Question 4 [20 marks] A small local florist that is open 7 days a week specialises in bouquets made from home-grown flowers and gourmet home-made chocolates. As they are a small business, the florist is aware that on any given day, they do not sell large quantities of either bouquets or chocolates. Based on past experience, the florist believes that, on any given day, the relationship between the number of bouquets they sell (Y) and the number of boxes of chocolates they sell (X) is described by the joint probability distribution below. The florist is also certain that sales of either bouquets or chocolates are independent from day to day. Bouquets (y) 4 5 8 10 2 Chocolates (x) 4 5 0.17 0.13 0.12 0.05 0.09 0.03 0.01 0.08 0.02 0.15 0.07 0.08 A. [3 marks] Calculate the variance of the number of boxes of chocolates that the florist sells on any given day. B. [4 marks] Calculate the variance of the quantity =3 for any given day. C. [4 marks] Over the next 120 days, find the approximate probability that on more than 63 days the number C. [4 marks] Over the next 120 days, find the approximate probability that on more than 63 days the number of bouquets sold was at least twice the number of boxes of chocolates sold. The florist is an excellent bookkeeper, and over the previous 300 days they have recorded the number of bouquets and boxes of chocolates sold on each day. The data is summarised in the table of counts below. Bouquets (y) 4 5 8 10 2 Chocolates (2) 4 5 40 44 43 19 28 10 4 4 27 17 35 29 D. [4 marks] Considering only days on which the florist sold 2 boxes of chocolates, test whether the true proportion of days the florist sold 5 bouquets is greater than 0.15. Clearly state your hypotheses and use a significance level of a = 1.1% E. [5 marks] Ultimately, the florist cares more about how many bouquets are sold than boxes of chocolates. Based on the data of counts, test whether this sample data supports the probability distribution for the number of bouquets sold on any given day that was initially proposed by the florist. Clearly state your hypotheses and use a significance level of a : 5% Question 4 [20 marks] A small local florist that is open 7 days a week specialises in bouquets made from home-grown flowers and gourmet home-made chocolates. As they are a small business, the florist is aware that on any given day, they do not sell large quantities of either bouquets or chocolates. Based on past experience, the florist believes that, on any given day, the relationship between the number of bouquets they sell (Y) and the number of boxes of chocolates they sell (X) is described by the joint probability distribution below. The florist is also certain that sales of either bouquets or chocolates are independent from day to day. Bouquets (y) 4 5 8 10 2 Chocolates (x) 4 5 0.17 0.13 0.12 0.05 0.09 0.03 0.01 0.08 0.02 0.15 0.07 0.08 A. [3 marks] Calculate the variance of the number of boxes of chocolates that the florist sells on any given day. B. [4 marks] Calculate the variance of the quantity =3 for any given day. C. [4 marks] Over the next 120 days, find the approximate probability that on more than 63 days the number C. [4 marks] Over the next 120 days, find the approximate probability that on more than 63 days the number of bouquets sold was at least twice the number of boxes of chocolates sold. The florist is an excellent bookkeeper, and over the previous 300 days they have recorded the number of bouquets and boxes of chocolates sold on each day. The data is summarised in the table of counts below. Bouquets (y) 4 5 8 10 2 Chocolates (2) 4 5 40 44 43 19 28 10 4 4 27 17 35 29 D. [4 marks] Considering only days on which the florist sold 2 boxes of chocolates, test whether the true proportion of days the florist sold 5 bouquets is greater than 0.15. Clearly state your hypotheses and use a significance level of a = 1.1% E. [5 marks] Ultimately, the florist cares more about how many bouquets are sold than boxes of chocolates. Based on the data of counts, test whether this sample data supports the probability distribution for the number of bouquets sold on any given day that was initially proposed by the florist. Clearly state your hypotheses and use a significance level of a : 5%