$2.(1+1+1+1+1=5$ pt$)$ Suppose you have a sample of $2000$ students from Syracuse University. Each student bought one tube of toothpaste in February $2023$ and one tube of toothpaste in March $2023.$ They bought brand A$,$ B$,$ C$,$ D$,$ or E$.$ The brand switching matrix is provided below:

Mar$-23$

Feb$-23$ A B C D E

A $30010001000$

B $0200000$

C $004500150$

D $10010003000$

E $00500150$

$2.($a$)$ How many students purchased D in February $2023?$

Answer:

$2.($b$)$ How many students purchased D in March $2023?$

Answer:

$2.($c$)$ How many students who bought D in February $2023$ switched to other brands in March $2023?($These are people who bought D in February, but bought some other brand in March.$)$

Answer:

$2.($d$)$ How many students did not buy D in February $2023,$ but bought D in March $2023?$

Answer:

$2.($e$)$ Based on the brand$-$switching matrix, which brand or brands, if any, are competitors of brand D$?$

Answer:

Note: A brand X is a competitor of brand D under all of the following conditions:

$$ DX $>0$ and XD $>0($two$-$way switching between D and X$)$

$$ DX $>0$ and XD $=0($one$-$way switching from D to X$)$

$$ DX $=0$ and XD $>0($one$-$way switching from X to D$)$

X is not a competitor of D if and only if DX $=0$ and XD $=0.$

$3.(3+3+3=9$ pt$)$ Suppose you have a sample of $40$ Syracuse area households. Each household visited exactly one grocery store on $1/5/2024,$ and exactly one grocery store on $1/12/2024.$ They only visited one of four stores: A $($Aldi$),$ B $($Price Chopper$),$ C $($Trader Joe$$s$),$ or D $($Wegmans$).$ The stores visited are provided below.

Household $1/5/20241/12/2024$ Household $1/5/20241/12/2024$

$1$ C C $21$ A B

$2$ D C $22$ B A

$3$ D D $23$ D D

$4$ A A $24$ C C

$5$ D D $25$ D D

$6$ C C $26$ D D

$7$ A A $27$ B B

$8$ A A $28$ D D

$9$ B B $29$ D D

$10$ A B $30$ D D

$11$ D D $31$ D D

$12$ C C $32$ D D

$13$ B B $33$ D C

$14$ D C $34$ A A

$15$ A A $35$ B B

$16$ C C $36$ B A

$17$ C C $37$ B B

$18$ D D $38$ B B

$19$ D C $39$ D D

$20$ D D $40$ A A

$3.($a$)$ Prepare a brand$-$switching matrix from the data given. $($This is the cross$-$tabulation where store visited on $1/5/2024$ is the row variable, and store visited on $1/12/2024$ is the column variable. You can prepare the matrix directly from the data given above, or using Excel.$)$

Answer:

$1/12/2024$

$1/5/2024$ A B C D

A

B

C

D

$3.($b$)$ Compute the market share of each store on $1/5/2024$ and on $1/12/2024$ for this sample of $40$ households. You can answer this question by using the brand$-$switching matrix from $3($a$),$ or by directly counting how many households visited a store on a given date directly from the data.

$($Compute the market share of a store on a given date as the fraction of the sample that visited the store on that date, multiplied by $100\%.$ For example, if $8$ households out of $40$ visited Store X on $1/12/2024,$ then the market share of Store X on $1/12/2024$ is $(8/40)\backslash$times $100\%=20\%.)$

Answer:

Store Number that went to store on $1/5/2024$ Market Share of store on $1/5/2024$ Number that went to store on $1/12/2024$ Market Share of store on

$1/12/2024$

A

B

C

D

$3.($c$)$ Based on the brand$-$switching matrix, what can you say qualitatively about the nature of competition among these stores?

$($For each store, examine if it is gaining customers from or losing customers to any other store. If it is gaining or losing, is the gain$/$loss one$-$way or two$-$way? If it is one$-$way switching, then which store is losing customers?$)$

Answer:

$4.(3+4+3+3=13$ points$)$ You have to prepare the answer to this question using Excel and the data given in the worksheet Problem $4$ in the Excel file $$Data for $754$S$24$ Assignment $1.$

A sample of $1000$ Syracuse University students was collected to examine smart phone usage and purchase intentions. In particular, each student has reported which brand of smart phone he$/$she currently owns $($that is$,$Current Brand$)$ and which brand he$/$she would most likely purchase the next time he$/$she desires to purchase a new phone $($that is$,$Next Brand$).$

$4($a$)$ Using the pivot table command in Excel, prepare a brand switching matrix. Either fill in the table below, or copy and paste the matrix from Excel.

Next Brand

Current Brand Apple LG None Other Samsung Grand Totals

Apple

LG

None

Other

Samsung

Grand Totals

$4($b$)$ Using the brand switching matrix from $4($a$),$ answer the following:

$$ What percentage of SU students currently own an Apple smartphone? $($This is the market share of Apple.$)$

$$ Consider now the SU students whose current brand is Apple smartphone. For this group of students, what percentage will select Apple as the next brand? $($This is the brand loyalty of Apple.$)$

$$ Consider the students who do not have a smartphone now. What percentage of these students want Apple as their next brand?

$$ Consider the students who have LG now. What percentage of these students want Apple as their next brand?

$4($c$)$ Using pivot table and gender as filter, prepare separate brand switching matrices for men and women.

Brand Switching matrix for men

Next BraPls