Unit 8 inference for means review this concludes Unit 8 let's summarize the focus of this module inference for means is inference for a population mean or a difference between two population means we begin this module with a discussion of the sampling distribution of sample means we then developed the probability model based on the sampling distribution we use the probability model with an actual sample mean to test a claim about population mean in a hypothesis test or to estimate a population mean with a confidence interal we then moved to inference for a difference in two population means or a treatment effect sampling distributions of means if we have a quantitative data set for from a population with a mean mu and standard deviation Sigma the model for the theoretical sampling distribution of the means of all random samples of size n has the following properties the mean of the sampling distribution of means is Mu the standard deviation of the sampling distribution is Sigma / the root of n notice that as n grows the center error for the sampling distribution of means shrinks that means that larger samples give more accurate estimates of a population mean for large enough sample size the sampling distribution of means is approximately normal even if population is not normal this is called the central limit theorem if a variable has a skew distribution for individuals in the population a larger sample size is needed to ensure that the sampling distribution has a normal shape the general rule is that if n is at least 30 then the sampling distribution of means is approximately normal however if the population is already normal then any sample size will produce a normal sampling distribution we practice finding a probability associated with a range of sample means which is similar to finding a P value in hypothesis testing the process is as follows convert a sample mean X into a zcore use technology to find the probability associated with a given range of zc scores confidence intervals the form a confidence interal approximates a population mean by giving us a range of values that likely contains the population mean mu the general form of the confidence interval is X plus or minus margin of error which is equal to X plus or minus critical value times the standard error we convert the following different types of confidence intervals we have a one samples Z interval we also have a one sample T interval for the one sample Z interval we have that Sigma is the population standard deviation when it's known for one sample T interval xar plus or minus t * s / theun of n where is s is the sample standard deviation used used because the population standard deviation is unknown the T model when the standard deviation of the population is unknown which is often the case we use the t- model to find the critical values when using the t- model to find critical values we need to select an appropriate number of degrees of freedom in the one sample case the number of degrees of freedom is one less than the sample size so degrees of freedom is equal to n minus one in the two independent sample case the degrees of freedom come from a complicated formula and we often use technology to find the degrees of freedom conclusions we say we are 95% confident that the population mean Falls within our confidence interval really means that about 95% of all confidence intervals computed in this way will capture the true population mean conditions the population must be normally distributed or the sample size must be large enough larger than 30 in the case of the two sample T intervals both population samples must meet these conditions in practice we use T procedures with smaller samples if the distribution of the the variable in the samples is not heavily skewed or is without outliers we take this as an indication that the variable has a fairly normal distribution in the population observations about confidence interval structure as we saw with other confidence intervals the width of of a confidence interval is twice the margin of error the smaller the margin of error the narrower the confidence interval and the more precise the estimate of the population parameter increasing the confidence level decreases the Pres Precision larger margin of error so wider interval decreasing the confidence interval level increases the Precision smaller margin of error so narrower interval confidence intervals are useful estimates only when they provide a good balance of confidence level and Precision in order to increase Precision without losing confidence we must increase the sample size in other words larger samples provide more precise estimates without sacrificing confidence hypothesis testing test for statistical significance the process of a hypothesis test consists of four basic steps the first one is to define the hypothesis second step is to collect the data we need random samples that are representative of the population for the two sample T tests the samples must be independent assess the evidence assessment includes checking appropriate conditions Computing test statistic and finding corresponding P values State the conclusion we compare the P value to Alpha decide whether or not to reject the null then State conclusion and context hypothesis the no hypothesis gives the value of the parameter we use to create the sampling distribution in this way the Noy hypothesis States what we assume to be true about the population the alternative hypothesis usually reflects the claim in the research question about the value of the parameter the alternative hypothesis says the parameter is greater than or less than or not equal to the value we assume to be true in the N hypothesis when the the alternative hypothesis is Mu less than mu or mu greater than mu the test is called a one tell test for the pair T Test the alternative hypothesis would look like mu greater than mu less than zero or mu greater than zero in this case in the case of a wantel test for the two sample T Test the alter alternative hypothesis would look like mu1 - mu2 less than 0 or mu1 - mu2 greater than 0 in the case of a one tell test when the alternative hypothesis is Mu not equal to Mu not the test is called the two test for the pair T Test the alternative hypothesis will look like mu not equal to zero in this case in the case of a two test for the two sample T Test the alternative hypothesis will look like mu1 minus mu2 not equal to zero in the case of a two test conditions the conditions that must be satisfied in order to carry out T procedures are as follows the population is normally distributed or the sample is large at least 30 this implies to both population for the two sample T tests the samples must be random in order to avoid bias the samples must be independent in the case of two in in the case of the two sample T Test test statistic the test T statistic is given by this formula we've learned about three different types of T tests one sample T Test pair T Test and we have different formulas for each we also have a two sample T Test where we have this formula here P values the P value is the probability of finding a random sample with a test statistic at least as Extreme as ours assuming that the no hypothesis is true we find P values by using the T distribution we to come to a conclusion about the no hypothesis We compare the P value to the significance level Alpha if p is less than or equal to Alpha we reject the null and conclude there is significant evidence to favor the alternative if p is greater than Al greater than or equal to Alpha we fail to reject the null and conclude the sample does not provide significant evidence in favor of the alternative error types hypothesis tests are based on random samples so the conclusions are really really statements about probabilities and it is possible for the conclusions to be wrong if our test results in rejecting a n hypothesis that is actually true it is called a type one error if our test result in failing to reject a no hypothesis that is actually false it is called a type two error you are now ready to put it all together and practice what you have learned in this unit