Warm-up7.1 Sampling Distributions

Ch. 7 beginning of Unit 4 - InferenceUnit 1: Data Analysis

Unit 2: Experimental Design

Unit 3: Probability

Unit 4: Inference

Ch. 1, 2, and 3

Ch. 4

Ch. 5 and 6

Ch. 7, 8, 9, and 10

Student of the day!Block 4

Student of the day!Block 5

Introduction In November 2005 the Harris Poll asked 889 U.S. adults

“Do you believe in ghosts?” 40% said they did.

At almost the same time CBS News polled 808 U.S. adults and asked the same question. 48% of the respondents professed a belief in ghosts.

Why the difference?

Sampling Distribution• Both the Harris Poll and CBS news sampled a certain

number of adults for the information. Both proportions gathered are statistics and should be

written as: • Imagine we have the actual parameter. And it is p = 0.45.• Both samples were not really wrong, the difference

between the statistics and the parameter is called sampling error or sampling variability.

48ˆ40ˆ pp

Sampling Distribution ModelThis is a histogram of 2000 simulations sampling 808 adults.

Characteristics of a fair sized sample• Symmetric (very Normal)• No outliers• Center p-hat will match p• The bigger the sample size the smaller the

spread

The sampling distribution model

In this case the p is the mean and what we normally considerthe standard deviation is called the standard error.What falls in the 95% is considered reasonably likely events.

Suppose we want to label the normal model with the information from the 2000 trials of 808 adults sampled about ghosts.

Assumptions and ConditionsWhen using a sampling distribution model we need twoassumptions: The Independence Assumption & Sample Size Assumption

Assumptions are difficult to check.Conditions to check before using the Normal Dist. to model thedistributions of sample proportions :1) Randomization Condition: Applies to experiments and surveys2) 10% Condition: The sample size n must be no larger than 10% of

the population3) Success/Failure Condition: Sample size must be big enough for at

least 10 expected success or at least 10 expected failures Example: CBS poll surveyed 808 adults.

Examining a sampling distribution modelThe Centers for Disease Control and Prevention report that 22%of 18-year old women in the U.S. have a BMI of 25 or more – avalue considered by the National Heart Lung and Blood Institute tobe associated with health risk.As part of a routine health check at a large college, the physicaleducation dept. usually requires students to come in to be measuredand weighed. This year, the department decided to try out a selfreport system. It asked 200 randomly selected female studentsto report their heights and weights (from which BMI was calculated).0nly 31 students had BMI higher than 25. How does p-hat compare to the actual proportion? (w/in SD from p)

Does sampling distribution model meet the conditions?

Comment on the numbers and this sampling distribution model.

Why use sampling distribution models?The textbook calls p-hat and standard error of p-hat pointestimators because you use these numbers to infersomething about the parameter of the population you aresampling.

Just like the conditions mentioned earlier, the distributionmodel must produce a statistics that is unbiased andprecise.

Homework/Class work• Complete E #4 and #5 4b. You can use the random # digit table in the back

of the book OR Randint(1, 50) under MATH -> -> -> PRB.

5. a, d, e require making a dotplot; I expect to see these 3 dotplots

Block 4

Block 5

Block 6