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 characteristics  p = proportion of population members in One Category  1-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)

Standard Deviation (Spread)

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 conditions 1. np and n(1-p) are both larger than 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) = 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.)

Rule For all normal distributions  Approx. 68% of all observations are between _________ and __________.  Approx. 95% of all observations are between _________ and __________.  Approx. 99.7% of all observations are between _________ and __________.

Rule for Sampling Distribution “all observations” – What does this mean?

Rule – Example #1

Rule – Example #2

Rule – Example #3

Using the Sampling Distribution Using the 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