Family Dinners

John has been arguing with his co‐owner that there tends to be fewer restaurants that cater for children and families in the city relative to those outside the city. His co‐owner, Danielle argues that it's more about the time of day that the restaurant is open. To help, Danielle asks you to test whether the occasion upon which the restaurant is open for dinner is independent of whether the restaurant caters for children/families or not. She has asked you to consider this in term of three possible dinner outcomes, categorising restaurants as: i) open exclusively at dinner only; ii) open at dinner, but also at another time of day (i.e., also open at breakfast and/or lunch); or, iii) not open at dinner. Danielle would like you to present several things: a) A joint frequency table of the three dinner outcomes against whether children/families are catered for or not. b) A table of expected frequencies of the three dinner outcomes against whether children/families are catered for or not assuming the two events are independent. c) A table representing your underlying calculations of a chi‐square test‐statistic to test the independence of the two events of interest. d) The critical value and a suitable description of how you generated this to conduct your test of independence. e) A conclusion of what these previous steps inform you with respect to Danielle’s question. f) Danielle asks you to present three conditional probabilities that illustrate her point. Specifically, determine the probability a restaurant caters for children/families, given the time it is open for dinner as defined above. Danielle also asks you to determine the marginal probability of catering for children/families regardless of opening time. g) Write a short summary for Danielle and John of what you have determined particularly focused on the outcomes of step (e) and illustrating this point using the outcomes of your calculations from step (f).                                                

Sample Solution