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5.7E: Excersies

5.7E: Excersies


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Exercise (PageIndex{1})

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

For the following exercises, the cylindrical coordinates (displaystyle (r,θ,z)) of a point are given. Find the rectangular coordinates (displaystyle (x,y,z)) of the point.

1) (displaystyle (4,frac{π}{6},3))

Answer

(displaystyle (2sqrt{3},2,3))

2) (displaystyle (3,frac{π}{3},5))

3) (displaystyle (4,frac{7π}{6},3))

Answer

(displaystyle −2sqrt{3},−2,3))

4) (displaystyle (2,π,−4))

For the following exercises, the rectangular coordinates (displaystyle (x,y,z)) of a point are given. Find the cylindrical coordinates (displaystyle (r,θ,z))of the point.

5) (displaystyle (1,sqrt{3},2))

Answer

(displaystyle (2,frac{π}{3},2))

6) (displaystyle (1,1,5))

7) (displaystyle (3,−3,7))

Answer

(displaystyle (3sqrt{2},−frac{π}{4},7))

8) (displaystyle (−2sqrt{2},2sqrt{2},4))

Exercise (PageIndex{2})

For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

9) [T] (displaystyle r=4)

Answer

A cylinder of equation (displaystyle x^2+y^2=16,) with its center at the origin and rulings parallel to the z-axis,

10) [T] (displaystyle z=r^2cos^2θ)

11) [T] (displaystyle r^2cos(2θ)+z^2+1=0)

Answer

Hyperboloid of two sheets of equation (displaystyle −x^2+y^2−z^2=1,) with the y-axis as the axis of symmetry,

12) [T] (displaystyle r=3sinθ)

13) [T] (displaystyle r=2cosθ)

Answer

Cylinder of equation (displaystyle x^2−2x+y^2=0,) with a center at (displaystyle (1,0,0)) and radius (displaystyle 1), with rulings parallel to the z-axis,

14) [T] (displaystyle r^2+z^2=5)

15) [T] (displaystyle r=2secθ)

Answer

Plane of equation (displaystyle x=2,)

16) [T] (displaystyle r=3cscθ)

Exercise (PageIndex{3})

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

17) (displaystyle z=3)

Answer

(displaystyle z=3)

18) (displaystyle x=6)

19) (displaystyle x^2+y^2+z^2=9)

Answer

(displaystyle r^2+z^2=9)

20) (displaystyle y=2x^2)

21) (displaystyle x^2+y^2−16x=0)

Answer

(displaystyle r=16cosθ,r=0)

22) (displaystyle x^2+y^2−3sqrt{x^2+y^2}+2=0)

Exercise (PageIndex{4})

For the following exercises, the spherical coordinates (displaystyle (ρ,θ,φ)) of a point are given. Find the rectangular coordinates (displaystyle (x,y,z)) of the point.

23) (displaystyle (3,0,π))

Answer

(displaystyle (0,0,−3))

24) (displaystyle (1,frac{π}{6},frac{π}{6}))

25) (displaystyle (12,−frac{π}{4},frac{π}{4}))

Answer

(displaystyle (6,−6,sqrt{2}))

26) (displaystyle (3,frac{π}{4},frac{π}{6}))

Exercise (PageIndex{5})

For the following exercises, the rectangular coordinates (displaystyle (x,y,z)) of a point are given. Find the spherical coordinates (displaystyle (ρ,θ,φ)) of the point. Express the measure of the angles in degrees rounded to the nearest integer.

27) (displaystyle (4,0,0))

Answer

(displaystyle (4,0,90°))

28) (displaystyle (−1,2,1))

29) (displaystyle (0,3,0))

Answer

(displaystyle (3,90°,90°))

30) (displaystyle (−2,2sqrt{3},4))

Exercise (PageIndex{6})

For the following exercises, the equation of a surface in spherical coordinates is given. Identify and graph the surface.

31) [T] (displaystyle ρ=3)

Answer

Sphere of equation (displaystyle x^2+y^2+z^2=9) centered at the origin with radius (displaystyle 3),

32) [T] (displaystyle φ=frac{π}{3})

33) [T] (displaystyle ρ=2cosφ)

Answer

Sphere of equation (displaystyle x^2+y^2+(z−1)^2=1) centered at (displaystyle (0,0,1)) with radius (displaystyle 1),

34) [T] (displaystyle ρ=4cscφ)

35) [T] (displaystyle φ=frac{π}{2})

Answer

The xy-plane of equation (displaystyle z=0,)

36) [T] (displaystyle ρ=6cscφsecθ)

Exercise (PageIndex{7})

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.

37) (displaystyle x^2+y^2−3z^2=0, z≠0)

Answer

(displaystyle φ=frac{π}{3}) or (displaystyle φ=frac{2π}{3};) Elliptic cone

38) (displaystyle x^2+y^2+z^2−4z=0)

39) (displaystyle z=6)

Answer

(displaystyle ρcosφ=6;) Plane at (displaystyle z=6)

40) (displaystyle x^2+y^2=9)

Exercise (PageIndex{8})

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φ

in radians rounded to four decimal places.

41) [T] (displaystyle (1,frac{π}{4},3))

Answer

(displaystyle (sqrt{10},frac{π}{4},0.3218))

42) [T] (displaystyle (5,π,12))

43) (displaystyle (3,frac{π}{2},3))

Answer

(displaystyle (3sqrt{2},frac{π}{2},frac{π}{4}))

44) (displaystyle (3,−frac{π}{6},3))

Exercise (PageIndex{9})

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

45) (displaystyle (2,−frac{π}{4},frac{π}{2}))

Answer

(displaystyle (2,−frac{π}{4},0))

46) (displaystyle (4,frac{π}{4},frac{π}{6}))

47) (displaystyle (8,frac{π}{3},frac{π}{2}))

Answer

(displaystyle (8,frac{π}{3},0))

48) (displaystyle (9,−frac{π}{6},frac{π}{3}))

Exercise (PageIndex{10})

For the following exercises, find the most suitable system of coordinates to describe the solids.

49) The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length (displaystyle a), where (displaystyle a>0)

Answer

Cartesian system, (displaystyle {(x,y,z)|0≤x≤a,0≤y≤a,0≤z≤a})

50) A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of (displaystyle a) and (displaystyle b), respectively, where (displaystyle b>a>0)

51) A solid inside sphere (displaystyle x^2+y^2+z^2=9) and outside cylinder (displaystyle (x−frac{3}{2})^2+y^2=frac{9}{4})

Answer

Cylindrical system, (displaystyle {(r,θ,z)∣r^2+z^2≤9,r≥3cosθ,0≤θ≤2π})

52) A cylindrical shell of height (displaystyle 10) determined by the region between two cylinders with the same center, parallel rulings, and radii of (displaystyle 2) and (displaystyle 5), respectively

Exercise (PageIndex{11})

53) [T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid (displaystyle z=x^2+y^2) and cone (displaystyle x^2+y^2−z^2=0.)

Answer

The region is described by the set of points (displaystyle {(r,θ,z)∣∣0≤r≤1,0≤θ≤2π,r^2≤z≤r}.)

54) [T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere (displaystyle x^2+y^2+z^2=4) and elliptical cone (displaystyle x^2+y^2−z^2=0.)

Exercise (PageIndex{12})

55) Washington, DC, is located at (displaystyle 39°) N and (displaystyle 77°) W (see the following figure). Assume the radius of Earth is (displaystyle 4000) mi. Express the location of Washington, DC, in spherical coordinates.

Answer

(displaystyle (4000,−77°,51°))

56) San Francisco is located at (displaystyle 37.78°N) and (displaystyle 122.42°W.) Assume the radius of Earth is (displaystyle 4000)mi. Express the location of San Francisco in spherical coordinates.

57) Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are (displaystyle (4000,−43.17°,102.91°).)

Answer

(displaystyle 43.17°W, 22.91°S)

58) Find the latitude and longitude of Berlin if its spherical coordinates are (displaystyle (4000,13.38°,37.48°).)

Exercise (PageIndex{13})

59) [T] Consider the torus of equation (displaystyle (x^2+y^2+z^2+R^2−r^2)^2=4R^2(x^2+y^2),) where (displaystyle R≥r>0.)

a. Write the equation of the torus in spherical coordinates.

b. If (displaystyle R=r,) the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is (displaystyle ρ=2Rsinφ.)

c. Use a CAS to graph the horn torus with (displaystyle R=r=2) in spherical coordinates.

Answer

(displaystyle a. ρ=0, ρ+R2−r2−2Rsinφ=0;)

c.

60) [T] The “bumpy sphere” with an equation in spherical coordinates is (displaystyle ρ=a+bcos(mθ)sin(nφ)), with (displaystyle θ∈[0,2π]) and (displaystyle φ∈[0,π]), where (displaystyle a) and (displaystyle b) are positive numbers and (displaystyle m) and (displaystyle n) are positive integers, may be used in applied mathematics to model tumor growth.

a. Show that the “bumpy sphere” is contained inside a sphere of equation (displaystyle ρ=a+b.) Find the values of (displaystyle θ) and (displaystyle φ) at which the two surfaces intersect.

b. Use a CAS to graph the surface for (displaystyle a=14, b=2, m=4,) and (displaystyle n=6) along with sphere (displaystyle ρ=a+b.)

c. Find the equation of the intersection curve of the surface at b. with the cone (displaystyle φ=frac{π}{12}). Graph the intersection curve in the plane of intersection.


Reebok B5.7e IWM Exercise Bike Review

The Reebok performance series of exercise bikes have been around for a few years and have proven to be a popular choice among home fitness enthusiasts. The B5.7e IWM is the newest model in the series and is the top of the range (with the exception of a black limited edition of which only 500 units have been made).

So, is this bike any good and what’s the IWM all about? Let’s start with the bike itself. The look & the design of the bike follows the same mold as the entry level performance bikes, but with higher quality components. The seat is well padded and comfortable and can be adjusted for both laterally and for height. The racing style handlebars are also adjustable and given the combination of the seat and handlebar adjustments possible, finding the perfect position should be possible for users of all heights.

The B5.7e IWM is very smooth to use, thanks to a decent 10kg flywheel and heavy duty components used in the construction. The resistance mechanism is Reebok’s M-Force EMS system. This is an electromagnetic system with no moving parts – the strength of the resistance is controlled by varying the amount of current being passed through some coils. This provides several advantages over the traditional servo assisted magnetic brakes: Changing the resistance level from the console is almost instant and completely silent and as there are no moving parts, the calibration won’t drift over an extended period of time.

As the top of the range Reebok bike, the B5.7e IWM is an ergometer with a resistance range of 25-400W. The top level of resistance is actually really high at 400w, but we can confirm that the pedalling action is still very smooth (though this was only tested for a few minutes as your reviewer was really struggling! :s )

The console is very clear with a big, blue LCD screen displaying all the key feedback metrics. One thing that we really like on the Reebok performance series of fitness equipment is the ability to setup multiple user profiles. For each profile, you can enter your age, gender, weight and height and this data is used by appropriate programmes (like IWM ones) as well as to provide more accurate estimates of calorie consumption.

In terms of programmes, there is a very good selection of 12 programmes that cover pretty much all the different methods of exercise that you’d do on an exercise bike. We’re pleased to see that this includes a heart rate control programme. In addition to the standard hand grip pulse sensors on the handlebars, accurate heart rate measurement is possible with the B5.7e IWM as the console has a Polar wireless heart rate receiver. You just need a standard uncoded polar chest strap (not included, but available for under £25 in most places) to take advantage.

IWM stands for Interactive Weight Management and it’s a rather unique feature of this exercise bike. The bike is supplied with a set of wireless scales and once you’ve setup a user profile on there, you press the IWM button on the console, stand on the scales and your weight (measured to the nearest 10 grammes) is transmitted wireless to the console. The measured weight data, along with your profile information is used by IWM to create a custom exercise programme.

We’ve worked out that the IWM technology is based on user’s body mass index and whilst the science being utilised here isn’t exactly good enough to match a human personal trainer monitoring progress and advising of changes to exercise routine, it does provide workout variety and we think it helps with the all important motivation!

In terms of warranty, it’s the standard Reebok 2 years onsite parts & labour, so should anything go wrong, there’s nothing to worry about.

The Reebok B5.7e IWM is not currently available from any retailers we're aware of in the UK.
It may be worth looking for a used machine on ebay

The Reebok B5.7e IWM is certainly a great bike with high specs that typically retails for £600. Whilst in absolute terms, it's an excellent exercise bike, it doesn't justify the price. However, with the promotional discount being offered direct from Reebok Fitness resulting in a sub £450 price, it's a bargain that we highly recommend!


5.7E: Excersies

Trapezoidal threads are codified by DIN 103. Although metric screw threads are generally more prevalent worldwide than imperial threads, the Acme thread is very common worldwide, and may be more widely used than the trapezoidal metric thread.

Trapezoidal threads are defined as follows by ISO standards: ISO 2901, ISO 2901, ISO 2903, ISO 2904 and ISO 103. Gages ISO metric trapezoidal screw thread are defined in the DIN 103-9 .

where Tr designates a trapezoidal thread, 8 is the nominal diameter in millimeters, and 1.5 is the pitch in millimeters. When there is no suffix it is a single start thread. If there is a suffix then the value after the multiplication sign is the lead and the value in the parentheses is the pitch. For example:

TR 60 x 18 (P9) LH would denoted two starts, as the lead divided by the pitch is two. The "LH" denotes a left hand thread.

The thread form and related equations for calculating Trapezoidal Screw Threads can be seen here:
External ISO Metric Trapezoidal Screw Threads Chart TR320 - TR1120


Reebok B5.7e Exercise Bike

Offering a fully adjustable seat and handlebar set up that provides outstanding comfort when in use, the Reebok B5.7e exercise bike provides you with the ideal training position. The console provides regular and accurate feedback on all key workout data and the hand grip pulse sensors guarantee you get consistent heart rate readings displayed on the console. All of this functionality provides you with complete personalisation and a customisable workout that suits your needs. It also suits different user types too, making certain all users get the most out of this excellent exercise bike.

The B5.7e is elegant, functional and easy to use and suits a range of users from the hardcore cyclist right the way through to the casual fitness user. The magnetic resistance system ensures consistent resistance throughout your entire session, while the large balanced flywheel guarantees you a smooth cycle action.

Product information

Description

Offering a fully adjustable seat and handlebar set up that provides outstanding comfort when in use, the Reebok B5.7e exercise bike provides you with the ideal training position. The console provides regular and accurate feedback on all key workout data and the hand grip pulse sensors guarantee you get consistent heart rate readings displayed on the console. All of this functionality provides you with complete personalisation and a customisable workout that suits your needs. It also suits different user types too, making certain all users get the most out of this excellent exercise bike.

The B5.7e is elegant, functional and easy to use and suits a range of users from the hardcore cyclist right the way through to the casual fitness user. The magnetic resistance system ensures consistent resistance throughout your entire session, while the large balanced flywheel guarantees you a smooth cycle action.

Key features

  • Flywheel Weight: 10Kg (22lbs)
  • Resistance: Electromagnetic resistance, 25-400W in 5W steps
  • Brake System: M-Force EMS electro induction
  • Belt Transmission: Poly-V belt
  • Computer: Blue backlit LCD display
  • Feedback: Speed, Time, Distance, Calories, Pulse, RPM, Watt
  • Programmes: 12 (5 programs that can individually be determined and saved)
  • User Profiles: 9
  • Heart Rate Measurement: ergonomic hand pulse sensors and an integrated Polar receiver with a chest transmitter (optional)
  • Seat Adjustment: vertical & horizontal
  • Handlebars: Adjustable
  • Pedals: Comfort pedals with straps
  • Floor Level Adjustment: Yes
  • Bottle Holder: Yes
  • Transport Wheels: Yes
  • Maximum User Weight: 150kg (330lbs)
  • Product Weight: 41kg (90lbs)
  • Dimensions: Length= 100cm (39.4”), Width= 55cm (21.6”), Height= 148cm (58.3”)
  • Warranty: Manufacturers’ on site 2 years parts and labour cover

Delivery

When showing as 'in stock', this item is delivered to you FREE of charge using a next day delivery service to all postcodes in mainland England Wales and Scotland (excludes Highlands and Islands as set out in the exceptions below).

The standard delivery is next working day ‘signed for’ service with deliveries, Monday to Friday from 7.30 – 17.30 to your front door. If you would like to delay your order to ensure that will be home or to request an upgraded delivery please request this using the ‘special instructions’ while making payment and a member of the team will be in touch to book in the delivery.

(Please note that if you are ordering this item with other items, we aim to deliver your entire order together where possible).

Unfortunately we are unable to deliver to the following locations:

Scottish Highlands & Islands (AB30-AB38, AB44-AB56, FK17-FK99, G83, HS1 - HS9, IV1- IV28, IV30 – IV39, IV41 - IV56, IV63, KA27- KA28, KW1 - KW17, PA20 – PA49,, PA60 - PA78, PH17 - PH26, PH30 - PH44, PH49 - PH50, ZE), Northern Ireland (All Northern Ireland postcodes).

Reviews

  • Flywheel Weight: 10Kg (22lbs)
  • Resistance: Electromagnetic resistance, 25-400W in 5W steps
  • Brake System: M-Force EMS electro induction
  • Belt Transmission: Poly-V belt
  • Computer: Blue backlit LCD display
  • Feedback: Speed, Time, Distance, Calories, Pulse, RPM, Watt
  • Programmes: 12 (5 programs that can individually be determined and saved)
  • User Profiles: 9
  • Heart Rate Measurement: ergonomic hand pulse sensors and an integrated Polar receiver with a chest transmitter (optional)
  • Seat Adjustment: vertical & horizontal
  • Handlebars: Adjustable
  • Pedals: Comfort pedals with straps
  • Floor Level Adjustment: Yes
  • Bottle Holder: Yes
  • Transport Wheels: Yes
  • Maximum User Weight: 150kg (330lbs)
  • Product Weight: 41kg (90lbs)
  • Dimensions: Length= 100cm (39.4”), Width= 55cm (21.6”), Height= 148cm (58.3”)
  • Warranty: Manufacturers’ on site 2 years parts and labour cover

When showing as 'in stock', this item is delivered to you FREE of charge using a next day delivery service to all postcodes in mainland England Wales and Scotland (excludes Highlands and Islands as set out in the exceptions below).

The standard delivery is next working day ‘signed for’ service with deliveries, Monday to Friday from 7.30 – 17.30 to your front door. If you would like to delay your order to ensure that will be home or to request an upgraded delivery please request this using the ‘special instructions’ while making payment and a member of the team will be in touch to book in the delivery.

(Please note that if you are ordering this item with other items, we aim to deliver your entire order together where possible).

Unfortunately we are unable to deliver to the following locations:

Scottish Highlands & Islands (AB30-AB38, AB44-AB56, FK17-FK99, G83, HS1 - HS9, IV1- IV28, IV30 – IV39, IV41 - IV56, IV63, KA27- KA28, KW1 - KW17, PA20 – PA49,, PA60 - PA78, PH17 - PH26, PH30 - PH44, PH49 - PH50, ZE), Northern Ireland (All Northern Ireland postcodes).


5.7E: Excersies

Trapezoidal threads are typically pecision rolled screw profiles and often used for lead screws and is similar to the Acme thread form, except the thread angle is 30°. It is codified by DIN 103. Although metric screw threads are generally more prevalent worldwide than imperial threads, the Acme thread is very common worldwide, and may be more widely used than the trapezoidal metric thread.

Trapezoidal threads are defined as follows by ISO standards: ISO 2901, ISO 2901, ISO 2903, ISO 2904 and ISO 103. Gages ISO metric trapezoidal screw thread are defined in the DIN 103-9.

These dimensions are determined by the following formula:

D = major diameter of internal thread
d = major diameter of external thread (nominal diameter)
D2 = pitch drameter of Internal thread
d2 = pitch diameter of external thread
D1 = minor diameter of internal thread
P = Pitch
H = height of fundamental triangle
H1 = height of basic profile
ac = crest clearance
es = fundamental deviation on external threads

Trapezoidal threads are defined as follows by ISO standards:

where Tr designates a trapezoidal thread, 8 is the nominal diameter in millimeters, and 1.5 is the pitch in millimeters. When there is no suffix it is a single start thread. If there is a suffix then the value after the multiplication sign is the lead and the value in the parentheses is the pitch. For example:

TR 60 x 18 (P9) LH would denoted two starts, as the lead divided by the pitch is two. The "LH" denotes a left hand thread.


Forecasting: Principles and Practice (3rd ed)

In practice, of course, we have a collection of observations but we do not know the values of the coefficients (eta_0,eta_1, dots, eta_k) . These need to be estimated from the data.

The least squares principle provides a way of choosing the coefficients effectively by minimising the sum of the squared errors. That is, we choose the values of (eta_0, eta_1, dots, eta_k) that minimise [ sum_^T varepsilon_t^2 = sum_^T (y_t - eta_ <0>- eta_ <1>x_ <1,t>- eta_ <2>x_ <2,t>- cdots - eta_ x_)^2. ]

This is called least squares estimation because it gives the least value for the sum of squared errors. Finding the best estimates of the coefficients is often called “fitting” the model to the data, or sometimes “learning” or “training” the model. The line shown in Figure 7.3 was obtained in this way.

When we refer to the estimated coefficients, we will use the notation (hateta_0, dots, hateta_k) . The equations for these will be given in Section 7.9.

The TSLM() function fits a linear regression model to time series data. It is similar to the lm() function which is widely used for linear models, but TSLM() provides additional facilities for handling time series.

Example: US consumption expenditure

A multiple linear regression model for US consumption is [ y_t=eta_0 + eta_1 x_<1,t>+ eta_2 x_<2,t>+ eta_3 x_<3,t>+ eta_4 x_<4,t>+varepsilon_t, ] where (y) is the percentage change in real personal consumption expenditure, (x_1) is the percentage change in real personal disposable income, (x_2) is the percentage change in industrial production, (x_3) is the percentage change in personal savings and (x_4) is the change in the unemployment rate.

The following output provides information about the fitted model. The first column of Coefficients gives an estimate of each (eta) coefficient and the second column gives its standard error (i.e., the standard deviation which would be obtained from repeatedly estimating the (eta) coefficients on similar data sets). The standard error gives a measure of the uncertainty in the estimated (eta) coefficient.

For forecasting purposes, the final two columns are of limited interest. The “t value” is the ratio of an estimated (eta) coefficient to its standard error and the last column gives the p-value: the probability of the estimated (eta) coefficient being as large as it is if there was no real relationship between consumption and the corresponding predictor. This is useful when studying the effect of each predictor, but is not particularly useful for forecasting.

Fitted values

Predictions of (y) can be obtained by using the estimated coefficients in the regression equation and setting the error term to zero. In general we write, [egin hat_t = hateta_ <0>+ hateta_ <1>x_ <1,t>+ hateta_ <2>x_ <2,t>+ cdots + hateta_ x_. ag <7.2>end] Plugging in the values of (x_<1,t>,dots,x_) for (t=1,dots,T) returns predictions of (y_t) within the training set, referred to as fitted values. Note that these are predictions of the data used to estimate the model, not genuine forecasts of future values of (y) .

The following plots show the actual values compared to the fitted values for the percentage change in the US consumption expenditure series. The time plot in Figure 7.6 shows that the fitted values follow the actual data fairly closely. This is verified by the strong positive relationship shown by the scatterplot in Figure 7.7.

Figure 7.6: Time plot of actual US consumption expenditure and predicted US consumption expenditure.

Figure 7.7: Actual US consumption expenditure plotted against predicted US consumption expenditure.

Goodness-of-fit

A common way to summarise how well a linear regression model fits the data is via the coefficient of determination, or (R^2) . This can be calculated as the square of the correlation between the observed (y) values and the predicted (hat) values. Alternatively, it can also be calculated as, [ R^2 = frac_ - ar)^2>-ar)^2>, ] where the summations are over all observations. Thus, it reflects the proportion of variation in the forecast variable that is accounted for (or explained) by the regression model.

In simple linear regression, the value of (R^2) is also equal to the square of the correlation between (y) and (x) (provided an intercept has been included).

If the predictions are close to the actual values, we would expect (R^2) to be close to 1. On the other hand, if the predictions are unrelated to the actual values, then (R^2=0) (again, assuming there is an intercept). In all cases, (R^2) lies between 0 and 1.

The (R^2) value is used frequently, though often incorrectly, in forecasting. The value of (R^2) will never decrease when adding an extra predictor to the model and this can lead to over-fitting. There are no set rules for what is a good (R^2) value, and typical values of (R^2) depend on the type of data used. Validating a model’s forecasting performance on the test data is much better than measuring the (R^2) value on the training data.

Example: US consumption expenditure

Figure 7.7 plots the actual consumption expenditure values versus the fitted values. The correlation between these variables is (r=0.877) hence (R^2= 0.768) (shown in the output above). In this case model does an excellent job as it explains 76.8% of the variation in the consumption data. Compare that to the (R^2) value of 0.15 obtained from the simple regression with the same data set in Section 7.1. Adding the three extra predictors has allowed a lot more of the variation in the consumption data to be explained.

Standard error of the regression

Another measure of how well the model has fitted the data is the standard deviation of the residuals, which is often known as the “residual standard error.” This is shown in the above output with the value 0.31.

It is calculated using [egin hat_e=sqrtsum_^>, ag <7.3>end] where (k) is the number of predictors in the model. Notice that we divide by (T-k-1) because we have estimated (k+1) parameters (the intercept and a coefficient for each predictor variable) in computing the residuals.

The standard error is related to the size of the average error that the model produces. We can compare this error to the sample mean of (y) or with the standard deviation of (y) to gain some perspective on the accuracy of the model.

The standard error will be used when generating prediction intervals, discussed in Section 7.6.


The Word on the Street

UPDATE:  The NordicTrack E 5.7 has been discontinued and replaced with all new, upgraded models. ਌heck out the latest reviews here.

The features on the NordicTrack E5.7 elliptical include a hefty 250-pound frame that measures 65 x 27 x 70 inches, plus transport wheels, and an Eddy Current Magnetic brake system with 20 levels of digital resistance.

The stride length can be adjusted from 18" - 20" and the ramp can be positioned manually from a flat level up to an incline of 20 degrees. It’s also a good looking machine with a platinum metallic color and offers a user weight capacity up to 300 pounds, plus it has iFit Live capability.

But that’s where the good news ends. While the specs listed above seem enticing, once you read a handful of user reviews you realize that this $599 trainer is not well designed or well built. When it works, the NordicTrack E5.7 offers a choppy elliptical feel while pedaling which taller riders will notice right away.

But the most alarming fact is that up to 50% of users who post online reviews report problems with the construction with welds failing, pedals and arms breaking, and other manufacturing defects.


5.6 Property rights, the rule of law, and the right to vote

Bruno thinks that the new rules, under which he makes an offer that Angela will not refuse, are not so bad after all. Angela is also better off than she had been when she had barely enough to survive. But she would like a share in the surplus.

Fairness—changing the law by democratic means

Angela and her fellow farm workers lobby for a new law that limits working time to 4 hours a day, while requiring that total pay is at least 4.5 bushels. They threaten to not work at all unless the law is passed.

Bruno Angela, you and your colleagues are bluffing. Angela No, we are not. We would be no worse off at our reservation option than under your contract, working the hours and receiving the small fraction of the harvest that you impose!

Angela and her fellow workers win, and the new law limits the working day to 4 hours.

Before the short-hours law, Angela worked for 8 hours and received 4.5 bushels of grain. This is point D in Figure 5.11. The new law implements the allocation in which Angela and her friends work for 4 hours, getting 20 hours of free time and the same number of bushels. Since they have the same amount of grain and more free time, they are better off. Figure 5.11 shows they are now on a higher indifference curve.

The effect of an increase in Angela’s bargaining power through legislation.

Figure 5.11 The effect of an increase in Angela’s bargaining power through legislation.

Before the short-hours law

Bruno makes a take-it-or-leave-it offer, gets grain equal to CD, and Angela works for 8 hours. Angela is on her reservation indifference curve at D and MRS = MRT.

Figure 5.11a Bruno makes a take-it-or-leave-it offer, gets grain equal to CD, and Angela works for 8 hours. Angela is on her reservation indifference curve at D and MRS = MRT.

What Angela receives before legislation

Angela gets 4.5 bushels of grain. She is just indifferent between working for 8 hours and her reservation option.

Figure 5.11b Angela gets 4.5 bushels of grain. She is just indifferent between working for 8 hours and her reservation option.

The effect of legislation

With legislation that reduces work to 4 hours a day and keeps Angela’s amount of grain unchanged, she is on a higher indifference curve at F. Bruno’s grain is reduced from CD to EF (2 bushels).

Figure 5.11c With legislation that reduces work to 4 hours a day and keeps Angela’s amount of grain unchanged, she is on a higher indifference curve at F. Bruno’s grain is reduced from CD to EF (2 bushels).

When Angela works for 4 hours, the MRT is larger than the MRS on the new indifference curve.

Figure 5.11d When Angela works for 4 hours, the MRT is larger than the MRS on the new indifference curve.

To sum up, the introduction of the new law has increased Angela’s bargaining power and Bruno is worse off than before. You can see that she is better off at F than at D. She is also better off than she would be with her reservation option, which means she is now receiving an economic rent.

Angela’s rent can be measured, in bushels of grain, as the vertical distance between her reservation indifference curve (IC1 in Figure 5.10) and the indifference curve she is able to achieve under the new legislation (IC2). We can think of the economic rent in two equivalent ways:

  • What she would give up to live under a better law: The rent is the maximum amount of grain per year that Angela would give up to live under the new law rather than in the situation before the law was passed.
  • What she would pay to pass a new law: Because Angela is obviously political, it is also the amount she would be willing to pay in order to have the law passed, for example, by lobbying the legislature or contributing to election campaigns.

Question 5.10 Choose the correct answer(s)

In Figure 5.11, D and F are the outcomes before and after the introduction of a new law that limits Angela’s work time to 4 hours a day while requiring a minimum pay of 4.5 bushels. Based on this information, which of the following statements are correct?

  • The change from D to F is a Pareto improvement.
  • The new outcome F is Pareto efficient.
  • Both Angela and Bruno receive economic rents at F.
  • As a result of the new law, Bruno has less bargaining power.
  • It is not a Pareto improvement, because Bruno is worse off (gets less grain) at F than at D.
  • At outcome F, where Angela works for 4 hours, MRT > MRS (compare the slopes of the feasible frontier and indifference curve). Therefore, it cannot be Pareto efficient. (For example, Bruno could be better off without making Angela worse off, if they could move to the left along IC2.)
  • At F, Angela is above her reservation indifference curve and is thus receiving an economic rent. Bruno’s reservation option is to receive nothing, so the grain he receives at F is an economic rent for him.
  • At D, Bruno obtained rent equal to CD, and Angela obtained no rent. At F, his rent is much lower—the law has increased Angela’s bargaining power and reduced Bruno’s.

Efficiency: Bargaining to an efficient sharing of the surplus

Angela and her friends are pleased with their success. She asks what you think of the new policy.

You Congratulations, but your policy is far from the best you could do. Angela Why? You Because you are not on the Pareto efficiency curve! Under your new law, Bruno is getting 2 bushels, and cannot make you work for more than 4 hours. So why don’t you offer to continue to pay him 2 bushels, in exchange for agreeing to let you keep anything you produce above that? Then you get to choose how many hours you work.

The small print in the law allows a longer work day if both parties agree, as long as the workers’ reservation option is a 4-hour day if no agreement is reached.

Now redraw Figure 5.11 and use the concepts of the joint surplus and the Pareto efficiency curve from Figure 5.10 to show Angela how she can get a better deal.

You Look at Figure 5.12. The surplus is largest at 8 hours of work. When you work for 4 hours, the surplus is smaller, and you pay most of it to Bruno. If you increase the surplus, you can pay him the same amount and your own surplus will be bigger—so you will be better off. Follow the steps in Figure 5.12 to see how this works.

Bargaining to restore Pareto efficiency.

Figure 5.12 Bargaining to restore Pareto efficiency.

The maximum joint surplus

The surplus to be divided between Angela and Bruno is maximized where MRT = MRS, at 8 hours of work.

Figure 5.12a The surplus to be divided between Angela and Bruno is maximized where MRT = MRS, at 8 hours of work.

But Angela prefers point F implemented by the legislation, because it gives her the same amount of grain but more free time than D.

Figure 5.12b But Angela prefers point F implemented by the legislation, because it gives her the same amount of grain but more free time than D.

Angela could also do better than F

Compared to F, however, she would prefer any allocation on the Pareto efficiency curve between C and G.

Figure 5.12c Compared to F, however, she would prefer any allocation on the Pareto efficiency curve between C and G.

At allocation H, Bruno gets the same amount of grain—CH = EF. Angela is better off than she was at F. She works longer hours but has more than enough grain to compensate her for the loss of free time.

Figure 5.12d At allocation H, Bruno gets the same amount of grain—CH = EF. Angela is better off than she was at F. She works longer hours but has more than enough grain to compensate her for the loss of free time.

A win–win agreement by moving to an allocation between G and H

F is not Pareto efficient because MRT > MRS. If they move to a point on the Pareto efficiency curve between G and H, Angela and Bruno can both be better off.

Figure 5.12e F is not Pareto efficient because MRT > MRS. If they move to a point on the Pareto efficiency curve between G and H, Angela and Bruno can both be better off.

The move away from point D (at which Bruno had all the bargaining power and obtained all the gains from exchange) to point H where Angela is better off consists of two distinct steps:

  1. From D to F, the outcome is imposed by new legislation: This was definitely not win–win. Bruno lost because his economic rent at F is less than the maximum feasible rent that he got at D. Angela benefitted.
  2. Once at the legislated outcome, there were many win–win possibilities open to them: They are shown by the segment GH on the Pareto efficiency curve. Win–win alternatives to the allocation at F are possible by definition, because F was not Pareto efficient.

Bruno wants to negotiate. He is not happy with Angela’s proposal of H.

Bruno I am no better off under this new plan than I would be if I just accepted the short-hours legislation. You But Bruno, Angela now has bargaining power, too. The legislation changed her reservation option, so it is no longer 24 hours of free time at survival rations. Her reservation option is now the legislated allocation at point F. I suggest you make her a counter offer. Bruno Angela, I’ll let you work the land for as many hours as you choose, if you pay me half a bushel more than EF.

They shake hands on the deal.

Because Angela is free to choose her work hours, subject only to paying Bruno the extra half bushel, she will work for 8 hours where MRT = MRS. Because this deal lies between G and H, it is a Pareto improvement over point F. Moreover, because it is on the Pareto-efficient curve CD, we know there are no further Pareto improvements to be made. This is true of every other allocation on GH—they differ only in the distribution of the mutual gains, as some favour Angela while others favour Bruno. Where they end up will depend on their bargaining power.

Question 5.11 Choose the correct answer(s)

In Figure 5.12, Angela and Bruno are at allocation F, where she receives 3 bushels of grain for 4 hours of work. From the figure, we can conclude that:

  • All the points on EF are Pareto efficient.
  • Any point in the area between G, H and F would be a Pareto improvement.
  • Any point between G and D would be a Pareto improvement.
  • They would both be indifferent between all points on GH.
  • Along EF, MRS < MRT. Therefore, EF is not Pareto efficient—there are other allocations where both would be better off.
  • In area GHF, Angela is on a higher indifference curve than IC2, and Bruno has more grain than EF, so both are better off.
  • Points on GD are Pareto efficient, but below G, Angela is on a lower indifference curve than at F, so she would be worse off.
  • Points on GH are all Pareto efficient, but Bruno and Angela are not indifferent. He prefers points nearer to G, and she prefers points nearer to H.

Seminar

The seminar this week is devoted to learning how to use the Synth package in R. This package has been developed to make it easier to implement synthetic control designs, though as you will see it does have a somewhat idiosyncratic coding style. You will need to install the package and load it as we have done in previous weeks:

6.0.1 The effect of Economic and Monetary Union on Current Account Balances – Hope (2016)

In early 2008, about a decade after the Euro was first introduced, the European Commission published a document looking back at the currency’s short history and concluded that the European Economic and Monetary Union was a “resounding success”. By the end of 2009 Europe was at the beginning of a multiyear sovereign debt crisis, in which several countries – including a number of Eurozone members – were unable to repay or refinance their government debt or to bail out over-indebted banks. Although the causes of the Eurocrisis were many and varied, one aspect of the pre-crisis era that became particularly damaging after 2008 were the large and persistent current account deficits of many member states. Current account imbalances – which capture the inflows and outflows of both goods and services and investment income – were a marked feature of the post-EMU, pre-crisis era, with many countries in the Eurozone running persistent current account deficits (indicating that they were net borrowers from the rest of the world). Large current account deficits make economies more vulnerable to external economic shocks because of the risk of a sudden stop in capital used to finance government deficits.

David Hope investigates the extent to which the introduction of the Economic and Monetary Union in 1999 was responsible for the current account imbalances that emerged in the 2000s. Using the sythetic control method, Hope evaluates the causal effect of EMU on current account balances in 11 countries between 1980 and 2010. In this exercise, we will focus on just one country – Spain – and evaluate the causal effect of joining EMU on the Spanish current account balance. Of the (J) countries in the sample, therefore, (j = 1) is Spain, and (j=2. 16) will represent the “donor” pool of countries. In this case, the donor pool consists of 15 OECD countries that did not join the EMU: Australia, Canada, Chile, Denmark, Hungary, Israel, Japan, Korea, Mexico, New Zealand, Poland, Sweden, Turkey, the UK and the US.

The hope_emu.csv file contains data on these 16 countries across the years 1980 to 2010. The data includes the following variables:

  1. period – the year of observation
  2. country_ID – the country of observation
  3. country_no – a numeric country identifier
  4. CAB – current account balance
  5. GDPPC_PPP – GDP per capita, purchasing power adjusted
  6. invest – Total investment as a % of GDP
  7. gov_debt – Government debt as a % of GDP
  8. openness – trade openness
  9. demand – domestic demand growth
  10. x_price – price level of exports
  11. gov_deficit – Government primary balance as a % of GDP
  12. credit – domestic credit to the private sector as a % of GDP
  13. GDP_gr – GDP growth in %

Use the read.csv function to load the data into R now. For this assignment, we will need the qualitative variables to be stored as character variables, rather than the factor encoding that R uses by default. For this reason, we will set the stringsAsFactors arugment in the read.csv function to be false.

Question 1. Plotting Spain’s current account balance

Plot the trajectory of the Spanish current account balance over time in red. Add other lines to the plot for the current account balance for 3 other countries (using the lines() function). Plot an additional dashed vertical line in 1999 to mark the introduction of the EMU (use the abline function, setting the v argument to the appropriate number). Would you be happy using any of them on their own as the control group?

None of these individual countries is a perfect approximation to the pre-treatment trend for Spain, although the US and the UK lines are clearly closer than the Japanese line. The goal of the synthetic control analysis is to create a weighting scheme which, when applied to all countries in the donor pool, creates a closer match to the pre-intervention treated unit trend than any of the individual countries do alone.

Question 2. Preparing the synthetic control

The Synth package takes data in a somewhat unusual format. The main function we will use to get our data.frame into the correct shape is the dataprep() function. Look at the help file for this function using ?dataprep . You will see that this function requires us to correctly specify a number of different arguments. I have summarised the main arguments you will need to use in the table below:

Argument Description
foo This is where we put the data.frame that we want to use for the analysis
predictors This argument expeects a vector of names for the covariates we would like to use to estimate the model. You will need to use the c() function, and enter in all the variable names that you will be using.
dependent The name of the dependent variable in the analysis (here, "CAB" )
unit.variable The name of the variable that identifies each unit (must be numeric)
unit.names.variable The name of the variable that contains the name for each unit (here, "country_ID" )
time.varaible The name of the variable that identifies each time period (must be numeric)
treatment.identifier The identifying number of the treatment unit (must correspond to the value for the treated unit in unit.variable )
controls.identifier The identifying numbers of the control units (must correspond to the values for the control units in unit.variable )
time.predictors.prior A vector indicating the time periods before the treatment
time.optimize.ssr Another vector indicating the time periods before the treatment
time.plot A vector indicating the time periods before and after the treatment

Use the dataprep() function to prepare the emu data. Try on your own first, and then look at the solution below.

Question 3. Estimating the synthetic control

Fortunately, though getting the data in the prep function correctly can be a pain, estimating the synthetic control is very straightforward. Use the synth() function on the dataprep_out object that you just created, remembering to assign the output to a new object.

Note: It can take a few minutes for this function to run, so be patient!

R prints some details when it finishes the estimation of the synthetic control, but these are a little difficult to interpret directly. Instead, we will move on to interpreting the types of plots that we saw in the lecture.

Question 4. Plotting the results

Use synth’s path.plot() and gaps.plot() functions to produce plots which compare Spain’s actual current account balance trend to that of the synthetic Spain you have just created. These function takes two main arguments, and then some additional styling arguments to make the plot look nice:

Argument Description
synth.res This is where we put the saved output of the synth() function (i.e. the estimated synthetic control object)
dataprep.res This is where we put the saved output of the dataprep() function (i.e. the data we used to estimate the synthetic control).
tr.intake A number to indicate the time of the treatment intake (here, 1999)
Xlab The name of the variable on the x-axis (here, "Time" )
Ylab The name of the variable on the x-axis (here, "Current account balance" )
Legend Optional text for the legend of the plot.
Ylim The range of the y-axis (here, c(-10,5) )

Look at the help file for more details.

Interpret these plots. What do they suggest about the effect of the introduction of EMU on the Spanish current account balance?

The synthetic version of Spain provides a reasonably good approximation to the pre-treatment trend of Spain, as there are only small differences in the Current Account Balance between real Spain and synthetic Spain before 1999.

In addition, it is clear that the trajectory of Spain and its synthetic control diverge significantly after the EMU is introduced in 1999. In particular, the actual Spanish current account balance deteriorated much more than the current account balances of the synthetic control unit in the post-EMU period. This therefore provides some empirical support for the hypothesis that the introduction of the EMU caused the current account balances of Spain to deteriorate.

Question 5. Interpreting the synthetic control unit

A crucial strength of the synthetic control approach is that it allows us to be very transparent about the comparisons we are making when making causal inferences. In particular, we know that the synthetic Spain that we created in question 2 is a weighted average of the 15 OECD non-EMU countries in our data. Let’s practice some of this transparency now by reporting the estimated vector of country weights in a nice table.

Look in the help file for ?synth , and read the “Value” section of that page. The value section will tell you all of the things that are returned by a function. You can access them by using the dollar sign operator that we have used in the past to extract variables from a data.frame.

a. What are the top five countries contributing to synthetic Spain?

As the table shows (I have contructed this table using the information from the lines of code above), the main contributors to synthetic Spain are Great Britain, Mexico, Australia and Japan, with a smaller contribution from Poland.

b. Fortunately, there is an easier way to extract this information as well as a) information on the weights assigned to each of the predictor variables in the model, and b) the balance on each predictor variable across the treated country and the synthetic country. Look at the help file for the synth.tab() function and apply that function to the output of the synth() and dataprep() functions from the questions above. Which variables contribute the most to the synthetic control? Is the synthetic control unit closer to the treated unit in terms of the covariates than the sample mean?

The country-weights are stored in synth.tables$tab.w , and contain the same information as the table that I constructed manually above.

synth.tables$tab.v contains the weights assigned to each of the predictor variables in the model. As this table shows, the highest weight is assigned to the x_price variable,which suggests that the price level of exports is an important predictor for matching the pre-treatment trend of current account balances in Spain to those of other countries. GDP per capita, the degree of domestic demand growth, and the degree of trade openness are all important for this reason as well.

synth.tables$tab.pred gives the mean of each of the predictor variables in the Treated unit (Spain), the Synthetic unit that is constructed by the synthetic control method, and for the entire sample. It is clear from the table that for all these variables, the synthetic unit is a much closer match for the treated unit than is the sample as a whole. This is, in fact, the whole point of synthetic control! It allows us to construct a control unit that is as similar as possible to our treatment unit.

Question 6. Estimate a placebo synthetic control treatment effect

One way to check the validity of the synthetic control is to estimate “placebo” effects – i.e. effects for units that were not exposed to the treatment. In this question we will replicate the analysis above for Australia, which did not join EMU in 1999.

a. In constructing synthetic Australia, we must exclude Spain – the actual treatment unit – from the analysis. Before you repeat the steps above for Australia, create a new data.frame that doesn’t include the Spanish observations.

Note: Here you will want to select all rows of the emu data for which the country_ID variable is not equal to "ESP" .

b. Now repeat the steps above to estimate the synthetic control for Australia.

c. What does the estimated treatment effect for Australia tell you about the validity of the design for estimating the treatment effect of the EMU on the Spanish current account balance?

The placebo test here supports the inferences drawn from the main synthetic control analysis. There is clearly no effect of the introduction of EMU on the current account balance of Australia. Of course, full permutation inference would require re-estimating the synthetic control for every unit in the donor pool, not just Australia, and comparing the distribution of these placebo treatment effects to the treatment effect for Spain. In the homework, you will be asked to complete this analysis.

d. Compare the treatment effects from the Australian synthetic control analysis and the Spanish synthetic control analysis in terms of the pre- and post-treatment root mean square error values.

We can calculate the root mean squared prediction error for the pre-and post-intervention periods for both Australia and Spain. Recall that the the RMSE measures the size of the gap between the outcome of interest in each country and its synthetic counterpart. Large values of the ratio of the pre- and post-RMSEs provides evidence that the treatment effect is large. (We take the ratio of these measures because a large post-treatment RMSE is not itself sufficient evidence of a large treatment effect, because the synthetic control may be a poor approximation to the unit of interest. We account for the quality of the synthetic control unit by diving the post-treatment RMSE by the pre-treatment RMSE).

The ratio of the RMSEs is much larger for Spain than for Australia, confirming the insight we took from the plots: the (null) placebo effect we estimated for Australia gives additional strength to our conclusion about the treatment effect we estimated for Spain.


Page History

27 November 2008 - Benchmark
25 January 2009 - Added information on British use
20 September 2009 - Added information for mounts on Hornet (CV-8) in 1942 and on Essex class (CVS-9) post-war
22 December 2009 - Added note on proposed Jean Bart reconstruction
14 January 2011 - Added information on USS Albany and USS Long Beach mountings, added cutaway sketch
27 February 2011 - Additional information on projectiles, including dyes and Chaff
13 October 2011 - Corrected typographical errors, added Mark 30 trunnion height
02 May 2012 - Added comment regarding USS Carpenter rearmament, added comments regarding missile cruiser conversions
17 October 2012 - Modified Mount / Turret note about HMS Delhi
25 October 2012 - Minor changes for clarity
24 February 2013 - Added rearmament data
05 July 2014 - Added information about crew for Mark 21 mounting and added crew diagram for Mark 21 Mounting
12 January 2016 - Added armor protection, train/elevation rates for Mark 21
03 June 2016 - Converted to HTML 5 format
07 June 2017 - Added note regarding barrel wear during Okinawa campaign
05 July 2017 - Added photograph of USS Independence and reorganized notes
20 August 2017 - Added photograph of USS Mansfield
20 February 2018 - Reorganized notes
29 September 2018 - Minor formatting changes
12 January 2019 - Added and redid mounting sketches
16 June 2019 - Added notes and sketch regarding those DEs fitted with Hedgehog ASW
31 August 2019 - Minor changes for clarity
09 August 2020 - Added use of Mark 24 mounting, added details and sketch for Mark 25 mounting, added notes regarding blast hood for mount captain and ejected cartridge cases and added photographs of USS Lamson (DD-367), USS Jarvis (DD-393) and USS Walke (DD-416)
23 October 2020 - Updated to latest template and added sketch of Mark 38 Mod 2 twin mounting
01 December 2020 - Added photograph of USS Farragut (DD-348)


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