 # Chapter 6 Hypotheses texts. Central Limit Theorem Hypotheses and statistics are dependent upon this theorem.

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Chapter 6 Hypotheses texts

Central Limit Theorem Hypotheses and statistics are dependent upon this theorem

Central Limit Theorem To understand the Central Limit Theorem we must understand the difference between three types of distributions…..

A distribution is a type of graph showing the frequency of outcomes:

Of particular interest is the “normal distribution”

Different populations will create differing frequency distributions, even for the same variable…

There are three types of distributions: 1. Population distributions

There are three types of distributions: 1. Population distributions

There are three types of distributions: 1. Population distributions

There are three types of distributions: 1. Population distributions 2. Sample distributions

There are three types of distributions: 1. Population distributions 2. Sample distributions

There are three types of distributions: 1. Population distributions 2. Sample distributions 3. Sampl ing distributions

There are three types of distributions: 1. Population distributions The frequency distributions of a population.

There are three types of distributions: 2. Sample distributions The frequency distributions of samples. The sample distribution should look like the population distribution….. Why?

There are three types of distributions: 2. Sample distributions The frequency distributions of samples.

There are three types of distributions 3. Sampl ing distributions The frequency distributions of statistics.

There are three types of distributions: 2. Sample distributions The frequency distributions of samples. The sampling distribution should NOT look like the population distribution….. Why?

Suppose we had population distributions that looked like these:

Say the mean was equal to 40, if we took a random sample from this population of a certain size n… over and over again and calculated the mean each time……

We could make a distribution of nothing but those means. This would be a sampling distribution of means.

Central Limit Theorem If samples are large, then the sampling distribution created by those samples will have a mean equal to the population mean and a standard deviation equal to the standard error.

Type I and Type TT errors –Type I ： reject the correct original Hypothesis ， called Producer's Risk –Type II ： accept the wrong original Hypothesis ， called Consumer’s Risk Population condition conclu sion H o true H a true AcceptH 0 correct type TT error conclusion RejectH 0 Type I error correct conclusion

We denote the probabilities of making the two errors as follows: α——the probability of making a Type I error β——the probability of making a Type TT error In practice ， the person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error ， called the level of significance for the test 。 Common choices for the level of significance are α=0.05 orα= 0.01 。 Type I and Type TT errors

Sampling Error = Standard Error

The sampling distribution will be a normal curve with: and

This makes inferential statistics possible because all the characteristics of a normal curve are known.

Errors:

Type I Error: saying something is happening when nothing is: p = alpha Type II Error: saying nothing is happening when something is: p = beta

Steps of Hypothesis Testing 1.Determine the null and alternative hypotheses. 2.Specify the level of significance . 3.Collect the sample data and calculate the test statistic. Using the p -Value 4.Use the value of the test statistic to compute the p - value. 5.Reject H 0 if p -value < .

Steps of Hypothesis Testing Using the Critical Value 4.Use  to determine the critical value for the test statistic and the rejection rule. 5.Use the value of the test statistic and the rejection rule to determine whether to reject H 0.