Homework 1 Due Date Time See Class Website 1 100 Points Total 1 2 144 1a Answer the following questions as well as you can LATE HOMEWORKS ARE NEVER ACCEPTED You may meet consult with colleagues in the class But the assignment you turn in needs to be your own work You should show some though not necessarily every bit of work for any substantial calculations Each part 5 points Suppose X N 30 144 That is X has a normal distribution with 30 and Find a transformation of X that will give it a mean of zero and a variance of one ie standardize X Let the transformed variable be named Z We desire z 0 2 One solution to this system of equations is b 1 12 and a 5 2 Of course if you recognized the fact that our standard Z transformation accomplishes precisely this z 1 This means 0 a b X and b2 2 X 1 you could write the transformation as 1b Find the probability that 18 x 36 1c 1d 2 For X 18 Z 1 For X 36 Z 5 The probability that Convert X into a standard normal via Z X is between 18 and 36 is thus equivalent to the probability of Z between 1 and 5 The latter term is F 5 F 1 6915 1587 5328 Supposing Y 10 5X find the mean of Y This is actually easier than 1a or 1b Y a b X 10 5 30 160 Find the variance of Y 2 X 52 144 3600 Y b2 2 Each part 4 points A bank has been receiving complaints from real estate agents that their customers have been waiting too long for mortgage confirmations The bank prides itself on its mortgage application process and decides to investigate the claims The bank manager takes a random sample of 20 customers whose mortgage applications have been processed in the last 6 months and finds the following wait times in days 5 7 22 4 12 9 9 14 3 6 5 8 10 17 12 10 9 4 3 13 Assume that the random variable X measures the number of days a customer waits for mortgage processing at this bank and assume that X is normally distributed 2a 2b 2c Find the sample mean of this data X Applying the formula for X gives 9 1 Find the sample variance of X Find the variance of X Applying the formula for S2 For c d and e use the appropriate t distribution Find the 90 confidence interval for the population mean X gives 24 31 The variance of X is S2 Xbar 24 31 20 1 215 The correct formula for a confidence interval with this information is use can be looked up from the t table and is 1 729 Making the interval The t value to Homework 1 Due Date Time See Class Website 2 Test the hypothesis that is equal to 7 at the 95 confidence level Should you do a one tailed or two tailed test here Two tailed Critical t values are 2 093 The test statistic from the information available is This is not in the critical region Do not reject the null hypothesis What is the approximate p value of this hypothesis According to the t table the probability of exceeding 1 904 is between 025 and 05 Because of the two sided nature of the test we double this value to account for the possibility of being in the other tail P value will therefore be between 05 and 1 Each part 4 points My nephew was born last summer He has 19 cousins on his father s side it s a big family I wish to know the mean x of the distribution of the ages of my nephew s cousins which is the variable X I take a sample of 4 with ages X 1 X 2 X 3 and X 4 These are all drawn from the same underlying population Instead of calculating the sample mean of these four I do the following calculation to create an estimator of which I call X X 0 1 X 1 0 2 X2 0 3 X3 0 4 X 4 3a 3b Explain what it means to say that X is unbiased It means that the expected value of X equals the population mean Show that X is unbiased 3c If the variance of X is 15 what is the variance of X Note that the covariance terms are zero because these are independent observations If the variance of X is 15 what is the variance of the X Which estimator is more efficient the sample mean X or X Explain The variance of X is larger than X Thus X is less efficient accurate than X 2d 2e 3 3d 3e Homework 1 Due Date Time See Class Website 3 4 Each part 4 points Suppose X N 30 144 and W N 40 225 4a If X and W are uncorrelated find the mean and variance of X 2W The value of the mean is 30 2 40 110 Since they are uncorellated the Correlation term is zero Thus the variance is 144 4 225 1044 4b Find the probability that X 2W 120 With the mean and variance calculated above the z value matching 120 is 120 110 32 31 31 The probability that z exceeds 31 is 3783 Henceforth suppose that X and W have a correlation coefficient 25 4c What is the covariance of X and W 4d Find the probability that X 2W 120 Using the formula from part a Var X 2W 864 making the z value 120 110 29 39 34 The probability of exceeding 34 is 3669 4e Find the probability that 50 X 2W 120 At 50 the z value is 60 29 39 2 04 The probability that z is less than this number is 0207 Since the probability of being less than 120 was implied in part d of being 6331 the probability of landing between the two is the difference 6124 5 Each part 5 points Our bank from Question 2 has decided to look more deeply into the matter of customer wait times In addition to information on the waiting times the bank has compiled information about the credit scores of the applicants That is the bank has 20 observation of the following 2 variables 5 Observation 1 Wait Time 12 5 Credit Score 740 730 550 700 65 0 3 22 2 7 4 4 7 9 6 9 660 630 600 760 730 8 14 10 6 9 3 Observation 11 Wait Time 5 Credit Score 700 12 8 620 13 10 600 14 17 580 15 12 650 16 10 670 17 9 670 18 4 790 20 19 3 13 750 610 5a Find the sample mean and variance of the Credit Score variable you can call this variable Y if you like 669 5 2 4405 5b Find the sample covariance and sample correlation coefficient of Wait Times and Credit Scores Covariance 288 4 Correlation 881 5c …