Unit 4 Foundation of Statistical Inference ll A sampling distribution is just the probability distribution of a sample statistic The sample mean is an example of a sample statistic so is the median so is the variance etc The sampling distribution of the sample mean is just the probability distribution of the sample mean We use letters from our familiar Roman alphabet to symbolize sample statistics We use Greek letters to symbolize the corresponding population parameters Note some important features of this distribution The mean of the distribution of sample means is near the population mean The distribution is roughly symmetrical It has a roughly normal shape It is less spread out than the population distribution itself The mean of the distribution of sample means is always near the population mean The distribution of sample means always looks roughly symmetrical indeed it looks Normal The distribution of sample means is less spread out than the population distribution But the spread of the distribution of sample means depends strongly on sample size the larger the sample the less spread out the distribution of sample means is For any variable that is normally distributed in the population with mean and standard deviation The sampling distribution of the sample mean is also Normal The mean of the sampling distribution of the sample mean is also The SD of of the sampling distribution of the sample mean is n where n is the sample size We refer to the SD of a sample statistic as the standard error of that statistic When our sample size is large enough the sampling distribution of the sample mean will be N n even if the population is not Normally distributed at all 95 of all sample means will be within roughly 2 n of the population parameter This interval around a sample mean is called the 95 confidence interval for the population mean It s only approximately true that 95 of a Normal distribution falls within 2 sds of the mean The exact value is 1 96 sds The CI gets smaller as the sample size gets bigger The CI gets wider as the sample size gets smaller The only thing that changes if we want a difference confidence level is the value that is multiplied by the standard error n This is known as the critical value of z or z so that the general formula for the CI is This part is called the standard error of the mean The standard error multiplied by the critical value of z is referred to as the margin of error As we increase the level of confidence the CI gets bigger As the population sd gets bigger the CI gets bigger In reality we use the sample sd s as an estimate of the population sd When the sample is large we can simply assume that s is equal to n 50 when n 50 we cannot assume that s is a perfectly good estimate of We adjust for the fact that s is not a perfectly good estimate of by making our CI wider than it would be if we knew for sure We do this by adjusting the critical value of z As the sample size gets smaller and smaller we must adjust the critical value of z more and more When we do this adjustment we use a critical value of t rather than z The little df notation stands for degrees of freedom and it is equal to n 1 np is at least 10 n 1 p is at least 10 the sampling distribution of the sampling proportion will be approximately Normal following the distribution But we re talking about constructing a confidence interval for the population proportion This is a situation in which we don t actually know p after all that s what we re trying to find out We use check whether we can use the Normal approximation by checking that in its place This symbol p hat denotes the sample proportion Thus we This gives us the following general formula for a CI for a population proportion