probability

Question 1

Var(X)

Answer 1

A measure of the variability of the randomly generated values.

Question 2

Population

Answer 2

Set of all eligible members of a group which we intend to study.

Question 3

Sample

Answer 3

A subset of the population which we select in order to make inferences about the population.

Question 4

Is generalizing from the sample to the population useful?

Answer 4

Will not be useful unless the sample is representative of the population.

Question 5

Population proportion (symbol and what is it)

Answer 5

p, proportion of individuals in the entire population possessing a particular attribute, and is constant.

Question 6

Sample proportion (symbol and what is it)

Answer 6

p̂, the proportion of individuals in a particular sample possessing the attribute, and varies from sample to sample.

Question 7

How to determine which var, var(x) or var(y) is larger?

Answer 7

The var that has higher probabilities for the extreme (low and high values)/ the var with probability more concentrated around the middle values is smaller.

Question 8

Describe the 3 rules of the Bernoulli sequences.

Answer 8

1. The trials must be independent.
2. Each trial should have exactly two outcomes: success or failure.
3. The probability of success or failure does not change for each trial.

Question 9

In a board game 4 six-sided dice are rolled and a “hit” is scored if at least one of the dice rolls a 1.

Will the probability of rolling at least six hits from ten sets of rolls be different from the previous answer? Explain why. (3 marks)
- probability of at least three hits occurring from five sets of rolls= 0.5332
- probability of a hit being scored= 0.5177

Answer 9

𝑌 ~ 𝐵𝑖(10 , 0.5177).
𝑃𝑟(𝑌 ≥ 6) = 0.4213 from calculator.
The two calculations give different answers.
Given p is the same for both distributions, they will have the same general shape.
For
X ~ Bi(n=5, p= 0.5177); x = 0,1, 2,3, 4,5 i.e. 6 possibilities.
Pr(X = 3)
includes probabilities from ‘centre’ to the right tail.
[i.e. 3 from 6, ½ the distribution]
For
Y ~ Bi(n =10, p = 0.5177); x = 0,1, 2,3,….., 9,10 i.e. 11 possibilities.
Pr(Y = 6)
includes probabilities from ‘the right of centre’ to the right tail.
[i.e. 5 from 11, less than ½ the distribution]
Hence, we expect
Pr(Y ≥ 6) < Pr(X ≥ 3)