1.   Seedsmen grow plants to harvest seeds which they then sell in packets to gardeners. Unsatisfied customers have no seeds develop when they plant them out from a packet. Assuming that all gardeners are equally competent and letting X be the number of successful germinations from a packet of n seeds of germination rate p, then

a)  Write an expression for the probability of an unsatisfied customer                                 [2]

b)  What is the probability of no germinations for a 50% germination rate and only four seeds in a packet?                                                                                                                                 [2]

The number of unsatisfied customers is directly related to the germination rate. The satisfaction level is the percentage of customers with some [not none] seeds germinating.

c  )Find the limiting germination rate to 3 sig.fig. for a 98% satisfaction rate.                          [3]

d)  Working at a 97% satisfaction level, show that there must be seven seeds in a packet if the germination rate is 40%. How low can this probability fall to and preserve this success rate?  [6]

2.Over a long period it is established that delphiniums have an expected germination success rate of 80%. There are usually 12 seeds to a packet

a)Find the most likely number of germinations and the probability of this number                 [4]

b)  If you do a test at the 2% significance level on a sample, what is the percentage probability, called the confidence level, of coming to the correct conclusion?                                          [1]

c)  A customer complains that only half her seeds grew, which is now referred to as the success rate for the seed packet. Test at the 5% level whether this is a justifiable complaint.            [5]

d  )For the same success rate of 50%, what is the most seeds that could be in a packet for the 95% confidence level [i.e. without justifying any complaints] ?                                               [4]

e)  A slightly more customer-friendly approach would aim at a packet success rate of 60-65% for that confidence level. How many seeds are required per packet?                                      [5]

3.  An expensive and difficult to propagate seed for a tropical plant has a germination rate sufficiently low to require special attention. Attempts are made to improve the customer directions so as to raise the germination rate as experience by customers. The seed company has a target figure of 85%, much as they achieve themselves. However, last year’s customers achieved only 30% according to their responses. Seeds were priced at £6 per half dozen.

a) Produce a table showing the distribution of the number of successful germinations last year. Show that 12% had no return on their money.                                                   [6]

b) Seedlings are thought to be worth about £10, so only one germination represents a loss of £2. Show the total expected gain is about £12.                                                  [5]

c) What is the expected gain if germination can be improved to 60%?              [6]

A small number of customers are selected to be encouraged to propagate sets of ten seeds under different direction. Mrs Trellis is one of these triallists.

d) What is the range of values that will convince the seed company that Mrs Trellis’s germination rate has risen above 30% this year?                                                                   [4]

e) If her hypothetical rate of germination was 60%, explain carefully the range of values that would cause you to reject this as an idea.                                                          [4] 

f) What is probability that the germination rate really has doubled but that the triallist’s figures fail to convince the seed company?                                                                   [3]

S1 on seeds     2004-03-22       [60]

© David Scoins 2017