Question

1. There are many simple experiments that can be performed to estimate r for a partic ular problem, and we encourage you to perform your own. The parts below discuss some experiments that we or our students have performed in the past. For each part, discuss conditions that are needed in order to satisfy the five assumptions for using the binomial distribution outlined in Section 1.1.1. (a) One of the first solid foods that infants can feed to themselves is the cereal Cheerios. Early on, infants often lack the dexterity in their fingers to pick up an individual Cheerio and put it into their mouth. In order to estimate the probability of success for one infant, an experiment was designed where a Cheerio would be set on an infant's food tray for 20 successive trials at the dinner table. If the Cheerio made it into the infant's mouth, the response for the trial was considered a success; otherwise, the response was considered a failure. Out of these 20 trials, there were 9 Cheerios that made it into the infant's mouth. (b) In most billiards games, the person who breaks a rack of balls gets another turn if at least one ball is sunk, excluding the cue ball. This is advantageous for this person because there are a fixed number of balls that need to be sunk in order to win a game. In order to estimate one person's probability of success of sinking a ball on the break, there were 25 consecutive breaks performed in 8-ball billiards. Out of these 25, 15 breaks had a ball sunk. ) The germination rate is the proportion of seeds that will sprout after being planted into soil. In order to estimate the germination rate for a particula of sweet corn, 64 seeds were planted in a 3' x 4' plot of land with fertile soil. The seed packet stated that sprouts should emerge within 7-14 and care guidelines given on the seed packet were followed as closely as possible. After waiting for three weeks, 48 seeds had sprouted out of the soil. days, and all planting

Answer 1

The assumptions underlying the Binomial distribution are:

1) There must be only two possible outcomes (success and failure).

2) The outcomes must be mutually exclusive (occurrence of one outcome prevents the occurrence of the other).

3) The number of trials must be finite.

4) The trials must be independent of each other (occurrence of one trial does not depend on the occurrence of any other trial).

5) The probability of success must be constant for each trial.

Let us now discuss the conditions required by the following experiments in order to satisfy the binomial distribution:

a) There can be only two possible outcomes of the experiment i.e., either the cheerios will reach the infant's mouth(success) or it will not (failure). It is obvious that outcomes are mutually exclusive. The experiment consists of 20 trials (which is finite). Since the outcome of one trial does not depend on the outcome of others, so the trials are independent too. The last assumption says that the probability of success must be constant for each trial, which can also be considered true in this case. Hence this experiment satisfies all the assumptions of a Binomial Distribution.

b) This experiment also satisfies all the conditions of a Binomial distribution as:

1) There are only two possible outcomes i.e., either a ball will sink at break (success) or it will not (failure).

2) These outcomes are mutually exclusive.

3) There are 25 trials, which is a finite number.

4) Each trial is independent of the other i.e., whether a ball sinks or not in any particular trial does not depend on whether it sunk in the previous trials.

5) Lastly, the probability that a ball will sink at break will remain constant at each trial, provided it is done by the same player.

c) Here, one seed can be considered as one trial. So there are a total of 64 trials of the experiment. This is a finite number. Now, there are only two possible outcomes - the seed will germinate or it will not. These two outcomes are obviously mutually exclusive. The germination of one seed does not depend on the germination of others. Hence the trials are independent. The probability of germination is the same for each seed as the soil fertility is kept same throughout and same planting and care guidelines were followed for all seeds. Thus, Binomial distribution can be used.

d) This problem also satisfies all the conditions of a binomial distribution. It can be checked in a similar manner as above. The outcomes are: a person using alternative fuel car (success) and a person not using alternative fuel car (failure).