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## Hypothesis Testing

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**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:**Different populations will create differing frequency**distributions, even for the same variable…**There are three fundamental types of distributions:**• Population distributions**There are three types of distributions:**• Population distributions**There are three types of distributions:**• Population distributions**There are three types of distributions:**• Population distributions**There are three types of distributions:**• Population distributions**There are three types of distributions:**• Population distributions • Sample distributions**There are three types of distributions:**• Population distributions • Sample distributions**There are three types of distributions:**• Population distributions • Sample distributions • Samplingdistributions**Population distributions**The frequency distributions of a population.**2. Sample distributions**The frequency distributions of samples. The sample distribution should look like the population distribution….. Why?**2. Sample distributions**The frequency distributions of samples.**3. Samplingdistributions**The frequency distributions of statistics.**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.**2. If the population mean was 40, how many**of the sample means would be larger than 40, and how many would be less than 40?**Regardless of the shape of the distribution**below, the sampling distribution would be symmetrical around the population mean of 40.**The means of all the samples will be closer**together (have less variance) if the variance of the population is smaller.**The means of all the samples will be closer**together (have less variance) if the size of each sample (n) gets larger.**n = number of samples**Sample**So the sampling distribution will have a mean**equal to the population mean, and a variance inversely proportional to the size of the sample (n), and proportional to the variance of the population. http://www.khanacademy.org/math/statistics/v/central-limit-theorem http://www.khanacademy.org/math/statistics/v/sampling-distribution-of-the-sample-mean**Central Limit Theorem**If samples are large, then the sampling distribution created by those samples will have a meanequal to the population mean and a standard deviation equal to the standard error.**This makes inferential statistics possible**because all the characteristics of a normal curve are known.**http://www.statisticalengineering.com/central_limit_theorem.htm**http://www.statisticalengineering.com/central_limit_theorem.htm A great example of the theorem in action….**https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem**https://www.khanacademy.org/math/probability/statistics-inferential/sampling_distribution/v/sampling-distribution-example-problem Another great example of the theorem in action….**Hypothesis Testing:**A statistic tests a hypothesis: H0**Hypothesis Testing:**A statistic tests a hypothesis: H0 The alternative or default hypothesis is: HA**Hypothesis Testing:**A statistic tests a hypothesis: H0 The alternative or default hypothesis is: HA A probability is established to test the “null” hypothesis.**Hypothesis Testing:**95% confidence: would mean that there would need to be 5% or less probability of getting the null hypothesis; the null hypothesis would then be dropped in favor of the “alternative” hypothesis.**Hypothesis Testing:**95% confidence: would mean that there would need to be 5% or less probability of getting the null hypothesis; the null hypothesis would then be dropped in favor of the “alternative” hypothesis. 1 - confidence level (.95) = alpha**Errors:**Type I Error: saying nothing is happening when something is: p = alpha