Midterm_solutions_Section B

MATH 2565 3.0 - Fall 2023 Test questions Question 1: Customers at Longos grocery store in Toronto can pay for purchases with cash, an ATM card, or a credit card. 55% of all customers use cash, and 38% use an ATM card. Careful research has shown that of those paying with cash, 75% use coupons; of those using an ATM card, 35% use coupons; and of those using a credit card, only 10% use coupons. Round answers to 3 decimals. a) If a random selected customer does not use coupons, what is the probability that the customer uses a credit card to pay for purchases? P (credit card | no coupons) = 0 . 141 b) If 3 customers are randomly selected, what is the probably that at least one of them uses coupons? P (at least 1 of 3 use coupons) = 0 . 910 1

Question 2: Consider the following data set x 1 , . . . , x 18 be 177 183 187 188 189 192 194 196 196 202 206 210 211 212 215 222 223 224 Round answers to 3 decimals. a) Using the following given information, n X i =1 x i = 3627 n X i =1 x 2 i = 734403 n X i =1 ( x i − ¯ x ) 2 = 3562 . 5 , find the sample mean and sample variance. ¯ x = 201 . 5 s 2 = 209 . 559 b) Use the IQR to identify outliers. Q 1 = 189 Q 3 = 212 IQR = 23 ( Q 1 − 1 . 5 IQR, Q 3 + 1 . 5 IQR ) = (154 . 5 , 246 . 5) Since all data are within (154.5, 246.5), therefore no outliers. 2

Question 3: Assume that online shopping browsing session durations of retail websites is normally distributed with a mean of 21.7 minutes and a standard deviation of 6.8 minutes, based on data from a market research firm. Round answers to 4 decimal places. a) Find the probability that a randomly selected online shopping browsing session has a duration less than 15 minutes. P ( X < 15) = 0 . 1611 b) Find the probability that a randomly selected online shopping browsing session has a duration between 15 to 20 minutes. P (15 < X < 20) = 0 . 2402 c) What durations make up the top 10% of all online shopping browsing durations for the retail websites? P ( Z < 1 . 28) = 0 . 90 1 . 28 = X − 21 . 7 6 . 8 X = 30 . 4 Thus, durations of 30.4 minutes or more make up the top 10%. d) Find the probability that a SRS of 10 online shopping browsing sessions has a mean duration less than 15 minutes. P ( ¯ X < 15) = 0 . 0009 4

Question 4: During the pandemic, it was common that many people worked from home. A multi-national company gathered data from random office locations worldwide to determine if there was a relationship between the number of cases of the virus at that office (Cases) and the percentage of employees who were working in the office at that location (InOffice). Furthermore, they wanted to determine if the number of cases could predict the percentage of employees who showed up to the office for work. Here is some R output from their analysis. > mean(Cases) [1] 25.39623 > mean(InOffice) [1] 25.38755 > cor(Cases,InOffice) [1] -0.7301128 > lm(InOffice ∼ Cases) Call: lm(formula = InOffice ∼ Cases) Coefficients : (Intercept) Cases 34.0818 -0.3423 Answer the following questions based on the R output. a) Which is the independent variable and which is the dependent variable? Write the linear regression equation. Independent variable is number of cases, dependent variable is percentage of employees ˆ y = 34 . 0818 − 0 . 3423 x b) Explain if the y -intercept make sense in this scenario. ˆ β 0 = 34 . 0818 is the percentage of employees in the office when there are no cases. It is possible to have 0 cases and thus the intercept makes sense. 5