Question

The correct size of a nickel is 21.21 millimeters. Based on that, the data can be summarized into the following table:

	Too Small	Too Large	Total
Low Income	25	15	40
High Income	23	12	35
Total	48	27	75

Based on this data: (give your answers to parts a-c as fractions, or decimals to at least 3 decimal places. Give your to part d as a whole number.)

a) The proportion of all children that drew the nickel too small is:   

Assume that this proportion is true for ALL children (e.g. that this proportion applies to any group of children), and that the remainder of the questions in this section apply to selections from the population of ALL children.

b) If 5 children are chosen, the probability that exactly 3 would draw the nickel too small is:

c) If 5 children are chosen at random, the probability that at least one would draw the nickel too small is:

d) If 120 children are chosen at random, it would be unusual if more than drew the nickel too small

Answer 1

a)

Out of 75 children, 48 drew the nickel too small. Therefore the proportion of all children who drew the nickel too small would be given by,

b)

Let the random variable X represent the number of children out of 5 who drew the nickel too small. Clearly X has a binomial distribution with N = 5 and p = 0.640

The required probability would be given by,

c)

Connsider X to be defined same as in previous part. The required probability would be given by,

d)

If the probability of an event is less than 0.05 then we consider it to be unusual. Since the sample size is greater in this case we can approximate the distribution of X as a normal distribution.

The mean of the distribution would be given by,

The standard deviation of the distribution would be given by,

For 1.645 z-score the distribution has a cumulative probability of 0.95 Hence the value of X corresponding to a Z-score of 1.645 will be the cutoff value.

Hence the cutoff value would be given by,