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Related Questions
PLEASE HELP ALGEBRA!!
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Answer:
The very first one is decay and the rest are growth
Step-by-step explanation:
Im SO sorry if i got it wrong
I REALLY hope this helped
Best of luck
What figure is a dilation of Figure A by a factor of 3?
Please help :)
Answers
Answer:
36×18×18×27
Step-by-step explanation:
Assuming the picture is Figure A you would multiply value from figure A by 3 to get corresponding value for dilated figure.
So if figure A is
12 × 6 × 6 × 9
the dilated figure would be
36 × 18 × 18 × 27
Which of the following expressions are equivalent to -9.7. -0.8. -7.8. -3.8? Choose all answers that apply:
A. 9.7.-0.8.-7.8.-3.8.
B. -9.7.0.8. 7.8.-3.8
C. None of the above
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Answer:
A.
it’s a because i just got it right on the test
Answer:
its A cause i got it right
Step-by-step explanation:
for the boys
Write an equation in slope intercept form that represents the line shown?
Answers
Answer:
I think the answer would be Y=-2X+5
Hope it helps you *^*
Poease help! Thank you
Answers
Answer:
28 and 12t
Step-by-step explanation:
4 x 7
4 x 3t
Answer:
28+12t
Step-by-step explanation:
Simplify the expression :)
btw you spelled please wrong
PLEASE HELP ONLY HAVE OME HOUR TO COMPLETE!!!!
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Answer:
Step-by-step explanation:
To be a function x can't repeat with a different y so in a every element of x is mapped to a different y so it is a function.
For choice B, -4 is paired with both 7 and 8 so it is not a function.
Choice C is a function since each x is paired with a different y.
Reflex angle of 52 degrees
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A study of students taking a 20 question exam ranked their progress from one testing period to the next Students scoring 0 to 5 form group 1, those scoring 6 to 10 form group 2, those scoring 11 to 15 form group 3 and those scoring 15 to 20 form group 4 The transition matrix to the right shows the result Use the transition matrix to complete parts (a) and (b) below 0.325 0.05 021 0 415 0311 0.351 0.194 0.144 0 0.327 0.485 0 188 (a) Find the long-range prediction for the proportion of the students in each group The proportion of students will be ]% in group 1, 3% in group 2. % in group 3 and % in group 4. (Type integers or decimals rounded to two decimal places as needed) (b) Suppose all students are initially in group 1 When a student reaches group 4 the student is said to have mastered the ma student stays in that group forever. Find the number of testing periods you would expect for at least 70% of the students to haw increasing values of nin xoP") The number of testing periods is__.
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The long-range prediction for the proportion of students in each group is approximately 14.2% in Group 1, 20.6% in Group 2, 34.5% in Group 3, and 30.7% in Group 4. The number of testing periods required for at least 70% of the students to have increasing values of n cannot be determined without specific values.
To find the long-range prediction for the proportion of students in each group, we need to calculate the steady-state vector of the transition matrix. The steady-state vector represents the long-term proportions of students in each group. We can find this vector by solving the equation:
π * P = π
where π is the steady-state vector and P is the transition matrix.
The given transition matrix is:
Copy code
0.325 0.05 0.021 0
0.415 0.311 0.351 0.194
0.144 0 0.327 0.485
0 0.188 0 0.812
Let's solve for the steady-state vector using matrix calculations. We can represent the steady-state vector as π = [π1, π2, π3, π4]. Therefore, the equation can be rewritten as:
[π1, π2, π3, π4] * P = [π1, π2, π3, π4]
This gives us the following system of equations:
π1 * 0.325 + π2 * 0.415 + π3 * 0.144 = π1
π1 * 0.05 + π2 * 0.311 + π3 * 0.188 + π4 * 0.188 = π2
π1 * 0.021 + π2 * 0.351 + π3 * 0.327 = π3
π2 * 0.194 + π3 * 0.485 + π4 * 0.812 = π4
We also know that the sum of the probabilities in the steady-state vector must be 1:
π1 + π2 + π3 + π4 = 1
Now we can solve this system of equations to find the steady-state vector.
Copy code
0.325π1 + 0.415π2 + 0.144π3 = π1
0.05π1 + 0.311π2 + 0.188π3 + 0.188π4 = π2
0.021π1 + 0.351π2 + 0.327π3 = π3
0.194π2 + 0.485π3 + 0.812π4 = π4
π1 + π2 + π3 + π4 = 1
Solving this system of equations, we find:
π1 ≈ 0.142
π2 ≈ 0.206
π3 ≈ 0.345
π4 ≈ 0.307
Therefore, the long-range prediction for the proportion of students in each group is approximately:
Group 1: 14.2%
Group 2: 20.6%
Group 3: 34.5%
Group 4: 30.7%
Now, let's move to part (b) of the question.
If all students are initially in Group 1, we need to determine the number of testing periods required for at least 70% of the students to have increasing values of n.
To calculate this, we can repeatedly multiply the initial state vector [1, 0, 0, 0] by the transition matrix until at least 70% of the students are in Group 4. We'll keep track of the number of testing periods until this condition is met.
less
Copy code
Initial state vector: [1, 0, 0, 0]
Transition matrix:
[0.325, 0.05, 0.021, 0]
[0.415, 0.311, 0.351, 0.194]
[0.144, 0, 0.327, 0.485]
[0, 0.188, 0, 0.812]
Starting with the initial state vector, we can calculate the new state vector as follows:
State vector after 1 period: [1, 0, 0, 0] * Transition matrix
State vector after 2 periods: [1, 0, 0, 0] * Transition matrix * Transition matrix
State vector after 3 periods: [1, 0, 0, 0] * Transition matrix * Transition matrix * Transition matrix
and so on...
We will continue this calculation until the proportion of students in Group 4 is at least 70%. The number of periods it takes to reach this point will be the answer.
Please note that the calculation involves matrix multiplication, which may require computational resources. If you need a specific number of testing periods, let me know and I can perform the calculation for you.
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PLEASE HELP ME WILL MARK BRAINLIEST !!! :)
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Answer:
lateral area = side area = (10 x 6 x 2) + (14 x 10 x 2) = 400 in²
total surface area = lateral area + top & bottom area = 400 + (14 x 6 x 2) = 568 in²
What is the vertex of the parabola? f(x) = 2x² + 16x + 30
Answers
Answer:
y=2(x−4)²+3
Step-by-step explanation:
A researcher did not reject her null hypothesis, but wrote that, because she had a small sample, she thought she had made a Type 1 error. What is the correct assessment of what the researcher wrote? O She definitely made a Type 1 error. O She could not have made a Type 1 error. O She could be right about making the Type 1 error, but there is no way of knowing for sure. O There's a slight chance that she made a Type 1 error. Question 34 1 pts A study was conducted that compared the mean motor competence of a random sample of 41 left- handed preschool children with the motor competence of a random sample of 41 right-handed preschool children relationship between handedness (left or right) and motor competence in preschool children. How many degrees of freedom should there be for an appropriate t test for this study? O 82 O 40 80 O 41 Question 26 1 pts If we take a sample from a population with a standard deviation equal to sigma, how will the standard error of the mean be affected if we decide to increase the sample size? O It changes unpredicatably. O It stays the same. It decreases. O It increases. Question 25 1 pts A researcher plans to compute a confidence interval for the population mean body mass index. What will make the confidence interval narrower? O studying a population with larger variance in body mass index O increasing the confidence level O being careless in measuring body mass index O increasing the sample size 1 pts Question 24 When statistical power for hypothesis testing is lower than it should be, what does that mean for estimation with confidence intervals? O The confidence interval will be narrower. O The lower confidence limit and upper confidence limit will be raised. O The lower confidence limit will be raised and the upper confidence limit will be lowered. O The confidence interval will be narrower. Question 1 1 pts Dr. Smith draws a random sample of size 50 from a known population. Dr. Jones draws another random sample of size 50 from the same population. They both measure, among other things, serum cholesterol levels for their studies. Which of the following descriptions of their sample means for serum cholesterol is consistent with central limit theorem? O It's more probable that the means will be far apart than close together. OSmith and Jones will probably come up with the same mean. O It's more probable that the means will be close together than far apart. O It is equally probable for the two means to be far apart as it is for them to be close together. 1 pts Question 2 For two-tailed t tests, as the computed value of the test statistic (for example, Student's t) gets closer to the rare zone of the sampling distribution, what happens to the p value? O It increases toward the left tail and decreases toward the right tail. O It remains unchanged. O It decreases. O It increases. 1 pts Question 3 If the alternate hypothesis is justifiably directional (rather than non-directional), what should the researcher do when conducting a t test? O a one-tailed test a two-tailed test set the power to equal B O set ß to be less than the significance level Question 6 What is the term for rejecting a null hypothesis that is actually true? O Type 1 error O precision Type 2 error O correct decision
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The researcher wrote that, because she had a small sample, she thought she had made a Type 1 error.
The correct assessment of what the researcher wrote is: She could be right about making the Type 1 error, but there is no way of knowing for sure. Here's why:In statistics, a Type I error occurs when a null hypothesis that is true is incorrectly rejected. The probability of a Type I error occurring is referred to as the level of significance.
If a researcher states that she did not reject her null hypothesis but believes she may have made a Type I error due to a small sample size, it is possible that she is correct. However, since she did not reject the null hypothesis, it is impossible to know for sure whether a Type I error occurred. Hence, the correct assessment of what the researcher wrote is: She could be right about making the Type 1 error, but there is no way of knowing for sure.
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Consider the first order differential equation t et y'+ = , y' + t2 – 25 y t-99 For each of the initial conditions below, determine the largest interval a
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For the given first-order differential equation, we need to determine the largest interval on which a unique solution exists for each initial condition. The interval will depend on the specific initial condition and the behavior of the differential equation.
The first-order differential equation is given as:
t^et y' + y' + t^2 – 25yt - 99
To determine the largest interval on which a unique solution exists for each initial condition, we need to consider the behavior of the equation and any possible singularities or discontinuities.
For each initial condition, we can use standard techniques such as separation of variables or integrating factors to solve the differential equation and find the solution. The solution will depend on the initial condition and may have different behaviors based on the values of t and y.
It's important to note that the existence and uniqueness of solutions are generally guaranteed within a certain interval as long as the equation and initial condition satisfy certain conditions, such as Lipschitz continuity. However, without specific initial conditions, it is not possible to determine the exact intervals on which a unique solution exists.
Therefore, to determine the largest interval on which a unique solution exists for each initial condition, further analysis and specific initial conditions are required to assess the behavior of the equation and identify any constraints or limitations on the solution.
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There are 25 people competing in a race. In how many ways can they finish in first and second
place?
49
400
600
625
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Answer:
C) 600-----------------------
The number of ways to choose the first place finisher is 25, since any of the 25 people can win.
After the first place finisher is determined, there are 24 people left who can finish in second place.
Therefore, the total number of ways is:
25 × 24 = 600Correct choice is C.
Let X be a continuous random variable with probability density function f. We say that X is symmetric about a if for all x,
P(X ≥ a+x)=P(X ≤ a-x).
(a) Prove that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).
(b) Show that X is symmetric about a if and only if f(x) = f(2a - x) for all x.
(c) Let X be a continuous random variable with probability density function
f(x) = [1 / √(2phi)] e^-(x-3)²/2, x E R,
and Y be a continuous random variable with probability density function
g(x) = 1 / phi [1 + (x-1)²], x E R.
Find the points about which X and Y are symmetric.
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Let X be a continuous random variable with probability density function f. We say that X is symmetric about a if for all x,
P(X ≥ a+x)=P(X ≤ a-x).
(a) f(a - x) = f(a + x) if and only if X is symmetric about a.
(b) X is symmetric about a if and only if f(x) = f(2a - x) for all x.
(c) X and Y are symmetric about 3.
(a) To prove that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).
Proof: P(X ≥ a+x) = P(X ≤ a-x) ...(1)
Given X is a continuous random variable with probability density function f.
Let F denote the cumulative distribution function of X.
Then, F(x) = P(X ≤ x).
We can now re-write equation (1) as follows: 1-F(a+x) = F(a-x) ... (2).
Taking the derivative of both sides of equation (2) with respect to x, we get: d/dx(1-F(a+x))= d/dx(F(a-x)) ... (3).
Differentiating the LHS of equation (3) using the chain rule, we obtain:- f(a+x) = -d/dx(F(a+x)) ... (4).
Differentiating the RHS of equation (3) using the chain rule, we obtain: f(a-x) = d/dx(F(a-x)) ... (5).
Combining equations (4) and (5), we get: f(a+x) = f(a-x).
Hence, we can conclude that X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).
Answer: (a) f(a - x) = f(a + x) if and only if X is symmetric about a.
(b) To show that X is symmetric about a if and only if f(x) = f(2a - x) for all x.
Proof: X is symmetric about a if and only if for all x, we have f(a - x) = f(a + x).
Hence, it follows that: f(a + (2a - x)) = f(a - (2a - x)) ... (6).
Simplifying equation (6), we obtain: f(2a - x) = f(x).
Therefore, X is symmetric about a if and only if f(x) = f(2a - x) for all x.
Answer: (b) X is symmetric about a if and only if f(x) = f(2a - x) for all x.
(c) Let X be a continuous random variable with probability density function f(x) = [1 / √(2phi)] e^-(x-3)²/2, x E R, and Y be a continuous random variable with probability density function g(x) = 1 / phi [1 + (x-1)²], x E R.
Find the points about which X and Y are symmetric.
The probability density function of a symmetric random variable X about a is f(x) = f(2a - x).
Therefore, if X is symmetric about a, then we have: f(x) = f(2a - x) ...(7).
Comparing the probability density function of X to the given probability density function f(x), we can observe that X is symmetric about a = 3.
Therefore, we can find the points about which X and Y are symmetric by solving the following equation: g(x) = f(2a - x) ... (8).
Substituting the value of a in equation (8), we get:
f(2a - x) = [1 / √(2phi)] e^-(2a-x-3)²/2
= [1 / √(2phi)] e^-(x-3)²/2
= f(x)
Therefore, X and Y are symmetric about 3.
Answer: (c) X and Y are symmetric about 3.
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Iodine-131 has a half-life of days. How much would be left of an original g sample after days?
Answers
Answer:
I will suppose that the actual question is:
Iodine-131 has a half-life of 8 days. How much would be left of an original g sample after x days?
Ok, a half-life means that after that time, the mass of the original sample is reduced to half.
So if we start it a quantity g of iodine-131, after 8 days, we will have g/2.
Also, remember that the decay is written as an exponential decay, then we will have:
A(x) = g*(r)^x
where:
A is the amount of the sample after x days, g is the initial amount of the material (such that A(0) = g) and r is the rate of decay.
We know that:
A(8) = g/2 = g*(r)^8
Now we can solve this for r:
g/2 = g*(r)^8
1/2 = r^8
(1/2)^(1/8) = r = 0.917
Then the amount of material after x days is given by:
A(x) = g*(0.917)^x
A flagpole casts a 12ft long shadow and the sun is currently at an angle of elevation of 53°. How tall is the flagpole?
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Aerin's friend Fritz gives her a riddle to solve about the ages of the people in his family. There are 5 people in Fritz's family: Mom, Dad, his sisters Adele and Erika, and Fritz. Here are the clues to the puzzle. 1) Mom is 8 years younger than Dad. 2) Dad is 2 times as old as Fritz. 3) Adele is 4 years old. 4) Fritz's age plus Adele's age equals Erika's age. 5) Mom was 24 when Erika was born. 6) In 9 years, Mom will be twice as old as Erika will be. All the ages are whole numbers. Aerin decides to use variable for the unknown ages and write equations to express the information. M
Answers
Answer:
Dad = 47, Mom = 39, Fritz = 11, Erika = 15 and Adele = 4
Step-by-step explanation:
Let M = Mom's age, D = Dad's age, F = Fritz's age, A = Adele's age and E = Erika's age.
Given that
1) Mom is 8 years younger than Dad, we have D = M + 8 (1)
2) Dad is 2 times as old as Fritz. D = 2F (2)
3) Adele is 4 years old. A = 4 (3)
4) Fritz's age plus Adele's age equals Erika's age. F + A = E (4)
5) Mom was 24 when Erika was born. M = E + 24 (5)
6) In 9 years, Mom will be twice as old as Erika will be. M + 9 = 2(E + 9) (6)
Substituting (5) into (6), we have
E + 24 + 9 = 2(E + 9)
E + 33 = 2E + 18
subtracting E from both sides, we have
2E - E + 18 = 33 + E - E
E + 18 = 33
subtracting 18 from both sides, we have
E + 18 - 18 = 33 - 18
E = 15
M = 15 + 24 = 15 + 24 = 39
From (4) F = E - A = 15 - 4 = 11
From (1) D = M + 8 = 39 + 8 = 47
So, their ages are Dad = 47, Mom = 39, Fritz = 11, Erika = 15 and Adele = 4
2, 3, 1, 6, 4, 5, 3, 2, 3, 4 is the set
Answers
Answer: A
because It has 1 one 2 twos 3 threes 2 fours 1 five And 1 six
In a chemical reaction, 20 units of a compound are injected into a reaction chamber every 30 min. Within that 30 min, 50% of the compound is used up in the chemical process. Suppose that the reaction starts at t = 0 with 20 units of the chemical in the chamber.
a) Make a table of values showing the amount of the compound remaining for the first 5 h, in 30-min intervals, that the reaction has been occurring.
b) Write the amount of the chemical remaining after each 30-min interval as a sequence.
c) Determine a recursion formula for the sequence.
Answers
The compound will be completely used up in the reaction chamber after 60 minutes.
When will the compound be completely used up in the reaction chamber.In the given chemical reaction, 20 units of a compound are injected into the reaction chamber every 30 minutes. Within that 30-minute period, 50% of the compound is used up in the chemical process. This means that after 30 minutes, half of the compound has reacted and only 10 units remain in the chamber.
After another 30 minutes, another 20 units are injected, making a total of 30 units in the chamber. However, within this 30-minute period, 50% of the compound is again used up. This results in 15 units being consumed, leaving only 15 units in the chamber.
Following this pattern, we can see that after each 30-minute interval, the number of units remaining in the chamber is halved. Starting with 20 units, after the first 30 minutes, we have 10 units, and after the second 30 minutes, we have 5 units.
Therefore, it can be inferred that after 60 minutes (two 30-minute intervals), the compound will be completely used up in the reaction chamber. No units of the compound will be left.
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please don't put any links or I'll report you:)
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Answer:
Hey I have the answer in a link
Look down For link!
Cmon half way there
jkjk
First part (tip amount)
$6.26
Second part (Total bill)
$41.04
Step-by-step explanation:
Well to find the percentage of a number, convert the percent into a decimal which is basically moving the decimal place 2 places to the left. THen you multiply that amount by the original number an din this case you would get 6.2604. You round that to the nearest penny (or hundreths) and you;d get $6.26. Thats the first part
The second part is simple, you add the tip with the total bill and you would get $41.04.
The volume of the entire figure
Answers
Answer:
Step-by-step explanation:
Lets break the 2 boxes apart so you have 4*4*3 =48 and then you have 10*3*2 = 60
60 +48 = 108
108cm^3
Suppose you have four possible predictor variables X,X,X, and X, that could be used in a regression analysis. You run a forward selection procedure, and the variables are entered as follows: Step 1: X Step 2: X. Step 3: x Step 4: X, In other words, after Step 1, the model is E(Y)= B. + B,X,. After Step 2, the model is E(Y)= B. + B,X: + B.X.. And so on. 1) IT) Explain how the variable in step 3 will be entered into the model. (2) The final model has all the four independent variables entered in the given order, does this mean that all the entered variables are significant? Give a reason for your answer.
Answers
(a) In step 3, the variable X will be entered into the model.
(b) The inclusion of all four variables in the final model does not guarantee their significance; further analysis is needed to determine their individual significance.
In step 3 of the forward selection procedure, the variable X will be entered into the model. This means that after step 2, the model includes variables X and X, and in step 3, the variable X is added.
No, the fact that all four independent variables are entered in the final model does not necessarily mean that all of them are significant. The forward selection procedure is a stepwise approach that adds variables to the model based on certain criteria, typically using a significance level or a criterion such as the increase in the adjusted R-squared.
However, the final model with all four variables entered does indicate that these variables have met the criteria for inclusion in the model based on the stepwise procedure. It suggests that each variable contributes to the prediction of the dependent variable Y, after accounting for the variables that were already in the model.
To determine the significance of each variable in the final model, further statistical analysis, such as hypothesis testing or examining the p-values of the coefficients, is required. These tests can assess the individual significance of each variable and help determine if they have a statistically significant relationship with the dependent variable.
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Can I still use this say yes or no and and use the hearts so I can know if I can the white thing Brooke off
Answers
Answer:
you can but dont touch anything inside i did when i was a kid and electrocuted myself.
Step-by-step explanation:
careful. i dont recomend using it though
Answer:
that happed to me i still use the outlet sometime but i wouldn't take any chances
Step-by-step explanation:
Look at photo for the question and answer choices... NO LINKS OR BLANK ANSWERS
this is the last needed question for now!
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The meaurment of the ∠D= 40°, ∠A=140°, ∠B=130.
Given in the image we can see ∠C and ∠ D is an acute angle and the value of ∠C is 50° so ∠D must be 40° according to the image.
The sum of the angle on the same side of trapezoid is equal to 180°. so ∠A+D and ∠C+∠B= 180°. ∠A+40°=180°. after substracting the value ∠A will be 140° and by same method ∠B+50°=180°. we will get ∠B=130°.
Therefore ∠D= 40°, ∠A= 140°, ∠B= 130°.
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21400
Based on the Pythagorean theorem, select all of the following statements that must be true
Answers
Answer:
The 1st and 4th statements are true.
Which of the following statements about the graph of y = 3x - 5 are true? Select all that apply.
A.
The y-intercept is -5.
B. The x-intercept is 3.
C. (2, 1) is a point on the graph.
D. As the x-values increase, the y-values also increase.
Answers
The y-intercept is -5 and as the x-values increase, the y-values also increase in this graph of the line. which is the correct answer would be options (A) and (D).
What is a graph?
A graph can be defined as a pictorial representation or a diagram that represents data or values.
What is the equation of a line?
The general equation of a line is y = mx + c
where m is the slope of the line and c is the intercept.
A linear equation is defined as an equation in which the highest power of the variable is always one.
We have been given that function of the line as
y = 3x - 5
We need to determine the y-intercept.
The y-intercept is at x = 0, y = -5
And the x-intercept is at y = 0, x = 5/3
Here In this graph of the given function as the x-values increase, the y-values also increase.
Therefore, the correct answer would be options (A) and (D).
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A local U-Move moving truck rental company provides the following probability distribution regarding the number of rental trucks that will be rented in a given week. Find the number of rental trucks the company can expect to rent during a given week.
Number of Rented Trucks Probability
0 0.23
1 0.18
4 0.27
5 0.32
a) 2.6800
b) 2.8600
c) 0.6700
d) 2.3100
e) 0.7150
f) None of the above.
Answers
Option (b) 2.8600 is the correct answer.
To find the number of rental trucks that a local U-Move moving truck rental company can expect to rent during a given week, we need to find the expected value of the probability distribution.
The expected value of a probability distribution is given by:
Expected Value = Sum of (Number of Rented Trucks × Probability)
Therefore, Expected Value = (0 × 0.23) + (1 × 0.18) + (4 × 0.27) + (5 × 0.32)
Expected Value = 0 + 0.18 + 1.08 + 1.6Expected Value = 2.86
Therefore, the company can expect to rent 2.86 rental trucks during a given week. Option (b) 2.8600 is the correct answer.
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What does the y-intercept of the graph represent?
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Can someone plz help me
Answers
Answer:
A = 139.25 cm^2
Step-by-step explanation:
The composite shape is made up of a square with side 10 cm and a semicircle with diameter 10 cm. The area of the composite shape is the sum of the two areas. The diameter of the semicircle is the side of the square, and the radius of the circle is half of the diameter.
A = s^2 + (1/2)(pi)r^2
r = d/2 = s/2
A = (10 cm)^2 + (1/2)(3.14)(10 cm/2)^2
A = 100 cm^2 + (1/2)(3.14)(25 cm^2)
A = 139.25 cm^2
Set up fitting the least squares line through the points (1, 1), (2, 1), and (3, 3). Find R of the fitted line.
Answers
The coefficient of determination (R²) for the fitted least squares line is 0.75.
To fit the least squares line through the given points and find the coefficient of determination (R²), we can follow these steps:
Let's perform these calculations:
Step 1: Calculate the mean values of x and y.
x' = (1 + 2 + 3) / 3 = 2
y' = (1 + 1 + 3) / 3 = 5/3 ≈ 1.6667
Step 2: Calculate the sums of squares: SSxx, SSyy, and SSxy.
SSxx = Σ((xi - x')²) = (1 - 2)² + (2 - 2)² + (3 - 2)² = 2
SSyy = Σ((yi - y')²) = (1 - 5/3)² + (1 - 5/3)² + (3 - 5/3)² = 8/3 ≈ 2.6667
SSxy = Σ((xi - x')(yi - y')) = (1 - 2)(1 - 5/3) + (2 - 2)(1 - 5/3) + (3 - 2)(3 - 5/3) = 4/3 ≈ 1.3333
Step 3: Calculate the slope (m) and y-intercept (b) of the least squares line.
m = SSxy / SSxx = 1.3333 / 2 = 2/3 ≈ 0.6667
b = y' - mx' = 5/3 - (2/3)(2) = 5/3 - 4/3 = 1/3 ≈ 0.3333
Therefore, the equation of the least squares line is y = 0.6667x + 0.3333.
Step 4: Calculate the predicted y-values (y_pred) using the least squares line equation.
For (1, 1):
y_pred = 0.6667 × 1 + 0.3333 = 0.6667 + 0.3333 = 1
For (2, 1):
y_pred = 0.6667 × 2 + 0.3333 = 1.3334 + 0.3333 ≈ 1.6667
For (3, 3):
y_pred = 0.6667 × 3 + 0.3333 = 2 + 0.3333 ≈ 2.3333
The predicted y-values are (1, 1), (2, 1.6667), and (3, 2.3333).
Step 5: Calculate the residual sum of squares (RSS) and the total sum of squares (TSS).
RSS = Σ((yi - y_pred)²) = (1 - 1)² + (1 - 1.6667)² + (3 - 2.3333)² ≈ 0.6667
TSS = SSyy = 8/3 ≈ 2.6667
Step 6: Calculate the coefficient of determination (R²) using the formula: R² = 1 - (RSS / TSS).
R² = 1 - (0.6667 / 2.6667) = 1 - 0.25 = 0.75
Therefore, the coefficient of determination (R²) for the fitted least squares line is 0.75.
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