Chapter Part 1 Sampling Distribution Models for

Sampling Distribution Models Population Sample Population Parameter? Sample Statistic Inference

Objectives Describe the sampling distribution of a sample proportion Understand that the variability of a statistic depends on the size of the sample Statistics based on larger samples are less variable

Review Chapter 12 – Sample Surveys Parameter (Population Characteristics)  (mean) p (proportion) Statistic (Sample Characteristics) (sample mean) (sample proportion)

Review Chapter 12 “Statistics will be different for each sample. These differences obey certain laws of probability (but only for random samples).” Chapter 14 Taking a sample from a population is a random phenomena. That means: The outcome is unknown before the event occurs The long term behavior is predictable

Example Who? Stat 101 students in Sections G and H. What? Number of siblings. When? Today. Where? In class. Why? To find out what proportion of students’ have exactly one sibling.

Example Population Stat 101 students in sections G and H. Population Parameter Proportion of all Stat 101 students in sections G and H who have exactly one sibling.

Example Sample 4 randomly selected students. Sample Statistic The proportion of the 4 students who have exactly one sibling.

Example Sample 1 Sample 2 Sample 3

What Have We Learned Different samples produce different sample proportions. There is variation among sample proportions. Can we model this variation?

Example Senators Population Characteristics p = proportion of Democratic Senators Take SRS of size n = 10 Calculate Sample Characteristics = sample proportion of Democratic Senators

Example Sample

SRS characteristics Values of and are random Change from sample to sample Different from population characteristics p = 0.50

Imagine Repeat taking SRS of size n = 10 Collection of values for and ARE DATA Summarize data – make a histogram Shape, Center and Spread Sampling distribution for

Sampling Distribution for Mean (Center) We would expect on average to get p. Say is unbiased for p.

Sampling Distribution for Standard deviation (Spread) As sample size n gets larger, gets smaller Larger samples are more accurate

Example 50% of people on campus favor current academic calendar. 1. Select n people. 2. Find sample proportion of people favoring current academic calendar. 3. Repeat sampling. 4. What does sampling distribution of sample proportion look like? n=2 n=5 n=10 n=25

Example 10% of all people are left handed. 1. Select n people. 2. Find sample proportion of left handed people. 3. Repeat sampling. 4. What does sampling distribution of sample proportion look like? n=2 n=10 n=50 n=100

Sampling Distribution for Shape Normal Distribution Two assumptions must hold in order for us to be able to use the normal distribution The sampled values must be independent of each other The sample size, n, must be large enough

Sampling Distribution for It is hard to check that these assumptions hold, so we will settle for checking the following conditions 10% Condition – the sample size, n, is less than 10% of the population size Success/Failure Condition – np > 10, n(1-p) > 10 These conditions seem to contradict one another, but they don't!

Sampling Distribution for Assuming the two conditions are true (must be checked for each problem), then the sampling distribution for is

Sampling Distribution for But the sampling distribution has a center (mean) of p (a population proportion) often times we don’t know p. Let be the center.

Example Senators Check assumptions (p = 0.50) 1. 10(0.50) = 5 and 10(0.50) = 5 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 U.S. adult population smoked cigarettes in Use the Rule to describe the sampling distribution for the sample proportion of smokers among 50 adults.

Example #1 Check assumptions: 1. np = (50)(0.264) = 13.2 > 10 nq = (50)(0.736) = 36.8 > n = 50, less than 10% of population Therefore, the sampling distribution for the proportion of smokers is

Example # 1 About 68% of samples have a sample proportion between 20.2% and 32.6% About 95% of samples have a sample proportion between 14% and 38.8% About 99.7% of samples have a sample proportion between 7.8% and 45%

Example #2 Information on a packet of seeds claims that the germination rate is 92%. What's the probability that more than 95% of the 160 seeds in the packet will germinate? Check assumptions: 1. np = (160)(0.92) = > 10 nq = (160)(0.08) = 12.8 > n = 160, less than 10% of all seeds?

Review - Standardizing You can standardize using the formula

Review Chapter 6 – The Normal Distribution Y~ N(70,3) Do you remember the Rule?

Example #2 Therefore, the sampling distribution for the proportion of seeds that will germinate is

Big Picture Population Sample Population Parameter? Sample Statistic Inference

Big Picture Before we would take one random sample and compute our sample statistic. Presently we are focusing on: This is an estimate of the population parameter p. But we realized that if we took a second random sample that from sample 1 could possibly be different from the we would get from sample 2. But from sample 2 is also an estimate of the population parameter p. If we take a third sample then the for third sample could possibly be different from the first and second s. Etc.

Big Picture So there is variability in the sample statistic. If we randomized correctly we can consider as random (like rolling a die) so even though the variability is unavoidable it is understandable and predictable!!! (This is the absolutely amazing part).

Big Picture So for a sufficiently large sample size (n) we can model the variability in with a normal model so:

Big Picture The hard part is trying to visualize what is going on behind the scenes. The sampling distribution of is what a histogram would look like if we had every possible sample available to us. (This is very abstract because we will never see these other samples). So lets just focus on two things:

Take Home Message 1. Check to see that A. the sample size, n, is less than 10% of the population size B. np > 10, n(1-p) > If these hold then can be modeled with a normal distribution that is:

Example #3 When a truckload of apples arrives at a packing plant, a random sample of 150 apples is selected and examined for bruises, discoloration, and other defects. The whole truckload will be rejected if more than 5% of the sample is unsatisfactory (i.e. damaged). Suppose that actually 8% of the apples in the truck do not meet the desired standard. What is the probability of accepting the truck anyway?

Example #3 What is the sampling distribution? 1. np = (150)(0.08) = 12>10 nq = (150)(0.92) = 138>10 2. n = 150 > 10% of all apples So, the sampling distribution is N(0.08,0.022). What is the probability of accepting the truck anyway?