Part I: Understanding Binomial Probability Distribution

First, let's try to understand binomial probabilities for events like coin tosses where there are only 2 possible outcomes. Find a coin to flip and below circle your outcomes for 20 coin tosses (circle H for head or T for tail after each toss).

1.H T2. H T3. H T4. H T5. H T6. H T7. H T8. H T9. H T10. H T

11.H T 12. H T13. H T14. H T15. H T16. H T17. H T18. H T 19. H T20. H T

1. How many heads did you get in total? (count and record):____ heads out of 20 coin flips

As each person in class reports their number of heads record an X above that number to make a histogram below.

2. What number do you think will be most common? ______

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Number of Heads Found in our Class

Part II: Understanding Probability and Sampling Distributions

A Sampling Distribution is a distribution obtained by selecting all of the possible samples of a specific size from a population. The Distribution of Sample Means is the collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population. There are three important points to remember regarding the Distribution of Sample Means:

1.Most of the sample means should be relatively close to the population mean.

2.The sample means should tend to form a normal distribution.

3.The larger the sample size, the closer the sample means should be to the population mean, .

Now open the following URL: http://onlinestatbook.com/stat_sim/sampling_dist/index.html

Click "Begin" on the top left of the web page to get started.

Click the "Animated" button. Five scores from a normal distribution will be sampled and plotted in a histogram. The mean of the sample will be computed and plotted in a second histogram. Repeat this 3 or 4 times, or until you understand the how the third graph "Distribution of Means" is created. The red line under graphs extends from the mean 1 standard deviation in each direction. The mean is a blue line.

Click the "10,000 samples" button to draw 10,000 random samples from the population (top histogram) with n=5 scores in each sample. The sampling distribution of the mean is the third histogram. With large number of reps (over 10,000) does the shape of the sample distribution look approximately normal? You can also tell normality by looking at skew and kurtosis. With normal distribution skew and kurtosis = 0. The mean of the population is 16.

1. What did you get for the mean of the sampling distribution? ________

2. What value did you find for skew? ______

3. What is the kurtosis value? _______

Standard Error (SE) provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (). SE is related to population SD () and sample size (n) as follows:

The distribution plotted in Question #2 above is the sampling distribution of the mean for a sample size of 5. For the fourth (bottom) graph, select "sd" and change the sample size in bottom histogram to n=10, n=16, etc... Click 10,000 samples. Note the shape and record the standard deviation (sd) of the sampling distributions below.

4. The standard error is the standard deviation of a sampling distribution. It is the "typical" amount you expect sample means to vary from the pop mean. Approximate the sampling distribution of the mean for N=5, 10, 16 and 25 (click 10,000 samples). Record the standard error of the mean for:

N= 5 _________ N=16 _________ N=10 _________ N=25 _________

5. What happens to the standard error of the mean as sample size increases?

________________________________________________________________________

6. What happens to the shape of the sampling distribution as sample size increases?

________________________________________________________________________

7. As the sample size decreases, the difference between the sample mean and the population mean

________________________________________________________________________

Understanding Bias. A statistic is unbiased if the mean of the sampling distribution of that statistic equals something very close to the population parameter and if the difference between them is equally likely to be higher or lower.

Note that any statistic you can compute from a sample has a sampling distribution (e.g. Median, Standard Deviation, Variance, and Range). Try exploring these by changing Mean to another statistic in the third graph.

But what if the population isn't normal? At the top, change the population distribution shape (instead of normal, try skewed and uniform distributions).

8. Is the sample mean an unbiased estimate of the population mean? ___________________

9. Is the sample variance an unbiased estimate of the population variance? ______________

10. What if the population isn't normal? At top change the population distribution shape-- instead of normal try skewed and uniform distributions. Is the sampling distribution of the mean still symmetrical?