Chapter 8 Section 2

Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 8 Section 2 – Slide 1 of 13

Chapter 8Section 2

Distributions of theSample Proportion


Chapter 8 – Section 2

● Learning objectives Describe the sampling distribution of a sample

proportion Calculate probabilities of a sample proportion

1

2





1

2



● In an election, polling companies wish to estimate the percent of people who will vote for each candidate

● This clearly is a situation for sampling as it is impractical to contact every single voter

● The desired results are proportions, for example that 59% of the voters (a proportion of 0.59) said that they will vote for candidate A



● We have the same questions for the sample proportion as we had for the sample mean What is the mean of the sample proportion?

● We have the same questions for the sample proportion as we had for the sample mean What is the mean of the sample proportion? What is the standard deviation of the sample

proportion?


proportion? What is the distribution of the sample proportion?


proportion? What is the distribution of the sample proportion? Can we apply the Central Limit Theorem to

approximate these with normal distributions?


proportion? What is the distribution of the sample proportion? Can we apply the Central Limit Theorem to

approximate these with normal distributions?

● The answer is yes …



● Sample proportions ● A random sample is take

Of size n Each individual either has or does not have a certain

characteristic In total, there are x individuals that have this

characteristic

● Then the sample proportion (p hat) is given byp̂

nx

p̂



● If a polling company polled 800 people to see if they supported a certain issue and 475 did, then we have a sample proportion problem with n = 800 x = 475 and a sample proportion of

590800475

.p̂


● If the population proportion is p, then the distribution of the sample proportion for a sample of size n Is approximately normal if np(1-p) ≥ 10 Has a mean of

Has a standard deviation of


pp̂

n)p(p

p̂ 1





1

2



● Example● Assume that 80% of the people taking aerobics

classes are female and a simple random sample of n = 100 students is taken


classes are female and a simple random sample of n = 100 students is taken What is the probability that at most 75% of the sample

students are female?


classes are female and a simple random sample of n = 100 students is taken What is the probability that at most 75% of the sample

students are female? If the sample had exactly 90 female students, would

that be unusual?



● The sample proportion of aerobics students who are female Has an approximately normal distribution Has a mean of 0.80 and a standard deviation of 0.04


● What is the probability that is 0.75 or less? 0.75 is 0.05 less than the mean of 0.80 0.05 is 1.25 standard deviations less than the mean

(i.e. the z-score is –1.25) The normal probability P(z ≤ –1.25) = .1056 Thus P( ≤ 0.75) = .1056

p̂

p̂

p̂





● What is the probability that is 0.90 or more? 0.90 is 0.10 more than the mean of 0.80 0.10 is 2.5 standard deviations more than the mean

(i.e. the z-score is 2.5) The normal probability P(z ≥ 2.5) = .0062 Thus P( ≥ 2.5) = .0062 … pretty unlikely

p̂

p̂

p̂


Summary: Chapter 8

● The sample proportion, like the sample mean, is a random variable If the sample size n is sufficiently large and the

population proportion p isn’t close to either 0 or 1, then this distribution is approximately normal

The mean of the sampling distribution is equal to the population proportion p

The standard deviation of the sampling distribution is equal to n/)p(p 1


Chapter 8 – Example 1

● According to a recent article, 98% of the people who travel on the New Jersey Turnpike exceed the 60 miles per hour speed limit. (Source: Lange JE, Johnson MB, Voas RB (2005) Testing the Racial Profiling Hypothesis for Seemingly Disparate Traffic Stops. Justice Quarterly (22)2.) Law enforcement officials are planning to covertly measure the speed of 1,000 cars on the Turnpike and will compute the proportion of drivers who exceed the posted speed limit.

a. Describe the sampling distribution of , the sample proportion of drivers who exceed the speed limit.

b. In a random sample of 1000 drivers, what is the probability that more than 97% of drivers will be speeding?

c. Would it be unusual if in a random sample of 1000 drivers, 97% were speeding?

d. What is the minimum number of drivers that must be sampled to be sure that is approximately normal?



● According to a recent article, 98% of the people who travel on the New Jersey Turnpike exceed the 60 miles per hour speed limit. (Source: Lange JE, Johnson MB, Voas RB (2005) Testing the Racial Profiling Hypothesis for Seemingly Disparate Traffic Stops. Justice Quarterly (22)2.) Law enforcement officials are planning to covertly measure the speed of 1,000 cars on the Turnpike and will compute the proportion of drivers who exceed the posted speed limit.

a. Describe the sampling distribution of , the sample proportion of drivers who exceed the speed limit. (Approximately normal with mean 0.98 and standard deviation 0.0044) pˆ

b. In a random sample of 1000 drivers, what is the probability that more than 97% of drivers will be speeding? (0.9881)

c. Would it be unusual if in a random sample of 1000 drivers, 97% were speeding? (Yes, since the probability that 97% or less would be speeding is 0.0119.)

d. What is the minimum number of drivers that must be sampled to be sure that is approximately normal? (511)



● The quality control director of a small company that manufactures exercise equipment knows that 10% of their products are defective. Market research shows that only 30% of their customers will use the product in the first year after the sale. So, the manufacturer expects that 3% of the units will be returned for service under a one-year full warranty. This company expects to sell 600 units next year. (This problem is based on data from an actual company.)

a. Describe the sampling distribution of , the proportion of the 600 units that will be returned for service.

b. What is the probability that more than 3% of the 600 units will be returned for service?

c. What is the probability that more than 5% of the 600 units will be returned for service?

d. Would it be unusual if in a random sample of 600 units, 5% were returned for service?



● The quality control director of a small company that manufactures exercise equipment knows that 10% of their products are defective. Market research shows that only 30% of their customers will use the product in the first year after the sale. So, the manufacturer expects that 3% of the units will be returned for service under a one-year full warranty. This company expects to sell 600 units next year. (This problem is based on data from an actual company.)

a. Describe the sampling distribution of , the proportion of the 600 units that will be returned for service. (Approximately normal with mean 0.03 and standard deviation 0.006964) pˆ

b. What is the probability that more than 3% of the 600 units will be returned for service? (0.5)

c. What is the probability that more than 5% of the 600 units will be returned for service? (0.0020)

d. Would it be unusual if in a random sample of 600 units, 5% were returned for service? (Yes)

Chapter 8 Section 2

Documents

Transcript of Chapter 8 Section 2