Introduction to Statistics
Introduction to Statistics CORE 143
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This 6 page Class Notes was uploaded by Mr. Rebekah Hackett on Monday October 5, 2015. The Class Notes belongs to CORE 143 at Colgate University taught by Chad Sparber in Fall. Since its upload, it has received 29 views. For similar materials see /class/219595/core-143-colgate-university in Liberal Arts Core Curriculum at Colgate University.
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Date Created: 10/05/15
Page 1 Name Ex Introduction to Statistics Midterm 2 Version 1 November 14 2008 100 Points Possible Score m Show your work for credit on short answer guestions Page 2 1 Multiple Choice Select one BEST answer for each question Look at your Chapter 7 lecture notes I said that the only two laws or theorems that would ever ask you to learn are the CLT and the Law of Large numbers They are fundamental to the statistical analysis we do in this course A The Central Limit Theorem implies that as you increase the number of observations 4 Points a b The sampling distribution becomes normal The sampling distribution becomes normal if the population distribution is normal 0 V The data distribution becomes normal if the population distribution is normal d The sample mean approaches the population mean B The Law of Large Numbers implies that as you increase the number of observations 4 Points 2 V The sampling distribution becomes normal The sampling distribution becomes normal if the population distribution is normal b O V The data distribution becomes normal if the population distribution is normal d The sample mean approaches the population mean Standard Error Make sure to use the proper notation in the following questions The standard error ofa sample statistic is fundamentally important for statistical inference We have used theseformulas repeatedly A What is the formula for the stande error of a sample mean 2 points SE U x B What is the formula for the stande error of a sample proportion 3 points a 1 55 M n C What formula do you use to estimate the standard error of a sample mean if parameters are unknown and have to be estimated 2 points 5 SE E D What formula do you use to estimate the standard error of a sample proportion if parameters are unknown and have to be estimated 3 points 1 Tl SE Page 3 3 Probability A V According to Table 5 in John Cawley s 2006 A Guide and Advice for Economists on the US Junior Academic Job Market 96 of new economics PhD graduates from secondtier universities received at least one job interview and 82 received at least one job offer An interview is required before a person receives a job offer De ne events A and B such that A A particular graduate from a secondtier university got at least one job interview B A particular graduate from a secondtier university got at least one job offer Also see question 1 from the sample exam orquot questions 3 anal 4 om H W 5 for prevzous examples ofsimilar questions A What does PB equal 3 points PB 82 B What does PAlB equal 3 points PA l B PA amp B PB but Since all who have ajob o er also have an interview PA l B 1 C What does PBlA equal 3 points PB lA PA amp B PA Anal since PA lB Jfrom above we know thatPA amp B PB 82 Thus PB lA 082096 85 D What does PA U B equal 3 points PA UB PA PB 7PA amp B PA UB 096 0827 082 096 Probability Distributions You run an experiment by rolling a single eigl1tsided die 40 times You roll two ls ve 2s seven 3s four 4s six 5s one 6 seven 7s and eight 8s See question 1 of H W 7for a previous example ofa similar question A Draw the population distribution for die rolls Make sure you label your axes and provide 3 pertinent numerical values on the axes if g 0125 possible 4 points 3 U39 2 D 7777 7 7 LL 1 2 3 4 395 6 7 3 X B Sketch the sampling distribution for die rolls Make sure you label your axes and provide pertinent numerical values on the axes if possible 4 points Xbar Mn 45 Page 4 5 Probability Distributions According to Table 2 in John Cawley s 2006 A Guide and Advice for Economists on the US Junior Academic Job Market 353 of the 930 new economics PhD degrees granted by US universities in 2001 were granted to US citizens Suppose an economics department at a particular school is comprised of 17 professors who earned their PhDs in a US university in 2001 Seven are not US Citizens A Assume hiring decisions were random How many professors would you expect have US citizenship Round t0 the second decimal place 4 points See question 4b of H W6 5b ofH W 7 for a previous example of a similar question De ne p as the probability ofbeing a citizen p 353930 038 un p17038646 Assume hiring decisions were random What is the standard deviation of the number of professors you would expect to have US citizenship Round t0 the second decimal place 4 points See question 40 of H W6 5c of H W7 or question 161 of H W8 for previous examples of a similar question 039 sqrtn p 1 7p sqrt7 038 062 2 If hiring decisions are citizenshipblind what was the probability that seven or fewer professors are nonUS citizens 4 points See question 10 of H W 8 for a previous example of a similar question binomalist7 17 062 true Or 1 7 binomalist9 17 038 true 6 64 If you were to make a histogram describing the population distribution of the number of professors who are US citizens what frequencies would you expect to nd for 7 8 9 and 10 4 points See question 4a of H W6 5a of H W7 for a previous example of a similar question binomalistX 17 038 false 739 1868 8 1 4 31 9 8 77 10 430 What is the standard error of the proportion of US citizen professors in the economics department 4 points See question 1 of H W8 for a previous example ofa similar question SE sqrt038 062 17 01177 Page 5 6 Z Scores For each of the following use Z scores to answer the question Please provide the Excel commands or Z scores that you used to answer the questions Read the questions carefully The population mean and standard deviation for men s heights in inches are u 70 and 6 5 respectively You sample 16 men and record their heights A What proportion of men are above 68 inches tall 3 points Also see question 5a on the sample exam for an example similar to this question 1 7 normdist68 70 5 true Or 1 7 normsdist 04 06554 B The middle quintile 20 of men fall between what two heights 6 points Also see question 5e on the sample exam for an example similar to this question The Z scores associated with the middle quintile are normsinv04 025 and normsinv0 6 025 Since 2 X7 Li am this context 025 X 7 70 5 implies X 6875 And 025 X 7 70 5 implies X 7125 C What is the proportion of samples that will nd a mean below 68 inches 3 points Also see question 5d on the sample exam for an example similar to this question NOW we need the standard error 5 sqrt16 125 normdist68 70 125 true Or normsdist 16 5 48 D What is the proportion of samples that will nd a mean 125 standard errors away from 70 inches 3 points Also see question 2 H W 6for an example similar to this question but using standard deviations Since 70 is the population mean we know that this is normsdist125 7 normsdist 125 7887 E What is the proportion of samples that will nd a mean 125 standard errors away from 68 inches 6 points A sample mean of68 has a Z score of687 70 125 16 Z scores 125 standard errors away from this are 167125 and 16125 285 and 035 Thus the probability is normsdist 035 7 normsdist 285 36 Page 6 7 Excel Go to httpnennle coloate edn c nar39 er J txt and save the le to your desktop This le contains data on National Football League NFL games played in the rst four weeks of the 2008 season from httpwwwfootballlockscom There are six variables in the dataset 1 The week the game was played 2 A number to identify a particular game 3 The team city favored to win the game 4 The number of points the favored team was expected to win by ie the Spread 5 The team expected to lose and 6 A categorical variable that identi es whether the favored team won by more points than they were expected to ie if they beat the spread You will only be using the last of these variables Also see questions 2 and 6 on the sample exam and the data acquisition homework for previous examples ofsimilar questions A The data in the text le is delimited by underscores 7 Put this data in Excel format and save it as an Excel le What steps did you take to put this in Excel format If you cannot complete this step I will give you the data in Excel format but you will not get the points for this part of the problem 6 points Copy and paste the data into Excel Highlight the column with the data then go to data gt text to columns Here choose delimited check the box for other and type an underscore 7 B Please provide the point estimate t statistic and standard error needed to construct a con dence interval for the proportion of teams that beat the spread at a 75 con dence level round each to THREE decimal places What is the mathematical formula not the Excel formula for this con dence interval 10 points It is easiest ifin a new column you type ifF2 Yes 10 to create a 10 variable instead of a Y esNo variable Then go to data analysis and descriptive statistics p hat 05 79 t tinv025 56 since degrees offreedom n 1 se 0 066 C1 p hat tSE C1 0579 1162 0066 C Gamblers often place bets on whether a team will beat the spread Casinos set the spread hoping that favored teams will beat it exactly 50 of the time If favored teams beat the spread more often rational gamblers will notice and start betting on favored teams to beat the spread The Casino will start to lose money Similarly if favored teams rarely beat the spread gamblers will bet against the favorites and the casinos will again lose money According to this data should the Casinos be concerned Why or why not 5 points Yes they should be concerned because 50 is outside of the 75 Con dence Interval 0502 0 656 A higher confidence level however may include this value