Hypothesis Test for Proportions

Section 10.3

One Sample

Remember: Properties of Sampling Distribution of Proportions

Approximately Normal if

pp

p

pq

n

5

5

np

nq

Test Statistic

pt. estimate - parameter

st. errorstatistic - parameter

st. dev.

z

z

p pz

pqn

Conditions

Educators estimate the dropout rate is 15%. Last year 38 seniors from a random sample of 200 seniors withdrew. At a 5% significance level, is there enough evidence to reject the claim?

p=true proportion of seniors who dropout

: 0.15

: 0.15o

A

H p

H p

Assumptions:

(1) SRS

(2) Approximately normal since np=200(.15)=30 and nq=200(.85)=270

(3) 10(200)=2000 {Pop of seniors is at least 2000}

Therefore the large sample Z-test for proportions may be used.

0.15(0.85)200

0.19 0.151.58

pqn

p pz

2(0.057) 0.114p val

Fail to reject Ho since p-value >α. There is insufficient evidence to support the claim that the dropout rate is not 15%. What type of error might we be making?

PHANTOMS P arameter H ypotheses A ssumptions N ame the test T est statistic O btain p-value M ake decision S tate conclusions in context

If the significance level is not stated – use 0.05.

Reject Ho

There is sufficient evidence to support the claim that …..

Fail to Reject Ho

There is insufficient evidence to support the claim that ….

A random sample of 270 CA lawyers revealed 117 who felt that the ethical standards of most lawyers are high. Does this provide strong evidence for concluding that fewer than 50% of all CA lawyers feel this way

Experts claim that 10% of murders are committed by women. Is there evidence to reject the claim if in a sample of 67 murders, 10 were committed by women. Use 0.01 significance.

A study on crime suggests that at least 40% of all arsonists were under 21 years old. Checking local crime statistics, we found that 30 out of 80 were under 21. Test at 0.10 significance.

A telephone company representative estimates that 40% of its customers want call-waiting. To test this hypothesis, she selected a sample of 100 customers and found that 37% had call waiting. At a 1% significance, is her estimate appropriate?

A statistician read that at least 77% of the population oppose replacing $1 bills with $1 coins. To see if this claim is valid, the statistician selected a sample of 80 people and found that 55 were opposed to replacing the $1 bills. Test at 1% level.