Assignment_1_suggested_solutions.doc

Speaking of Statistics(12) - Assignment 1 Suggested Solution1. Below we list several variables. Which of these variables are quantitative and which are qualitative? Explain.a. The dollar amount on an accounts receivable invoice. Quantitativeb. The net profit for a company in 2005. Quantitativec. The stock exchange on which a companys stock is traded. Qualitatived. The national debt of the United States in 2005. Quantitativee. The advertising medium (radio, television, or print) used to promote a product. Qualitative2. Classify each of the following qualitative variables as ordinal or nominative. Explain your answers.Qualitative Variables The Corresponding CategoriesStatistics course letter grade A B C D F OrdinalPersonal computer ownership Yes No NominalRestaurant rating * * * * * OrdinalIncoming tax filing status Married filing jointly; Married filing separately; Single; Head of household; Qualifying widow(er) Nominal3. Forbes magazine publishes the Forbes Platinum 400 a list of the “Best Big Companies in America” as selected by the magazines writers and editors. Table above gives the best companies in the retailing industry as given in this list on the Forbes website on March 16, 2005. (Data in the last page). b. Organize data of the profit margin percentages into a frequency distribution. Number of class K=6 (Since where n is the total number of data = 35.)The class length is calculated as ClassFrequencyRelative FrequencyPercentage Frequency1.1-4.9140.40 40.0%5.0-8.8170.49 48.6%8.9-12.730.09 8.6%12.8-16.600.00 0.0%16.7-20.500.00 0.0%20.6-24.410.03 2.9%a. Construct a stem-and leaf display of the return on capital percentages for the retailers in Table. Find the median and then describe the distribution of the return.The median is the 18th measurement, 15.4.The data are widely spread out, with majority falling in the range 8.3, 22.8. The distribution is somewhat symmetric, with extreme values at both ends.c. Construct a histogram of the profit margin percentages. 4. In order to control costs, a company wishes to study the amount of money its sales force spends entertaining clients. The following is a random sample of six entertaining expenses (dinner costs for four people) from expense reports submitted by members of the sales force $157 $132 $109 $145 $125 $139a. Show the equivalence of the following two formulas: s2=i=1n(xi-x)2n-1 , s2=1n-1i=1nxi2-(i=1nxi)2n.Solutions:b. Calculate , , and s for the expense data, use the two different formulas in (a) to calculate.Solutions , Or using another formula:5. On February 15, 2005, the Gallup Organization released the results of a Gallup UK poll regarding Internet usage in Great Britain. Each of 1,009 randomly selected UK adults was asked to respond to the following question:How much time, if at all, do you personally spend using the Internet-more than an hour a day, up to one hour a day, a few times a week, a few times a month or less, or never?The polls results were as follows: More than an hour a day (22%); Up to an hour a day (14%); A few times a week (15%); A few times a month or less (10%); Never (39%). Use these data to construct a bar chart and a pie chart. Bar Chart: Pie Chart:6. Thirteen internists in the Midwest are randomly selected, and each internist is asked to report last years income. The incomes obtained (in thousands of dollars) are 127132138141144146152154162171177192241Find:There are 13 measurements in total (n=13). They are already in ascending order.a. The 90th percentile.Calculate the index of the 90th percentile: Round 11.7 up to 12, hence the 90th percentile is the 90th measurement 192.b. The median.The median is equivalent to the 50th percentile: Round 6.5 up to 7, hence the median is the 7th measurement 152.c. The first quartile.We know the index of the first quartile can be found as: Round 3.25 up to 4, hence the first quartile is the 4th measurement 141.d. The third quartile We know the index of the third quartile can be found as: Round 9.75 up to 10, hence the third quartile is the 10th measurement 171.e. The 10th percentile.Calculate the index of the 10th percentile: Round 1.3 up to 2, hence the 10th percentile is the 2nd measurement 132.f. The interquartile range.Interquartile range is the difference between first and third quartile g. Develop a five-number summary and a box-and-whiskers display.7. According to the M website, the 2004 total return percentages for several popular funds were as follow:Fund2004Total Return%Vanguard 500 Index10.7Wasatch Core Growth21.7Fidelity Stock Selector9.9Fidelity Dividend Growth6.0Janus Worldwide5.2Suppose that an investor had $100,000 invested in the Vanguard 500 Index fund, $500,000 invested in the Wasatch Core Growth fund, $500,000 invested in the Fidelity Stock Selector fund, $200,000 invested in the Fidelity Dividend Growth fund, and $50,000 invested in the Janus Worldwide fund.a. Compute a weighted mean that measures the 2004 average total return for the investors portfolio.Compare your weighted mean with the un-weighted mean of the five total return percentages. Explain why they differ.The weighted mean is And the un-weighted mean is The average total return calculated using weighted mean is higher because more money was invested in funds with larger gains, such as Wasatch Core Growth.8. As given on the Morningstar website, the mean return for Fidelity Leveraged Company Stock is 17.98 percent with a standard deviation of 10.02 percent, while the mean return for Baron Opportunity Retail Stock is 16.43 percent with a standard deviation of 12.23 percent. Which stock do you buy? (Which is riskier than the other one?)The mean return for Fidelity Leveraged Company Stock is higher than of Baron Opportunity Retail Stock. When we compute and compare the coefficient of variation for these two stocks using , we find that the coefficient of variation of Fidelity Leverage () is lower than the one of Baron Opportunity (), meaning the variation in returns for the Baron Opportunity is higher, so investing in Baron Opportunity is riskier.Therefore investing in Fidelity Leveraged Company Stock is more advisable. 9. The card game of Euchre employs a deck that consists of all four of each of the aces, kings, queens, jacks, tens, and nines (one of each suitclubs, diamonds, spades, and hearts). Find the probability that a randomly selected card from a Euchre deck isThere are a total of 24 cards (4 aces, 4 kings, 4 queens, 4 jacks, 4 tens and 4 nines.a. A jack (J)The probability of selecting a jack is b. A spade (S)The probability of selecting a spade card is c. A jack or an ace (A)The probability of selecting a jack OR an ace is d. A jack or a spade.The probability of selecting a jack OR a spade card is e. Are the events J and A mutually exclusive? J and S? Why or Why not?The events J and A are mutually exclusive because one cannot draw a Jack AND an Ace at the same time in one single draw. The events J and S are NOT mutually exclusive because one can draw a jack of spade under one single draw, hence J and S could occur simultaneously. 10. Fifteen percent of the employees in a company have managerial positions, and 25 percent of the employees in the company have MBA degrees. Also, 60 percent of the managers have MBA degree. a. What proportion of employees are managers and have MBA degrees.9% of employees are managers AND have MBA degreesb. What proportion of MBAs are managers.36% of MBAs are managersc. Are the events being a manager and having an MBA independent? Justify your answer.No. Because 11. In a survey of 100 insurance claims, 40 are fire claims (FIRE), 16 of which are fraudulent (FRAUD). Also, there are a total of 40 fraudulent claims.a. Construct a contingency table summarizing the claims data. Use the pairs of events FIRE and , FRAUD and .Contingency Table:Solution: a. FireTotalFraud162440Fraud243660Total4060100b. What proportion of the fire claims are fraudulent?40% of the fire claims are fraudulentc. Are the events a claim is fraudulent and a claim is a fire claim independent? Use your probability of part b to prove your answer.Yes. Because 12. Each month a brokerage house studies various companies and rates each companys stock as being either “low risk” or “moderate to high risk.” In a recent report, the brokerage house summarized its findings about 15 aerospace companies and 25 food retailers in the following table:Company TypeLow RiskModerate to High RiskAerospace company 6 9Food retailer 15 10If we randomly select one of the total 40 companies, findCompleting the contingency table as follow:Company TypeLow RiskModerate to High RiskTotalAerospace company 6 915Food retailer 15 1025Total211940 a. The probability that the companys stock is moderate to high risk given that the firm is an aerospace company.P(ModToHigh | Areo) = 9/15=0.6b. The probability that the companys stock is moderate to high risk given that the firm is a food retailer.P(ModToHigh | Food) = 10/25=0.4c. Determine if the company type is independent of the level of risk of the firms stock. If the company type is independent of the level of risk of the firms stock, the we should find equalities s