1. WHAT IS A SAMPLING DISTRIBUTION? Textbook Section 7.1

2. PARAMETERS AND STATISTICS As we start using sample data to draw conclusions about the population at large, we need to clearly define some vocabulary A Parameter is a number that describes some characteristic of the population of interest. The value of a parameter is often unknown because we cannot examine the entire population (too expensive, time-consuming, impractical, etc.) A Statistic is a number that describes some characteristic of a sample. We use a well designed sample and study to draw conclusions about the population. Notation PopulationMeasureSample pProportion (percent) μ Mean σ Standard Deviations

3. SAMPLING VARIABILITY How can a sample of less than 2% of the American population give us an accurate picture of the entire population? Wouldn’t another sample select different people and then have different results? – YES!! This inherent difference among different samples is called Sampling Variability. If we take many, many samples and calculate the mean of each sample, those numbers will likely all be different. BUT if we put those sample means together into a distribution, we can draw some POWERFUL conclusions from the Sampling Distribution.

4. CREATE A SAMPLING DISTRIBUTION The height of young women varies approximately according to the N(64.5, 2.5) distribution. The random variable X = the height of a randomly selected young woman. Let’s simulate a sample of young women to make a sampling distribution. Place your cursor on top of List 1. Press MATH, arrow over to PRB, then choose 6:randNorm Enter randNorm(64.5, 2.5, 100) to generate a random list of 100 young women’s heights Find the Mean of your sample ( 1-Var-Stats ), and write it on a post- it note. Add your post-it note to the appropriate place on the chalkboard. Repeat two more times.

5. DESCRIBING SAMPLING DISTRIBUTIONS Center: Biased and Unbiased estimators How close does the center of your sampling distribution come to the ACTUAL population? If your center is exactly the same as the population, we call it UNBIASED. Realistically, each sample’s center will be different from the population center in some way – it’s the COLLECTION of samples that allow us to approximate the population parameter. Larger samples lead to more unbiased estimators – but ONLY if the samples are put together using valid sampling techniques Low variability is better – The smaller your standard deviation, the more confident you can be about the unbiased nature of your estimator.

6. CHECK YOUR UNDERSTANDING The histogram above shows the intervals (in minutes) between eruptions of the Old Faithful geyser for all 222 recorded eruptions during a particular month. For this population, the median is 75 minutes. We used software to take 500 SRSs of size 10 from the population. The 500 values of the sample median are displayed in the histogram above right. The mean of these 500 values is 73.5. 1.Is the sample median an unbiased estimator of the population median? 2.Suppose we had taken samples of size 20 instead of 10. Would the spread of the sampling distribution be larger, smaller or about the same? 3.Describe the shape of the sampling distribution.

7. BIAS & VARIABILITY p. 500