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Proportions &
With hypothesis testing, we assess the probability of an outcome (e.g., assuming the null hypothesis to be true). If the outcome is significantly unusual, then we reject the null hypothesis. Every statistical test involves proportion and probability.
A proportion ranges from zero to one. When flipping a coin, we can expect that the proportion of times the coin will land heads up is .50 of the time. The probability of an outcome matches its proportion.
Lets say we cut a pie into four pieces, and put a candle onto just one slice. Each piece of pie represents a proportion of 0.25. The probability of your randomly picking the slice with the candle is also 0.25.
The normal distribution, which is a bell shaped curve, can be used to calculate the probability of an event. Imagine that the height of 90% of women is between five feet and five feet eight inches. For our normal distribution, if we shaded in the proportion corresponding to women with a height between five feet and five feet eight inches, that would be 0.90 of the distribution. The shaded area represents a proportion of 0.90 of the distribution of women's height. The probability of randomly selecting a woman whose height is between five feet and five feet eight inches would be 0.90.
The following three games use images visualizing proportions and the normal distribution.
Gain experience visualizing proportions of a distribution. Note that based upon the Empirical Rule, it is expected that approximately:
0.68 of the population is within one standard deviation of the mean (i.e., 68%)
0.95 of the population is within two standard deviations of the mean (i.e., 95%)
0.997 of the population is within three standard deviations of the mean (i.e., 99.7%)
We expect most of the values within a distribution to be "typical" - and found at the center of the distribution. This game focuses on proportions centered at the middle of the distribution.
To get started, click on the 'Open Game' button.
The more extreme a set of values, the smaller the proportion it represents. In the middle of the distribution are the "typical" values. In either tail are the extreme values.
A value is said to be "representative" of a distribution if it is somewhat similar to the mean. On the other hand, a value found in the extreme tails of a distribution would be considered non-representative (e.g., from a proportion of .05 or less, divided between the two tails).
When focusing on height, if we were to consider the shortest and tallest people in the world (with the left tail inclusive of the shortest .025 of the population and the right tail inclusive of the tallest .025 of the population), we might not consider either group representative of the typical height of people.
To get started, click on the 'Open Game' button.
What proportion of people are six feet tall or shorter? What proportion are six feet tall or larger? In both cases, the proportion goes from a given value (e.g., six feet tall) to one of the tails of the distribution.
When the focus is on just one tail of a distribution, if a set of values represents just .05 or less of the distribution, that set is generally considered non-representative of the distribution.
When focusing on height, if we considered the tallest people in the world (making up .05 of the population), we might consider them non-representative of the typical height of people.
To get started, click on the 'Open Game' button.
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