1
Change in value of x = x2 -x1
y2 = final value of y
y 1 = initial value of y
x2 = final value of x
x1 = initial value of x
The line passes through the points (1,-5) and (-3,7)
y2 = 7
y1 = - 5
x2 = - 3
x1 = 1
Slope,m = (7 - - 5)/(- 3 - 1) = 12/- 4 =
- 3
To determine the intercept, we would substitute x = - 3, y = 7 and m= - 3,
7 = - 3 × - 3 + c = 9 + c
c = 7 - 9 = - 2
The equation becomes
y = - 3x - 2
Let A be a 5 x 3 matrix, let y be a vector in R3, and let z be a vector in R5. Suppose Ay = z. What fact allows you to conclude that the system Ax = 5z is consistent?
Answer:
Step by step approach is as shown
Step-by-step explanation:
Consider the system Ax = 5z .............(1)Recalling that z = AySubstitute (Ay) for z in equation (1)therefore, Ax = 5(Ay) ....................... (2) Hence the equation can also be written as Ax = A(5y) ................ (3)recalling from commutative law that A + B = B + A and since A is a scalar, and from scalar multiplication of matrix.
From equation (3) ; Ax = A(5y), it implies that x = 5y from comparison and as such if we compare with equation (2) where z = Aytherefore equation (2) can then be written as Ax = 5z, since there is consistency as such the the equation will also have a solution.The health of the bear population in a park is monitored by periodic measurements taken from anesthetized bears. A sample of the weights of such bears is given below. Find a 95% confidence interval estimate of the mean of the population of all such bear weights.
Answer:
[tex]182.167-2.03\frac{114.05}{\sqrt{36}}=143.580[/tex]
[tex]182.167+2.03\frac{114.05}{\sqrt{36}}=220.754[/tex]
So on this case the 95% confidence interval would be given by (143.580;220.754)
Step-by-step explanation:
Assuming the following dataset:
77, 349,417,349, 167 , 225, 265, 360,205
145,335,40,139, 177,108, 163, 202, 22
123,439, 125,135, 86,43, 217,49, 156
119,178, 151, 61, 350, 312, 91, 89,89
We can calculate the sample mean with the followinf formula:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}= 182.167[/tex]
And the sample deviation with:
[tex] s = \sqrt{\frac{\sum_{i=1}^n (X_i-\bar X)^2}{n-1}}=114.05[/tex]
The sample size on this case is n =36.
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X=182.167[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=114.05 represent the sample standard deviation
n=36 represent the sample size
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
The point estimate of the population mean is [tex]\hat \mu = \bar X =182.167[/tex]
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=36-1=35[/tex]
Since the Confidence is 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,35)".And we see that [tex]t_{\alpha/2}=2.03[/tex]
Now we have everything in order to replace into formula (1):
[tex]182.167-2.03\frac{114.05}{\sqrt{36}}=143.580[/tex]
[tex]182.167+2.03\frac{114.05}{\sqrt{36}}=220.754[/tex]
So on this case the 95% confidence interval would be given by (143.580;220.754)
The question is asking for a 95% confidence interval for the mean weight of bears in a park. The confidence interval is a range of values, derived from the data collected, that is estimated to contain the true population mean. 95% of such confidence intervals are expected to contain the true value.
Explanation:In statistics, we often
use sample data to make generalizations
about an unknown population. This part of statistics is known as
inferential statistics
. The sample data help us to make an estimate of a population parameter. We realize that the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, which are often called confidence intervals.
A confidence interval is a type of estimate but, instead of being just one number, it is an interval of numbers. The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
In this case, you've been asked to calculate a 95% confidence interval for the mean weight of bears in a park. From the provided data, you would calculate the sample mean and standard deviation, and use a statistical formula to calculate the confidence interval. For example, if the confidence level is 95 percent, then we say, 'We estimate with 95 percent confidence that the true value of the population mean is between x and y.'.
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Show that each of these conditional statements is a tautology by using truth tables.
a) (p ∧ q) → p
b) p → (p ∨ q)
c) ¬p → (p → q)
d) (p ∧ q) → (p → q)
e) ¬(p → q) → p f ) ¬(p → q) → ¬q
By using truth tables, we can prove that given conditional statements are tautologies. The logic used in the example can also be applied to other statements. Every possible combination of truth values for the components of a tautology yields a true statement.
Explanation:To show that these conditional statements are tautologies, we can generate truth tables for each one. Tautologies are logical statements that are always true regardless of the truth values of their variable components.
For example, let's examine the statement (p ∧ q) → p:
When both p and q are true, the whole statement is true since 'true AND true' is true, and 'true implies true' is also true.When p is true and q is false, the 'AND' clause yields a false, but because 'true implies false' is still true, the statement remains true.When p is false (regardless of q's value), the 'AND' clause is false, as is the p in the 'implies' clause. However, 'false implies false' is true, and overall, the statement is also true.
This logic can similarly be applied to the remaining statements to prove that they're tautologies.
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If an annuity makes an infinite series of equal payments at the end of the interest periods, it is called a perpetuity. If a lump sum investment of An is needed to result in n periodic payments of R when the interest rate per period is i, then the sum is represented by the following.
An = R 1 - (1 + i)^-n
i
Evaluate the following limit to find a formula for the lump sum payment for a perpetuity.
lim An
n?
Answer:
[tex]lim_{n \to \infty} A_n = \frac{R}{i}[/tex]
Step-by-step explanation:
For this case we have this expression:
[tex] A_n = R [\frac{1 -(1+i)^{-n}}{i}][/tex]
The lump sum investment of An is needed to result in n periodic payments of R when the interest rate per period is i.
And we want to find the:
[tex] lim_{n \to \infty} A_n[/tex]
So we have this:
[tex] lim_{n \to \infty} A_n = lim_{n \to \infty}R [\frac{1 -(1+i)^{-n}}{i}] [/tex]
Then we can do this:
[tex] lim_{n \to \infty} A_n = lim_{n \to \infty} R [\frac{1 -\frac{1}{(1+i)^n}}{i}][/tex]
[tex]lim_{n \to \infty} A_n = R lim_{n \to \infty} [\frac{1 -\frac{1}{(1+i)^n}}{i}][/tex]
And after find the limit we got:
[tex] lim_{n \to \infty} A_n = R [\frac{1-0}{i}][/tex]
Becuase : [tex] \frac{1}{(1+i)^{\infty}} =0[/tex]
And then finally we have this:
[tex]lim_{n \to \infty} A_n = \frac{R}{i}[/tex]
f(x)={5−1 if 1≤x<7, if 7≤x≤13. Evaluate the definite integral by interpreting it as signed area.
Answer:
\int\limits^7_1 {f(x)} \, dx =24
\int\limits^13_7 {f(x)} \, dx =24
Step-by-step explanation:
From Exercise we have f(x)=5-1 , we get f(x)=4.
We calculate integral, if 1≤x<7, we get
\int\limits^7_1 {f(x)} \, dx =\int\limits^7_1 {4} \, dx =4[x]\limits^7_1=4(7-1)=4·6=24
We calculate integral, if 7≤x<13, we get
\int\limits^13_7 {f(x)} \, dx =\int\limits^13_7 {4} \, dx =4[x]\limits^13_7=
=4(13-7)=4·6=24
Therefore, we conclude that the given two integrals are the same.
2. The beam of a lighthouse can be seen for up to 20 miles. You are on a ship that is 10 miles east and 16 miles north of the lighthouse. Write an inequality to describe the region lit by the lighthouse beam. Can you see the lighthouse beam on your ship?
Answer:
The person on the ship can see the lighthouse
Step-by-step explanation:
The Circle Function
A circle centered in the point (h,k) with a radius r can be written as the equation
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Any point (x,y) can be known if it's inside of the circle if
[tex](x-h)^2+(y-k)^2\leq r^2[/tex]
The question is about a beam of a lighthouse than can be seen for up to 20 miles. If we assume the lighthouse is emitting the beam as the shape of a circle centered in (0,0), then its radius is 20 miles. Thus any person riding a ship inside the circle can see the lighthouse. This means that
[tex]x^2+y^2\leq 20^2[/tex]
[tex]x^2+y^2\leq 400[/tex]
The ship's coordinates respect to the lighthouse are (10,16). We should test the point to verify if the above inequality stands
[tex]10^2+16^2\leq 400[/tex]
[tex]356 \leq 400[/tex]
The inequality is true, so the person on the ship can see the lighthouse
The inequality is:
√(x^2 + y^2) ≤ 20 mi.
And you can see the lighthouse beam from your ship.
How to write the inequality?
First, let's define our coordinate system, North will be the positive y-axis and east will be the positive x-axis. Such that the origin is the lighthouse.
The distance between a point (x, y) and the lighthouse is given by:
d = √(x^2 + y^2).
And we know that the lighthouse beam can be seen for up to 20 miles, so you can only see the lighthouse if your position (x, y) is such that the inequality is true.
√(x^2 + y^2) ≤ 20 mi.
Now, can you see the lighthouse beam from your ship?
Your ship's position is: (10 mi, 16 mi)
Replacing that in the inequality we get:
√((10mi)^2 + (16mi)^2) ≤ 20 mi
18.87 mi ≤ 20 mi
The inequality is true, then you can see the beam from the ship.
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The diameter of a brand of tennis balls is approximately normally distributed, with a mean of 2.63 inches and a standard deviation of 0.03 inch. If you select a random sample of 9 tennis balls,
a. What is the sampling distribution of the mean?
b. What is the probability that the sample mean is less than 2.61 inches?
c. What is the probability that the sample mean is between 2.62 and 2.64 inches?
d. The probability is 6-% that the sample mean will be between what two values symmetrically distributed around the population mea
Answer:
a) sample mean = 2.63 inches
sample standard deviation = [tex]\frac{standard \hspace{0.15cm} deviation}{\sqrt{n} } = \frac{0.03}{\sqrt{9} } = \frac{0.03}{3} = 0.01[/tex]
b) P(X < 2.61) = 0.0228
c.) P(2.62 < X < 2.64) = 0.6827
d.) Therefore 0.06 = P(2.6292 < X < 2.6307)
Step-by-step explanation:
i) the diameter of a brand of tennis balls is approximately normally distributed.
ii) mean = 2.63 inches
iii) standard deviation = 0.03 inches
iv) random sample of 9 tennis balls
v) sample mean = 2.63 inches
vi) sample standard deviation = [tex]\frac{standard \hspace{0.15cm} deviation}{\sqrt{n} } = \frac{0.03}{\sqrt{9} } = \frac{0.03}{3} = 0.01[/tex]
vii) the sample mean is less than 2.61 inches = P(X < 2.61) = 0.0228
viii)the probability that the sample mean is between 2.62 and 2.64 inches
P(2.62 < X < 2.64) = 0.6827
ix) The probability is 6-% that the sample mean will be between what two values symmetrically distributed around the population measure
Therefore 0.06 = P(2.6292 < X < 2.6307)
a) The sampling distribution of the mean is 2.63 inches and sample standard deviation = 0.01
b) P(X < 2.61) = 0.0228
c) P(2.62 < X < 2.64) = 0.6827
d) 0.06 = P(2.6292 < X < 2.6307)
Let's solve this step by step:
Step 1: The diameter of a brand of tennis balls is approximately normally distributed.
Given:
Mean = 2.63 inches
Standard deviation = 0.03 inches
For random sample of 9 tennis balls
Step 2: Sample mean = 2.63 inches
Sample standard deviation = [tex]\frac{\text{standard deviation}}{\sqrt{n} } =\frac{0.03}{\sqrt{9} } =\frac{0.03}{3} =0.01[/tex]
Step 3: The sample mean is less than 2.61 inches = P(X < 2.61) = 0.0228
Step 4: The probability that the sample mean is between 2.62 and 2.64 inches = P(2.62 < X < 2.64) = 0.6827
Step 5: The probability is 6-% that the sample mean will be between what two values symmetrically distributed around the population measure
Therefore 0.06 = P(2.6292 < X < 2.6307)
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A sample of 60 women is obtained, and their heights ( in inches ) and pulse rates ( in beats per minute ) are measured. The linear correlation coefficient is 0.234 and the equation of the regression line is ^y= 17.5 + 0.850x, where x represents height. The mean of the 60 heights is 63.4 in and the mean of the 60 pulse rates is 75.6 beats per minute. Find the best predicted pulse rate of a woman who is 74 inches tall. Use a significance level of a=0.01.
Answer:
The best predicted pulse rate of a woman who is 74 inches tall is 80.4 (beats per minute).
Step-by-step explanation:
The best predicted pulse rate of a woman who is 74 inches tall is obtained below:
Let x denotes height (in inches) and y denotes pulse rates (in beats per minute).
From the information, the regression equation is, the total number of women is 60 and the linear correlation coefficient between height and pulse rates is 0.234 and the mean of heights is 63.4 and the mean of pulse rates is 75.6. The estimated regression line is
^y= 17.5 + 0.850x
The required best predicted pulse rates is,
=17.5 + (0.850×74)
=17.5 + 62.9
= 80.4 beats per minute
Suppose that A A is a set containing 13 13 elements. Find the number of different subsets of A.
Answer:
number of subsets of a set with 13 elements are: [tex]2^{13}[/tex]
Step-by-step explanation:
In order to solve this intuitively, we can start by a set with lesser elements. This will reveal a pattern that will be used to solve for the subsets of the 13 element set.
If we start with a set B. which contains only 3 elements.
[tex]B = \{1,2,3\}[/tex]
how many subsets of B are there? well we can count them. [the set containing {1,2} and {2,1} are the same, arrangement doesn't matter]
[tex]B_{0} = \{\}\\B_{1a}=\{1\}\\B_{1b}=\{2\}\\B_{1c}=\{3\}\\B_{2a}=\{1,2\}\\B_{2b}=\{2,3\}\\B_{2c}=\{3,1\}\\B_{3a}=\{1,2,3\}\\[/tex]
there are a total of 9 subsets here.
Similarly, if you try a with a subset with only two elements you'll find that it has a total of 4 subsets.
We can see that combinatorics is at play here.
for the set B. the number of subsets can be written as:
[tex]\text{\# of subsets of B} = ^3C_0+^3C_1+^3C_2+^3C_3\\\text{\# of subsets of B} = 1+3+3+1\\\\text{\# of subsets of B} = 8[/tex]
if we try with a 2-element set:
[tex]\text{\# of subsets} = ^2C_0+^2C_1+^2C_2\\\text{\# of subsets} = 1+2+1\\\ \text{\# of subsets} = 4[/tex]
We can use the same technique to find the number of subsets of the 13 element set.
But if you recognize a pattern here that this sets of combinations are actually part of the pascal triangle, the sum of each row of the triangle is 2^{the row's number}. hence.
[tex]\text{\# of subsets of B} = 2^3\\\ \text{\# of subsets of B} = 8[/tex]
So finally, the subsets of a 13-element set A will be
[tex]\text{\# of subsets of A} = ^{13}C_0+^{13}C_1+^{13}C_2+^{13}C_3\cdots+^{13}C_{12}+^{13}C_{13}\\OR\\\text{\# of subsets of A} = 2^{13}\\\text{\# of subsets of A} = 8192[/tex]
If the set A has 13 elements, the number of different subsets is [tex]2^{13}=8192[/tex]
All the possible subsets that can be formed from any given set is called the Power set of that set. Generally, if we had a set [tex]H[/tex] such that
[tex]|H|=k[/tex]
Where [tex]|H|[/tex] denotes the cardinality, or number of elements, in [tex]H[/tex], the power set of [tex]H[/tex], denoted by [tex]P(H)[/tex], has the following formula
[tex]P(H)=2^k\text{ elements}[/tex]
So, given the set [tex]A[/tex] such that
[tex]|A|=13[/tex]
the power set of [tex]A[/tex] will have [tex]2^{13} \text{ or } 8192 \text{ elements}[/tex]
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Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1, 0, −1), B(5, −3, 0), C(1, 5, 2)
∠CAB =
∠ABC =
∠BCA =
Answer:
angle CAB = 113.8 degree
angle ABC = 35.6 degree
angle BCA = 30.6 degree
Step-by-step explanation:
Given data:
A(1, 0, −1),
B(5, −3, 0),
C(1, 5, 2)
calculate the length of side by using the distance formula
so
AB = (5,-3,0) - (1,0,-1) = (4,-3,1)
AC= (1,5,2) - (1,0,-1) = (0,5,3)
|AB|
|AC| =[tex]\sqrt {(0 + 5^2+3^2)} = \sqrt{34}[/tex]
From following formula, calculate the angle between the two side i.e Ab and AC
AB.AC = |AB|*|AC| cos ∠CAB
(4,-3,1).(0,5,3)
4*0 -3*5 +1*3
-12 =
cos ∠CAB = - 0.404
angle CAB = 113.8 degree
BA =B- A = (1,0,-1) - (5,-3,0) = (-4,3,-1)
BC = (1,5,2)-(5,-3,0) = (-4,8,2)
|BA| = \sqrt{(26)}
|BC| [tex]= \sqrt {(4^2 + 8^2 + 2^2)} = \sqrt{(84)}[/tex]
BA.BC = |BA|*|BC|* cosABC
(-4,3,-1).(-4,8,2) =[tex]\sqrt{(26)} * \sqrt{(84)} *cosABC[/tex]
16+24-2
cos ∠ABC = 0.813
angle ABC = 35.6 degree
we know sum of three angle in a traingle is 180 degree hence
sum of all three angle = 180
angle BCA + 35.6 + 113.8 = 180
angle BCA = 30.6 degree
In the general population, what is the probability that an individual will have the birth defect, assuming that maternal and paternal genes
Answer:
0.0625
Step-by-step explanation:
The prevalence of gene a = 25 %, P (a) = 0.25
birth defect occurs when both parents have prevalence of gene a.
P (Defect) = P ( Both parents have gene a)
If both parents inherit the gene a independently, the the individual will have a birth defect when both parents have gene a.
P ( Father having gene a) = 0.25
P ( Mother having gene a) = 0.25
Hence,
P (Birth Defect) = P ( Both parents have gene a) = 0.25 * 0.25 = 0.0625
The right-hand endpoint of the kth subinterval is denoted x∗k. What is x∗k (in terms of k and n)? x∗k =_________.
In this problem you will calculate ∫302+4 by using the formal definition of the definite integral: ∫()=lim→∞[∑=1(∗)Δ].
(a) The interval [0,3] is divided into equal subintervals of length Δ. What is Δ (in terms of )? Δ =
(b) The right-hand endpoint of the th subinterval is denoted ∗. What is ∗ (in terms of and )? ∗ =
Answer:
a) Δ= [tex]\frac{3}{n}[/tex]
b) [tex]x^{*}_{k} = \frac{3k}{n}[/tex]
Step-by-step explanation:
a) If the interval [0,3] , i.e let a = 0 , b =3 and n=n.
So [0,3] divide into n equal subintervals;
Therefore, the length Δ= [tex]\frac{b-a}{n}[/tex]
Δ= [tex]\frac{3-0}{n}[/tex]
Δ= [tex]\frac{3}{n}[/tex]
b) To calculate [tex]x^{*}_{k}[/tex];
[tex]x^{*}_{k}[/tex] = a + k . Δ (where n= 0, Δ = [tex]\frac{3}{n}[/tex])
= 0 + k . [tex]\frac{3}{n}[/tex]
[tex]x^{*}_{k}[/tex] = [tex]\frac{3}{k}[/tex]
The right-hand endpoint of the kth subinterval, denoted x∗k, can be expressed as a function of both k and n (the total number of subintervals). The formula x∗k = k/n can be used in this context, where k denotes the position of the subinterval and n represents the total number of subintervals.
Explanation:In the context of subintervals, x∗k represents the right-hand endpoint of the kth subinterval. When a range is divided into n subintervals, the right-hand endpoint of the kth subinterval could be represented as a function of both k and n. Thus, x∗k can be expressed as k/n.
To illustrate, suppose we have a range from 0 to 1 (inclusive) and want to split it into 4 subintervals (n=4). The right endpoint of the 1st subinterval (k=1) would be 1/4 = 0.25. For the 2nd subinterval (k=2), the endpoint would be 2/4 = 0.5. And so on.
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supposee you start saving today for a $8000 down payment that you plan to make on a condo in 4 years. Assume that you make no deposits into the account after your initial deposit. The account has quarterly compounding and an APR of 6%. How much would you need to deposit now to reach your $8000 goal in 4 years
To reach a $8000 down payment in 4 years with an account that compounds quarterly at an APR of 6%, you would need to make an initial deposit of approximately $6304.05 today.
Explanation:To calculate the initial deposit needed to save for a $8000 down payment in 4 years with an account that offers quarterly compounding at an APR of 6%, we use the compound interest formula:
[tex]P = A / (1 + r/n)^{(nt)[/tex]
Where:
P is the principal amount (initial deposit)A is the future value of the investment/loan, including interestr is the annual interest rate (decimal)n is the number of times that interest is compounded per yeart is the time the money is invested/borrowed for, in yearsGiven:
A = $8000r = 6/100 = 0.06 (as a decimal)n = 4 (quarterly compounding)t = 4 yearsNow we can calculate the initial deposit:
[tex]P = $8000 / (1 + 0.06/4)^(4*4)P = $8000 / (1 + 0.015)^(16)P = $8000 / (1.015)^16P \approx $8000 / 1.26824179P \approx $6304.05[/tex]
Therefore, you would need to deposit approximately $6304.05 today to have $8000 in 4 years in the account with the given interest rate and compounding frequency.
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To reach an $8000 goal in 4 years with a 6% APR compounded quarterly, you need to deposit approximately $6307.17 now.
To determine how much to deposit now to reach your $8000 goal in 4 years with quarterly compounding and an APR of 6%, you can use the formula for compound interest:
[tex]P = A / (1 + r/n)^(nt)[/tex]
Where:
P = initial deposit (what we're solving for)A = future value ($8000)r = annual interest rate (6% or 0.06)n = number of times interest is compounded per year (4 for quarterly)t = number of years (4)Now, substitute the given values into the formula:
P = [tex]$8000 / (1 + 0.06/4)^(4*4)[/tex]
P = $8000 / [tex](1 + 0.015)^(16)[/tex]
P = $8000 / [tex](1.015)^(16)[/tex]
P = $8000 / 1.2682 (approximately)
P = $6307.17 (approximately)
So, you would need to deposit approximately $6307.17 now to reach your $8000 goal in 4 years with quarterly compounding at a 6% APR.
6-27. Erin started with one corner of a figure located at (−4, 5) and translated it to end at (6, 8). To find
out how far the shape moved horizontally, she decided to find the difference between the two xcoordinates. She wrote: 6 − (−4).
Answer:
The horizontal move of the shape was 10 units to the right,Explanation:
The problem requires that you find how far the shape was moved horizontally.
To find the horizontal translation of the shape, you must subtract the x-coordinate of the original corner (the pre-image) from the x-coordinate of the final corner (the image)
Final corner (image):
Coordinates: (6,8)x-coordinate: 6Original corner (preimage):
Coordinate: (-4,5)x-coordinate: -4Translation:
Difference = 6 - (-4), which is what Erin wroteSimplify: 6 + 4 = 10Thus, the horizontal move of the shape was 10 units to the right.
Patrice Patriot has dimes and uarters in a piggy bank. She has a total of 20 coins for atotal of $4.25. How many dimes does she have?
Answer:
see the picture of work shown
Answer:she has 5 dimes and 15 quarters.
Step-by-step explanation:
A dime is worth 10 cents. Converting to dollars, it becomes
10/100 = $0.1
A quarter is worth 25 cents. Converting to dollars, it becomes
25/100 = $0.25
Let x represent the number of dimes that she has.
Let y represent the the number of quarters that she has
Patrice Patriot has a total of 20 coins. It means that
x + y = 20
The total worth of dimes and quarters that she has in a piggy bank is $4.25. It means that
0.1x + 0.25y = 4.25 - - - - - - - - - -1
Substituting x = 20 - y into equation 1, it becomes
0.1(20 - y) + 0.25y = 4.25
2 - 0.1y + 0.25y = 4.25
- 0.1y + 0.25y = 4.25 - 2
0.15y = 2.25
y = 2.25/0.15
y = 15
x = 20 - y = 20 - 15
x = 5
Suppose that the number of gallons of milk sold per day at a local supermarket are normally distributed with mean and standard deviation of 436.6 and 23.23, respectively. What is the probability that on a given day the supermarket will sell below 449 gallons of milk?
Answer:
70.19% probability that on a given day the supermarket will sell below 449 gallons of milk.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 436.6, \sigma = 23.23[/tex]
What is the probability that on a given day the supermarket will sell below 449 gallons of milk?
This is the pvalue of Z when X = 449. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{449 - 436.6}{23.23}[/tex]
[tex]Z = 0.53[/tex]
[tex]Z = 0.53[/tex] has a pvalue of 0.7019.
So there is a 70.19% probability that on a given day the supermarket will sell below 449 gallons of milk.
Construct a frequency distribution and a relative frequency distribution for the light bulb data with a class width of 20, starting at 800. Copy and paste your distribution tables here. 2. Construct a histogram based on this frequency distribution table for the light bulb data. Copy and paste your histogram here. Describe the shape of the histogram. (Is it unimodal, bimodal, skewed, etc.?) skewed 3. Now, construct a frequency distribution table and a relative frequency distribution table for the light bulb data with a class width of 100, starting at 800. Copy and paste your distribution tables here. 5. Construct a histogram based on this frequency distribution table for the data. Copy and paste your histogram here. Describe the shape of the histogram. (Is it unimodal, bimodal, skewed, etc.?)
819
836
888
897
903
907
912
918
942
943
952
959
962
986
992
994
1004
1005
1007
1015
1016
1018
1020
1022
1034
1038
1072
1077
1077
1082
1096
1100
1113
1113
1116
1153
1154
1174
1188
1230
Answer:
Step-by-step explanation:
Hello!
You have the information about light bulbs (i believe is their lifespan in hours) And need to organize the information in a frequency table.
The first table will be with a class width of 20, starting with 800. This means that you have to organize all possible observations of X(lifespan of light bulbs) in a class interval with an amplitude of 20hs and then organize the information noting their absolute frequencies.
Example
1) [800;820) only one observation classifies for this interval x= 819, so f1: 1
2)[820; 840) only one observation classifies for this interval x= 836, so f2: 1
3)[840;860) no observations are included in this interval, so f3=0
etc... (see attachment)
[ means that the interval is closed and starts with that number
) means that the interval is open, the number is not included in it.
fi: absolute frequency
hi= fi/n: relative frequency
To graph the histogram you have to create the classmark for each interval:
x'= (Upper bond + Lower bond)/2
As you can see in the table, there are several intervals with no observed frequency, this distribution is not uniform least to say symmetric.
To check the symmetry of the distribution is it best to obtain the values of the mode, median and mean.
To see if this frequency distribution has one or more modes you have to identify the max absolute frequency and see how many intervals have it.
In this case, the maximal absolute frequency is fi=6 and only one interval has it [1000;1020)
[tex]Mo= LB + Ai (\frac{D_1}{D_1+D_2} )\\[/tex]
LB= Lower bond of the modal interval
D₁= fmax - fi of the previous interval
D₂= fmax - fi of the following interval
Ai= amplitude of the modal interval
[tex]Mo= 1000 + 20*(\frac{(6-3)}{(6-3)+(6-4)} )=1012[/tex]
This distribution is unimodal (Mo= 1012)
The Median for this frequency:
Position of the median= n/2 = 40/2= 20
The median is the 20th fi, using this information, the interval that contains the median is [1000;1020)
[tex]Me= LB + Ai*[\frac{PosMe - F_{i-1}}{f_i} ][/tex]
LB= Lower bond of the interval of the median
Ai= amplitude of the interval
F(i-1)= acumulated absolute frequency until the previous interval
fi= absolute frequency of the interval
[tex]Me= 1000+ 20*[\frac{20-16}{6} ]= 1013.33[/tex]
Mean for a frequency distribution:
[tex]X[bar]= \frac{sum x'*fi}{n}[/tex]
∑x'*fi= summatory of each class mark by the frequency of it's interval.
∑x'*fi= (810*1)+(230*1)+(870*0)+(890*2)+(910*4)+(930*0)+(950*4)+(970*1)+(990*3)+(1010*6)+(1030*4)+(1050*0)+(1070*3)+(1090*2)+(1110*4)+(1130*0)+(1150*2)+(1170*1)+(1190*1)+(1210*0)+(1230*1)= 40700
[tex]X[bar]= \frac{40700}{40} = 1017.5[/tex]
Mo= 1012 < Me= 1013.33 < X[bar]= 1017.5
Looking only at the measurements of central tendency you could wrongly conclude that the distribution is symmetrical or slightly skewed to the right since the three values are included in the same interval but not the same number.
*-*-*
Now you have to do the same but changing the class with (interval amplitude) to 100, starting at 800
Example
1) [800;900) There are 4 observations that are included in this interval: 819, 836, 888, 897 , so f1=4
2)[900;1000) There are 12 observations that are included in this interval: 903, 907, 912, 918, 942, 943, 952, 959, 962, 986, 992, 994 , so f2= 12
etc...
As you can see this distribution is more uniform, increasing the amplitude of the intervals not only decreased the number of class intervals but now we observe that there are observed frequencies for all of them.
Mode:
The largest absolute frequency is f(3)=15, so the mode interval is [1000;1100)
Using the same formula as before:
[tex]Mo= 1000 + 100*(\frac{(15-12)}{(15-12)+(15-8)} )=1030[/tex]
This distribution is unimodal.
Median:
Position of the median n/2= 40/2= 20
As before is the 20th observed frequency, this frequency is included in the interval [1000;1100)
[tex]Me= 1000+ 100*[\frac{20-16}{15} ]= 1026.67[/tex]
Mean:
∑x'*fi= (850*4)+(950*12)+(1050*15)+(1150*8)+(1250*1)= 41000
[tex]X[bar]= \frac{41000}{40} = 1025[/tex]
X[bar]= 1025 < Me= 1026.67 < Mo= 1030
The three values are included in the same interval, but seeing how the mean is less than the median and the mode, I would say this distribution is symmetrical or slightly skewed to the left.
I hope it helps!
Topic: Associative, Commutative, and Distributive Properties
Which property is not used to simplify the following expression?
2*(x+5)+7x=(2x+10)+7x
=(10+2x)+7x
=10 +(2x+7x)
=10 + 9x
=9x + 10
A. associative property of addition
B. commutative property of addition
C. distributive property
D. commutative property of multiplication
Answer:
D
Step-by-step explanation:
cumulative property of multiplication
Final answer:
The distributive property is not used in the simplification of the expression 2×(x+5)+7x. The expression is simplified using the associative and commutative properties of addition.
Explanation:
The property that is not used to simplify the following expression 2×(x+5)+7x=(2x+10)+7x=(10+2x)+7x=10 +(2x+7x)=10 + 9x=9x + 10 is C. distributive property. Let's look at how each property is applied in the simplification:
Associative property of addition: This property is used when the expression goes from 2×(x+5)+7x to (2x+10)+7x and again from (10+2x)+7x to 10 +(2x+7x).
Commutative property of addition: This property is used when the terms 2x and 10 are rearranged as (10+2x) and also when the final step changes 10 + 9x to 9x + 10.
The distributive property is not used at any step in this problem. This property would involve multiplication across a sum, such as a(b + c) = ab + ac, which is not seen in the simplification process.
The commutative property of multiplication is also not used, as there is no rearrangement of multiplication terms.
Find the inverse of the function âf(x)equals 18x .Verify that f (f Superscript negative 1 Baseline (x ))equalsxand f Superscript negative 1 Baseline (f (x ))equalsx.f Superscript negative 1 Baseline (x )equalsnothing
Answer:
inverse f-1 (x) = x/18
Step-by-step explanation:
To find the inverse of a function f-1 (x), we represent f(x) by y or let f(x) = y
then make x the subject of the formula,
f(x) = 18x
Let f(x) = y
hence Y = 18x, make x the subject of the formula, do that by dividing both sides by 18,
Y/18 = x or x = Y/18
Interchanging or swapping x and y, therefore f-1 (x) = x/18
therefore, F(x) = 18x, the inverse f-1 (x) = x/18
A firm producing socks has a fixed cost of $20,000 and variable cost of $2 per pair of socks. Let x = the number of pairs of socks. Find the break-even point if the socks sell for $4.50 per pair.
Answer:
break even point = 8000 socks produced or $36000 in costs
Step-by-step explanation:
the cost function of the firm is
total cost = fixed cost + variable cost = $20000 + $2*Q
where Q= number of socks
the revenue from sales is
sales = Price* Q = $4.50*Q
the break even point is reached when the net profit is = 0 ( that is, the total cost is equal to the revenue from sales) , then
total cost = sales
$20000 + $2*Q =$4.50*Q
Q= $20000/($4.50-$2) = 8000 socks
that represents
total cost = $20000 + $2*8000 = $36000
then
break even point = 8000 socks produced or $36000 in costs