Question

a restaurant chain that has 3 locations in portland is trying to determine which of their 3 locations they should keep open on new years eve. they survey a random sample of customers at each location and ask each whether or not they plan on going out to eat on new years eve. the results are below. run a test for independence to decide if the proportion of customers that will go out to eat on new years eve is dependent on location. use α = 0.05.

	nw location	ne location	se location
wont go out	20	25	20

enter the test statistic - round to 3 decimal places.

Explanation:

Step1: Calculate row and column totals

Total customers who will go out: $66 + 40+45=151$
Total customers who won't go out: $20 + 25+20 = 65$
Total customers at NW location: $66 + 20=86$
Total customers at NE location: $40 + 25 = 65$
Total customers at SE location: $45+20 = 65$
Total number of customers: $151+65 = 216$

Step2: Calculate expected frequencies

Expected frequency formula for a cell in a contingency - table is $E_{ij}=\frac{R_i\times C_j}{n}$
For the cell of NW location and will go out: $E_{11}=\frac{151\times86}{216}\approx60.231$
For the cell of NW location and won't go out: $E_{12}=\frac{65\times86}{216}\approx25.769$
For the cell of NE location and will go out: $E_{21}=\frac{151\times65}{216}\approx45.347$
For the cell of NE location and won't go out: $E_{22}=\frac{65\times65}{216}\approx19.653$
For the cell of SE location and will go out: $E_{31}=\frac{151\times65}{216}\approx45.347$
For the cell of SE location and won't go out: $E_{32}=\frac{65\times65}{216}\approx19.653$

Step3: Calculate the chi - square test statistic

The chi - square test statistic formula is $\chi^2=\sum\frac{(O - E)^2}{E}$
For the first cell: $\frac{(66 - 60.231)^2}{60.231}=\frac{33.28}{60.231}\approx0.553$
For the second cell: $\frac{(20 - 25.769)^2}{25.769}=\frac{33.28}{25.769}\approx1.291$
For the third cell: $\frac{(40 - 45.347)^2}{45.347}=\frac{28.58}{45.347}\approx0.630$
For the fourth cell: $\frac{(25 - 19.653)^2}{19.653}=\frac{28.58}{19.653}\approx1.454$
For the fifth cell: $\frac{(45 - 45.347)^2}{45.347}=\frac{0.12}{45.347}\approx0.003$
For the sixth cell: $\frac{(20 - 19.653)^2}{19.653}=\frac{0.12}{19.653}\approx0.006$
$\chi^2=0.553 + 1.291+0.630 + 1.454+0.003+0.006=3.937$

Answer:

$3.937$