HOMEWORK SETS FOR PRINCIPLES OF STATISTICS I (ECONOMICS 261) Lewis Karstensson, Ph.D. Department of Economics University of Nevada, Las Vegas

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1 HOMEWORK SETS FOR PRINCIPLES OF STATISTICS I (ECONOMICS 261) Lewis Karstensson, Ph.D. Department of Economics University of Nevada, Las Vegas 2005

2 INTRODUCTION This packet contains the Homework Sets for the ten topics covered in this course. Each set contains (1) a list the Terms to Know, and (2) a set of Problems and Interpretation Questions for a given topic. It is suggested that you use the pertinent homework exercises, together with the text readings and lecture notes, to help you prepare for each course Exam. You should practice defining the terms accurately, noting examples of each where appropriate, work through the problems correctly, and interpret the answers correctly with as much repetition as is necessary to know the material. The process of learning statistics is really no different from that of learning many other things: To learn how to play tennis well, you have to practice playing good tennis; to learn how to play the piano well, you have to practice on the piano a lot; and to learn how to do statistical analysis correctly, you have to practice, and practice, and practice, doing statistical analysis correctly, that is, learning the language of statistics, working problems, and interpreting results correctly. 0.1

3 Course Topics The principal topics considered in this course are the following: Topic 1: Introduction to Statistics 1. The Probabilistic World 2. Some Beginning Terms 3. Some Arithmetic Operations Topic 2: Data Presentation 1. Table Presentations 2. Graph Presentations 3. Using Excel, The Chart Wizard, Function Wizard, Histogram Tool Topic 3: Descriptive Statistics 1. Measures of Central Tendency 2. Measures of Variation 3. Other Measures 4. The Excel Descriptive Statistics Tool, Correlation Tool EXAM 1 Topic 4: Probability 1. The Meaning of Probability 2. Some Probability Terms 3. Probability Rules and Problems Topic 5: Probability Distributions 1. Random Variables and Distributions 2. The Binomial Distribution 3. The Normal Distribution EXAM 2 Topic 6: Random Samples and their Distributions 1. Random Sampling Techniques 2. Sampling Distributions 3. The Central Limit Theorem 0.2

4 0.3 Topic 7: Estimation 1. The Process of Estimation 2. Confidence Intervals 3. Sample Size Estimation Topic 8: Hypothesis Testing 1. The Process of Hypothesis Testing 2. Hypothesis Test for a Mean 3. Hypothesis Test for a Proportion EXAM 3 Topic 9: Small Sample Statistics 1. Student's t-statistic 2. Confidence Interval for a Mean 3. Hypothesis Test for a Mean Topic 10: Regression Analysis 1. Simple Regression Analysis 2. Interpretation of Excel Regression Output EXAM 4 (Final Exam)

5 TOPIC 1 INTRODUCTION TO STATISTICS Terms to Know: Statistics Descriptive statistics Inferential statistics Population Parameter Sample Statistic Variable Quantitative variable Qualitative variable Continuous variable Discrete variable Dummy variable Problems and Interpretation Questions: 1. Identify the following notations: (a) µ (b) σ 2 (c) σ 2. Identify the following notations: _ (a) X (b) S 2 (c) S 3. A researcher observes that average player salaries in professional basketball, baseball, and football in the United States in 1990 (in thousands of current dollars) were $817, $598, and $350, respectively. This is an example of what type of statistics? 4. A researcher is interested in determining the average income for families in Nye County, Nevada. To accomplish this, she takes a random sample of 400 families from the county and uses the data gathered from these families to estimate the average income for families in the entire county. This is an example of what type of statistics? 1.1

6 The owner of a fleet of forty taxis in Las Vegas is trying to estimate his costs for next year's operations. One major cost item is fuel. He measures the gas mileage for eight taxis. The results (in miles per gallon of gasoline) are as follows: 18.1, 13.6, 20.1, 17.5, 17.6, 16.8, 19.0, 19.3 (a) What is the population of interest in this analysis? (b) What is an example of a parameter in this case? (c) What is the sample in this analysis? (d) What is an example of a statistic in this case? 6. Given: X = 1, 2, 3, 4. Perform the following summations: (a) ΣX (b) ΣX 2 (c) Σ(X-2) (d) (ΣX) 2 (e) Σ(X-2) 2 7. Evaluate the following powers: (a) 2 3 (b) 2-3 (c) 2 1 (d) Find the following factorials: (a) 4! (b) 1! (c) 0!

7 TOPIC 2 DATA PRESENTATION Terms to Know: Frequency distribution Relative frequency distribution Cumulative frequency distribution Contingency table Histogram Frequency polygon Ogive Stem and leaf display Line chart Bar chart Column chart Pie chart Scatter diagram (plot) Problems and Interpretation Questions: 1. For practice in data presentation and interpretation, complete the following Solved Problems in Kazmier: 2.23, 2.28, 2.29, 2.32, 2.34, and

8 TOPIC 3 DESCRIPTIVE STATISTICS Terms to Know: Measures of central tendency Arithmetic mean Weighted mean Median Mode Measures of variation Range Variance Sum of squares Standard deviation Coefficient of variation Z-value Empirical rule Chebyshev's theorem Skewness Kurtosis Correlation coefficient Outlier Problems and Interpretation Questions: 1. The number of accidents which occurred during a given month in the 10 manufacturing departments of an industrial plant was: 2, 0, 0, 3, 3, 12, 1, 0, 5, 0 Calculate and interpret the following parameters for this population data set: (a) arithmetic mean. (b) median. (c) mode. (d) range. (e) sum of squares. (f) variance. (g) standard deviation. 2. The weights of a sample of outgoing packages in a mail room, weighed to the nearest ounce, are found to be: 21, 18, 30, 12, 14, 17, 28, 10, 16, 25 Calculate and interpret the following statistics for this sample data set: (a) arithmetic mean. (b) median. (c) mode. (d) range. (e) sum of squares. (f) variance. (g) standard deviation. 3.1

9 Suppose the profit rates for firms A, B, and C were 10, 12, and 15 percent, respectively. The assets of firm A were $2 billion whereas those for the other two firms were $1 billion each. Calculate the: (a) arithmetic mean rate of profit for the firms. (b) asset weighted mean rate of profit for the firms. 4. (a) Calculate coefficients of variation for the distributions given in problems 1 and 2 above. (b) Based on the standard deviation, which distribution shows the greater variation? (c) Which distribution exhibits the greater relative variation? 5. (a) Transform the data given in problem 1 above into Z-values. Interpret the observation, X = 1. (b) Calculate the mean and standard deviation of the distribution of Z-values. Interpret the results. 6. Suppose the Nevada National Bank is reviewing its service charge and interest policies on checking accounts that it holds. The Bank has found that the average daily balance on personal checking accounts is normally distributed around a mean of $ with a standard deviation of $ Between what two amounts will the average balance fall 95 percent of the time? 7. Suppose the rates of return last year on the common stocks in a large portfolio were normally distributed, with a mean of 20 percent and a standard deviation of 10 percent. (a) What proportion of the stocks had a return of between 10 percent and 30 percent? (b) What proportion of the stocks had a positive return? 8. (a) What proportion of the observations in any type of distribution will fall within two standard deviations of the mean? (b) Does this hold for the distribution given in problem 1 above? Explain. 9. A firm manufactures metal rods which must be rejected if they are not between and inches in diameter. While the shape of the distribution of rod sizes is not known, a recent sample of rods indicates a mean diameter of inches and a standard deviation of.025 inches. Estimate the proportion of rods that can be expected to be rejected. 10. Calculate the Pearson coefficient of skewness for the distribution given in problem 1 above. Interpret the result.

10 TOPIC 4 PROBABILITY Terms to Know: Probability Experiment Event Classical approach Relative frequency approach Subjective approach Sample space Elementary events Mutually exclusive events Collectively exhaustive events Composite events Complementary events Independent events Marginal probability Union probability Joint probability Conditional probability Problems and Interpretation Questions: 1. The following contingency table gives admitted headcount student enrollment data for UNLV for the Fall Semester of 2002: Women Men Totals Undergraduate 10,403 8,231 18,634 Graduate 3,024 2,028 5,052 Totals 13,427 10,259 23,686 (a) Construct a probability matrix for these data. (b) What is the probability that a randomly selected student is a woman? (c) What is the probability that a randomly selected student is a woman or undergraduate? (d) What is the probability that a randomly selected student is a woman and undergraduate? (e) What is the probability that a randomly selected woman student is undergraduate? 4.1

11 Because of a firm's growth, it is necessary to transfer one of its employees to one of its branch stores. Three of the nine employees are women and each of the nine employees is equally qualified for the transfer. If the person to be transferred is chosen at random, what is the probability that the transferred person is a woman? 3. Of 100 students, 24 are economics majors, 18 are computer science majors, and 8 are majoring in both economics and computer science. If a student is picked at random, what is the probability that the selected student will be an economics major or a computer science major or both? 4. Suppose 30 percent of American adults own stocks, 20 percent own bonds, and 10 percent own both stocks and bonds. If an investor is one who owns stocks and/or bonds, what proportion of American adults are investors? 5. A firm is considering three possible locations for a new factory. The probability that site A will be selected is 0.30 and the probability that site B will be selected is If only one location will be chosen, what is the probability that: (a) site A or B will be chosen? (b) neither site A nor site B will be chosen? 6. During a given quarter, the probability that GNP will increase, stay the same, or decrease is estimated to be 0.60, 0.10, and 0.30, respectively. What is the probability that GNP will either increase or stay the same during the given quarter? 7. A company estimates that 30% of the population has seen its commercial and that if a person sees its commercial there is a 20% probability that the person will buy its product. What is the probability that a person chosen at random from the population will have seen the commercial and bought its product? 8. If 10% of all light bulbs a company manufactures are defective, the probability of any one bulb being defective is.10. What is the probability that two bulbs drawn independently from the company's stock will be defective? 9. Suppose the probability that a prospect will make a purchase when he is contacted by a salesman is If a salesman selects two prospects randomly from a file and makes contact with them, what is the probability that both prospects will make a purchase? 10. A store manager is asked to make three different yes-no decisions that have no relation to each other. Because he is impatient to leave work, he flips a coin for each decision. If the correct decision in each case was yes, what is the probability that: (a) all decisions were correct? (b) none of the decisions were correct? (c) two or more of the decisions were correct?

12 TOPIC 5 PROBABILITY DISTRIBUTIONS Terms to Know: Random variable Discrete random variable Continuous random variable Discrete probability distribution Probability density curve Expected value Bernoulli process Binomial distribution Normal distribution Standard normal variable Standard normal probability distribution Normal approximation of the binomial distribution Problems and Interpretation Questions: 1. Consider the following probability distribution for the discrete random variable, X: X P(X) (a) Find the expected value of the random variable. (b) Find the standard deviation of the random variable. 2. Suppose there is a.97 probability that no accident will occur at a particular power plant during each day; the probability of one accident is.02; and there is a.01 probability of two accidents. (a) Find the expected number of accidents per day. (b) Find the standard deviation for the number of accidents. 3. The Gamma Corporation is equally likely to sell 0, 1, 2, 3, or 4 bicycles in a day. (a) Find the expected number of bicycles sold per day. (b) Find the standard deviation for the number of bicycles sold. 5.1

13 The arrival of customers during randomly chosen 10-minute intervals at a drive-in facility specializing in photo development and film sales has the following discrete probability distribution: Arrivals Probability X P(X) (a) Find the expected value of arrivals. (b) Find the standard deviation for the arrivals. 5. Solve the following problems using the binomial formula: (a) P(x=4*n=8, p=0.30) (b) P(x<2*n=5, p=0.50) 6. Solve the following problems using the binomial table: (a) P(x=4*n=8, p=0.30) (b) P(x<2*n=5, p=0.50) 7. Mary Johnson owns stock in five companies. There is a 0.50 probability that each stock will rise in price this year. (a) Construct a probability distribution for the number of rising stocks from the binomial table. (b) Draw a probability histogram for this distribution. (c) Calculate the (1) expected value, (2) variance, and (3) standard deviation for this distribution. (d) What is the probability that all five stocks will increase in price? (e) What is the probability that none of the stocks will increase in price? (f) What is the probability that at least two of the stocks will increase in price? 8. Suppose that 40 percent of the hourly employees in a large firm are in favor of union representation, and a random sample of 10 employees are contacted and asked for an anonymous response. What is the probability that a majority of the respondents will be in favor of union representation?

14 The Maroni Corporation bids on 10 jobs, believing that its chances of getting each one is What is the probability that the firm will get one or more of the bids? 10. Find the area under the standard normal curve which lies between the Z-values of: (a) 0 and 1.82 (b) and 0 (c) and 1.08 (d) and Find the probability that the standard normal variable, Z, lies: (a) above 2.3 (b) below -3.0 (c) between 1 and 2 (d) between -1 and Find the value of Z if the area under the standard normal curve: (a) between 0 and Z is.1985 (b) between -Z and 0 is.0910 (c) to the left of Z is.8051 (d) between -Z and Z is Suppose the annual sales of a given firm is a normally distributed random variable with a mean of $300 billion and a standard deviation of $60 billion. What is the probability that the sales for this firm for the year will: (a) be less than $280 billion? (b) exceed $350 billion? (c) be between $185 billion and $265 billion? (d) be between $305 billion and $375 billion? 14. The amount of time required for a given type of automobile transmission repair at a service garage is normally distributed with a mean of 45 minutes and a standard deviation of 8.0 minutes. The service manager plans to have work begin on the transmission of a customer's car 10 minutes after the car is dropped off, and he tells the customer that the car will be ready within 1 hour total time. What is the probability that he will be wrong?

15 Suppose wage increases in a given industry are normally distributed around a mean increase of $1.00 per hour with a standard deviation of $0.30 per hour. While union negotiators are now asking for a raise of $1.45 per hour, they expect to get something less than their request. They, however, do hope to get a raise of no less than $0.90 per hour. What is the probability that the wage increase will: (a) be more than $1.45 per hour? (b) fall within the interval between $0.90 and $1.45? (c) be less than $0.90 per hour? 16. A factory's rate of electric power consumption per day is normally distributed with a mean consumption rate of 8,000 kilowatts and a standard deviation of 1,000 kilowatts. What is the probability that the power consumption on any given day will be: (a) at least 6,500 kilowatts? (b) greater than 10,000 kilowatts? 17. New MBA students at UNLV must take the GMAT examination. Suppose the scores achieved by incoming students are normally distributed with a mean of 500 and a standard deviation of 50. If UNLV gives a scholarship to the top 15 percent of the students, what score must be achieved in order to get a scholarship? 18. A Myrtle Beach resort hotel has 120 rooms. Hotel room occupancy is approximately 75%. What is the probability that: (a) at least half the rooms are occupied on a given day? (b) 100 or more rooms are occupied on a given day? (c) 80 or fewer rooms are occupied on a given day?

16 TOPIC 6 RANDOM SAMPLES AND THEIR DISTRIBUTIONS Terms to Know: Random Sample Simple random sample Systematic random sample Stratified random sample Cluster (area) sample Sampling error Sampling distribution of sample means Sampling distribution of sample proportions Central limit theorem Standard error of the mean Standard error of the proportion Problems and Interpretation Questions: 1. A national tire manufacturer claims that the average lifetime of their premium tire is 50,000 miles. The standard deviation of the lifetime of these tires is 2,000 miles. Suppose all possible samples of size 100 are taken from this population of tires. Calculate the standard error of the mean for this sampling distribution. What does your answer mean? 2. A local firm claims that their tires last 45,000 miles on average with a standard deviation of 2,500 miles. Assuming the firm's claim is correct, suppose a random sample of 100 of these tires is tested. What is the probability that the sample mean found in the test will be equal to or less than 44,500 miles? 3. Studies have shown that the total number of points scored by both teams in National Football League (NFL) games over several seasons has a mean of 41 points with a standard deviation of 14 points. What is the probability of obtaining a mean total score equal to or less than 48 points in a random sample of 30 NFL games? 4. The family income distribution in St. Paul, Minnesota, is skewed to the right. The latest census reveals that the mean family income is $32,000 and that the standard deviation is $4,000. If a simple random sample of 75 families is drawn, what is the probability that the sample mean family income will differ from St. Paul's mean income by more than $500? 5. It is estimated that there are 1,400 automobile dealers in the Chicago area and that the average dollar sales per dealer per month is $750,000. A random sample of 50 dealers is selected, and the mean and standard deviation are calculated. If the standard deviation is equal to $95,000, what is the probability that the sample mean is between $740,000 and $765,000? 6.1

17 A recent survey suggests that the average annual starting salary for economists with a bachelors degree is $34,000 with a standard deviation of $2,500. If a sample of 50 first-year economists is selected randomly, what is the probability that the mean starting salary for this sample will be at least $33,500? 7. Sears claims that seven percent of all video games purchased during the Christmas season are defective and returned. Suppose all possible samples of size 100 are taken from Sears inventory of video games and tested for defects. Calculate the standard error of the proportion for this sampling distribution. What does your answer mean? 8. The unemployment rate in the New Orleans area for a recent month was 9.6%. What is the probability that the percentage unemployed in a random sample of 600 people is over 10%? 9. A mortgage company knows that 8% of its home loan recipients default within the first five years. What is the probability that out of 350 loan recipients, less than 25 will default within the next five years? 10. From past experience, a company knows that 55 percent of the surveys that they send out will be completed and returned. What is the probability that they will have at least 50 percent returned of 125 surveys mailed?

18 TOPIC 7 ESTIMATION Terms to Know: Estimation Estimator Estimate Point estimate Interval estimate Confidence coefficient Sample size estimation Problems and Interpretation Questions: 1. For a large bank having several thousand customers, five checking account balances, selected at random, are: $400, $850, $180, $240, and $160. What is the point estimate for the mean (F) of all checking account balances for this bank? 2. In a survey of 224 large companies conducted by the Conference Board, 45 companies said they give three to five months notice of plant closings. What is the point estimate of the population proportion (π) of companies that give such notice of plant closings? 3. A nursery sells trees of different types and heights. Suppose that 75 pine trees are sold for planting at City Hall. These 75 trees average 60 inches in height with a standard deviation of 16 inches. Calculate the standard error of the mean for this sample. 4. Suppose the population standard deviation on the weight of aluminum ingots is known to be 20 pounds. A random sample of 100 ingots at a given aluminum plant yielded a mean weight of 602 pounds. Construct a 95 percent confidence interval for the population mean weight of aluminum ingots. 5. An economist wishes to estimate the mean population elasticity of supply of poultry farmers at their respective production levels. A random sample of 100 producers yields an average elasticity of supply of 1.9 and a standard deviation of 1.0. Construct a 90 percent confidence interval for the population mean elasticity of supply. 6. Fifty cans of dog food were randomly sampled for cereal content. The mean cereal content was 6.0 oz. with a standard deviation of.05 oz. Construct a 99 percent confidence interval for the population mean cereal content. 7. A sample of 100 new home owners were asked if they were satisfied with the services provided by their real estate agents. In this sample, 92 reported that they were satisfied. Calculate the standard error of proportion for this sample. 7.1

19 A breakfast food company wants to estimate the proportion of cornflake eaters who prefer the flakes soggy when eaten. A sample of 100 cornflake eaters reveals that 30 prefer soggy flakes. Construct a 90 percent confidence interval for the proportion of cornflake eaters in the population who like soggy flakes. 9. A survey was taken from a random group of employees of a large firm. These employees were asked whether or not they were happy with their jobs. Of the 150 employees surveyed, 90 said they were content with their current jobs. Calculate a 95 percent confidence interval for the proportion of employees who were happy with their jobs. 10. A prospective purchaser wishes to estimate the mean dollar amount of sales per customer at a toy store located at an airline terminal. Based on data from other similar airports, the standard deviation of such sales amounts is estimated to be about $3.20. What size random sample should she collect, as a minimum, if she wants to estimate the mean sales amount within $1.00 and with 99 percent level of confidence? 11. A research firm has been asked to estimate the proportion of all restaurants in the state of Ohio that serve alcoholic beverages. The firm wants to be 90 percent confident of its results, but has no idea what the actual proportion is. The firm would like to report an error of no more than How large a sample should it take?

20 TOPIC 8 HYPOTHESIS TESTING Terms to Know: Null hypothesis Alternative hypothesis Type I error Type II error One-tail test Two-tail test Acceptance region Rejection region Level of significance Problems and Interpretation Questions: 1. The Life Insurance Institute, based on a projection from last year's figures, claims that the mean face value of the life insurance policies sold this year is $35,000. A random sample of 49 of this year's policies has an average face value of $38,000 with a sample standard deviation of $20,000. Can the Institute's claim be accepted at the.05 level of significance? 2. The mean size of the stock purchases by customers of Merrill Lynch last month was $2,800. This month Merrill Lynch has reduced its sales commissions hoping to induce larger purchases. A sample of 100 purchases this month had a mean size of $2,900 with a sample standard deviation of $1,500. At the.01 significance level, can it be concluded that the reduced commissions increased average stock purchases? 3. The manufacturer of a new compact car claims that the car will average at least 35 miles per gallon in general highway driving. For 40 test runs, the car averaged 34.5 miles per gallon with a standard deviation of 2.3 miles per gallon. Can the manufacturer's claim be rejected at the 5 percent level of significance? 4. The average annual income for graduates in their first job after completing business school was thought to be $28,000. A survey of 144 recent business school graduates found that the average salary was $28,500 with a standard deviation of $1,200. Test the null hypothesis that the starting salary of business school graduates is $28,000 at the 1 percent level of significance. 5. Greyhound claims that at least 90 percent of the packages it delivers arrive on time. One hundred packages are sent at differing times from differing locations via Greyhound with the result that eighty packages arrive on time. Can Greyhound's claim be accepted at the.10 significance level? 6. The cable comedy channel claims that at least 30 percent of the homes in Las Vegas watch South Park. An advertiser randomly samples 60 homes and finds that 15 are watching South Park. Should the comedy channel's claim be rejected at the 5 percent level of significance? 8.1

21 During the 1996 National Football League (NFL) preseason a total of 62 games were played to conclusion. Of this total, 40 games were won by the home team while the remaining 22 were won by the visiting team. Test the hypothesis that the population proportion (π) of home team wins during the NFL preseason is equal to 50 percent using a.05 level of significance. Interpret the result.

22 TOPIC 9 SMALL SAMPLE STATISTICS Terms to Know: Student's t-distribution Degrees of freedom Problems and Interpretation Questions: 1. An analyst in a personnel department randomly selects the records of 16 hourly employees and finds that the mean wage rate per hour is $9.50. The wage rates in the firm are assumed to be normally distributed. If the standard deviation of the wage rates is known to be $1.00, estimate the mean wage rate in the firm using a 95 percent confidence interval. 2. A retailer wishes to estimate the mean time it takes for a wholesaler to fill an order. From a sample of 10 orders, the retailer finds the mean time is 15 days with a standard deviation of 4 days. Construct a 90 percent confidence interval for the mean time it takes to get an order filled by the wholesaler. 3. Burger Chemical of Newark claims that the mean daily amount of pollutants being emitted from one of its stacks is 880 kilograms. Newark monitors the stack on 25 randomly selected days in order to see if the mean is greater or less than 880 kilograms per day. For the 25 sampled days, the mean was 910 kilograms with a sample standard deviation of 200 kilograms. At the.05 significance level, can Newark conclude that there has been a change in the mean amount of pollutants emitted from the stack? 4. The mean pollution index in Denver was 176 last winter. A random sample of 12 days this winter yielded a mean of 158 with a standard deviation of 40. At the 1 percent level of significance, can it be concluded that the pollution index has declined? 9.1

23 TOPIC 10 REGRESSION ANALYSIS Terms to Know: Deterministic model Stochastic model Simple regression Dependent variable Independent variable Scatter plot Least squares criterion Constant term Slope coefficient Standard error of estimate Residual Coefficient of determination Correlation coefficient Homoscedasticity Problems and Interpretation Questions: 1. A railroad is interested in an analysis of how many meals are demanded on runs of its trains from New York to Florida. Four runs have been sampled yielding the following data on the number of passengers and the number of meals demanded: Passengers (X) Meals (Y) (a) Determine the least squares regression equation for this data set. Explain the equation. (b) Calculate the standard error of estimate. What does the standard error of estimate measure? (c) Test the null hypothesis that the slope of the regression line is zero using a 10 percent level of significance. (d) Calculate the coefficient of determination and the correlation coefficient. What do the obtained values mean? 10.1

24 A large meat packing company has done a study on the relationship between the number of cattle grazing and in feed lots as of May and the supermarket price of beef in the following November. The following has been found for the last four years: Cattle (X) Price (Y) 800 $1.60 1, , , Solve (a) through (d) given in problem Explain the assumptions underlying linear regression. 4. Process the data in problems 1 and 2 above through the Regression Analysis Tool in the Excel Analysis ToolPak: (1) Input the data into a spreadsheet and run the regression. (2) Print the regression output. (3) Answer questions (a) through (d) using the regression printout.