a. Explanation Calculation: Linear regression model: A linear regression model is given as y = b 0 + b 1 x where y ^ be the predicted values of response variable and x be the predictor variable. b 1 be the slope and b 0 be the intercept of the line. The estimated least square line is y ^ = β ^ 0 + β ^ 1 x where y ^ be the predicted values of response variable and x be the predictor variable. The β ^ 1 be the estimate of slope and β ^ 0 be the estimate of intercept of the line. Expectation of y i : The expectation of least square line y i = β 0 + β 1 x i is, E ( y i ) = E ( β 0 + β 1 x i ) = β 0 + β 1 x i Thus, the expectation of y i is β 0 + β 1 x i . Expectation of y ¯ : The expectation of averaged values of least square line y ¯ = β 0 + β 1 x ¯ is, E ( y ¯ ) = E ( β 0 + β 1 x ¯ ) = β 0 + β 1 x ¯ Thus, the expectation of y ¯ is β 0 + β 1 x ¯ . Expectation of ( y i − y ¯ ) : The expectation of the term E ( y i − y ¯ ) is, E ( y i − y ¯ ) = E ( y i ) − E ( y ¯ ) = ( β 0 + β 1 x i ) − ( β 0 + β 1 x ¯ ) = ( β 1 x i − β 1 x ¯ ) = β 1 ( x i − x ¯ ) Thus, the expectation of ( y i − y ¯ ) is β 1 ( x i − x ¯ ) . Least squares estimate of slope β ^ 1 : By the principle of estimation of least squares, the value of b 1 which minimizes the error function f ( b 0 , b 1 ) = ∑ i = 1 n ( y i − ( b 0 + b 1 x i ) ) 2 is, β ^ 1 = ∑ i ( x i − x ¯ ) ( y i − y ¯ ) ∑ i ( x i − x ¯ ) 2 b. To determine Prove that V ( β ^ 1 ) = σ 2 ∑ i ( x i − x ¯ ) 2 .

Bundle: Probability and Statistics for Engineering and the Sciences, 9th + WebAssign Printed Access Card for Devore's Probability and Statistics for ... and the Sciences, 9th Edition, Single-Term

9th Edition

ISBN: 9781305779372

Author: Jay L. Devore

Publisher: Cengage Learning

See similar textbooks

Concept explainers

2-Proportion Test

In a 2-proportion test, two separate proportions are measured at the same time, and their similarity is evaluated. It is also known as a 2-proportion z-test, and a comparison of two different proportions is done in this method.

Videos

Question

Chapter 12.3, Problem 41E

To determine

Prove that β^1 is an unbiased estimator of β1.

To determine

Prove that V(β^1)=σ2∑i(xi−x¯)2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12.3, Problem 40E

Chapter 12.3, Problem 42E

Students have asked these similar questions

1 - Multiple Regression Equations and Predictions with XLMiner Analysis ToolPak (Structured) Video The owner of Showtime Movie Theaters, Inc., would like to predict weekly gross revenue as a function of advertising expenditures. Historical data for a sample of eight weeks are entered into the Microsoft Excel Online file below. Use the XLMiner Analysis ToolPak to perform your regression analysis in the designated areas of the spreadsheet. Due to a recent change by Microsoft you will need to open the XLMiner Analysis ToolPak add-in manually from the home ribbon. Screenshot of ToolPak X Open spreadsheet a. Develop an estimated regression equation with the amount of television advertising as the independent variable (to 2 decimals). JAN 27 Revenue = × TVAdv + b. Develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables (to 2 decimals). Revenue = + TVAdy + NewsAdv c. Is the estimated regression equation coefficient for…

Question 2: When John started his first job, his first end-of-year salary was $82,500. In the following years, he received salary raises as shown in the following table. Fill the Table: Fill the following table showing his end-of-year salary for each year. I have already provided the end-of-year salaries for the first three years. Calculate the end-of-year salaries for the remaining years using Excel. (If you Excel answer for the top 3 cells is not the same as the one in the following table, your formula / approach is incorrect) (2 points) Geometric Mean of Salary Raises: Calculate the geometric mean of the salary raises using the percentage figures provided in the second column named “% Raise”. (The geometric mean for this calculation should be nearly identical to the arithmetic mean. If your answer deviates significantly from the mean, it's likely incorrect. 2 points) Hint for the first part of question 2: To assist you with filling out the table in the first part of the question,…

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the range, interquartile range, variance, and standard deviation (to a maximum of 2 decimals, if decimals are necessary). Range Interquartile range Variance Standard deviation

Chapter 12 Solutions

Bundle: Probability and Statistics for Engineering and the Sciences, 9th + WebAssign Printed Access Card for Devore's Probability and Statistics for ... and the Sciences, 9th Edition, Single-Term

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.

Prove that β ^ 1 is an unbiased estimator of β 1 .

Solution Summary: The author demonstrates that stackrelbeta_1 is an unbiased estimator of