Explanation Calculation: Least squares estimate: In a linear equation y = b 0 + b 1 x i the constant b 1 be the slope and b 0 be the y -intercept form and x is the independent variable and y is the independent variable. The error function for the equation is, f ( b 0 , b 1 ) = ∑ i = 1 n ( y i − ( b 0 + b 1 x i ) ) 2 Normal equations of Y i = β 0 + β 1 x i + ε i : The first normal equations of the error function will be obtained by finding the first derivative of f ( b 0 , b 1 ) with respect to b 0 and equating it to 0. Hence, the first normal equation is n b 0 + ( ∑ x i ) b 1 = ∑ y i The second normal equations of the error function will be obtained by finding the first derivative of f ( b 0 , b 1 ) with respect to b 1 and equating it to 0. Hence, the second normal equation is ( ∑ x i ) b 0 + ( ∑ x i 2 ) b 1 = ∑ x i y i In the linear equation the point estimates of the b 0 and b 1 are β ^ 0 and β ^ 1 . These point estimates are the values that minimize the equation f ( b 0 , b 1 ) . That is, f ( β ^ 0 , β ^ 1 ) ≤ f ( b 0 , b 1 ) . Then the values of β ^ 0 and β ^ 1 are the least squares estimates and the equation y ^ = β ^ 0 + β ^ 1 x i Slope: In a linear equation y ^ = β ^ 0 + β ^ 1 x the constant b 1 be the slope and b 0 be the y -intercept form and x is the independent variable and y is the independent variable. The general formula to obtain slope is, b 1 = β ^ 1 = S x y S x x = ∑ ( x i − x ¯ ) ( y i − y ¯ ) ∑ ( x i − x ¯ ) 2 Y-intercept: In a linear equation y ^ = β ^ 0 + β ^ 1 x the constant b 1 be the slope and b 0 be the y -intercept form and x is the independent variable and y is the independent variable. The general formula to obtain y- intercept is, b 0 = β ^ 0 = y ¯ − β ^ 1 x ¯ = ∑ y i − β ^ 1 ∑ x i n Here, the objective is to prove that b 1 = β ^ 1 = S x y S x x and b 0 = β ^ 0 = y ¯ − β ^ 1 x ¯ satisfies the normal equations of Y i = β 0 + β 1 x i + ε i . First normal equation: To prove that the value n b 0 + ( ∑ x i ) b 1 is ∑ y i with the substitution of b 1 = β ^ 1 = S x y S x x and b 0 = β ^ 0 = y ¯ − β ^ 1 x ¯ . The computation procedure goes as follows: n b 0 + ( ∑ x i ) b 1 = n β ^ 0 + ( ∑ x i ) β ^ 1 = n × ( y ¯ − β ^ 1 x ¯ ) + ( ∑ x i ) × S x y S x x = n y ¯ − n S x y S x x x ¯ + n × x ¯ × S x y S x x = n y ¯ = ∑ y i Thus, the first normal equation is satisfied by b 1 = β ^ 1 = S x y S x x and b 0 = β ^ 0 = y ¯ − β ^ 1 x ¯

Bundle: Probability and Statistics for Engineering and the Sciences, Loose-leaf Version, 9th + WebAssign Printed Access Card, Multi-Term Courses

9th Edition

ISBN: 9780357099797

Author: DEVORE, Jay L.

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of tw…

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency o…

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow compar…

Videos

Question

Chapter 12.2, Problem 25E

To determine

Show that b1=β^1=SxySxx and b0=β^0=y¯−β^1x¯ satisfies the normal equations of Yi=β0+β1xi+εi.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.2, Problem 24E

Chapter 12.2, Problem 26E

Students have asked these similar questions

iid 1. The CLT provides an approximate sampling distribution for the arithmetic average Ỹ of a random sample Y₁, . . ., Yn f(y). The parameters of the approximate sampling distribution depend on the mean and variance of the underlying random variables (i.e., the population mean and variance). The approximation can be written to emphasize this, using the expec- tation and variance of one of the random variables in the sample instead of the parameters μ, 02: YNEY, · (1 (EY,, varyi n For the following population distributions f, write the approximate distribution of the sample mean. (a) Exponential with rate ẞ: f(y) = ß exp{−ßy} 1 (b) Chi-square with degrees of freedom: f(y) = ( 4 ) 2 y = exp { — ½/ } г( (c) Poisson with rate λ: P(Y = y) = exp(-\} > y! y²

2. Let Y₁,……., Y be a random sample with common mean μ and common variance σ². Use the CLT to write an expression approximating the CDF P(Ỹ ≤ x) in terms of µ, σ² and n, and the standard normal CDF Fz(·).

math

Chapter 12 Solutions

Bundle: Probability and Statistics for Engineering and the Sciences, Loose-leaf Version, 9th + WebAssign Printed Access Card, Multi-Term Courses

Ch. 12.1 - The efficiency ratio for a steel specimen immersed...Ch. 12.1 - The article Exhaust Emissions from Four-Stroke...Ch. 12.1 - Bivariate data often arises from the use of two...Ch. 12.1 - The accompanying data on y = ammonium...Ch. 12.1 - The article Objective Measurement of the...Ch. 12.1 - One factor in the development of tennis elbow, a...Ch. 12.1 - The article Some Field Experience in the Use of an...Ch. 12.1 - Referring to Exercise 7, suppose that the standard...Ch. 12.1 - The flow rate y (m3/min) in a device used for...Ch. 12.1 - Suppose the expected cost of a production run is...

Ch. 12.1 - Suppose that in a certain chemical process the...Ch. 12.2 - Refer back to the data in Exercise 4, in which y =...Ch. 12.2 - The accompanying data on y = ammonium...Ch. 12.2 - Refer to the lank temperature-efficiency ratio...Ch. 12.2 - Values of modulus of elasticity (MOE, the ratio of...Ch. 12.2 - The article Characterization of Highway Runoff in...Ch. 12.2 - For the past decade, rubber powder has been used...Ch. 12.2 - For the past decade, rubber powder has been used...Ch. 12.2 - The following data is representative of that...Ch. 12.2 - The bond behavior of reinforcing bars is an...Ch. 12.2 - Wrinkle recovery angle and tensile strength are...Ch. 12.2 - Calcium phosphate cement is gaining increasing...Ch. 12.2 - a. Obtain SSE for the data in Exercise 19 from the...Ch. 12.2 - The invasive diatom species Didymosphenia geminata...Ch. 12.2 - Prob. 25E Ch. 12.2 - Show that the point of averages (x,y) lies on the...Ch. 12.2 - Prob. 27E Ch. 12.2 - a. Consider the data in Exercise 20. Suppose that...Ch. 12.2 - Consider the following three data sets, in which...Ch. 12.3 - Reconsider the situation described in Exercise 7,...Ch. 12.3 - During oil drilling operations, components of the...Ch. 12.3 - Exercise 16 of Section 12.2 gave data on x =...Ch. 12.3 - During oil drilling operations, components of the...Ch. 12.3 - For the past decade, rubber powder has been used...Ch. 12.3 - Refer back to the data in Exercise 4, in which y =...Ch. 12.3 - Misi (airborne droplets or aerosols) is generated...Ch. 12.3 - Prob. 37E Ch. 12.3 - Refer to the data on x = liberation rate and y =...Ch. 12.3 - Carry out the model utility test using the ANOVA...Ch. 12.3 - Prob. 40E Ch. 12.3 - Prob. 41E Ch. 12.3 - Verify that if each xi is multiplied by a positive...Ch. 12.3 - Prob. 43E Ch. 12.4 - Fitting the simple linear regression model to the...Ch. 12.4 - Reconsider the filtration ratemoisture content...Ch. 12.4 - Astringency is the quality in a wine that makes...Ch. 12.4 - The simple linear regression model provides a very...Ch. 12.4 - Prob. 48E Ch. 12.4 - You are told that a 95% CI for expected lead...Ch. 12.4 - Prob. 50E Ch. 12.4 - Refer to Example 12.12 in which x = test track...Ch. 12.4 - Plasma etching is essential to the fine-line...Ch. 12.4 - Consider the following four intervals based on the...Ch. 12.4 - The height of a patient is useful for a variety of...Ch. 12.4 - Prob. 55E Ch. 12.4 - The article Bone Density and Insertion Torque as...Ch. 12.5 - The article Behavioural Effects of Mobile...Ch. 12.5 - The Turbine Oil Oxidation Test (TOST) and the...Ch. 12.5 - Toughness and fibrousness of asparagus are major...Ch. 12.5 - Head movement evaluations are important because...Ch. 12.5 - Prob. 61E Ch. 12.5 - Prob. 62E Ch. 12.5 - Prob. 63E Ch. 12.5 - The accompanying data on x = UV transparency index...Ch. 12.5 - Torsion during hip external rotation and extension...Ch. 12.5 - Prob. 66E Ch. 12.5 - Prob. 67E Ch. 12 - The appraisal of a warehouse can appear...Ch. 12 - Prob. 69SE Ch. 12 - Forensic scientists are often interested in making...Ch. 12 - Phenolic compounds are found in the effluents of...Ch. 12 - The SAS output at the bottom of this page is based...Ch. 12 - The presence of hard alloy carbides in high...Ch. 12 - The accompanying data was read from a scatterplot...Ch. 12 - An investigation was carried out to study the...Ch. 12 - Prob. 76SE Ch. 12 - Open water oil spills can wreak terrible...Ch. 12 - In Section 12.4, we presented a formula for...Ch. 12 - Show that SSE=Syy1Sxy, which gives an alternative...Ch. 12 - Suppose that x and y are positive variables and...Ch. 12 - Let sx and sy denote the sample standard...Ch. 12 - Verify that the t statistic for testing H0: 1 = 0...Ch. 12 - Use the formula for computing SSE to verify that...Ch. 12 - In biofiltration of wastewater, air discharged...Ch. 12 - Normal hatchery processes in aquaculture...Ch. 12 - Prob. 86SE Ch. 12 - Prob. 87SE

Knowledge Booster

$Statistics$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Show that b 1 = β ^ 1 = S x y S x x and b 0 = β ^ 0 = y ¯ − β ^ 1 x ¯ satisfies the normal equations of Y i = β 0 + β 1 x i + ε i .

Solution Summary: The author explains how the least squares estimators b_1=stackrelbetayxxx satisfies the normal equations

Concept explainers

Videos

Show that b1=β^1=SxySxx and b0=β^0=y¯−β^1x¯ satisfies the normal equations of Yi=β0+β1xi+εi.

Want to see the full answer?

Chapter 12 Solutions