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In a study of copper bars, the relationship between shear

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 13E Chapter 7.3

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 13E

In a study of copper bars, the relationship between shear stress in ksi \((x)\) and shear strain in % \((y)\) was summarized by the least-squares line \(y=-20.00+2.56 x\). There were a total of \(n=17\) observations, and the coefficient of determination was \(r^{2}=0.9111\). If the total sum of squares was \(\sum\left(y_{i}-\bar{y}\right)^{2}=234.19\), compute the estimated error variance \(s^2\).

Equation Transcription:

Text Transcription:

(x)

(y)

y=-20.00+2.56x

n=17

r^2=0.9111

Sigma(yi-overline{y})^2=234.19

s^2

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Study Guide EC202 Midterm (Midterm is on 4/27 at regular class time) 1. Overview of Macroeconomics ○ Economy as a whole ○ Consider overall quantities ○ Government policy as a tool to influence the state of the economy ○ We will think of the overall quantities as having been formed by the interactions of individual agents and firms ○ All markets interact simultaneously 2. Supply/Demand/GDP ○ Demand ■ Demand curve: relationship between the price of a good and the quantity demanded of that good ■ Law of Demand: price and quantity demanded and negatively related ■ Types of demand: ● Individual: demand for a specific good ban individual ● Market: demand for a specific good by​ll individuals ● Aggregate: total demand for all goods and servic in an economy ○ Supply ■ Supply curve: relationship between the price of a good and the quantity supplied of that good ■ Law of Supply: price and quantity supplied and positively related ■ Types of supply: ● Firm: supply of a specific good b​ firm ● Market: supply of a specific good ball firms ● Aggregate: total supply ofall goods and servic produced within an economy ○ Equilibrium ■ Occurs when the price level is such that the quantity demanded is equal to the quantity supplied (when the demand curve crosses the supply curve) ○ 3 Main Macro Variables: ■ Output (GDP, industrial production, etc.) ■ Prices (CPI, PCE, etc.) ■ Employment (# of people employed, # of hours worked) ○ Growth Rates ■ Percentage change in a variable ■ Growth rate= ((value in later period­value in earlier period)/ value in earlier period) * 100% alt form: ((new­old)/old)*100% ○ Gross Domestic Product (GDP): ■ The market value of all final goods and services produced in a country in a given time period ■ AKA aggregate output, aggregate income ■ Calculating GDP (expenditure approach): adding up all the things that are purchased minus the imports that weren’t made in the country (GDP=C+I+G+X­M) ● Things that are purchased: ○ C­consumption by households (buying goods) ○ I­investment by firms (factory upgrade) ○ G­government spending (public defense) ○ X­exports to other countries (making goods in the country and selling them to another country) ○ M­imports from other countries (buying goods made in another country) ■ “All sales income has to end up in someone’s pocket eventually” ○ Two types of goods: ■ Intermediate good: a good used in the production of another good ■ Final good: a good consumed by the consumer 3. Measuring Economic Performance ○ Measuring Welfare (or standard of living): ■ Total aggregate income (GDP) ● Remember there is possibility of one person having most of the wealth in a region ■ Average aggregate income (GDP per person) ■ The ​ distribution of aggregate income (considers rich vs. poor disparities) ■ Measuring Happiness ● Culture matters: what country you grow up in influences how “happy” you are ● Demographics: women are “happier” than men, conservatives are “happier” than liberals ● Genetics matter: sometimes genetic makeup plays a factor in “happiness” ○ Recession vs. Expansion: ■ Recession: a recession is a period in which real GDP decreases. ● The ​ growth rate is negative for at least two successive periods ● Recession lasts from the peak to the trough ■ Expansion: an expansion is a period during in which real GDP increases. ● This is the entire time from atrough (lowest point) to peak (highest point) ■ Turning point: when an economy hits the trough or peak, we say that it has reached a turning point. ○ Prices: ■ The price level is the average level of prices and the value of money ■ A persistently rising level is calleinflation (positive growth). ■ A persistently falling price level is calldeflation (negative growth). ○ Consumer Price Index (CPI): ■ The CPI measures the average of the prices paid by urban consumers for a “fixed” basket of consumer goods and services ● They are used for the purposes of comparison across time ● The values of two different indexes cannot be compared ● The values of a price index are not in terms of dollar(there are NO units) ● An index is ​defined to equal 100 in threference base period ● The reference base period could in theory be any period ■ Constructing the CPI: ● 1) Selecting the CPI basket (which goods/services to look at) ● 2) Conducting a monthly price survey ● 3) Calculating the CPI ○ Take all the items and determine the quantity and price of specific items and total them ■ Biases in the CPI: the CPI might ​ overstatethe true inflation rate for four reasons: ● New goods: new types of goods are often more expensive than comparable goods (i.e. computers vs. typewriters) ● Quality change: some goods improve their quality over time and should be more expensive (2016 TV vs. 1950 TV) ● Commodity substitution: people substitute away from more expensive goods, keeping the price of the bundle low ● Outlet substitution: people substitute away from more expensive stores, keeping the price of their bundle low ○ Nominal VS. Real ■ It is important to distinguish between the two because nominal values are influenced by inflation and real values aren’t. ■ Real value = Nominal value * (prices in base period/prices in the current period) Nominal Real Variable is a variable in terms of the is a variable in terms of the prices in thcurrent​ year prices in a​ommon base year (we will be focusing on “real” because it is more accurate to view inflation and such) GDP is the value of goods and is the value of final goods services produced during a and services produced in a given year when valued at given year valued at the prices that prevailed in that he prices of a reference base year same year Wages Wages valued at the prices Wages valued at the prices that prevailed in that same of a reference base year year ■ Inflation Rate: ● the percentage change in the price level from one year to the next ● inflation rate = ((CPI this year ­ CPI last year)/ CPI last year)*100% alt form: ((new­old)/old)*100% ● Core inflation: is the CPI inflation rate excluding the volatile elements (of food and fuel) 4. Unemployment ○ Definitions: ■ Employed: those who have a job ■ Unemployed: those who do not have a job, ​ but would like to have one ■ Labor force: those who want to work, ​ whether or not they have a job (employed + unemployed) ■ Out of labor force: those who do not have a job, and do not want one, but could have one ■ Young and institutionalized: those who cannot work (the young and the elderly) ○ Measuring unemployment: ■ Recall that in this section the focus is measuring macroeconomic conditions ■ 3 labor market indicators: ● The unemployment rate: the percentage of the labor force that is unemployed ○ =(#of people unemployed/labor force)*100 ○ Note that the number of unemployed and the unemployment rate are measuring two different things ● The employment­to­population ratio: the percentage of the working­age population who have jobs ○ = (employment/working­age population)*100 ● The labor force participation rate: the percentage of the working­age population who are members of the labor force ○ = (labor force/working­age population)*100 ■ Types of unemployment: ● Frictional: unemployment that arises from normal labor market turnover ● Structural: is unemployment created by changes in technology and foreign competition that change the skills needed to perform jobs or the location of jobs (there’s no way to avoid this) ○ Example: buggy whip makers were unemployed when cars were invented and got more popular, because buggies weren’t being made any more so neither were buggy whips. ● Cyclical: is the higher than normal unemployment at a business cycle trough and lower than normal unemployment at a business cycle peak. ■ Full employment: (important!) ● Natural unemployment= frictional + structural ○ The natural unemployment rate is natural unemployment as a percentage of the labor force ○ Full employment is defined as the situation in which the unemployment rate equals the natural unemployment rate (when cyclical is at 0) *this is great! ○ Unemployment is ​ below the natural rate in expansions, and above​ the natural rate in recessions ■ Unemployment and output ● Potential GDP is the quantity of real GDP produced at full employment ● Real GDP minus potential GDP is the ​output​gap ○ Output gap= ​ Real GDP­Potential GDP ● The output gap is positiv during expansions, andnegative during recessions. 5. Economic Growth ○ Economic growth: the sustained expansion of production possibilities measured as the increase in real GDP over a given period ■ Business cycles: fluctuations​round potential GDP ■ Economic growth: an ​ increase in potential GDP itself (increasing the sustainability) ○ GDP per Capita: ■ When we want to exclude the effects of increased population ● GDP per capita= GDP/population (dollars per person) ● Note: this could either be nominal GDP or real GDP per capita ○ Rule of 70: ■ The rule of 70 states that the number of years it takes for the level of a variable to double approximately 70 divided by the annual percentage growth rate of the variable ○ Potential GDP: the sustainable level of output an economy can produce, and is influenced by many things: ■ Natural resources (land)* ■ The level of capital (factories, education, etc)* ■ “Entrepreneurship” (hard to define)* ● *all fixed in short run ■ Labor ○ Potential GDP is defined to be the level of real GDP that is achieved when the economy is operating atfull employment ■ Look at: ● An aggregate production function (how much produced) ● An aggregate labor market (what’s available) ○ Economic Model: ■ Attempts to formally describe some aspect of the real world ■ Simplifies as much as possible, while still capturing the most important features of reality ■ Often this means that we will not want to describe all features of reality in order to have a simpler model ○ Aggregate Production Function: ■ A description of the relationship between real GDP and quantity of hours worked. For simplicity, we ignore the fact that there are many different types of work. ● Features: ○ If there is no labor, there is no production ○ Adding more labor always increases real GDP ○ There are decreasing returns to scale in labor: each additional hour of hour of work increases real GDP less than the previous hour did ○ Starting at 0 and increases in a diminishing marginal return rate ■ Aggregate Labor Market: describes the supply and demand for labor in the economy ● Firms ​ demand ​ labor ● Individuals ​ supply labor ● The ​ real wage​ individuals receive the price of labor ○ Nominal: the wage you are paid in terms of dollars ○ Real: the wage you are paid in terms of purchasing power ○ Real Wage=nominal wage* (CPI base year/CPI current year) ■ Want to use Real wage! ■ Supply of Labor: ● The relationship between the total number of hours individuals want to work and the real wage rate ○ It matters what the real wage rate is ○ Law of Supply: as the real wage increases, people are willing to work more hours ■ Demand for Labor: ● The relationship between the total number of hours firms want to hire individuals to work and the real wage rate ○ Firms are willing to pay workers according to their productivity level ○ Law of Demand: as the real wage decreases, firms want to hire more workers ■ Equilibrium in the Labor Market ● At the intersection of supply and demand ● There’s no pressure for the real wage to change ■ Labor Surplus: ● Equilibrium real wages is too high ○ There will be a surplus of labor ○ There is pressure for the real wage to fall ○ Will bring it back to equilibrium ■ Labor Shortage: ● Equilibrium real wage is too low ○ There will be a shortage of labor ○ There is pressure for the real wage to rise ○ Will bring it back to equilibrium ○ Full Employment ■ Growth of labor supply ● Probable causes: ○ Individuals want to work more ○ Larger population joins the labor force ○ The population increases ■ Shift the labor supply curve outwards ■ Movement along the production function ○ Labor Productivity ■ Amount of additional output created by an additional hour of work ■ Labor Productivity = real GDP/aggregate labor hours ■ Growth: ● Increase in capital ● Increase in education ● Increase in technology ○ If worker productivity increases, then firms will want to hire more workers at any wage ○ This causes a shift right in the labor demand function ■ Effect of an increase in labor productivity on potential GDP ● There is a shift in the production function upwards ● This directly increases potential GDP ● Because productivity is higher, the labor demand curve shifts to the right ● The shift in labor demand increases equilibrium real wage and the full employment level of hours worked ● The increase in labor further increases potential GDP ○ Theories of Growth: ■ Classical: the view that the growth of real GDP per person is temporary and that when it rises above the subsistence level, a population explosion eventually brings real GDP per person back to the subsistence level. Prediction: growth will eventually stop, and individual standards of living will eventually fall back to subsistence. ■ Neoclassical (Solo model): the view that real GDP per person grows because of increases physical capital per person ○ Because there are diminishing returns to capital, eventually capital per person stops growing. ○ At that point the economy no longer grows, but it also does not fall back to subsistence. Prediction: growth will eventually stop, individual standards of living will not fall ■ New: holds that real GDP per person grows because: ● People continually innovate produce new technologies and increases profits ● Technological improvements are a public good (can be enjoyed by everyone) ● Knowledge is ​ not ​ubject to diminishing returns Prediction: growth can potentially continue forever, and individual standards of living will continue to rise ○ Aggregate Production 6. Finance, Saving, and Investment ○ Terms to know: ■ Income: the quantity of wage and other income that people receive per unit time ■ Wealth: the value of all the things that people own at a given time ■ Saving: the amount of income that is not paid in taxes or spent on consumption goods and services, ​er unit time ■ Interest: price of borrowing money ■ Gross Investment ● Investment: specifically means the purchase of physical capital by firms ● The total quantity of physical capital purchased by firms in a year ● As firms invest, labor productivity increases ■ Depreciation ● Whereas investment increases the level of physical capital, we also know that over time physical capital can break or become obsolete ● This general reduction in the level of physical capital is depreciation ■ Net Investment ● The overall change in the quantity of physical capital when taking into account both gross investment and depreciation ● Net investment=gross investment ­ depreciation ○ Financial asset ■ an exchange of funds between a lender​ and aborrower​ along with a contract for repayment, potentially including interest ■ Borrower’s perspective ● A financial asset involves ○ Receiving a lump of funds immediately ■ Lender’s perspective ● A financial asset involves: ○ A good that is ​purchased ○ At a given p​ ric, and which ○ Provides a known ​ return ■ The price of a financial asset is the amount that is loaned, and the return is the amount that needs to be repaid: ● return=price + interest ■ The interest rate is the growth rate of your principal ● Interest rate= (return­price)/price *100% ● Interest rate= (price + interest ­ price)/price *100% ● interest/price *100% ○ Nominal Interest Rate ■ The interest expressed as a percentage of the loan amount ○ Real Interest Rate ■ The nominal interest rate adjusted to remove the effects of inflation ■ Real interest rate = nominal interest rate­inflation rate *Just as firms and workers care about real wage rather than nominal wage, they care about real interest rate rather than nominal interest rate. ○ Savings and Lending ■ Because individuals save​, they have funds available for lending. There are three primary type ofinancial markets​ in which individuals lend funds. ● Loan markets ● Bond markets ● Stock markets ■ Market for Loanable Funds ● The market for loanable funds is the aggregate of all the individual financial markets ● Demand for Loanable funds: the demand for loanable funds is the relationship between the quantity of loanable funds demanded (by firms, for investment) and the real interest rate. ○ The quantity of loanable funds demanded depends on ■ The real interest rate ■ Expected profit ○ The effect of real interest rates on demand for loanable funds: ■ When the real interest rate is higher, it is more costly to borrow money. ■ Thus the quantity of loanable funds demanded falls (and vice­versa). ■ This is a m​ovement along ​ the demand curve. ○ The effect of increased expected profits on demand for loanable funds: ■ When expected profit is higher, firms will want to invest more at any real interest rate ■ When firms want to invest more, their demand for loanable funds increases ■ This shifts the demand curve to the right ● Supply of Loanable Funds: the relationship between the quantity of loanable funds supplied (by individuals) and the real interest rate ○ The quantity of loanable funds depends on: ■ The real interest rate ■ Disposable income ■ Expected future income ■ Wealth ■ Default risk ○ Effect of real interest rates on the supply of loanable funds: ■ When the real interest rate is higher, lenders receive more interest for each loan. ■ Thus the quantity of loanable funds supplied increases (and vice­versa). ■ This is a m​ovement along ​ the supply curve. ○ Effect of an increase in disposable income on the supply of loanable funds: ■ When there is an increase in disposable income, individuals tend to save more ■ The increase in saving means an increase in lending at any real interest rate ■ This the supply of loanable funds supplied increases ■ This a ​ shif of the supply curve to the right. ○ Effect of an increase in expected future income or wealth on the supply of loanable funds: ■ When there is an increase in expected future income or wealth, individuals tend to save less ■ The decrease in saving means a decrease in lending at any interest rate ■ Thus the supply of loanable funds supplied decreases. ■ This is a ​shiftof the supply curve to the left. ○ Effect of an increase in default risk on supply of loanable funds: ■ Default risk is the risk that the person you lent money to does not pay you back. ■ When there is an increase in default risk, individuals are less likely to lend at any real interest rate. ■ Thus the supply of loanable funds supplied decreases. ■ This is a ​shiftof the supply curve to the left. ● Remember that e ​quilibrium ​is when market supplied equals market demanded (where the two curves cross) 7. Markets ○ Money: any commodity or token that is generally acceptable as a means of payment ■ Functions of money ● Means of payment: a means of payment is a method of settling a debt ● Medium of exchange ○ A medium of exchange is an object that is generally accepted in exchange for goods and services ○ In the absence of money, people would need to exchange goods and services directly, which is called barter ○ Barter requires a double coincidence of wants, which is rare, so barter is costly. ■ Unit of account: an agreed measure for stating the prices of goods and services ■ Store of value: as a store of value, money can be held for a time and later exchanged for goods and services ■ Definition of money: money in the US consists of: ● Currency: the notes and coins held by households and firms ● Deposits at banks and other depository institutions ■ Measuring money: ● The two main official measure of money in the US are M1 and M2 ○ M1: consists of currency and traveler’s checks and checking deposit accounts owned by individuals and businesses. ○ M2: consists of M1, and saving deposits, money market mutual funds, and other deposits. ● Are M1 and M2 really money ○ All the items in M1 are means of payment, They are money. ○ Some savings deposits in M2 are note means of payments­they are called liquid assets ■ Liquidity is the property of being instantly convertible into a means of payment with little loss of value. ● Things that aren’t money: ○ Checks: not money, it is an instruction to a bank to transfer money ○ Credit cards: takes out a loan and the loan must be repaid ● Depository Institutions ○ A firm that takes deposits from households and firms and makes loans to other households and firms ○ Banks are the more prominent type ■ How do banks work: 1. To generate profit, the interest rate at which a bank lends must be higher that the interest rate it pays on deposits 2. They have to balance.. a. Loans generate profit b. Depositors must be able to obtain their funds when they want them 3. To make sure that banks do not make too many loans, they are required to keep a portion of all deposits a. The ​reserve ratio is the percentage of deposits that cannot be loaned out ■ Benefits of banks (they are important): 1. Create liquidity 2. Pool risk: the make things less risky by having several different types of people using the bank 3. Lower the cost of borrowing 4. Lower the cost of monitoring borrowers ■ Federal Reserve System ● Is the ​central bank​ of the US­ it’s the bank for all other banks ● The public authority that regulates a nation’s depository institutions and controls the quantity of money­ controls the supply of money ● The Fed’s goals: ○ Keep inflation in check ○ Maintain full employment ○ Moderate the business cycle ○ Contribute toward achieving long­term growth **Really just want to keep the economy in check as a whole ● Policy Tools: ○ Open market operations:​ the federal reserve can increase or decrease the money supply ○ Last resort loans:​ commercial banks are guaranteed to be able to get loans from the Federal Reserve ○ Required reserve ratios:​ control how much reserves commercial banks must hold. ● Federal Funds Rate: the interest rate that commercial banks charge each other on overnight loans or reserves ○ Very short term borrowing from other banks is done all the time to make sure banks hold the required amount of reserves. ○ The federal funds rate is the price of a very short term loan between banks ● Structure of the Fed: ○ The Board of Governors: 7 members with 12 year terms ○ The regional Federal Reserve banks: there are 12 regional Federal Reserve banks ○ The Federal Open Market Committee ● Open Market Operations: ○ The purchase or sale of government bonds by the Fed from or to a commercial bank or the public ■ When the de buys securities, it pays for them with newly created money, s​o the money supply increases ■ When the fed sells securities, they are paid for with money held by banks,​ so the money supply decreases. ***​ o open market operations influence bank’s reserves ● Federal Open Market Committee: ○ Who controls the open market operations ○ The FOMC is the main policy­making group in the Federal Reserve System ○ The FOMC meets every six weeks to formulate m​onetary policy, ○ Money Market ■ Describes the supply and demand relationship for money ■ Terms: ● Money vs Loanable funds ○ Individuals demand​ money, andsupply loanable funds (through individuals that are saving) ○ Loanable funds ​pay interes, where money does not. ○ Loanable funds are ​less liqu than money ● Real vs Nominal Interest Rates ○ Nominal interest rate is the given interest rate on a loan ○ Real interest rate is the nominal interest rate adjusted to remove the effects of inflation on the purchasing power of money ○ Real interest rate = nominal interest rate ­ inflation ● Real vs Nominal Wage ○ Nominal money: money in terms of dollars ○ Real money: money in terms of the purchasing power of dollars ○ Real money = nominal money * (CPI base year/CPI current year ****We will focus on the supply and demand of real money ■ Demand for money ● The quantity of money that people plan to hold depends on… ○ The ​ price level:higher prices make us hold more money ○ The ​ nominal interest rate:​ this is the​rice of holding money ○ Real GDP​ : higher GDP (which is higher income) increases expenditures, so people need more money ○ Financial innovation​ : financial innovation makes interest­bearing assetsmore liquid​, so people hold less money ■ Supply of Money ● Is dictated by the Federal reserve ● Depends on: ○ The monetary base ○ The required reserve ration ○ The currency drain ratio All Equations: ● Growth rate= ((value in later period­value in earlier period)/ value in earlier period)*100 ○ Try to remember: ((new­old)/old)*100 ● GDP=C+I+G+X­M ■ C­consumption by households (buying goods) ■ I­investment by firms (factory upgrade) ■ G­government spending (public defense) ■ X­exports to other countries (making goods in the country and selling them to another country) ■ M­imports from other countries (buying goods made in another country) ● GDP= income ● Real value = Nominal value * (prices in base period/prices in the current period) ○ Real Wage=nominal wage* (CPI base year/CPI current year) ○ Real money = nominal money * (CPI base year/CPI current year) ● Inflation rate = ((CPI this year ­ CPI last year)/ CPI last year)*100% ○ Try to remember: ((new­old)/old)*100% ● The unemployment rate=(#of people unemployed/labor force)*100 ● The employment­to­population ratio= (employment/working­age population)*100 ● The labor force participation rate= (labor force/working­age population)*100 ● Output gap​ = ​eal GDP­Potential GDP ● GDP per capita= GDP/population (dollars per person) ● Labor Productivity = real GDP/aggregate labor hours ● Interest rate= (return­price)/price *100% ○ Interest rate= (price + interest ­ price)/price *100% ○ interest/price *100% ● Real interest rate = nominal interest rate­inflation rate Exam Outline: (Chapters 4­8) *all the material covered up to the end of class Wednesday (4/20) ● True or False Statements ● Multiple Choice Questions ● Explaining Situations ● Graphing There is a practice exam on Canvas­answers will be available at the end of class Monday (4/25)

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Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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In a study of copper bars, the relationship between shear