# Hypotheses

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Share with your peers the null and alternative hypotheses for a decision that is relevant to your life. This can be a personal item or something at work. Be sure that it is mathematical in nature.  Additionally, identify the Type I and Type II Errors that could occur with your decisionmaking process. Be sure to quantify your hypotheses as much as possible and identify your variables.

For example, suppose a newspaper article stated that the average weight of cats is 5 lbs. Suppose you think that the average weight of cats is more than 5 lbs. Then the hypotheses are:

H0: μ < 5
Ha:  μ >5
Here  μ is the population average weight of cats. This would be an example of an Upper Tail test.

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A way of understanding hypothesis testing is to think of a court case:
Suppose Joe is accused of stealing an expensive diamond. Then there are 2 hypotheses:

H0: Joe is innocent (H0 is the assumption, the status quo, innocent until proven guilty)
H1: Joe is guilty (the alternative hypothesis)

The goal of the prosecutor is to collect evidence so that the judge switches over from H0 to H1.

You are assuming H0 is true. Thus, the goal of the prosecutor is to collect convicting evidence so compelling that it is very unlikely for an innocent person to have.

For example, it is very unlikely that an innocent person is found with the stolen diamond in hand, a video camera showing Joe near the scene, a map in Joe’s home of the Jewelry store,…,etc.

Since this is very unlikely to happen to an innocent person, the judge switches over by rejecting H0, innocent.

You want to the probability of convicting an innocent person to be small.
i.e. you want to minimize the probability of rejecting H0 given that H0 is true (Minimize the probability of a Type I error).

This can be written as the following conditional probability:
small= P( rejecting H0 | H0 is true) = the probability of convicting an innocent person

The probability above is called the probability of a Type I error. It is denoted with the Greek letter alpha.

P(Type I error) = P( rejecting H0 | H0 is true) = alpha
The statistician decides what alpha will be.

For example, the judge might draw the line at alpha=.001.

i.e.
.001= P( rejecting H0 | H0 is true) = the probability of convicting an innocent person.

Alpha is the borderline probability between innocent and guilty.

i.e. If Joe is assumed to be innocent and the evidence collected against him has a probability of less than .001 occurring, then the judge will reject H0 and accept Ha (guilty).

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