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Anticipating Patterns, Part 2

Question 1: Show that if ? is a Normally distributed random variable with ?=? and ?=?2,?=?-?2?~?0,1.

Answer 1: Proof: Recalling that ?=?, where ? is a constant, ?±?=?(?)±? and?=?1?=1?=?=1?=1?=?(?)=1?=?, then?-?2?=1?2?-?=1?2?-?(?)=1?2?-?=0Recalling that ?=?2?, ?±?=?+?, and ?=0, where ? is a constant,?-?2?=1?2?-?=?2?+?=?2?2?+0=1

There are lots of good resources about Anticipating Patterns that you can find available.

Question 2: Define the sampling distribution of a sample proportion ?/? with a sample space S={A,B}, ? is the number of events A in the sample, and ? is size of the sample.

Answer 2: A sample proportion of a sample space S={A,B} is the fraction of the sample that is either event A. For example, if S={female, male}, a sample proportion could be the fraction of people that are female. The value ? would distribute as a binomial distribution, ?(?,?,?) where ? the number of females, ? is the number of people in the sample, ? is the probability that a person is female, and ?/? is an estimate of ? and is referred to as the sample proportion of females.For a binomial distribution, ?=? and ?=?(1-?), so recalling that ?=?,?=1?=1?=?,and recalling that ?=?2?, ?=1?2?=1?2?1-?=?1-?.For two independent sample proportions,?1=?1?1 and ?2=?2?2,?1=?1, ?2=?2, ?1=?11-?1?1, ?2=?21-?2?2.

Question 3: Given two independent sample proportions ?1/?1 and ?2/?2 which are estimates ?1 and ?2 of probabilities ?1 and ?2 for the binomial distributions ?1,?1,?1 and ?1,?1,?1, find ?(?1-?2) and ?1-?2 and show that even though ?1/?1 and ?2/?2 are not normally distributed,?1?1-?2?2-?1-?2?11-?1?1+?21-?2?2 converges in distribution as ?1, ?2?8 to ?0,1.

Answer 3: If the true population proportions are not known, ?-?-?-?-?+?-? converges in distribution to a normal distribution as ?1, ?2?8 according to the Central Limit Theorem. It has average zero and variance 1 in the limit,?1?1-?2?2-?1-?2?11-?1?1+?21-?2?2=?1?1-?2?2-?1-?2?11-?1?1+?21-?2?2=?1-?2-?1-?2?11-?1?1+?21-?2?2=0?1?1-?2?2-?1-?2?11-?1?1+?21-?2?2=?1?1-?2?2-?1-?2?11-?1?1+?21-?2?2=?11-?1?1+?21-?2?2-0?11-?1?1+?21-?2?2=1Therefore, the Z-Table (Standard Normal Table) may be used if ?1 and ?2 are large enough.?=?za/2/[0.21p], za/2/[0.21(1-p)]. For a 95 confidence level, after rounding up to nearest integer, ?,?1-?= 1.96/0.21=9.33?10.zza4zz

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