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Unformatted text preview: More PastPaper: http://ihome.ust.hk/~cs_gxx ISMT111 Business Statistics Final Examination For sections L3, L4, only 15th December 2006 Directions 1) Answer ALL SIX questions. Marks are shown in square brackets. 2) There are 4 pages in this examination paper including the cover page. Check to make sure you have a complete set and notify the invigilator immediately if part of it is missing. 3) Key formulas and Statistical tables are provided separately. 4) Calculator may be used in this examination. 1 ht tp :// ih om e. us t.h k/ ~c s_ gx x/ 5) You are given THREE HOURS to complete this examination. Do not begin until you are told to do so. More PastPaper: http://ihome.ust.hk/~cs_gxx Question 1: [18 Marks] Data on the monthly service fee in dollars when a customer’s account falls below the minimum required $5000 balance for a sample of 26 banks yields a sample mean of 71.5 and a sample standard deviation of 30.55. (a) Please find a 95% confidence interval for the population mean monthly service fee. (b) Can we say that with probability 0.95 the interval obtained in (a) will include the population mean? If not, then what is the correct interpretation? Explain briefly. (c) Data on 10 more banks were added so that the sample size increased to 36 and we obtained a 95% confidence interval [59.55, 80.45]. What is the mean of the sample of 36? (d) Continue from (c). What is the standard error for the sample mean? (e) Assume that the estimated population standard deviation in (d) equals the true population s.d. What is the minimum sample size required so that the length of the 95% interval is less than 16? (f) Continue from (c). If there are four banks in the sample of 36 do not charge monthly service fee when the balance fall below $5000. Please find a 95% confidence interval for the proportion of banks that do not charge monthly service fee. Question 2: [ 16 Marks] One way to measure the economic outlook is the percentage of employers hiring new employees. A survey conducted by the government last year showed that 66 of the 150 employers recruiting new employees. A similar survey this year reveals that 90 of 180 employers hiring new employees. (a) An economist would like to test the hypothesis that the economic outlook this year is better than last year using the data on hiring new employees. State the null and alternative hypotheses. (b) Carry out the test and draw your conclusion using 5% significance level. (c) What is the p-value of the test in (b)? (d) Suppose the economist later found out a very accurate estimate 0.44 is available for the proportion p1 of employers hiring new employees last year. Therefore he assumes p1 =0.44 and ˆ decide to use { p 2 > 0.5} as the rejection region for the null hypothesis H 0 : p 2 = 0.44 . What is the probability of type I error for this rejection region? Question 3: [16 Marks] A statistics professor would like to demonstrate the concept of sampling distributions to a group of students. He used the computer to generate 1000 samples of size 25 each from the normal distribution with mean 3 and standard deviation 1. For each sample he calculated the sample mean x . (a) Please find the proportion of samples with sample means between 2.9 and 3.3. (b) For each sample he formed the interval x ± 0.36 . What is the proportion of intervals enclosing 3? x −3 for each sample, where s is the sample standard deviation. s/5 gx x/ (c) He also calculated the statistic k/ ~c s_ Please find the proportion of the preceding statistic taking values between 1.71 and 1.32? 2 ht tp :// ih om e. us t.h (d) For each sample, he looked at the sample proportion of observations greater than 4. What is the proportion of samples with sample proportions greater than 0.2? More PastPaper: http://ihome.ust.hk/~cs_gxx Question 4: [18 Marks] A large company employs several thousand people in the manufacture of keyboards, equipment cases, and cables for the small-computer industry. The personnel manager of the company would like to find ways to forecast the absentee rate among the company employees. An effective method of forecasting would greatly strengthen the ability to plan properly. He took a sample of 15 employees and recorded the number of absent days (Y) during the last fiscal year along with employee age (X). The computer output of a regression analysis is as follows. The regression equation is absent days = - 4.28 + 0.254 age Predictor Constant age Coef -4.277 0.25379 S = 1.10807 SE Coef 1.116 0.02850 R-Sq = 85.9% T - 3.83 8.91 P 0.002 0.000 R-Sq(adj) = 84.8% Analysis of Variance Source Regression Residual Error Total DF 1 13 14 SS 97.372 15.962 113.333 MS 97.372 1.228 F 79.30 P 0.000 (a) An employee, John, is 30 years old. According to the regression equation, what is his expected number of absent days in the coming fiscal year? (b) Test the regression coefficient b1 of age is larger than 0.2 using 5% significance level. (c) Find a 95% confidence interval for the regression coefficient of age. (d) The sample mean and sample standard deviation for age are 37.87 and 10.39, respectively. Find a 95% confidence interval for the mean absent days of 30 years old employees. Question 5: [16 Marks] In the following questions about regression, X is the independent variable, and Y is the dependent variable. (a) Suppose that you can choose the X- values at which Y- values are obtained. To estimate the slope and intercept of the regression line accurately, would you choose X- values so that they have larger or smaller sample variance? Please explain your answer briefly. (b) Let S x , S y denote the standard deviations of X and Y, respectively; b1 , the least-square slope estimate, r, the correlation coefficient. Then the following relation holds S x b1 = rS y (no need to derive this formula). Based on the preceding formula, if we increase the X-value by one standard deviation, what is the average change in Y? 3 ht tp :// ih om e. us t.h slope and intercept of the least square regression line? Explain your answer briefly. k/ ~c s_ ~ ~ ~ ~ X i = ( X i − X ) / S x , Yi = (Yi − Y ) / S y . If we run the regression of Y on X , what is the gx x/ (c) Continue from (b), let the correlation coefficient r = 0.6. Standardize the X and Y by subtracting the corresponding mean and dividing by the corresponding standard deviation, that is http://ihome.ust.hk/~cs_gxx Question 6: [16 Marks] :// ih om e. us t.h k/ ~c s_ gx x/ . 4 ht tp More PastPaper: ...
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This note was uploaded on 02/27/2010 for the course ISOM ISOM111 taught by Professor Anthonychan during the Spring '09 term at HKUST.

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