chapter 18 – part i sampling distribution for sample proportion statistic

Chapter 18 – Part I

Sampling Distribution for Sample Proportion Statistic

Review

Categorical Variable One Category vs. All Other Categories

Examples

What political party is the U.S. senator?Ex. Democrat, All Other Parties

Who will you vote for in 2008 election?Hillary Clinton, All Other Candidates

What color are your eyes?Blue, All Other Eye Colors

Review

Population characteristicsp = proportion of population members in One

Category1-p = proportion of population members in All

Other Categories

Examples

p = proportion of U.S. senators who are Democrats.

p = proportion of voters who will vote for Hillary Clinton.

p = proportion of people with blue eyes.

Review

SRS characteristics_____ = proportion of sample members in

One Category_____ = proportion of sample members in All

Other Categories

Examples

_____ = proportion of U.S. senators in sample of size 10 who are Democrats.

_____ = proportion of voters in sample of size 2000 who will vote for Hillary Clinton.

_____ = proportion of people in sample of size 5 with blue eyes.

Randomness of Sampling

Random event = ___________________.

Before taking sample ________________.

Repeated Sampling

Repeat taking SRS of size n.Example: 2002 Senate.______ = proportion of Democratic Senators

in sample of size _______.

What happens to values of ________?

Repeated Sampling by Hand

Repeated Sampling Using Computer

Sampling Distribution

Many, many possible samples. Many, many possible values of ________. These ___________ are DATA.

Summarize DATA values!!

Mean (Center)

pp )ˆ(

Standard Deviation (Spread)

n

ppp

)1()ˆ(

Example

50% of registered voters plan to vote for Hillary Clinton. Select n people. (2, 5, 10, 25) Find sample proportion of registered voters planning to

vote for Clinton. Repeat sampling. What does sampling distribution of sample proportion

look like?

Example

10% of all people are left handed. Select n people. (2, 10, 50, 100) Find sample proportion of left handed people. Repeat sampling. What does sampling distribution of sample

proportion look like?

Shape

Two conditions1. np and n(1-p) are both larger than 10.

2. sample size n is less than 10% of population size.

Shape is approximately a ________________________!!!

Sampling Distribution for _____ Check for two conditions

Example

U.S. Senators Check assumptions (p = 0.54)

1. 10(0.54) = 5.4 and 10(0.46) = 4.6

2. n = 10 is 10% of the population size. Assumption 1 does not hold. Sampling Distribution of ___________

Example #1

Public health statistics indicate that 26.4% of the adult U.S. population smoke cigarettes. Describe the sampling distribution for the sample proportion of smokers among a random sample of 50 adults.

Example #1 (cont.)

Check assumptions

Example #1 (cont.)

Example #2

A student in Statistics 101 flipped a fair coin 200 times. What is the sampling distribution of the proportion of heads flipped?

Example #2 (cont.)

Check assumptions

Example #2 (cont.)

Example #3

Information on a packet containing 160 seeds that the germination rate of the seeds is 92%. Assume the seeds are a random collection of all seeds produced. What is the sampling distribution of the proportion of seeds in the packet that will germinate?

Example #3 (cont.)

Check assumptions

Example #3 (cont.)

68-95-99.7 Rule

For all normal distributionsApprox. 68% of all observations are between

_________ and __________.Approx. 95% of all observations are between

_________ and __________.Approx. 99.7% of all observations are

between _________ and __________.

68-95-99.7 Rule for Sampling Distribution “all observations” – What does this mean?

68-95-99.7 Rule – Example #1

68-95-99.7 Rule – Example #2

68-95-99.7 Rule – Example #3

Using the Sampling Distribution

Using the 68-95-99.7 rule, we know what to expect for the values of _______ for a given value of ________.

Can we use the normal distribution to find the probability of Having a __________ value below a certain amount? Having a __________ value above a certain amount?

Example #1

What is the probability of getting a random sample of 50 adults with 18 or more smokers?

Example #1

Example #2

What is the probability that in flipping a fair coin 200 times, we would get 80 or fewer flips with heads?

Example #2

Example #3

What is the probability that in a packet of 160 seeds we would see 138 or fewer seeds germinate?

Example #3