$.(3+3+3=9$ 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 $6200$

B $2600$

C $0060$

D $00414$

$3.($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: