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Hypothesis Testing of Single Mean and Single Proportion: Rare Events

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

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Suppose you make an assumption about a property of the population (this assumption is the null hypothesis). Then you gather sample data randomly. If the sample has properties that would be very unlikely to occur if the assumption is true, then you would conclude that your assumption about the population is probably incorrect. (Remember that your assumption is just an assumption - it is not a fact and it may or may not be true. But your sample data is real and it is showing you a fact that seems to contradict your assumption.)

For example, Didi and Ali are at a birthday party of a very wealthy friend. They hurry to be first in line to grab a prize from a tall basket that they cannot see inside because they will be blindfolded. There are 200 plastic bubbles in the basket and Didi and Ali have been told that there is only one with a $100 bill. Didi is the first person to reach into the basket and pull out a bubble. Her bubble contains a $100 bill. The probability of this happening is 1200=0.0051200=0.005. Because this is so unlikely, Ali is hoping that what the two of them were told is wrong and there are more $100 bills in the basket. A "rare event" has occurred (Didi getting the $100 bill) so Ali doubts the assumption about only one $100 bill being in the basket.


A statement about the value of a population parameter. In case of two hypotheses, the statement assumed to be true is called null hypothesis (notation H0H0 size 12{H rSub { size 8{0} } } {}) and contradictory statement is called alternate hypothesis (notation HaHa size 12{H rSub { size 8{a} } } {}).

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