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# Theoretical mean of p ˆ theoretical sd of p ˆ n 25

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Theoretical mean of p ˆ Theoretical SD of p ˆ n = 25, S = 0.5 n = 100, S = 0.25 n = 100, S = 0.80 How do the theoretical means and standard deviations compare to the simulated values (see the graphs before part (m)) and to each other? P²ºV P²V P P±V P±ºV V V f ( x )

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Chance/Rossman, 2015 ISCAM III Investigation 1.7 69 (o) Now suppose we had taken samples of size n = 5 candies instead (from a process with success probability 0.80). Predict how the distribution of sample proportions will behave (shape, center, variability). Shape: Mean: SD: (p) Use the applet to check your predictions. Discuss your observations. Discussion: It is very important to keep in mind that this normal probability model is not always a valid approximation for the distribution of sample proportions. Whether it is valid will be determined by a combination of the sample size n (larger samples result in more symmetric distributions) and the value of S (values closer to 0 or 1 result in less symmetric distributions). A common guideline is to assume symmetry when n × S > 10 and n × (1 ± S ) > 10. (q) Explain how these guidelines are consistent with your observations above. This result about the long-run pattern of variation in sample proportions is more formally called the Central Limit Theorem (CLT) and is one of the most important results in Statistics. It states that if the sample size is large enough, then the sampling distribution of the sample proportion p ˆ will be well modeled by a normal distribution with mean E( p ˆ ) equal to S , the process probability, and standard deviation SD( p ˆ ) equal to n ) 1 ( S S ± . The convention is to consider the sample size large enough if n × S t 10 and n × (1 ± S ) t 10. Binomial( n = 100, S = 0.80) / 100 vs. Normal (mean = 0.80, SD = 0.040) Discussion: Use of the normal probability model to approximate the sampling distribution of sample proportions is largely historical (back when they didn’t have computers, the binomial distribution calculations were a bit of a pain but much easier with the normal distribution). But you will see in the next few investigations that some calculations are still more convenient with the normal distribution.
Chance/Rossman, 2015 ISCAM III Investigation 1.7 70 For example, we stated in Investigation A that we usually begin to think an observation is unusual when it lies more than two standard deviations above or below the mean of the distribution. This comes from a very special property of the normal probability model. (r) Reset the Reese’s Pieces applet, specify S = 0.25 and n = 100, and check the Exact Binomial and Normal Approximation boxes. Now use the theoretical mean and SD values from (n) to calculate the value of p ˆ that is two standard deviations below the mean. Enter this value in the As extreme as box, toggle to “less than” and press Count. Mean: SD: Mean 2SD: Probability below Binomial model: Normal model: (s) How do the probabilities compare?

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