Chapter 18 Sampling Distribution Models The histogram above is a simulation of what we'd get if we could see all the proportions from all possible samples. The distribution has a special name. It's called the sampling distribution of the proportions. 1
Chapter 18 Sampling Distribution Models p true proportion of successes in the population model parameter observed proportion of successes in a sample estimate of p q 1 p true proportion of failures in the population observed proportion of failures in the sample Each comes from a different independent sample. If sampled values are independent and sample size is large enough, the sampling distribution of is modeled by a Normal model with: 2
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For sample proportions: Normal model the larger the sample size, the better the model works. Basic concept Proportions from random samples are random quantities and we can say something specific about their distribution. Proportion A random quantity that has a distribution no longer just a number computed from a set of data. This distribution is called the sampling distribution model. 4
Sampling Distribution Model 1. The sampling distribution of any proportion is approximately Normal. 2. The sampling distribution is centered at the population proportion, p. 3. Tells you about the amount of variation you should expect when sampling. The standard deviation goes down by the square root of the sample size, n. 4. The sampling distribution is based on repeated samples we might have taken (simulations) and serves as a bridge between the real world of data and the simulated model of the statistic. 5. By looking at what might happen if you draw many samples from the same population, you can learn a lot about how one particular sample will behave. 5
For the model for the distribution of sample proportions, there are two assumptions: 1. The sampled values must be independent of each other. 2. The sample size, n, must be large enough. Because the assumptions are often impossible to check, we assume them. We check the following conditions before using the Normal to model the distribution of sample proportions: 1. Random Condition This condition is about the CENTER. Ensures that sampling is unbiased. If true, the sampling distribution (all possible samples of that size) will have exactly p as its mean. 6
2. 10% Condition This condition is about SPREAD. Ensures that there is (almost) independence. As long as the population is at least 10 times larger than the sample size, the probability of picking every element of the sample without replacement will not change to much. This means we can use the SD formula: 3. Success/Failure Condition This condition is about SHAPE. Sampling distribution for proportions is ~ normal and approaches the Normal Distribution for large values of n. np 10 and nq 10 is a rule of thumb that ensures that the shape of the distribution will be close to normal within 3 standard deviations. 7
5. Coin tosses. In a large class of introductory Statistics students, the professor has each person toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results and the professor plots a histogram of these several proportions. a. What shape would you expect this histogram to be? b. Where do you expect the histogram to be centered? c. How much variability would you expect among these proportions? d. Explain why a Normal model would not be used here. 8
7. More coins. Suppose the class in Exercise 5 repeats the coin tossing experiment, but with 25 coin tosses. a. Confirm that you can use a Normal model here. b. The students toss the coins 25 times each. Use the 68 95 99.7 Rule to describe the sampling distribution model. c. The increase the number of tosses to 64 each. Draw and label the appropriate sampling distribution model. Check the appropriate conditions to justify your model. d. Explain how the sampling distribution model changes as the number of tosses increases. 9