1+3+5+7+...+999 =
= 1+2+3+4+...+500
+1+2+3+...+499
= 2·(1+2+3+...+499) + 500
= 2·(499·500)/2 + 500
= 499·500 + 500
= 500·(499 + 1)
= 500·500
= 250.000
To find the sum of the series 1 + 3 + 5 + 7 + ... + 999, we recognize this is the sum of the first 500 odd numbers. Using the property that the sum of the first n odd numbers is n², we find the sum to be 500², which is 250000.
Summation of Odd Numbers:
To find the sum of the series 1 + 3 + 5 + 7 + ... + 999 using Gauss's method, we first recognize that this sequence is a series of the first 500 odd numbers.
An interesting property to note is that the sum of the first n odd numbers is n².
For example:
→ 1 = 1 (which is 1¹)
→ 1 + 3 = 4 (which is 2²)
→ 1 + 3 + 5 = 9 (which is 3³)
And so on...
In general, the [tex]n_{th[/tex] odd number can be expressed as 2n - 1.
For 999, to find its position in the series, note that 999 is the 500th odd number (since 2*500 - 1 = 999).
Thus, the sum of the first 500 odd numbers is:
→ Sum = 500²
= 250000
→ So, the sum of the series 1 + 3 + 5 + 7 + ... + 999 is 250000.
2. Is the bottled water you’re drinking really purified water? A 4-year study of bottled water brands conducted by the Natural Resources Defense Council found that 25% of bottled water is just tap water packaged in a bottle. Consider a sample of five bottled water brands, and let Y equal the number of these brands that use tap water. A. What is the type of probability distribution for Y. B. Find P(Y=2) C. Find P(Y<1).
Answer:
a.
The random variable has binomial distribution with parameters p = 0.25, n=5.
b.
[tex]P(Y=2) = {5 \choose 2} (0.25)^2(0.75)^3 = 0.264\\P(Y<1 ) = P(Y = 0 ) = (0.75)^5 = 0.237[/tex]
Step-by-step explanation:
a.
Remember what a random variable with binomial distribution is. It is a random variable that counts the number of successes and in n trials of an experiment.
In this case, if you have 0,1,2,3,4,5 bottles of water, your success is that the bottle has tap water, and the probability of that success is p = 0.25. and the number of trials is n = 5.
b.
Using the formula for the binomial distribution you get that
[tex]P(Y=2) = {5 \choose 2} (0.25)^2(0.75)^3 = 0.264\\P(Y<1 ) = P(Y = 0 ) = (0.75)^5 = 0.237[/tex]
(1 point) Supppose A is an invertible n×n matrix and v⃗ is an eigenvector of A with associated eigenvalue 8. Convince yourself that v⃗ is an eigenvector of the following matrices, and find the associated eigenvalues. The matrix A4 has an eigenvalue . The matrix A−1 has an eigenvalue . The matrix A+4In has an eigenvalue . The matrix 5A has an eigenvalue .
By definition, if [tex]\vec v[/tex] is an eigenvector of [tex]\mathbf A[/tex] (with associated eigenvalue 8), then
[tex]\mathbf A\vec v=8\vec v[/tex]
Notice that[tex]\mathbf A^4\vec v=\mathbf A^3\cdot\mathbf A\vec v=8\mathbf A^3\vec v[/tex]
Rinse and repeat to find
[tex]\mathbf A^4\vec v=8^4\vec v[/tex]
so that [tex]\mathbf A^4[/tex] has a corresponding eigenvalue of 4096.
Let [tex]\lambda[/tex] be the eigenvalue of [tex]\mathbf A^{-1}[/tex] corresponding to [tex]\vec v[/tex]. Then[tex]\mathbf A^{-1}\vec v=\lambda\vec v\implies\mathbf A\cdot\mathbf A^{-1}\vec v=\mathbf A\cdot\lambda \vec v\implies\vec v=8\lambda\vec v[/tex]
For this to hold, we require [tex]8\lambda=1[/tex], or [tex]\lambda=\frac18[/tex]. So [tex]\mathbf A^{-1}[/tex] has a corresonding eigenvalue of 1/8.
Expanding gives[tex](\mathbf A+4\mathbf I_n)\vec v=\mathbf A\vec v+4\mathbf I_n\vec v=8\vec v+4\vec v=12\vec v[/tex]
so that [tex]\mathbf A+4\mathbf I_n[/tex] has an associated eigenvalue of 12.
You know the drill:[tex]5\mathbf A\vec v=5\cdot8\vec v=40\vec v[/tex]
so the eigenvalue of [tex]5\mathbf A[/tex] is 40.
The matrix [tex]A^{4}[/tex] has eigen value of 4096.
The matrix [tex]A^{-1}[/tex] has eigen value of [tex]\frac{1}{8}[/tex]
The eigen value of (A+4I) is 12.
The eigen value of [tex]5Av[/tex] is 40.
Eigen values of matrix:Eigenvalues are the set of scalar values that is associated with the set of linear equations most probably in the matrix equations.
The eigenvectors are also termed as characteristic roots.
If v is an eigen vector of matrix A associated with eigen value [tex]\lambda[/tex].
Then, [tex]Av=\lambda v[/tex]
it is given that, v is an eigen vector of A with eign value 8.
so taht, [tex]Av=8v[/tex]
We have to find eigen value for [tex]A^{4}[/tex].
[tex]A^{4}v=A^{3}(8v)=8^{4}v=4096v[/tex]
Thus, [tex]A^{4}[/tex] has eigen value of 4096.
If A has eigen value of 8 . Then [tex]A^{-1}[/tex] has eigen value of [tex]\frac{1}{8}[/tex]
The eigen value of (A+4I) is computed as,
[tex](A+4I_{n})v=Av+4v=8v+4v=12v[/tex]
The eigen value of (A+4I) is 12.
The eigen value of [tex]5Av[/tex] is,
[tex]5Av=5*8v=40v[/tex]
The eigen value of [tex]5Av[/tex] is 40.
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A random variable follows the continuous uniform distribution between 160 and 340. Calculate the following quantities for the distribution. a) P(220less than or equalsxless than or equals290) b) P(160less than or equalsxless than or equals250) c) P(xgreater than190) d) What are the mean and standard deviation of this distribution?
Answer:
a) 0.3889
b) 0.5
c) 0.8333
d) The mean is 250 and the standard deviation is 51.96.
Step-by-step explanation:
An uniform probability is a case of probability in which each outcome is equally as likely.
For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.
The probability of finding a value of X higher than x is:
[tex]P(X > x) = 1 - \frac{x - a}{b-a}[/tex]
The probability of finding a value of X between c and d is:
[tex]P(c \leq X \leq d) = \frac{d - c}{b - a}[/tex]
The mean and the standard deviation are, respectively:
[tex]M = \frac{a+b}{2}[/tex]
[tex]S = \sqrt{\frac{b-a}^{2}{12}}[/tex]
A random variable follows the continuous uniform distribution between 160 and 340.
This means that [tex]a = 160, b = 340[/tex]
a)
[tex]P(220 \leq X \leq 290) = \frac{290 - 220}{340 - 160} = 0.3889[/tex]
b)
[tex]P(160 \leq X \leq 250) = \frac{250 - 160}{340 - 160} = 0.5[/tex]
c)
[tex]P(X > 190) = 1 - \frac{190 - 160}{340 - 160} = 0.8333[/tex]
d)
[tex]M = \frac{160 + 340}{2} = 250[/tex]
[tex]S = \sqrt{\frac{340 - 160}^{2}{12}} = 51.96[/tex]
The mean is 250 and the standard deviation is 51.96.
Find the output, k, when the input, t, is 3.
k = 13t – 2
k=[
S
Answer:
k = 37
Step-by-step explanation:
[tex]k = 13t - 2 \\ \therefore \: k = 13 \times 3 - 2 \\ \therefore \: k = 39 - 2 \\ \huge \red{ \boxed{ \therefore \: k = 37}}[/tex]
A market surveyor wishes to know how many energy drinks teenagers drink each week. They want to construct a 80%80% confidence interval with an error of no more than 0.070.07. A consultant has informed them that a previous study found the mean to be 3.63.6 energy drinks per week and found the variance to be 1.441.44. What is the minimum sample size required to create the specified confidence interval? Round your answer up to the next integer.
Answer:
481
Step-by-step explanation:
Given:
Variance = 1.44
S.d = √1.44 = 1.2
C.I = 80%
Margin of error, E = 0.07
Mean = 3.6
Using Z table, the Z score for 80% confidence interval, Zc = 1.28
To find the sample size, n, we have:
[tex] n= [\frac{Z_c * s.d}{E}]^2 [/tex]
Substituting figures in the formula, we have:
[tex] n = [\frac{1.28 * 1.2}{0.07}]^2 [/tex]
n = 21.94286²
n = 481.49
Approximately, n = 481
A stack of four cards contains two red cards and two black cards. I select two cards, one at a time, and do not replace the first card selected before selecting the second card. Consider the events:A = the first card selected is red.B = the second card selected is red.The events A and B are:conditionals.independent.disjoint.None of the answer options are correct.
Final answer:
The events A and B are mutually exclusive. The events A and C are not mutually exclusive. Two events A and B are independent events.
Explanation:
The events A and B are mutually exclusive because they cannot happen at the same time. P(A AND B) = 0.
No, A and C are not mutually exclusive because they can occur at the same time. In fact, C includes all of the outcomes of A; if the card chosen is blue it is also (red or blue). P(A AND C) = P(A) = 20.
Two events A and B are independent events if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are not independent, then we say that they are dependent events. Sampling may be done with replacement or without replacement. With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.
A boat is moving due east at 15 miles per hour when it encounters a current LaTeX: \textbf{C} = 2\textbf{i}+17\textbf{j}C = 2 i + 17 j. What is the path of the boat in this current if the boater keeps it pointed due east? What direction should the boater steer in order to go due east?
Answer:
a) [tex]\vec v_{B} - \vec v_{C} = 3\, i - 17\,j[/tex], b) [tex]\theta = 79.992^{\textdegree}[/tex] (clockwise)
Step-by-step explanation:
The resultant velocity of the boat must be:
[tex]\vec v_{B} = 15\,i[/tex]
Likewise, the velocity of the current is:
[tex]\vec v_{C} = 2\,i + 17\,j[/tex]
a) The intended velocity of the boat is:
[tex]\vec v_{B} - \vec v_{C} = (15-2)\,i + (0-17)\,j[/tex]
[tex]\vec v_{B} - \vec v_{C} = 3\, i - 17\,j[/tex]
b) The direction of the boat is:
[tex]\theta = \tan^{-1}\left(\frac{17}{3} \right)[/tex] (clockwise)
[tex]\theta = 79.992^{\textdegree}[/tex] (clockwise)
Please help I can’t fail after all the hard work I have put in
Answer:180cm^2
Step-by-step explanation:
A1=25×20=500
A2=16×20=320
A1-A2
500-320=180
The Wisconsin Dairy Association is interested in estimating the mean weekly consumption of milk for adults over the age of 18 in that state. To do this, they have selected a random sample of 300 people from the designated population. The following results were recorded: Given this information, if the leaders wish to estimate the mean milk consumption with 90 percent confidence, what is the approximate margin of error in the estimate? Question 16 options: ±12.996 ounces ±0.75 ounce ±0.456 ounce z = 1.645
Answer:
[tex] ME = 1.653 *\frac{7.9}{\sqrt{300}}= \pm 0.75[/tex]
±0.75 ounce
Step-by-step explanation:
Assuming this complete question: The Wisconsin Dairy Association is interested in estimating the mean weekly consumption of milk for adults over the age of 18 in that state. To do this, they have selected a random sample of 300 people from the designated population. The following results were recorded: xbar=34.5 ounces, s=7.9 ounces Given this information, if the leaders wish to estimate the mean milk consumption with 90 percent confidence, what is the approximate margin of error in the estimate?
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=34.5[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=7.9 represent the sample standard deviation
n=300 represent the sample size
Solution to the problem
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)
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=300-1=299[/tex]
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,199)".And we see that [tex]t_{\alpha/2}=1.653[/tex]
And the margin of error is given by:
[tex] ME = 1.653 *\frac{7.9}{\sqrt{300}}= \pm 0.75[/tex]
±0.75 ounce
Answer:
The approximate margin of error in the estimate is ±0.75 ounces.
Step-by-step explanation:
The question is incomplete:
The following results were recorded: xbar=34.5 ounces, s=7.9 ounces.
The sample size is n=300.
We will use the sample standard deviation to estimate the population standard deviation, so we will use the t-statistic.
To develop a confidence interval, we first have to calculate the degrees of freedom, and then look up in a t-students distribution table the critical value for a 90% confidence interval.
The degrees of freedom are:
[tex]df=n-1=300-1=299[/tex]
The critical value for a 90% CI is t=1.65.
Now, we can calculate the margin of error of the confidence interval as:
[tex]E=t\cdot s/\sqrt{n}=1.65*7.9/\sqrt{300}=13.035/17.32=0.75[/tex]
The lower and upper bounds of the confidence interval will be:
[tex]LL=\bar x-t\cdot s/\sqrt{n}=34.5-0.75=33.75\\\\UL=\bar x+t\cdot s/\sqrt{n}=34.5+0.75=35.25[/tex]
The confidence interval is (33.75, 35.25)
Solve the inequality 4x- 7 < 5
Answer:
x < 3
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the greater than sign from both sides of the inequality :
4*x-7-(5)<0
Step 1 :
Pulling out like terms :
1.1 Pull out like factors :
4x - 12 = 4 • (x - 3)
Equation at the end of step 1 :
Step 2 :
2.1 Divide both sides by 4
Solve Basic Inequality :
2.2 Add 3 to both sides
x < 3
By solving the given inequality "[tex]4x-7<5[/tex]", we get the answer "[tex]x <3[/tex]". A complete solution is below.
The given equation of inequality is:
[tex]4x-7<5[/tex]Now,
By adding "7" both sides of the equation, we get
→ [tex]4x-7+7<5+7[/tex]
→ [tex]4x< 12[/tex]
→ [tex]x < \frac{12}{4}[/tex]
→ [tex]x < 3[/tex]
Thus the above solution is correct.
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To solve the system of linear equations 3 x minus 2 y = 4 and 9 x minus 6 y = 12 by using the linear combination method, Henry decided that he should first multiply the first equation by –3 and then add the two equations together to eliminate the x-terms. When he did so, he also eliminated the y-terms and got the equation 0 = 0, so he thought that the system of equations must have an infinite number of solutions. To check his answer, he graphed the equations 3 x minus 2 y = 4 and 9 x minus 6 y = 12 with his graphing calculator, but he could only see one line. Why is this? because the system of equations actually has only one solution because the system of equations actually has no solution because the graphs of the two equations overlap each other because the graph of one of the equations does not exist
Answer: C. Because the graphs of the two equations overlap each other.
Step-by-step explanation: I took the test its right! Hope this helps
Type the noun form of the verb.
contribute
Answer:
Contribution
Step-by-step explanation:
The verb would be used as:
I like to contribute to discussions.
The noun would be used as:
My contribution to the discussion was alright.
I hope this helped! (Sorry for the dry examples, I couldn't think of anything else)
Find the length of the bold arc. Round to the nearest tenth and use 3.14 or the pi button on your calculator for pi.
Answer:
8π yards
Step-by-step explanation:
A circle subtends a total angle of 360 ° from its center.The length of an arc is directly proportional to the angle it subtends from the circle's center. The arc's length can therefore be calculated as:
C=πd=2πr
Where C is circumference, d is diameter and r is radius. Given r as 16 yards then the arc length which is equivalent to circumference is given as
[tex]C=\frac {90}{360}*2*\pi*16=8\pi[/tex]
Answer:
2,144.7 i just answered the other one is wrong
Step-by-step explanation:
Which of the following functions will only lie in the third quadrant?
y = 2 − 3
y = − − 4
None of the above.
Both A and B.
b.
If it's a (+,+) it's the 1st quadrant. If it's a (-,+) it's the 2nd quadrant. If its a (-,-) it's the 3rd quadrant. And if it's a (+,-) it's the 4th quadrant
Answer:
B
Step-by-step explanation:
Which system of equations has exactly one solution?
A.) 2 x minus 4 y = 8. x + y = 7.
B.) 3 x + y = negative 1. 6 x + 2 y = negative 2.
C.) 3 x + 3 y = 3. Negative 6 x minus 6 y = 3.
D.) 3 x minus 2 y = 4. Negative 3 x + 2 y = 4.
Answer:
A.) 2 x minus 4 y = 8. x + y = 7.
Step-by-step explanation:
A.) 2 x minus 4 y = 8. x + y = 7.
2x-4y = 8
x+y=7
This has one solution
B.) 3 x + y = negative 1. 6 x + 2 y = negative 2.
3x+y = -1
Multiply by 2
6x +2y = -2
This is the second equation so it has infinite solutions
C.) 3 x + 3 y = 3. Negative 6 x minus 6 y = 3.
3x+3y =3
Multiply by -2
-6x-6y = -6
The equation is the same but the constant, which means they are parallel lines, they have zero solutions
D.) 3 x minus 2 y = 4. Negative 3 x + 2 y = 4.
3x -2y = 4
Multiply by -1
-3x +2y = -4
The equation is the same but the constant, which means they are parallel lines, they have zero solutions
Answer:
A.) 2 x minus 4 y = 8. x + y = 7.
Step-by-step explanation:
A.) 2 x minus 4 y = 8. x + y = 7.
2x - 4y = 8
x - 2y = 4
x = 4 + 2y
x + y = 7
4 + 2y + y = 7
3y = 3
y = 1
x = 4 + 2
x = 6
Only solution: (6,1)
A tire company sends an email survey to all customers who purchase new tires each month. The previous month, 201 people purchased new tires. Surveys were sent to 100 of these people, chosen at random, and 51 people responded to the survey. Identify the population and the sample.
Answer:
The population consists of the 201 people who bought tires last month.
The sample consists of the 51 people who responded.
Step-by-step explanation:
In statistical analysis, the population is a set of all values that can be considered for an experiment. It is a set of observations from which a sample is selected. The sample is a subset of the population.
In this case, it is provided that a tire company sends an email survey to all customers who purchase new tires each month.
The number of tires sold last month was N = 201.
The email survey was sent to 100 of these 201 people and 51 responded.
In this case, since the survey is done every month, the population consists of all the people who bought tires the previous month.
So, the population consists of the 201 people who bought tires last month.
The number of people who responded was, n = 51.
The sample consists of these 51 people because the data from these 51 people would be used to conduct and draw conclusion about the survey.
So, the sample consists of the 51 people who responded.
What is the answer for 8b - 2b -4
Simplfied Answer: [tex]6b-4[/tex]
Combine Like Terms
[tex]8b+-2b+-4\\(8b+-2b)+(-4)\\6b+-4[/tex]
Answer:
[tex]8b - 2b - 4 \\ 6b - 4 \\ = 2(3b - 2)[/tex]
Scores on the SAT college entrance test in a recent year were roughly Normal with mean 1012.5 and standard deviation 184.9. You choose an SRS of 90 students and average their SAT scores. If you do this many times, the mean of the average scores you get will be close to
Answer:
By the Central Limit Theorem, the mean of the average scores you get will be close to 1012.5.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean score of the population:
1012.5
If you do this many times, the mean of the average scores you get will be close to
By the Central Limit Theorem, the mean of the average scores you get will be close to 1012.5.
Can someone please help me fill this out?
Given:
The height of the the tree from where the tree was broken = 13 ft
The distance from the foot of the tree to the broken top = 36 ft
To find the height of the tree.
Formula
By Pythagoras theorem we get,
h² = l²+b²,
where, h be the hypotenuse
b be the base and
l be the other side of the triangle along the right angle.
Now,
Putting, l =13 and b = 36 we get,
[tex]h^{2} =36^{2} +13^{2}[/tex]
or, [tex]h^{2} = 1296+169[/tex]
or, [tex]h^{2} =1465[/tex]
or, [tex]h=\sqrt{1465}[/tex]
or, [tex]h = 38.3[/tex]
Therefore,
The height of the tree is about = 13+38.3 ft = 51.3 ft
Hence,
The height of the tree is about 51.3 ft.
mackenzie made 33 out of 50 throws. what is the experimental probability that mackenzie will not make the next free throw she attempts
answer:
1.94117647058
Step-by-step explanation:
33/50 her free throw % is 66% at the moment based on how many she missed you'd do 33 / by 17 you get 1.94117647058
Determine the intercepts of the line.
y = -7x +3
y-intercept:
t-intercept: (
-intercept:
,
)
A quadratic function has an equation of f(x) = a(x + 3)2 – 4 and goes through
the point (-2, 2). What is the a-value of the equation?
Answer:
6
Step-by-step explanation:
Put the given point into the equation and solve for a.
f(-2) = a(-2 +3)² -4 = a -4 = 2
a = 6 . . . . . add 4
The a-value in the equation is 6.
If a pool measures 30 feet by 66 feet, what is the scale of the drawing shown? 1 in =
To determine the scale of a drawing, you need the dimensions on the drawing or map as well as the actual dimensions of the object. The scale is then calculated as the ratio of these two sets of dimensions. However, without the dimensions of the drawing in this case, it is not possible to determine the exact scale.
Explanation:The scale of a drawing is a ratio that compares the dimensions of the physical object to the dimensions on the drawing or map. In this case, the physical object is a pool that measures 30 feet by 66 feet. To find the scale, you would need additional information such as the dimensions of the pool in the drawing.
For example, if the pool is drawn as 2 inches by 4.4 inches on the paper, then the scale of the drawing would be 1 inch = 15 feet (for length) and 1 inch = 15 feet (for width), as 30 feet / 2 inches = 15 feet per inch and 66 feet / 4.4 inches = 15 feet per inch respectively.
Without this additional information, it is not possible to provide the exact scale of the drawing.
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The scale for the drawing shown is 1 in = 6 feets
How to determine the Scale of the drawingUsing the parameters given ;
Actual dimension :
length = 66 feets width = 30 feetsDrawing dimension :
Length = 11 feets width = 5 feetsTo calculate the scale of the drawing : Take the ratio of equivalent sides of the two drawings ;
Actual length / Drawing length = Actual width / drawing widthNow we have ;
Scale = 66 / 11 = 6Hence the scale drawing is :
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How much is 1.7÷.01 (step-by-step)
Answer:
170
Step-by-step explanation:
Hello!
Let's represent a decimal as a fraction. That will help.
Since division is multiplication by the reciprocal, we can represent 0.01 as 1/100 which will be 100 when we divide.
1.7 * 100 = 170.0 (when we move one spot over.)
Thus, we see that [tex]\boxed{170}[/tex] is the answer.
Hope this helps!
find and list the slope and y-intercept of the equation below. -2y=6(2x-2)
[tex]\text{Solve the equation to make it into slope intercept form}\\\\\text{Slope intercept form: y = mx + b}\\\\\text{Solve:}\\\\-2y=6(2x-2)\\\\\text{Distribute the 6 to the variables inside the parenthesis}\\\\-2y=12x-12\\\\\text{Divide both sides by -2}\\\\y=-6x+6\\\\\text{The slope would be -6 and the y intercept would be 6}\\\\\boxed{\text{Slope: -6 y-intercept: 6}}[/tex]
A husband and wife, Ed and Rina, share a digital music player that has a feature that randomly selects which song to play. A total of 3476 songs have been loaded into the player, some by Ed and the rest by Rina. They are interested in determining whether they have each loaded different proportions of songs into the player. Suppose that when the player was in the random-selection mode, 35 of the first 53 songs selected were songs loaded by Rina. Let p denote the proportion of songs that were loaded by Rina.
State the null and alternative hypotheses to be tested. How strong is the evidence that Ed and Rina have each loaded a different proportion of songs into the player? Make sure to check the conditions for the use of this test. (Round your test statistic to two decimal places and your P-value to four decimal places. Assume a 95% confidence level.) Hypotheses:A) H0: p = 0.5 Ha: p < 0.5B) H0: p = 0.5 Ha: p ≠ 0.5C) H0: p = 0.5 Ha: p > 0.5
The correct hypotheses to be tested are:
H0: p = 0.5 (The proportion of songs loaded by Rina is 0.5, meaning Ed and Rina loaded an equal proportion of songs.)
Ha: p = 0.5 (The proportion of songs loaded by Rina is not equal to 0.5, meaning Ed and Rina loaded different proportions of songs.)
To determine how strong the evidence is that Ed and Rina have each loaded a different proportion of songs into the player, we can perform a hypothesis test. The test statistic for a proportion in a binomial setting is given by:
[tex]\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the sample proportion of successes (in this case, the proportion of songs by Rina),
- [tex]\(p_0\)[/tex] is the hypothesized proportion under the null hypothesis (0.5),
- [tex]\(n\)[/tex] is the sample size (53 songs).
Given that 35 out of 53 songs were by Rina, we calculate [tex]\(\hat{p}\)[/tex] as:
[tex]\[ \hat{p} = \frac{35}{53} \approx 0.660 \][/tex]
Now we can calculate the test statistic [tex]\(z\)[/tex] :
[tex]\[ z = \frac{0.660 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{53}}} = \frac{0.160}{\sqrt{\frac{0.25}{53}}} \approx \frac{0.160}{\sqrt{0.004717}} \approx \frac{0.160}{0.0686} \approx 2.33 \][/tex]
The P-value for a two-tailed test is the probability of observing a test statistic as extreme as or more extreme than the one observed, under the assumption that the null hypothesis is true. We can find the P-value by looking up the z-score in a standard normal distribution table or using a calculator:
[tex]\[ P\text{-value} = 2 \times P(Z > |2.33|) \approx 2 \times 0.0099 \approx 0.0198 \][/tex]
Since the P-value (0.0198) is less than the significance level of 0.05, we reject the null hypothesis in favour of the alternative hypothesis. This means there is strong evidence to suggest that Ed and Rina have each loaded a different proportion of songs into the player.
Before we conclude, we must check the conditions for the use of this test:
1. The samples are independent. Each song selection is made independently of the others.
2. The number of successes and failures are each at least 10. In this case, Rina's songs (35) and Ed's songs (18) are both greater than 10, so this condition is satisfied.
3. The sample size is less than 10% of the population size. The population size is 3476 songs, and the sample size is 53 songs, which is much less than 10% of the population, so this condition is also satisfied.
Since all conditions are met, we can confidently conclude that there is strong evidence that Ed and Rina have each loaded a different proportion of songs into the player.
Calibrating a scale:
Making sure that the scales used by businesses in the United States are accurate is the responsibility of the National Institute for Standards and Technology (NIST) in Washington, D.C. Suppose that NIST technicians are testing a scale by using a weight known to weigh exactly 1000 grams. The standard deviation for scale reading is known to be σ = 2.3. They weigh this weight on the scale 49 times and read the result each time. The 49 scale readings have a sample mean of x = 999.0 grams. The calibration point is set too low if the mean scale reading is less than 1000 grams.
1. The technicians want to perform a hypothesis test to determine whether the calibration point is set too low. Use the α = 0.01 level of significance and the P-value method with the TI-84 calculator.
Answer:
We conclude that the calibration point is set too low.
Step-by-step explanation:
We are given that NIST technicians are testing a scale by using a weight known to weigh exactly 1000 grams. The standard deviation for scale reading is known to be σ = 2.3. They weigh this weight on the scale 49 times and read the result each time. The 49 scale readings have a sample mean of x = 999.0 grams.
The calibration point is set too low if the mean scale reading is less than 1000 grams.
Let [tex]\mu[/tex] = mean scale reading
So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu \geq[/tex] 1000 grams {means that the calibration point is not set too low}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 1000 grams {means that the calibration point is set too low}
The test statistics that would be used here One-sample z test statistics as we know about the population standard deviation;
T.S. = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\bar X[/tex] = sample mean = 999 grams
[tex]\sigma[/tex] = population standard deviation = 2.3 grams
n = sample of scale readings = 49
So, test statistics = [tex]\frac{999-1000}{\frac{2.3}{\sqrt{49} } }[/tex]
= -3.04
The value of z test statistics is -3.04.
Now, P-value of the test statistics is given by the following formula;
P-value = P(Z < -3.04) = 1 - P(Z [tex]\leq[/tex] 3.04)
= 1 - 0.99882 = 0.00118
Since, the P-value is less than the level of significance as 0.01 > 0.00118, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.
Therefore, we conclude that the calibration point is set too low.
(−8k+1)(−8k+1) standard form
Answer:
64k^2 - 16k +1
Step-by-step explanation:
We can rewrite this as
(-8k+1) ^2
We know that (a+b)^2 = a^2 +2ab +b^2
Let a = -8k and b = 1
(-8k+1) = (-8k)^2 +2*(-8k)(1) + 1^2
=64k^2 - 16k +1
Answer:
64k² - 16k + 1
Step-by-step explanation:
(−8k+1)(−8k+1)
64k² - 8k - 8k + 1
64k² - 16k + 1
Whats the answer of this question:
Question: 1+1
Answer:
2
Step-by-step explanation:
Answer: 9?
Step-by-step explanation: I think.
4 vd
6 yd
20 yd
Surface Area =