Consider the following regression model: Yi = αXi + Ui , i = 1, .., n (2)

The error terms Ui are independently and identically distributed with E[Ui |X] = 0 and V[Ui |X] = σ^2 .

1. Write down the objective function of the method of least squares.

2. Write down the first order condition and derive the OLS estimator αˆ.

Suppose model (2) is estimated, although the (true) population regression model corresponds to: Yi = β0 + β1Xi + Ui , i = 1, .., n with β0 different to 0.

3. Derive the expectation of αˆ, E[ˆα], as a function of β0, β1 and Xi . Is αˆ an unbiased estimator for β1? [Hint: Derive first E[ˆα|X].]

4. Derive the conditional variance of αˆ, V[ˆα|X], as a function of σ^2 and Xi .

An electric vehicle has the following parameter values:

m_v = 800kg

C_D =0.2

A_f = 2.2 m²

ρ = 1.18 kg/m

f r = 0.008 + 0.6×10^-6×v² (v: vehicle speed in m/s)

The vehicle is on level road. It accelerates from 0 to 100 km/h in 10s such that its velocity profile is given by v (t) =0.29055t² for 0 < t < 10s. (The mass factor is assumed unit)

a. Define the traction force expression F_t

b. Sketch F_t versus time

c. Define the instantaneous traction power expression

d. Calculate the energy consumed during the acceleration (0 < t < 10s)

e. Calculate the energy lost for non-conservative forces (wind and rolling resistance)

f. Find the change in kinetic energy and the change in tractive energy during acceleration.

Partial Differential Equations

(a) Find the general solution to the given partial differential equation and (b) use it to find the solution satisfying the given initial data.

Exercise 1. 2∂u ∂x − ∂u ∂y = (x + y)u

u(x, x) = e −x 2

Exercise 2. ∂u ∂x = −(2x + y) ∂u ∂y

u(0, y) = 1 + y 2

Exercise 3. y ∂u ∂x + x ∂u ∂y = 0

u(x, 0) = x 4

Exercise 4. ∂u ∂x + 2y ∂u ∂y = e −x − u

u(0, y) = arctan y

Exercise 5. ∂u ∂x+v ∂u ∂y = −ru

(here r and v 6= 0 are constants) u(x, 0) = sin x x

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Instructions:

Watch the following videos, then answer ONE of the questions below: (150 WORDS)

https://www.youtube.com/watch?time_continue=15&v=9Os7LDOOJao

https://www.youtube.com/watch?time_continue=1&v=BzWWL2LXoNk

Discussion Questions:

How was America affected by the Cold War?

Why did the policy of “containment” develop and what were its goals?

How did the fear of Communism and the “red scare” affect American society during the 1950s?

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 40 of his hens to each of the three treatments.

H₀:

At least one group has a different mean cholesterol level.

All groups have different mean cholesterol levels.    

All groups have the same mean cholesterol level.

The all-vegetarian diet group has a higher mean cholesterol level.

The all-vegetarian diet group has a lower mean cholesterol level.

H_a:

At least one group has a different mean cholesterol level.

The all-vegetarian diet group has a lower mean cholesterol level.    

The all-vegetarian diet group has a higher mean cholesterol level.

All groups have the same mean cholesterol level.

All groups have different mean cholesterol levels.

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 97 responses, of which 30 were from students who attend varsity football or basketball games only, 18 were from students who also attend other varsity competitions, and 49 who did not attend any varsity games.

H₀:

At least one group has a different mean rating.

The group of students who do not attend games has a higher mean rating.

     All groups have the same mean rating. All groups have different mean ratings.

The group of students who do not attend games has a lower mean rating.

H_a:

The group of students who do not attend games has a lower mean rating. The group of students who do not attend games has a higher mean rating.     All groups have the same mean rating. At least one group has a different mean rating. All groups have different mean ratings.

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 45 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

H₀:

At least one group has a different mean quiz score.

All groups have different mean quiz scores.

The group taught by the oldest TA has a lower mean quiz score.

All groups have the same mean quiz score.

The group taught by the oldest TA has a higher mean quiz score.

H_a:

At least one group has a different mean quiz score.

The group taught by the oldest TA has a higher mean quiz score.    

All groups have the same mean quiz score

. All groups have different mean quiz scores.

The group taught by the oldest TA has a lower mean quiz score.

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 30 of his hens to each of the three treatments.

H₀:

The all-vegetarian diet group has a lower mean cholesterol level.

The all-vegetarian diet group has a higher mean cholesterol level.   

At least one group has a different mean cholesterol level.

All groups have the same mean cholesterol level.

All groups have different mean cholesterol levels.

H_a:

The all-vegetarian diet group has a lower mean cholesterol level.

The all-vegetarian diet group has a higher mean cholesterol level.   

All groups have different mean cholesterol levels.

All groups have the same mean cholesterol level.

At least one group has a different mean cholesterol level.

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 95 responses, of which 31 were from students who attend varsity football or basketball games only, 17 were from students who also attend other varsity competitions, and 47 who did not attend any varsity games.

H₀:

The group of students who do not attend games has a higher mean rating.

The group of students who do not attend games has a lower mean rating.   

All groups have different mean ratings.

All groups have the same mean rating.

At least one group has a different mean rating.

H_a:

The group of students who do not attend games has a higher mean rating.

The group of students who do not attend games has a lower mean rating.   

All groups have different mean ratings.

At least one group has a different mean rating.

All groups have the same mean rating.

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 42 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

H₀:

At least one group has a different mean quiz score.

All groups have the same mean quiz score.   

The group taught by the oldest TA has a lower mean quiz score.

All groups have different mean quiz scores.

The group taught by the oldest TA has a higher mean quiz score.

H_a:

At least one group has a different mean quiz score.

The group taught by the oldest TA has a higher mean quiz score.   

The group taught by the oldest TA has a lower mean quiz score.

All groups have different mean quiz scores.

All groups have the same mean quiz score.

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 35 of his hens to each of the three treatments.

H₀:

All groups have the same mean cholesterol level.

At least one group has a different mean cholesterol level.

The all-vegetarian diet group has a higher mean cholesterol level.

All groups have different mean cholesterol levels.

The all-vegetarian diet group has a lower mean cholesterol level.

H_a:

The all-vegetarian diet group has a lower mean cholesterol level.

At least one group has a different mean cholesterol level.

All groups have different mean cholesterol levels.

The all-vegetarian diet group has a higher mean cholesterol level.

All groups have the same mean cholesterol level.

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 97 responses, of which 30 were from students who attend varsity football or basketball games only, 19 were from students who also attend other varsity competitions, and 48 who did not attend any varsity games.

H₀:

The group of students who do not attend games has a lower mean rating.

The group of students who do not attend games has a higher mean rating.

All groups have the same mean rating.

All groups have different mean ratings.

At least one group has a different mean rating.

H_a:

All groups have different mean ratings.

The group of students who do not attend games has a lower mean rating.

All groups have the same mean rating.

At least one group has a different mean rating.

The group of students who do not attend games has a higher mean rating.

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 45 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

H₀:

All groups have the same mean quiz score.

The group taught by the oldest TA has a lower mean quiz score.

At least one group has a different mean quiz score.

The group taught by the oldest TA has a higher mean quiz score.

All groups have different mean quiz scores.

H_a:

All groups have different mean quiz scores.All groups have the same mean quiz score.    At least one group has a different mean quiz score.The group taught by the oldest TA has a higher mean quiz score.The group taught by the oldest TA has a lower mean quiz score.

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 25 of his hens to each of the three treatments.

H₀:

All groups have the same mean cholesterol level.

All groups have different mean cholesterol levels.    

The all-vegetarian diet group has a higher mean cholesterol level.

At least one group has a different mean cholesterol level.

The all-vegetarian diet group has a lower mean cholesterol level.

H_a:

All groups have different mean cholesterol levels.

All groups have the same mean cholesterol level.

   The all-vegetarian diet group has a higher mean cholesterol level.

At least one group has a different mean cholesterol level.

The all-vegetarian diet group has a lower mean cholesterol level.

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 99 responses, of which 30 were from students who attend varsity football or basketball games only, 18were from students who also attend other varsity competitions, and 51 who did not attend any varsity games.

H₀:

At least one group has a different mean rating.

The group of students who do not attend games has a lower mean rating.  

  The group of students who do not attend games has a higher mean rating.

All groups have different mean ratings.

All groups have the same mean rating.

H_a:

All groups have the same mean rating.

At least one group has a different mean rating.    

The group of students who do not attend games has a lower mean rating.

The group of students who do not attend games has a higher mean rating.

All groups have different mean ratings.

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 45 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

H₀:

At least one group has a different mean quiz score.

All groups have the same mean quiz score.   

The group taught by the oldest TA has a higher mean quiz score.

All groups have different mean quiz scores.

The group taught by the oldest TA has a lower mean quiz score.

H_a:

The group taught by the oldest TA has a lower mean quiz score.

At least one group has a different mean quiz score.  

  The group taught by the oldest TA has a higher mean quiz score.

All groups have different mean quiz scores.

All groups have the same mean quiz score.

For each of the following situations, give the degrees of freedom for the group (DFG), for error (DFE), and for the total (DFT). State the null and alternative hypotheses, H₀ and H_a, and give the numerator and denominator degrees of freedom for the F statistic.

(a) A poultry farmer is interested in reducing the cholesterol level in his marketable eggs. He wants to compare two different cholesterol-lowering drugs added to the hen's standard diet as well as an all-vegetarian diet. He assigns 20 of his hens to each of the three treatments.

H₀:

All groups have different mean cholesterol levels.All groups have the same mean cholesterol level.     At least one group has a different mean cholesterol level.The all-vegetarian diet group has a higher mean cholesterol level.The all-vegetarian diet group has a lower mean cholesterol level.

H_a:

At least one group has a different mean cholesterol level.All groups have different mean cholesterol levels.     The all-vegetarian diet group has a higher mean cholesterol level.The all-vegetarian diet group has a lower mean cholesterol level.All groups have the same mean cholesterol level.

(b) A researcher is interested in students' opinions regarding an additional annual fee to support non-income-producing varsity sports. Students were asked to rate their acceptance of this fee on a seven-point scale. She received 94 responses, of which 31 were from students who attend varsity football or basketball games only, 15 were from students who also attend other varsity competitions, and 48 who did not attend any varsity games.

H₀:

At least one group has a different mean rating.The group of students who do not attend games has a lower mean rating.     All groups have different mean ratings.The group of students who do not attend games has a higher mean rating.All groups have the same mean rating.

H_a:

At least one group has a different mean rating.All groups have different mean ratings.     The group of students who do not attend games has a higher mean rating.The group of students who do not attend games has a lower mean rating.All groups have the same mean rating.

(c) A professor wants to evaluate the effectiveness of his teaching assistants. In one class period, the 45 students were randomly divided into three equal-sized groups, and each group was taught power calculations from one of the assistants. At the beginning of the next class, each student took a quiz on power calculations, and these scores were compared.

H₀:

At least one group has a different mean quiz score.The group taught by the oldest TA has a higher mean quiz score.     The group taught by the oldest TA has a lower mean quiz score.All groups have the same mean quiz score.All groups have different mean quiz scores.

H_a:

At least one group has a different mean quiz score.The group taught by the oldest TA has a higher mean quiz score.     The group taught by the oldest TA has a lower mean quiz score.All groups have different mean quiz scores.All groups have the same mean quiz score.