CAS MA 115 Lecture Notes - Lecture 3: Point Estimation, Frequency Distribution, Regional Policy Of The European Union

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1 May 2018

School

Boston University

Department

Mathematics & Statistics

Course

CAS MA 115

Professor

Alvard Arazyan

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CHAPTER 3 – NUMERICALLY SUMMARIZING DATA

Section 3.1 – Measures of Central Tendency (what is happening to data on average)

Objective 1: Determine the Arithmetic Mean of a Variable from Raw Data

• Arithmetic Mean (of a variable) – computed by adding all the values of the variable in the

data set and dividing by the number of observations

o Population Arithmetic Mean (μ) – computed by using all the individuals in a

population

▪ Important to note that this is a parameter

▪ If X1 + X2 +…+XN are N observations of a variable from a population,

then the population mean:

▪ μ = 

 = 



o Sample Arithmetic Mean (x

̄) – computed by using sample data

▪ Important to note that this is a statistic

▪ If X1 + X2 +…+Xn are n observations of a variable from a population, then

the population mean:

▪ x

̄ = 

 = 



▪ Point Estimate – a single value used to estimate the population arithmetic

mean

• Not entirely accurate but a good base point

• Sample Size and Interval Length are interrelated in determining the

point estimate

• When to Use: When data is quantitative and the frequency distribution is roughly

symmetric

Objective 2: Determine the Median of a Variable from Raw Data

• Median (of a variable) (M) – the value that lies in the middle of the data when arranged in

ascending order

o Steps to Find the Median:

▪ 1. Arrange the data in ascending order

▪ 2. Determine the number of observations (n)

▪ 3. Determine the observation in the middle of the data set

o If the data set is odd, the median is the observation in the 

 position

o If the data set is even, the median is the observations in the 

 + 1position

• When to Use: When data is quantitative and the frequency distribution is skewed

left/right due to outliers

Objective 3: Explain What It Means for a Statistic to be Resistant

• Resistant Numerical Summary of Data – if extreme values (very large/small) relative to

the data do not substantially affect its value (ex: resistant – range, IQR, median v. non-

resistant – standard deviation, variance, mean)

• Relation Between Mean/Median/Distribution Shape

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o skewed left (smaller values) = mean is substantially smaller than the median

▪ The data set contains drastically smaller observations

o symmetric bell-shaped = mean is roughly equal to the median

o skewed right (larger values) = mean is substantially larger than the median

▪ The data set contains drastically larger observations

• The mean will always be more affected than the median (because the mean accounts for

individual values while the median accounts for the total count)

• When to Use: When data is skewed, use the median. When data is symmetric, use the

mean.

Objective 4: Determine the Mode of a Variable from Raw Data

• Mode (of a variable) – the most frequent observation of the variable that occurs in the

data set

o Data can have no/one/more than one mode

▪ ex: none of the numbers occur more than once = no mode

▪ ex: all of the numbers occur three times = all the numbers are the mode

• When to Use: When the most frequent observation is needed or if data is qualitative

Section 3.2 – Measures of Dispersion (what is happening to data with outliers)

Objective 1: Determine the Range of a Variable from Raw Data

• Range (of a variable) (R) – the difference between the largest and smallest data values

o Formula = (largest data value – smallest data value)

Objective 2: Determine the Standard Deviation of a Variable from Raw Data

• Population Standard Deviation (of a variable) (σ) – the square root of the sum of squared

deviations about the population mean divided by the number of observations in the

population

o Formula =

o Computational Formula =

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Document Summary

Section 3. 1 measures of central tendency (what is happening to data on average) Important to note that this is a parameter. If x1 + x2 + +xn are n observations of a variable from a population, then the population mean: = x(cid:2869) + x(cid:2870) + +xn. = (cid:3046)(cid:3048)(cid:3040) (cid:3042)(cid:3033) (cid:3039)(cid:3039) (cid:3049)(cid:3039)(cid:3048)(cid:3032)(cid:3046: sample arithmetic mean (x ) computed by using sample data. Important to note that this is a statistic. If x1 + x2 + +xn are n observations of a variable from a population, then the population mean: x = x(cid:2869) + x(cid:2870) + +xn. Objective 2: determine the median of a variable from raw data: median (of a variable) (m) the value that lies in the middle of the data when arranged in ascending order, steps to find the median, 1. Determine the number of observations (n: 3. Section 3. 2 measures of dispersion (what is happening to data with outliers)

Related Questions

Just need the code for (a) I haven't had any luck. Thanks inadvance!

you must use matlab or octave program. others are not allowed T.T. please use only above the two program to get solution.

Problem 3: The Inverse Transformation Method A general method for simulating a RV having a continuous distribution called the inverse transformation method is based on the following result (see also Section 10.2 of the textbook and Quiz 2) Proposition 2.1. The Inverse Transformation Method). L UN U(0,1). For any continuous et distribution function F, X F-1(U) has distribution function F. (F-1(ar) is defined to equal that value y for which F(y) r.) In order to simulate X n Exp(1), for example, generate an U U(0,1) and let X log(1 -U) (see Quiz 2(c)). In fact, X log(U) is also Exp(1), since 1 U (0,1). Since X Exp(A) if X Exp(1) (why?), it follows that log(U)A has an Exp(1) distribution. This can be easily illustrated in Octave xi -log Grand (100,1)); 100 random samples from Exp (1) x2 -log (rand (100, 1))/2; 100 random samples from Exp(2) x3 -log Grand (100, 1))*2; 100 random samples from Exp(0.5) [mean (x1) mean(x2) mean (x3)] check their mean values In addition, Y X Gamma(n, A) if X Xn are i.i.d. Exp(A). Therefore, if Un are i.i.d. U(0, 1), log (Ui) 1 log Ui Gamma(n, A) (1) i-1 i-1 (a) Write an Octave code to generate 100 random samples from Gamma(10, A) by using the inverse transformation method (1). Let A 0.5, 1, 2. Plot histograms for the Gamma random samples. Octave also has a built-in gamma RV generator gamrnd (google it for reference!). Generate 100 random samples of Gamma(10, A) using gamrnd, make histograms, and compare two results. (b) As a function of U U(0, 1), create a new RV W with CDF F(w) 1 e-ti (w 0). (This is Quiz 2 (b).) Write an Octave code to generate 100 random samples of W F using the inverse transformation method (c) Consider a method for generating Weibull(a,B) RVs that have the following CDF F(z) exp(-ar 0 z oo, a Note that W in (b) is in fact Weibull(1,2). Write an Octave function myweibrnd.m that will generate n random samples for given parameters a and B. It should have the following format function CRJ myweibrnd n, alpha, beta) this function receives sample size n and parameters alpha and beta as input arguments and return R Ca vector of n random samples) (put your code here) Using your myweibrnd function, generate 1000 simulated values when (i) (a, B) (1,2) and (ii) (a,B) (2,1), respectively. Plot histograms and estimate their mean and variance

Show transcribed image text Problem 3: The Inverse Transformation Method A general method for simulating a RV having a continuous distribution called the inverse transformation method is based on the following result (see also Section 10.2 of the textbook and Quiz 2) Proposition 2.1. The Inverse Transformation Method). L UN U(0,1). For any continuous et distribution function F, X F-1(U) has distribution function F. (F-1(ar) is defined to equal that value y for which F(y) r.) In order to simulate X n Exp(1), for example, generate an U U(0,1) and let X log(1 -U) (see Quiz 2(c)). In fact, X log(U) is also Exp(1), since 1 U (0,1). Since X Exp(A) if X Exp(1) (why?), it follows that log(U)A has an Exp(1) distribution. This can be easily illustrated in Octave xi -log Grand (100,1)); 100 random samples from Exp (1) x2 -log (rand (100, 1))/2; 100 random samples from Exp(2) x3 -log Grand (100, 1))*2; 100 random samples from Exp(0.5) [mean (x1) mean(x2) mean (x3)] check their mean values In addition, Y X Gamma(n, A) if X Xn are i.i.d. Exp(A). Therefore, if Un are i.i.d. U(0, 1), log (Ui) 1 log Ui Gamma(n, A) (1) i-1 i-1 (a) Write an Octave code to generate 100 random samples from Gamma(10, A) by using the inverse transformation method (1). Let A 0.5, 1, 2. Plot histograms for the Gamma random samples. Octave also has a built-in gamma RV generator gamrnd (google it for reference!). Generate 100 random samples of Gamma(10, A) using gamrnd, make histograms, and compare two results. (b) As a function of U U(0, 1), create a new RV W with CDF F(w) 1 e-ti (w 0). (This is Quiz 2 (b).) Write an Octave code to generate 100 random samples of W F using the inverse transformation method (c) Consider a method for generating Weibull(a,B) RVs that have the following CDF F(z) exp(-ar 0 z oo, a Note that W in (b) is in fact Weibull(1,2). Write an Octave function myweibrnd.m that will generate n random samples for given parameters a and B. It should have the following format function CRJ myweibrnd n, alpha, beta) this function receives sample size n and parameters alpha and beta as input arguments and return R Ca vector of n random samples) (put your code here) Using your myweibrnd function, generate 1000 simulated values when (i) (a, B) (1,2) and (ii) (a,B) (2,1), respectively. Plot histograms and estimate their mean and variance

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CAS MA 115 Lecture Notes - Lecture 3: Point Estimation, Frequency Distribution, Regional Policy Of The European Union

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