MAST10011 Chapter Notes - Chapter 8: Confidence Interval, Test Statistic

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magentamuskrat643

12 Nov 2018

School

University of Melbourne

Department

Mathematics and Statistics

Course

MAST10011

Professor

Dr Rheanna Mainzer

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Document Summary

Regression: used for prediction, when you want to use x to predict y. Correlation: used to measure association between x and y. The average increase in y when x is increased by one. A(cid:374)d are esti(cid:373)ated, (cid:449)e ha(cid:448)e a fitted (cid:373)odel. The fitted model is used to estimate the average value of y for a given value of x. It is also used to predict a future observation of y for a given value of x. Estimating and : least squares (cid:2010) =(cid:1870)(cid:1871)(cid:3052)(cid:1871)(cid:3051), (cid:2009) =(cid:1877) (cid:2010) (cid:1876) Variances should be equal - look for trends in a residuals vs fitted plot should be randomly scattered. Should be normally distributed check with qq plot residual sum of squares: (cid:1856)(cid:2870)= ((cid:1877) (cid:2009) (cid:2010) (cid:1876))(cid:2870) error mean squared (to estimate (cid:2870)): (cid:1871)(cid:2870)= 2 (cid:2870)= (cid:2869) (cid:2870)(cid:1871)(cid:3052)(cid:2870)(cid:4666)(cid:883) (cid:1870)(cid:2870)(cid:4667) H0: = 0 (cid:894)testi(cid:374)g the utilit(cid:455) of the (cid:373)odel: is (cid:454) of a(cid:374)(cid:455) use i(cid:374) predi(cid:272)ti(cid:374)g (cid:455)? (cid:895).

Related Questions

Researchers measured the blood pressures of 22 students in

two situations: when the students were relaxed and when the students were taking an

important examination. The file

systolic.txt

lists the systolic and diastolic pressures for

each student in each situation. Dataset is available at

systolic_url = "https://mcs.utm.utoronto.ca/~nosedal/data/systolic.txt"

Use R

to answer each of the following questions. You have to show

all your work

to get

full credit.

Answers, even if correct, with no justifications will not receive any

marks

Using R

, compute the change in systolic pressure by subtracting systolic pressure when

relaxed from systolic pressure during the exam; call this variable

Using R

, repeat part a) for diastolic pressure. Call the resulting variable

Using R

, make a scatterplot of

versus

, fit a regression line to the data and add

this line to your plot. Use: 2 c) your last name as the main title for your plot. (In my

case it would be: 2 c) Nosedal).

Important note. Use

as your response variable and label horizontal and vertical axes.

d) What is the fitted regression model? Provide slope and y-intercept

, make a residual plot from the regression in part c). Does this plot call into

question the use of a linear model? Why?

Note. Use:

2 e) your last name

as the main title for your plot. (In my case it would

be:

2 e) Nosedal

f) Note the outlier in the residual plot (and on the scatterplot from part c)).

Using

, delete the outlier from the data set. Then repeat parts c) and e). Note. Use:

2 f )

your last name

as the main title for your plots. (In my case it would be:

2 f ) Nosedal

g) What is the fitted regression model (after the outlier has been removed)? Provide slope

Modeling Data

The endpoints of the interval over which distinct vision is possible are called the near point and far point of the eye. With increasing age, these points normally change. The table shows the approximate near points (in inches) for various ages (in years). (Source: Ophthalmology & Physiological Optics)

Age, x	16	32	44	50	60
Near Point, y	3.0	4.7	9.8	19.7	39.4

(a) Find a rational model for the data by taking the reciprocals of the near points to generate the points ( x , 1 / y ) . Use the regression capabilities of a graphing utility to find the least squares regression line for the revised data. The resulting line has the form . Solve for .

(b) Use a graphing utility to plot the data and graph the model.

Average price per square foot of floor area in new one-family houses sold. Price per square foot of floor area excludes value of improved lot Source: Year 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Price per square foot $55.18 $57.41 $60.21 $62.45 $64.38 $66.81 $67.83 $67.70 $70.43 $71.93 $75.68 $79.21 $85.13 $90.63 $91.99 Instead of working on portions of the scatter plots, we want to approximate the data by a line we can use to predict prices in the future. The line that will best fit the given data is called the linear regression. The approximation will be acceptable whether the determina- tion coefficient R (r**2) is close to 1 or the correlation coefficient r is close to -1 or +1.

a) using your entered data to find the equation y = ax + b of the best line through the data points and give the values of coefficients a and b, and of the determination coefficient R or correlation coefficient r.

b. Use Excel to plot the scatter plot and the regression line on the same graph.

c. Is the linear regression a visually good approximation of the data? Sustain your argument by a mathematical proof.

d. use the linear regression to predict the price per square foot area in 2009 e. In what year will the price per square foot be greater than $199?

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MAST10011 Chapter Notes - Chapter 8: Confidence Interval, Test Statistic

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