A Zener deck of cards has cards that show one of five...

A Zener deck of cards has cards that show one of five different shapes with equal representation, so that the probability of selecting any particular shape is $0.20 .$ A card is selected randomly, and a person is asked to guess which card has been chosen. The graph below shows a computer simulation of experiments in which a "person" was asked to guess which card had been selected for a large number of trials. (If the person does not have ESP, then his or her proportion of successes should be about $0.20$, give or take some amount.) Each dot in the dotplots represents the proportion of success for one person. For instance, the dot in Figure A farthest to the right represents a person with an $80 \%$ success rate. One dotplot represents an experiment in which each person had 10 trials; another shows 20 trials; and a third shows 40 trials. Explain how you can tell, from the widths of the graphs, which has the largest sample $(n=40)$ and which has the smallest sample $(n=10)$

A Zener deck of cards has cards that show one of five different shapes with equal representation, so that the probability of selecting any particular shape is $0.20 .$ A card is selected randomly, and a person is asked to guess which card has been chosen. The graph below shows a computer simulation of experiments in which a "person" was asked to guess which card had been selected for a large number of trials. (If the person does not have ESP, then his or her proportion of successes should be about $0.20$, give or take some amount.) Each dot in the dotplots represents the proportion of success for one person. For instance, the dot in Figure A farthest to the right represents a person with an $80 \%$ success rate. One dotplot represents an experiment in which each person had 10 trials; another shows 20 trials; and a third shows 40 trials.
Explain how you can tell, from the widths of the graphs, which has the largest sample $(n=40)$ and which has the smallest sample $(n=10)$

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Key Concepts

Effect of Sample Size on Variability

This concept explains that larger samples tend to produce more precise estimates, as evidenced by a reduction in the spread or width of the distribution of sample statistics. In the experiment described, the dotplot corresponding to the largest sample size (n=40) would have the narrowest spread of success proportions, while the smallest sample (n=10) would exhibit the widest variability due to a larger standard error.

Standard Error

Standard error measures the variability or dispersion of a sample statistic (such as the sample proportion) from the true population parameter. It is calculated based on both the intrinsic variability of the data and the sample size. The formula SE = ?(p(1-p)/n) shows that as the sample size (n) increases, the standard error decreases, resulting in narrower sampling distributions.

Sampling Distribution

This concept refers to the probability distribution of a statistic (like the sample proportion) computed from a large number of samples drawn from a population. In the context of the problem, the sampling distribution of the proportion of successes (guesses) illustrates how the observed proportions vary from sample to sample, even when they are drawn from a population with a known probability (0.20 in this case).

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