Advanced Math Topics

9.7 The Estimation of Proportions

40% of all customers at McDonald’s supersize their meal. You sampled 50 customers and found that 38% of them supersized their meal.

The variable p stands for the population proportion.

p = 40 or .40

The variable (p hat) stands for the sample proportion.

= 38 or .38

p̂

p̂

It is known that 70% of all airplane tickets sold by a certain company are round-triptickets. A random sample of 100 passengers is taken. What is the probability that at least 75% of these passengers have round trip tickets?

There is a large population in which a proportion of the population has a certaincharacteristic. We select random samples of size n and determine the proportionin each sample with the characteristic. Then the sample proportions will be approximately normally distributed with mean p (the population proportion)and standard deviation…

n

ppp

)1(ˆ

100

)70.01(70.0ˆ

p

046.0ˆ p

p = 70%p = 75%

0.3621

p

ppz

ˆ

ˆ

046.0

70.075.0 z

z = 1.09

.5 – 0.3621 = 0.1379 = 13.79%

Similar to our z formula…

A union member reported that 80 out of 120 workers interviewed supported someform of work stoppage to further its demands for a shorter workweek. Find a 95%confidence estimate of the true proportion of workers supporting the union’s standon work stoppage.

The following is a 95% confidence interval for p.

n

pppp

)ˆ1(ˆ96.1ˆ

3

2

120

80ˆ p

A 90% confidence interval would be the same but replace 1.96 with 1.645.

120

)32

1(32

96.13

2 p 0.5823 to 0.7510

There is a 95% chance that the true proportion of union worker’sin favor of the union’s stand is between 58.23% and 75.10%

5) Police department agents claim that 80% of drivers on the highway do not obey thespeed limit. If a random sample of 60 cars on the highway is selected, find the probability that fewer than 75% of them do not obey the speed limit.

From the HW P. 464

16.60%

6) A random sample of 390 students taking a math course found that 260 of them believed that math finals should be optional. Find a 90% confidence interval for the true proportion of students who think math finals should be optional.

From the HW P. 464

62.74% - 70.59%

P. 464 #1-7, 9, 11 Makeups tomorrow after school

From the HW P. 306