1. (10 Marks) The lifetime (hours) of 20 lightbulbs were recorded and the sample data are shown in the following stem-and-leaf plot. Stem (in 10) Leaf (in 1) 34567 2 4 B 6532 1555 A 59266 880 8 8 (a) Suppose that the interquartile range of the sample data is 20, the range of the sample data is 45, and the median of the sample data is 61. Find A, B, and C. (b) Construct a box-and-whisker diagram for the sample data. is 45, and the median of the sample data is 61. Find A, B, and C. (b) Construct a box-and-whisker diagram for the sample data. 2. (20 Marks) (a) Company A is an organization whose members possess IQ that are in the top 5% of population in a country. Assuming that IQ of a randomly selected citizen in that country is normally distributed with a mean of 100 and a standard deviation of 15. Find the minimum IQ needed to be a member of Company A. (b) There are 16 parking spaces for the 20 executives who own cars in a company. It is known that 80% of them will drive to office independently on a weekday. In a certain working week starting on Monday, what is the probability that the parking lot is full for the third time on Friday that week? 3. (16 Marks) A company manufactures steel shafts for use in engines. Assume that the length of the steel shaft is normally distributed. A random sample of 10 shafts produced in the following measurements of their lengths (in centimetres): 3. (16 Marks) A company manufactures steel shafts for use in engines. Assume that the length of the steel shaft is normally distributed. A random sample of 10 shafts produced in the following measurements of their lengths (in centimetres) 20.2 18.9 196 20.7 201 198 190 20.5 198 21.2 (a) Suppose first that the population standand deviation of the length of the shaft is known. In addition, based on this population standard deviation, the margin of error at a confidence level of 95% for the mean length of the shaft for the sample of size 25 is 0.50. Find and interpret an estimate for the mean length of the shaft based on the above sample of size 10 with 99% confidence. (b) Assume now that the population standed deviation of the length of the shaft is unknows Based on the above sample of size 10, the margin of error for the mean length of the shad is 0.51564. Find the critical value and significance level associated with this margin of mr. In addition, find unit interpret the corresponding confidence interval estimate for the mean length of the shaft State clearly the level of confidence. (c) One method for judging inconsistencies in the production process to determine the variance of the length of the shaft Find and impect an estimane for the variance of the lengths of the shaft with 90% confidence based on the above sample of size 10. 4. (30 Marks) (a) A group of researchers randomly sampled and recorded the monthly returns (in %) of Funds 1 and 2 for 50 months each. The mean and the standard deviations of the monthly returns of Funds 1 and 2 are as follows: Fund 1: 6.5761, $ = 6.1737, Fund 2: 23.7232, $ = 6.5833. Can we conclude at the 5% significance level that the mean of the monthly returns (in %) of Fund 1 exceeds that of Fund 2? Assume that the two populations of monthly returns are normally distributed. (b) In addition to Funds 1 and 2 in (a), the same group of researchers randomly sampled and recorded the monthly returns (in %) of Funds 3 and 4 for 50 months each. The mean and the standard deviations of the monthly returns of Funds 3 and 4 are as follows: Fund 3: 4.7852, Fund 4: 47.0224, S3 = 6.0538, S = 6.2643. Can we conclude at the 5% significance level that there are differences among the means of the monthly returns (in %) of Funds 1-4? Assume that the four populations of monthly returns are normally distributed and that these populations have equal variances. 5. (24 Marks) A research department of an American automobile company wants to develop a model to predict gasoline mileage (measured in MPG) of the company's vehicles by using their horsepower and weights (measured in pounds). To do this, it took a random sample of 50 vehicles to perform a regression analysis as follows: 3 Regression State Multiple R 0.565689 R Square 0.749417 Adjusted R Square 0.738754 Standard F 4.176602 Observations 50 ANOVA of 35 MS F Regression 2451.973702 1225.987 d Residual b 819.8680976 Total 49 3271.8418 Standard Coefficiens Emir Seat Intercept 58.15708 2658248208 21.87797 Horsepower Weighe 40.11753 0.032643428 -3.60028 000687 0.001401171 490349 (a) State the multiple regressum option Interpret the meanings of the coefficients for horepower and weight d) Test the validity of this multiple regression equation at the significance level of the Show your reasoning (e) The search department claims that the weight of the vehicle is negatively linearly related to in gasoline mileage. Test its claim at the significance level of 1% 934