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	19	21	40
High Income	19	16	35
Total	38	37	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: 38/75

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 100 children are chosen at random, it would be unusual if more than ____ drew the nickel too small

Answer 1

a.
p = 38/75 = 0.5067

b.
n = 5
x = 3
P(x) = nCx * (p^x) * ((1-p)^(n-x))

probability that exactly 3 would draw the nickel too small is
P(3) = 5C3 * (0.5067^3) * (0.4933^2)
P(3) = 10 * (0.5067^3) * (0.4933^2)
= 0.3166

c.
probability that at least one would draw the nickel too small is
P(X>=1) = 1 - P(0)
= 1 - (5C0 * 0.5067^0 * 0.4933^5)
= 1 - (1 * 0.5067^0 * 0.4933^5)
= 0.9708

d.
n = 100
p = 0.5067
using binomial approximation
mean = np = 50.67
standard deviation = sqrt(np(1-p)) = sqrt(50.67*(1-0.5067)) = 5

It would be unusual if the probability is less than 0.05
P(X>k) = 0.05
P(X P(Z<(k-50.67)/5) = 0.95

Since 0.95 = P(Z<1.645)
(k-50.67)/5 = 1.645
k = 58.895
it would be unusual if more than 58 drew the nickel too small.

Answer 2

a) Proportion Low Income Too Large = (28/75)*100 = 37.3%

b) Proportion High Income Too Large = (8/75) *100 = 10.7%

c) Proportion All Children Too Large = (36/75) * 100 = 48.0%

d) Conditional Probability = (28/75)/(36/75) = 0.778 (77.8%)

Answer 3

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 12 28 40
High Income 27 8 35
Total 39 36 75


Based on this data: (give your answers as fractions, or decimals to at least 3 decimal places)

a) The proportion of children from the low income group that drew the nickel too large is:

b) The proportion of children from the high income group that drew the nickel too large is:

c) The proportion of all children that drew the nickel too large is:

d) If a child is picked at random, what is the probability they are in the low income group, given they drew the nickel too large?

answer

a) Proportion Low Income Too Large = (28/75)*100 = 37.3%

b) Proportion High Income Too Large = (8/75) *100 = 10.7%

c) Proportion All Children Too Large = (36/75) * 100 = 48.0%

d) Conditional Probability = (28/75)/(36/75) = 0.778 (77.8%)

Answer 4

a) Proportion Low Income Too Large = (28/75)*100 = 37.3%

b) Proportion High Income Too Large = (8/75) *100 = 10.7%

c) Proportion All Children Too Large = (36/75) * 100 = 48.0%

d) Conditional Probability = (28/75)/(36/75) = 0.778 (77.8%)