17 17 1. (15 points) For a data sample of 17 observations, we have X x; = 34 and X x? = 468. i=1 i=1 Calculate: (a) (5 points) the sample mean (b) (5 points) the sample standard deviation (c) (5 points) the coefficient of variation2. (5 points) Given the following data: 4 2 1 1 5 3 0 1 O Calculate: (a) (3 points) the sample median (b) (2 points) the sample range 3. (10 points) An insurance company has collected the following data on the gender and marital status of 300 customers. Single ( S) | Married ( M ) | Divorced ( D ) | Totals Female (F) Male (F) Totals A customer is selected at random. Calculate the probability that the selected customer is: a married female not single married given that the customer is male Are marital status and gender independent? Explain using probabilities. (a) (2 points (b) (2 points () ( (d e r 2 points ) (4 points ~ 4. (10 points) Consider the follow probabilities for events A and B. P[A|B] =025 P[A|B']=0.5 P[B] =0.35 Calculate P[A] [HINT: YOU NEED TO USE BAYES' THEOREM TO CALCULATE THIS PROBABILITY, AND DON'T FORGET ABOUT LAW OF TOTAL PROBABILITY]. 5. (10 points) Consider a Normal random variable X with mean p =5 and standard deviation o = 2.5. (a) (5 points) Calculate P[X 6.25]. 6. (5 points) Customers arrive randomly to a barber shop with the following probability distribution. 0.177 when n=0 0.221 when n=1 P,=:0.276 when n=2 0.230 when n=3 0.096 when n=4 Here, P, is the probability that there are n number of customers in the barber shop. What is the minimum number of stools needed to accommodate all of the customers at least 70% of the time? 7. (15 points) Assume the time required to pass through security at a particular airport follows the continuous uniform distribution with a minimum time of 5 minutes and maximum time of 30 minutes. (a) (5 points) Calculate the value of f(x). (b) (5 points) What are the mean and standard deviation of this distribution? (c) (5 points) What is the probability that the next passenger will require less than 25 minutes to pass through security? 8. (30 points) Dr. Gazi Igbal is studying the relationship between monthly expenses and monthly income for NYIT QANT 501 Spring 2024 class students. He has collected data from some students. Below is a scatter plot showing an approximately linear relationship between month expenses and monthly income. Limited regression output and some calculations are shown. Yo, = 20651, Y a? = 10671181, 3y, = 13116, 342 = 6632500, i t=1 g=1 i=1 i=1 where n = 29 Monthly Expense 400 500 600 700 300 400 600 800 1000 1200 Monthly Income DF Sum 5q Mean Sq F Value Monthly_Income 158519 Residuals 541931 Total Residual standard error: 141.7 on 27 degrees of freedom Coefficients: Estimate Std. Error Intercept 231.2281 82.9398 Monthly_Income ; .1 (i) [15 points] Complete the ANOVA table in the regression output. Show your work in computing SSR, SSTO, MSE, and F-value.(i) [5 points] For a monthly income of $750, a 95% prediction interval for monthly expense is ($147.753, $739.2133). Interpret this interval in a sentence. (iii) [10 points|] Comment on the following figures . Based on these two plots, which model assumptions appear violated and which ones appear satisfied? Explain. Residual o o o 100 -100 0 -300 o o o o o o o o o 2 08 _ o o o o o o o o I T 1 1 T 400 600 800 1000 1200 Monthly Income : Residual vs. Monthly Income \f