For the line of best fit in the least-squares method is: b) the sum of the squares of the residuals has the least possible value
How to find the line of best fit in regression?The regression line is sometimes called the "line of best fit" because it is the line that best fits when drawn through the points. A line that minimizes the distance between actual and predicted results.
The best-fit straight line is usually given by the following equation:
ŷ = bX + a,
where:
b is the slope of the line
a is the intercept
Now, least squares in regression analysis is simply the process that helps find the curve or line that best fits a set of data points by reducing the sum of squares of the offsets of the data points (residuals). curve.
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The sum of two numbers is 18 and their difference is 6.
What are the two numbers?
Larger number
Smaller number
Answer:
Large number=12
Smaller number=6
Step-by-step explanation:
Let the two numbers be x and y
x+y=18
x-y=6
Which of the following sets of numbers could not represent the three sides of a right
triangle?
Answer:
{48, 64, 81}
Step-by-step explanation:
Drag the operations signs to make the number sentence true. Use each operation sign once. +–×÷ 4 (3 2) 6 1 = 14
Answer: its 4x(3+2)-6÷ 1=14
Step-by-step explanation: in the picture
two points on a parabola are (-3,5) and (11,5) what is the equation of the axis of symmetry
Answer:
I don't know how to do it the subject
An asteroid is heading for the planet Zorgon (diameter of 2600 miles). When it hits, it will create a blast zone that extends 150 miles in all directions from the point of impact. What is the probability that Astronaut Joe, who is on Zorgon currently, will be affected by this impact?
The probability that Astronaut Joe, who is currently on Zorgon, will be affected by the impact is 0.667%.
To determine the probability that Astronaut Joe will be affected by the impact of the asteroid on planet Zorgon, we need to consider the area of the blast zone in relation to the total area of the planet.
An asteroid is heading for the planet Zorgon with a diameter of 2600 miles. When it hits, it will create a blast zone that extends 150 miles in all directions from the point of impact.
What is the probability that Astronaut Joe, who is currently on Zorgon, will be affected by this impact?
The total surface area of Zorgon is given by:2 * 3.14 * (1,300 miles)2 = 10.6 million sq Miles
The blast zone covers an area of:3.14 * (150 miles)2 = 70,685 sq Miles
To calculate the probability that Astronaut Joe will be affected by the impact, we divide the area of the blast zone by the total surface area of the planet.
That is,70,685 / 10.6 million = 0.00667 = 0.667%.
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Margaret's garden is 36 feet long and has a perimeter of feet. She is wanting to plant flowers along the diagonal of the garden. How long in the diagonal of her
Answer: Where is the rest of the question?
Step-by-step explanation:
b) Abigail (A) and Balan (B) want to share a pizza of size 1. Suppose both agents have the same utility function u(x) = 2 for pizza, Abigail discounts with 8A = 1/2 and Balan discounts with 88 1/2. Abi- gail moves first. Calculate the Rubinstein solution of the bargaining problem. c) Why does Abigail get a larger share of the pizza?.
Solution: Given, Abigail (A) and Balan (B) want to share a pizza of size
1. Both agents have the same utility function u(x) = 2 for pizza, Abigail discounts with 8A = 1/2. Balan discounts with 88 1/2 Abigail moves first.
We have to calculate the Rubinstein solution of the bargaining problem. Bargaining Solution Using Rubinstein's alternating offers model, the bargaining solution is:
Take x as the size of the pizza that Abigail gets.
Hence, Balan gets 1 - x.
The possible utility that they get are:
Abigail: 2x(1/2) + 0(1/2) = x
Balan: 2(1 - x)(88 1/2) + 0(11 1/2) = 177 - 177x The minimum utility that they both need to be satisfied is:
minimum value = 2 x 177/2 = 177
The bargaining range is [0,1], thus there are infinite pairs that satisfy the minimum utility requirement. However, Rubinstein assumes that the final solution should be somewhere between both parties' ideal points, so we can restrict the bargaining range to [1/2, 1]. Abigail gets x and Balan gets 1 - x. Now, we have to see what happens if Balan rejects this offer.
When Balan rejects, the bargaining range is [0, x) for Abigail and (x, 1] for Balan. In this range,
Abigail's ideal point is 2x(1/2) + 0(1/2) = x and
Balan's ideal point is 2(1 - x)(88 1/2) + 0(11 1/2) = 177 - 177x.
The bargaining range is again restricted to [1/2, 1]. When Balan rejects, the bargaining range is [0, x) for Abigail and (x, 1] for Balan. In this range, Abigail's ideal point is 2x(1/2) + 0(1/2) = x and Balan's ideal point is 2(1 - x)(88 1/2) + 0(11 1/2) = 177 - 177x.The bargaining range is again restricted to [1/2, x) and (x, 1].
Now, if Abigail rejects, then the bargaining range is (0, x) for Abigail and [x, 1] for Balan. In this range, Abigail's ideal point is 2x(88 1/2) + 0(11 1/2) = 177x and Balan's ideal point is 2(1 - x)(1/2) + 0(1/2) = 1 - x. The bargaining range is again restricted to (1/2, x) and (x, 1/2 + 1/176). In the next step, if Balan rejects, then the bargaining range is [0, x) for Abigail and (x, 1] for Balan. In this range, Abigail's ideal point is 2x(1/2) + 0(1/2) = x and Balan's ideal point is 2(1 - x)(1/2) + 0(1/2) = 1 - x. The bargaining range is again restricted to [1/2, x) and (x, 1/2 + 1/176).
Repeating the steps,
the solution is: x = 2/3, 177 - 177x = 59.
After calculating the Rubinstein solution of the bargaining problem, we can see that Abigail gets 2/3 of the pizza and Balan gets 1/3 of the pizza. There are two reasons why Abigail gets a larger share of the pizza: Abigail moves first, so she has an advantage because she can propose a deal that is more favorable to her. This is why the bargaining range is initially restricted to [1/2, 1]. Abigail has a lower discount rate than Balan, so she is willing to wait longer for a deal. This means that Abigail can drive a harder bargain because she has a higher reservation utility.
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PLEASE HELP I’m bad at math
A.Find the greatest common factor GCF of 42 and 12
B.Use the GCF to factor 42 + 12
Please be quick if you can
Answer:
6
Step-by-step explanation:
The GCF of 42 and 12 is 6
the area of a rhombus is 24 square inches. What is the are of a similar rhombus that is 7 times as big?
Answer:
324
...................
Water treatment plant receives a 5% polymer solution. Calculate how much polymer should be mixed with water to produce 350 gallons of a 0.5% solution.
Answer:
Polymer to be mixed with water to produce 350 gallons of a 0.5% solution is 1.75 gallons.
Step-by-step explanation:
Solution weight-age 100% = 350 gallons
Polymer weight-age = 0.5 % = ?
Water weight-age = 99.5 % = ?
100 =99.5 w + 0.5 p
350 = ? + ?
Using ratios
100 350
0.5 p
Applying the cross product rule
p = 350 *0.5/100= 1.75 gallons
Polymer to be mixed with water to produce 350 gallons of a 0.5% solution is 1.75 gallons
Using ratios
100 350
99.5 w
Applying the cross product rule
w = 350*99.5 /100= 348.25 gallons
Water to be mixed with water to produce 350 gallons of a 0.5% solution is 348.25 gallons
Need help due in 30 mins Find the area of each. Round to the nearest tenth. Do not include units in your answer (ie. ft, in, km, etc)(triangle-FR)
Your answer
Answer:
1)
Triangle formula:
A = 1/2bh
A = 1/2(3)(5.9)
A = 8.85
2)
Parallelogram Area:
A = bh
A = (8.9)(6)
A = 53.4
The AIC strikes a balance between:
The AIC, or the Akaike Information Criterion, strikes a balance between model complexity and goodness of fit.
In statistical modeling, it is crucial to find a balance between the complexity of a model and its ability to accurately capture the underlying patterns in the data. On one hand, a complex model with numerous parameters may be able to fit the data very closely, resulting in a low error or residual.
However, such a model runs the risk of overfitting, meaning it may become too specific to the training data and perform poorly when applied to new, unseen data.
On the other hand, a simpler model with fewer parameters may not capture all the nuances of the data and may have a higher error or residual. This is known as underfitting, as the model fails to capture the underlying complexity of the data.
The AIC addresses this trade-off by considering both the goodness of fit and the complexity of the model. It penalizes models with a higher number of parameters, encouraging a balance between model complexity and goodness of fit.
The AIC takes into account the residual sum of squares (RSS) or the likelihood of the model, and adjusts it based on the number of parameters used. The goal is to select the model with the lowest AIC value, indicating a good compromise between complexity and fit.
By striking this balance, the AIC provides a reliable criterion for model selection, allowing researchers and statisticians to choose the most appropriate model for their data while avoiding both overfitting and underfitting.
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A SHS student conducted a survey to test the claim that "less than half of all the adults are annoyed by the violence on television" . Suppose that from a poll of 2,400 surveyed adults, 1,152 indicated their annoyance with television violence. Test this claim using 0.10 level of significance.
The null hypothesis (H0) assumes that the proportion of adults annoyed by television violence is equal to or greater than 0.5, while the alternative hypothesis (Ha) assumes that the proportion is less than 0.5.
In this case, the sample proportion is calculated as the number of adults indicating annoyance divided by the total sample size: 1,152/2,400 = 0.48.
Next, we can calculate the test statistic, which follows a standard normal distribution under the null hypothesis. The test statistic formula is z = (p - P) / sqrt(P(1-P)/n), where p is the sample proportion, P is the hypothesized proportion under the null hypothesis (0.5 in this case), and n is the sample size.
Using the given values, we can calculate the test statistic:
z = (0.48 - 0.5) / sqrt(0.5(1-0.5)/2400) ≈ -1.67.
Finally, we compare the test statistic to the critical value. At a significance level of 0.10, the critical value for a one-tailed test is approximately -1.28. Since the test statistic (-1.67) is smaller than the critical value, we reject the null hypothesis. This suggests that there is evidence to support the claim that less than half of all adults are annoyed by violence on television.
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The heights of the female adults in a country can be represented by a random variable that follows the normal distribution N(170,30) Answer these questions:
1. To enter the tallest 20% of the female adults, a man must be at least [....] cm tall.
2. To enter the tallest 1% of the female adults, a man must be at least [....] cm tall.
Given the heights of the female adults in a country is represented by a random variable that follows the normal distribution N(170,30)1. To enter the tallest 20% of the female adults, a man must be at least 184.87 cm tall.
Solution:It is given that, the heights of the female adults in a country can be represented by a random variable that follows the normal distribution N(170,30)Let X be the height of female adults, then X ~ N(170, 30)
Let P be the probability of the tallest 20% female adults.To find the value of x we need to use the standard normal distribution formula which is given byz = (x - μ) / σWhere,z = standard score or z-scorex = the raw scoreμ = the meanσ = the standard deviation
Now, the probability of the tallest 20% female adults is P = 0.20 or 20%We know that the total area under the normal curve is 1 which means P(X < μ) = 0.5So, P( X > μ) = 1 - P(X < μ) = 1 - 0.5 = 0.5Therefore, 0.5 = P(Z < z) at z = 0.84 from standard normal distribution table,0.84 = (x - μ) / σOn substituting the values,0.84 = (x - 170) / 30x - 170 = 0.84 x 30x - 170 = 25.2x = 195.2So, to enter the tallest 20% of the female adults, a man must be at least 184.87 cm tall.2.
To enter the tallest 1% of the female adults, a man must be at least 201.17 cm tall.
Solution: It is given that, the heights of the female adults in a country can be represented by a random variable that follows the normal distribution N(170,30)Let X be the height of female adults, then X ~ N(170, 30)
Let P be the probability of the tallest 1% female adults.
Now, the probability of the tallest 1% female adults is P = 0.01 or 1%We know that the total area under the normal curve is 1 which means P(X < μ) = 0.5So, P( X > μ) = 1 - P(X < μ) = 1 - 0.5 = 0.5Therefore, 0.5 = P(Z < z) at z = 2.33 from standard normal distribution table,2.33 = (x - μ) / σOn substituting the values,2.33 = (x - 170) / 30x - 170 = 2.33 x 30x - 170 = 69.9x = 239.9 cm
So, to enter the tallest 1% of the female adults, a man must be at least 201.17 cm tall.
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To enter the tallest 1% of the female adults, a man must be at least 104.1 cm tall.
The heights of female adults in a country can be represented by a random variable that follows the normal distribution N(170,30).
The questions and their solutions are:
1. To enter the tallest 20% of female adults, a man must be at least [ ] cm tall.
To solve this, we can use the standard normal distribution table.
Let Z be the standard normal distribution.
To find the corresponding Z-score to the 20th percentile, we use the standard normal distribution table.
P(Z < z) = 0.20, where P(Z < z) is the area under the standard normal distribution curve to the left of z.
z = -0.84 (rounded to 2 decimal places).
Using the formula z = (X - µ) / σ, we can solve for X, the height of the woman:
[tex]z = (X - µ) / σX = σz + µX = 30(-0.84) + 170X = 147.8[/tex] (rounded to the nearest tenth of a cm)
Therefore, to enter the tallest 20% of the female adults, a man must be at least 147.8 cm tall.
2. To enter the tallest 1% of female adults, a man must be at least [ ] cm tall.
P(Z < z) = 0.01
z = -2.33 (rounded to 2 decimal places).
Using the formula z = (X - µ) / σ, we can solve for X, the height of the woman:
[tex]z = (X - µ) / σX = σz + µX = 30(-2.33) + 170X = 104.1[/tex] (rounded to the nearest tenth of a cm)
Therefore, to enter the tallest 1% of the female adults, a man must be at least 104.1 cm tall.
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46 people are going to the beach. Nine people can ride in each van. How many vans are needed?
Answer:
6
Step-by-step explanation:
No. of people going to beach = 46 .
Using Unitary Method ,
9 people can go in one van .
1 person can go in 1/9
46 people can go in 1/9 * 46 = 5.1
Since vans can't be in fraction , 1 extra van for 1 person is need that is for 46th person .
So total number of vans = 5 + 1 = 6
Assume that a sample is used to estimate a population mean u. Find the margin of error M.E. that corresponds to a sample of size 23 with a mean of 37.6 and a standard deviation of 16.1 at a confidence level of 95%.
Report ME accurate to one decimal place because the sample statistics are presented with this accuracy. M.E. ______
Answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.
Based on the illustration above, the value of margin of error M.E is 6.961
Margin of error (M.E) is calculated as the product of critical value (CV) and standard error (SE) of sample mean.
The formula for standard error of sample mean is:
SE = σ/√n
where σ is the population standard deviation and n is the sample size. The formula for margin of error is:
M.E. = CV x SE
where CV is the critical value.
The critical value for a 95% confidence level with 22 degrees of freedom (sample size 23 - 1) is 2.074 (rounded to 3 decimal places).
The sample mean is 37.6 and the population standard deviation is 16.1.
Sample size, n = 23.
Using the formula,
SE = σ/√n
SE = 16.1/√23
SE = 3.365 (rounded to 3 decimal places)
Now, using the calculated value of SE and CV,
ME = CV x SE
ME = 2.074 × 3.365
ME = 6.961 (rounded to 1 decimal place)
Therefore, the margin of error (M.E.) is 6.961 (rounded to 1 decimal place).
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Five-sixths of the students at a nearby college live in dormitories. If 6000 students at the college live in dormitories, how many students are there in the college?
The number of students that are in college is 7200 if Five-sixths of the students at a nearby college live in dormitories. If 6000 students at the college live in dormitories.
What is a fraction?Fraction number consists of two parts, one is the top of the fraction number which is called the numerator and the second is the bottom of the fraction number which is called the denominator.
It is given that:
Five-sixths of the students at a nearby college live in dormitories. If 6000 students at the college live in dormitories
Let x be the number of students that are in college.
Then from the question:
The value of x can be found as follows:
x = (6000/5)×6
x = (1200)×6
x = 7200
Thus, the number of students that are in college is 7200 if Five-sixths of the students at a nearby college live in dormitories. If 6000 students at the college live in dormitories.
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Plsss helppp ASAP
Will mark brainleist
Answer:
B
Step-by-step explanation:
The average rate of change in the closed interval [ a, b ] is
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
Here [ a , b ] = [ 15, 35 ]
and f(b) = f(35) = 400 ← from graph
f(a) = f(15) = 200 ← from graph
Then average rate of change is
[tex]\frac{400-200}{35-15}[/tex] = [tex]\frac{200}{20}[/tex] = 10 m/ s → B
An arithmetic sequence has first term (a) and the common difference (d). The sum of the first 25 terms is 15 times the sum of the first 4 terms. Find (a)
Answer:
a = 12.
Step-by-step explanation:
Sum of n terms = n/2[2a + d(n-1)]
For 25 terms
S25 = 12.5(2a + 24d)
S25 = 25a + 300d
For 4 terms
S4 = 2(2a + 3d)
So:
S25 = 15*2(2a + 3d)
S25 = 60a + 90d
25a + 300d = 60a + 90d
35a = 210d
a = 6d
Take a to be 12 and d to be 2:
25th term = 12.5(2*12 + 24 * 2) = 900
4th term = 2(24 + 6) = 60
900 = 15 * 60 so a = 12.
What is the greatest number of obtuse angles that a triangle can have?
Answer:
1
Step-by-step explanation:
dba algebra 2 module 2 flvs what's on it
Answer:
Uh i dont really get it
Step-by-step explanation:
best of luck though
What is the answer for 3743x453
Answer:
1695579
Step-by-step explanation:
A pressure vessel has a design pressure of 50 bar. However the safety case for the chemical plant on which it is to be used requires that the pressure vessels have a 95% probability of surviving a pressure of 70 bar. Computer codes have generated an estimate of only 0.80 for the probability that any such pressure vessel, picked at random, will survive at 70 bar. However, they have also calculated that of the 20% of the pressure vessels that will not survive a pressure of 70 bar, 40% will fail under a pressure of 58 bar or less, while 80% will fail under a pressure of 65 bar or less. It is decided that an over-pressure test needs to be used to give reassurance on the behaviour of this particular pressure vessel. This test may be carried out at either 58 bar or 65 bar. The lower pressure test is considerably less difficult and cheaper to administer. (i) Suppose that you are brought in as a consultant. By calculating the probability of the pressure vessel being able to support the 70 bar maximum pressure if the over-pressure test is passed, advise on which over-pressure test should be administered.
It is advised that lower pressure test should be administered.
Probability of survival of pressure vessel the pressure vessel is tested under the lower pressure test at 58 bar, the probability of survival is given by the sum of the probability of survival if the vessel is one of the 60% that will survive 70 bar and the probability of survival if the vessel is one of the 40% that will fail at 70 bar but will survive 58 bar or less. If P1 represents the probability of survival of the vessel if it is one of the 60% that will survive 70 bar, and P2 represents the probability of survival if it is one of the 40% that will fail at 70 bar but will survive 58 bar or less, then the probability of survival of the vessel, if it is tested under the lower pressure test at 58 bar, is given by:
P = 0.60 x 1 + 0.40 x (1 - 0.60) = 0.76
If the pressure vessel is tested under the higher pressure test at 65 bar, the probability of survival is given by the sum of the probability of survival if the vessel is one of the 60% that will survive 70 bar, and the probability of survival if the vessel is one of the 40% that will fail at 70 bar but will survive 65 bar or less. If P3 represents the probability of survival of the vessel if it is one of the 40% that will fail at 70 bar but will survive 65 bar or less, then the probability of survival of the vessel, if it is tested under the higher pressure test at 65 bar, is given by:
P = 0.60 x 1 + 0.40 x (1 - 0.80 x P3)
The condition that the probability of survival of the vessel, if it is tested under the higher pressure test at 65 bar, is at least 0.95 is therefore:
0.60 + 0.40 x (1 - 0.80 x P3) ≥ 0.95
This simplifies to: P3 ≤ 0.625
Using the above values for P1, P2, and P3, it is clear that the probability of the vessel surviving if tested at the lower pressure of 58 bar is greater than 0.95. Therefore, the lower pressure test should be carried out.
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divide 32x3 48x2 − 40x by 8x. 4x2 − 6x 5 4x2 6x − 5 4x3 − 6x2 5 4x3 6x2 − 5
The division of 32x^3 - 48x^2 - 40x by 8x results in the quotient 4x^2 - 6x - 5 on solving the given equation.
To divide 32x^3 - 48x^2 - 40x by 8x, we divide each term of the dividend by the divisor, 8x.
Dividing 32x^3 by 8x gives us 4x^2, as x^3/x = x^2 and 32/8 = 4.
Dividing -48x^2 by 8x gives us -6x, as -48x^2/8x = -6x.
Dividing -40x by 8x gives us -5, as -40x/8x = -5.
Combining these results, the quotient is 4x^2 - 6x - 5.
The quotient represents the result of dividing the dividend by the divisor, resulting in a polynomial expression without any remainder. Therefore, when dividing 32x^3 - 48x^2 - 40x by 8x, the quotient is 4x^2 - 6x - 5.
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I need help with this question can you guys help me
Answer:
$18.5
Step-by-step explanation:
You just add up all the costs and divide them by two and then you would get your answer
If arc QT = (27x + 3) arc RT = (9x – 5) and RST = (102 – 2) find arc RT.
Answer:
Step-by-step explanation:
x = 6
arc RT = 49°
What is tangent?"It is a line that intersects the circle exactly at one point."
What is secant?"It is a line that intersects circle at two points."
For given example,
arc QT = (27x + 3)°
arc RT = (9x – 5)°
∠RST = (10x – 2)°
From figure we can observe that line ST is tangent and line SQ is secant.
∠RST is the angle subtended by tangent ST and secant SQ
We know, the angle subtended by the tangent and the secant is half the difference of the measures of the intercepted arcs.
⇒ ∠RST = (QT - RT)/2
⇒ 10x - 2 = [(27x + 3) - (9x - 5)] /2
⇒ 2(10x - 2) = 27x + 3 - 9x + 5
⇒ 20x - 4 = 18x + 8
⇒ 20x - 18x = 8 + 4
⇒ x = 6
So, arc RT would be,
⇒ 9x - 5 = 9(6) - 5
⇒ 9x - 5 = 49°
Therefore, arc RT = 49°
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Find the profit function if cost and revenue are given by C(x) = 150 +5.8x and R(x) = 9x -0.01x?
The profit function is; P(x) = -0.01·x² + 3.2·x + 150
What is a profit?
Profit is the amount gained following a business transaction, which is the difference between the amount received as payment for doing a business transaction, within a specified period, known as the revenue and the amount amount spent or invested in doing the business, including the fixed and variable expenses, which is the cost of the business.
Therefore; Profit = Revenue - CostThe cost and the revenue functions, obtained from a similar question on the internet are;
The cost function is; C(x) = 150 + 5.8·x
The profit function is; R(x) = 9·x - 0.01·x²
The profit function, P(x), is therefore; P(x) = 9·x - 0.01·x² - (150 + 5.8·x) = -0.01·x² + 3.2·x + 150
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you randomly select 100 drivers ages 16 to 19 from example 4. what is the probability that the mean distance traveled each day is between 19.4 and 22.5 miles?
Given that we randomly select 100 drivers ages 16 to 19 from example 4. We are to determine the probability that the mean distance traveled each day is between 19.4 and 22.5 miles. The probability that the mean distance traveled each day is between 19.4 and 22.5 miles is approximately 1.00.
Probability distribution is a function which represents the probabilities of all possible values of a random variable.
When the probability distribution of a random variable is unknown, we can use the Central Limit Theorem (CLT) to estimate the mean of the population.
Let X be the mean distance traveled each day by the 100 drivers ages 16 to 19.
Then, the distribution of X is approximately normal with the mean μ = 20.4 miles and the standard deviation σ = 3.8 miles.
Therefore, we can calculate the z-score as follows; z = (X - μ) / (σ / √n), where X = 19.4 and n = 100.
z₁ = (19.4 - 20.4) / (3.8 / √100)
z₁ = -2.63 and
z₂ = (22.5 - 20.4) / (3.8 / √100)
z₂ = 5.53
Hence, the probability that the mean distance traveled each day is between 19.4 and 22.5 miles is;
P(19.4 < X < 22.5) = P(z₁ < z < z₂).
Using the z-table, the probability is found to be; P(-2.63 < z < 5.53) ≈ 1.00.
Therefore, the probability that the mean distance traveled each day is between 19.4 and 22.5 miles is approximately 1.00.
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Given v = 4i - j, and w = 3i + 2j, find the angle between v and w. (Type your answer in degrees. Do not round until the final answer. Then round to the nearest tenth as necessary.)
Given [tex]v=4i-j[/tex], and [tex]w=3i+2j\\[/tex], the angle between v and w is 47.7°.
To find the angle between vectors v and w, we can use the dot product formula:
v · w = |v| |w| cos(θ)
where v · w represents the dot product of v and w, |v| and |w| represent the magnitudes of vectors v and w, and θ represents the angle between the vectors.
First, let's calculate the magnitudes of vectors v and w:
|v| = √(4² + (-1)²) = √(16 + 1) = √17
|w| = √(3² + 2²) = √(9 + 4) = √13
Next, let's calculate the dot product of v and w:
v · w = (4)(3) + (-1)(2) = 12 - 2 = 10
Now, we can substitute the values into the dot product formula to find the angle θ:
10 = (√17)(√13) cos(θ)
cos(θ) = 10 / (√17)(√13)
cos(θ) = 10 / (√(17 * 13))
cos(θ) = 10 / (√221)
θ = cos⁻¹ (0.6717)
θ = 47.7°.
Therefore, the angle between vectors v and w is approximately 47.7° .
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