Statistics – Economics 250 Introductory Statistics Exercise 2

Economics 250Introductory StatisticsExercise 2Due Thursday 3 March in class and on paperInstructions: There is no drop box and this exercise can be submitted only in class. Nolate submissions are possible. Show intermediate steps (such as formulas) in your workboth for ease of review and part marks in case of error. Do the exercise on your own andsubmit only your own work. Write in pen (or type) and not in pencil. If you are late toclass please submit your work at the end of the meeting.1. Suppose that we know that, in a population, spells of unemployment are normallydistributed with a mean of 220 days and a standard deviation of 6 days.(a) What is the probability that a spell lasts more than 226 days?(b) Researchers collect a random sample of 4 unemployment spells. What is the probabilitythat the average spell lasts more than 226 days?2. Suppose that researchers devise an indicator of business cycles with 0 indicating arecession and 1 indicating no recession. In two adjacent years, the indicators are jointlydistributed like this:This Year01Next Year010.50.10.10.3(a) Find the marginal distribution for the indicator next year. Then find the mean andstandard deviation using that distribution. (Sometimes these are called the unconditionalmean and standard deviation.)(b) Suppose that you observe that the indicator takes the value 1 this year. Find theconditional distribution for the indicator next year. Then find the conditional mean andstandard deviation.(c) Are the two random variables independent? Briefly justify your answer.3. Suppose that 2% of athletes in a competition are cheating by taking banned stimulants.A test is available that yields evidence of cheating (i.e. a positive test) 8% of the time forathletes who are not cheating and 70% of the time for athletes who are cheating.(a) What is the probability of a randomly selected athlete testing positive?(b) If an athlete tests positive what is the probability she or he was cheating?1 4. Suppose that a training program succeeds in raising any worker’s wage with a probability of 0.60.(a) If 8 workers participate in the program then what is the probability that at least halfof them (i.e. 4 or more) experience a wage increase?(b) What is the probability that at most 3 of them experience a wage increase?(c) If 80 workers participate then what is the probability that at least half of them experience a wage increase? (Hint: Think of this sample size as large and do not use thebinomial formula.)(d) The agency funding the program does not yet know its budget or what the sample sizewill be. Given the success rate in the population, describe the distribution of the samplesuccess rate (or proportion of successes) for a sample size n (assuming n is large).5. Statistics are central to managing an investment portfolio. Imagine an investor ischoosing how to divide each dollar between two investments, labelled 1 and 2. Investment1 has a return with mean 1 and standard deviation 0.5, investment 2 has a return withmean 3 and standard deviation 2, and the correlation between the two returns is 0.6. Theportfolio invests a fraction ω in investment 1 and a fraction 1 − ω in investment 2.(a) Find formulas for the mean of the portfolio return, labelled µp , and the variance,2labelled σp , as functions of ω.(b) The investor would like a high return on average but also would like low variability (ameasure of risk). Suppose the investor seeks the highest value of2µp − σp .Find the value of ω that meets this objective.6. Use the data set schoolexp from the Projects web page for this course. For a sample of30 school districts in Indiana, it records the number of students (in thousands) and annualexpenditures (in millions of dollars).(a) Use Excel to produce and properly label a scatter plot of expenditures (on the verticalaxis) plotted against the number of students (on the horizontal axis).Next run a linear regression like this:y = a + bx,In Excel, go to the ‘data’ tab then to ‘data analysis” and ‘regression’.(b) Report the coefficient on the x-variable (i.e. the number of students) and its standarderror. Also report the R-square (R2 ) statistic.2 (c) Sections 2.3–2.4 of the textbook tell us how to interpret these statistics. Based onthese statistics is there much evidence of a relationship between the number of students ina school district and its expenditures?(d) A new school district is being established with an enrolment of 80 thousand students.Predict its annual expenditures.7. Use the data set tornadoes, where the first two columns give the year from 1953 to 2008and the number of tornadoes reported in the US each year.(a) Use Excel to produce and properly label a scatter plot of the number of tornadoes (onthe vertical axis) plotted against the year (on the horizontal axis).(b) Run a linear regression of the number of tornadoes on the year. Report the coefficienton the x-variable (i.e. the year) and its standard error. Report the R-square (R2 ) statistic.(c) Is there evidence of a statistical relationship?(d) Predict the number of tornadoes in 2009.3

Economics 250Introductory StatisticsExercise 2Due Thursday 3 March in class and on paperInstructions: There is no drop box and this exercise can be submitted only in class. Nolate submissions are possible. Show intermediate steps (such as formulas) in your workboth for ease of review and part marks in case of error. Do the exercise on your own andsubmit only your own work. Write in pen (or type) and not in pencil. If you are late toclass please submit your work at the end of the meeting.1. Suppose that we know that, in a population, spells of unemployment are normallydistributed with a mean of 220 days and a standard deviation of 6 days.(a) What is the probability that a spell lasts more than 226 days?(b) Researchers collect a random sample of 4 unemployment spells. What is the probabilitythat the average spell lasts more than 226 days?2. Suppose that researchers devise an indicator of business cycles with 0 indicating arecession and 1 indicating no recession. In two adjacent years, the indicators are jointlydistributed like this:This Year01Next Year010.50.10.10.3(a) Find the marginal distribution for the indicator next year. Then find the mean andstandard deviation using that distribution. (Sometimes these are called the unconditionalmean and standard deviation.)(b) Suppose that you observe that the indicator takes the value 1 this year. Find theconditional distribution for the indicator next year. Then find the conditional mean andstandard deviation.(c) Are the two random variables independent? Briefly justify your answer.3. Suppose that 2% of athletes in a competition are cheating by taking banned stimulants.A test is available that yields evidence of cheating (i.e. a positive test) 8% of the time forathletes who are not cheating and 70% of the time for athletes who are cheating.(a) What is the probability of a randomly selected athlete testing positive?(b) If an athlete tests positive what is the probability she or he was cheating?1 4. Suppose that a training program succeeds in raising any worker’s wage with a probability of 0.60.(a) If 8 workers participate in the program then what is the probability that at least halfof them (i.e. 4 or more) experience a wage increase?(b) What is the probability that at most 3 of them experience a wage increase?(c) If 80 workers participate then what is the probability that at least half of them experience a wage increase? (Hint: Think of this sample size as large and do not use thebinomial formula.)(d) The agency funding the program does not yet know its budget or what the sample sizewill be. Given the success rate in the population, describe the distribution of the samplesuccess rate (or proportion of successes) for a sample size n (assuming n is large).5. Statistics are central to managing an investment portfolio. Imagine an investor ischoosing how to divide each dollar between two investments, labelled 1 and 2. Investment1 has a return with mean 1 and standard deviation 0.5, investment 2 has a return withmean 3 and standard deviation 2, and the correlation between the two returns is 0.6. Theportfolio invests a fraction ω in investment 1 and a fraction 1 − ω in investment 2.(a) Find formulas for the mean of the portfolio return, labelled µp , and the variance,2labelled σp , as functions of ω.(b) The investor would like a high return on average but also would like low variability (ameasure of risk). Suppose the investor seeks the highest value of2µp − σp .Find the value of ω that meets this objective.6. Use the data set schoolexp from the Projects web page for this course. For a sample of30 school districts in Indiana, it records the number of students (in thousands) and annualexpenditures (in millions of dollars).(a) Use Excel to produce and properly label a scatter plot of expenditures (on the verticalaxis) plotted against the number of students (on the horizontal axis).Next run a linear regression like this:y = a + bx,In Excel, go to the ‘data’ tab then to ‘data analysis” and ‘regression’.(b) Report the coefficient on the x-variable (i.e. the number of students) and its standarderror. Also report the R-square (R2 ) statistic.2 (c) Sections 2.3–2.4 of the textbook tell us how to interpret these statistics. Based onthese statistics is there much evidence of a relationship between the number of students ina school district and its expenditures?(d) A new school district is being established with an enrolment of 80 thousand students.Predict its annual expenditures.7. Use the data set tornadoes, where the first two columns give the year from 1953 to 2008and the number of tornadoes reported in the US each year.(a) Use Excel to produce and properly label a scatter plot of the number of tornadoes (onthe vertical axis) plotted against the year (on the horizontal axis).(b) Run a linear regression of the number of tornadoes on the year. Report the coefficienton the x-variable (i.e. the year) and its standard error. Report the R-square (R2 ) statistic.(c) Is there evidence of a statistical relationship?(d) Predict the number of tornadoes in 2009.3

Economics 250Introductory StatisticsExercise 2Due Thursday 3 March in class and on paperInstructions: There is no drop box and this exercise can be submitted only in class. Nolate submissions are possible. Show intermediate steps (such as formulas) in your workboth for ease of review and part marks in case of error. Do the exercise on your own andsubmit only your own work. Write in pen (or type) and not in pencil. If you are late toclass please submit your work at the end of the meeting.1. Suppose that we know that, in a population, spells of unemployment are normallydistributed with a mean of 220 days and a standard deviation of 6 days.(a) What is the probability that a spell lasts more than 226 days?(b) Researchers collect a random sample of 4 unemployment spells. What is the probabilitythat the average spell lasts more than 226 days?2. Suppose that researchers devise an indicator of business cycles with 0 indicating arecession and 1 indicating no recession. In two adjacent years, the indicators are jointlydistributed like this:This Year01Next Year010.50.10.10.3(a) Find the marginal distribution for the indicator next year. Then find the mean andstandard deviation using that distribution. (Sometimes these are called the unconditionalmean and standard deviation.)(b) Suppose that you observe that the indicator takes the value 1 this year. Find theconditional distribution for the indicator next year. Then find the conditional mean andstandard deviation.(c) Are the two random variables independent? Briefly justify your answer.3. Suppose that 2% of athletes in a competition are cheating by taking banned stimulants.A test is available that yields evidence of cheating (i.e. a positive test) 8% of the time forathletes who are not cheating and 70% of the time for athletes who are cheating.(a) What is the probability of a randomly selected athlete testing positive?(b) If an athlete tests positive what is the probability she or he was cheating?1 4. Suppose that a training program succeeds in raising any worker’s wage with a probability of 0.60.(a) If 8 workers participate in the program then what is the probability that at least halfof them (i.e. 4 or more) experience a wage increase?(b) What is the probability that at most 3 of them experience a wage increase?(c) If 80 workers participate then what is the probability that at least half of them experience a wage increase? (Hint: Think of this sample size as large and do not use thebinomial formula.)(d) The agency funding the program does not yet know its budget or what the sample sizewill be. Given the success rate in the population, describe the distribution of the samplesuccess rate (or proportion of successes) for a sample size n (assuming n is large).5. Statistics are central to managing an investment portfolio. Imagine an investor ischoosing how to divide each dollar between two investments, labelled 1 and 2. Investment1 has a return with mean 1 and standard deviation 0.5, investment 2 has a return withmean 3 and standard deviation 2, and the correlation between the two returns is 0.6. Theportfolio invests a fraction ω in investment 1 and a fraction 1 − ω in investment 2.(a) Find formulas for the mean of the portfolio return, labelled µp , and the variance,2labelled σp , as functions of ω.(b) The investor would like a high return on average but also would like low variability (ameasure of risk). Suppose the investor seeks the highest value of2µp − σp .Find the value of ω that meets this objective.6. Use the data set schoolexp from the Projects web page for this course. For a sample of30 school districts in Indiana, it records the number of students (in thousands) and annualexpenditures (in millions of dollars).(a) Use Excel to produce and properly label a scatter plot of expenditures (on the verticalaxis) plotted against the number of students (on the horizontal axis).Next run a linear regression like this:y = a + bx,In Excel, go to the ‘data’ tab then to ‘data analysis” and ‘regression’.(b) Report the coefficient on the x-variable (i.e. the number of students) and its standarderror. Also report the R-square (R2 ) statistic.2 (c) Sections 2.3–2.4 of the textbook tell us how to interpret these statistics. Based onthese statistics is there much evidence of a relationship between the number of students ina school district and its expenditures?(d) A new school district is being established with an enrolment of 80 thousand students.Predict its annual expenditures.7. Use the data set tornadoes, where the first two columns give the year from 1953 to 2008and the number of tornadoes reported in the US each year.(a) Use Excel to produce and properly label a scatter plot of the number of tornadoes (onthe vertical axis) plotted against the year (on the horizontal axis).(b) Run a linear regression of the number of tornadoes on the year. Report the coefficienton the x-variable (i.e. the year) and its standard error. Report the R-square (R2 ) statistic.(c) Is there evidence of a statistical relationship?(d) Predict the number of tornadoes in 2009.3

Economics 250Introductory StatisticsExercise 2Due Thursday 3 March in class and on paperInstructions: There is no drop box and this exercise can be submitted only in class. Nolate submissions are possible. Show intermediate steps (such as formulas) in your workboth for ease of review and part marks in case of error. Do the exercise on your own andsubmit only your own work. Write in pen (or type) and not in pencil. If you are late toclass please submit your work at the end of the meeting.1. Suppose that we know that, in a population, spells of unemployment are normallydistributed with a mean of 220 days and a standard deviation of 6 days.(a) What is the probability that a spell lasts more than 226 days?(b) Researchers collect a random sample of 4 unemployment spells. What is the probabilitythat the average spell lasts more than 226 days?2. Suppose that researchers devise an indicator of business cycles with 0 indicating arecession and 1 indicating no recession. In two adjacent years, the indicators are jointlydistributed like this:This Year01Next Year010.50.10.10.3(a) Find the marginal distribution for the indicator next year. Then find the mean andstandard deviation using that distribution. (Sometimes these are called the unconditionalmean and standard deviation.)(b) Suppose that you observe that the indicator takes the value 1 this year. Find theconditional distribution for the indicator next year. Then find the conditional mean andstandard deviation.(c) Are the two random variables independent? Briefly justify your answer.3. Suppose that 2% of athletes in a competition are cheating by taking banned stimulants.A test is available that yields evidence of cheating (i.e. a positive test) 8% of the time forathletes who are not cheating and 70% of the time for athletes who are cheating.(a) What is the probability of a randomly selected athlete testing positive?(b) If an athlete tests positive what is the probability she or he was cheating?1 4. Suppose that a training program succeeds in raising any worker’s wage with a probability of 0.60.(a) If 8 workers participate in the program then what is the probability that at least halfof them (i.e. 4 or more) experience a wage increase?(b) What is the probability that at most 3 of them experience a wage increase?(c) If 80 workers participate then what is the probability that at least half of them experience a wage increase? (Hint: Think of this sample size as large and do not use thebinomial formula.)(d) The agency funding the program does not yet know its budget or what the sample sizewill be. Given the success rate in the population, describe the distribution of the samplesuccess rate (or proportion of successes) for a sample size n (assuming n is large).5. Statistics are central to managing an investment portfolio. Imagine an investor ischoosing how to divide each dollar between two investments, labelled 1 and 2. Investment1 has a return with mean 1 and standard deviation 0.5, investment 2 has a return withmean 3 and standard deviation 2, and the correlation between the two returns is 0.6. Theportfolio invests a fraction ω in investment 1 and a fraction 1 − ω in investment 2.(a) Find formulas for the mean of the portfolio return, labelled µp , and the variance,2labelled σp , as functions of ω.(b) The investor would like a high return on average but also would like low variability (ameasure of risk). Suppose the investor seeks the highest value of2µp − σp .Find the value of ω that meets this objective.6. Use the data set schoolexp from the Projects web page for this course. For a sample of30 school districts in Indiana, it records the number of students (in thousands) and annualexpenditures (in millions of dollars).(a) Use Excel to produce and properly label a scatter plot of expenditures (on the verticalaxis) plotted against the number of students (on the horizontal axis).Next run a linear regression like this:y = a + bx,In Excel, go to the ‘data’ tab then to ‘data analysis” and ‘regression’.(b) Report the coefficient on the x-variable (i.e. the number of students) and its standarderror. Also report the R-square (R2 ) statistic.2 (c) Sections 2.3–2.4 of the textbook tell us how to interpret these statistics. Based onthese statistics is there much evidence of a relationship between the number of students ina school district and its expenditures?(d) A new school district is being established with an enrolment of 80 thousand students.Predict its annual expenditures.7. Use the data set tornadoes, where the first two columns give the year from 1953 to 2008and the number of tornadoes reported in the US each year.(a) Use Excel to produce and properly label a scatter plot of the number of tornadoes (onthe vertical axis) plotted against the year (on the horizontal axis).(b) Run a linear regression of the number of tornadoes on the year. Report the coefficienton the x-variable (i.e. the year) and its standard error. Report the R-square (R2 ) statistic.(c) Is there evidence of a statistical relationship?(d) Predict the number of tornadoes in 2009.3

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