STAT1201 Module 5 Notes Revision - Sample mean o Sample mean · Normal distribution o Mean u and standard deviation /Vn - In normal distribution about 95% of observations occur within 1.96 standard deviations of mean o Therefore in 95% of samples, the sample mean will be within 1.96 ?/vn of ? . ? is the mean of the sample means Reversing this, in 95% of samples u will be within 1.96 g/n of x This tells us when we use sample mean to estimate population mean, we can also get idea of how far away the actual population mean is from our estimate We are 95% confident the population mean is 0 x+1.96 Vn Or that it is in interval x-1.96 ?, x+1.96- ? x is the sample mean. This is the confidence interval for population mean. This allows us to make a concrete statement about population based on the sample It relies on the population standard deviation ( ? ) 5.1.2 Student's t Distribution Revision If X is a random variable with mean u and standard deviation ? then X-u Z =- Has approx the standard Normal distribution. Standard deviation of sample mean ( o/n ) - Allows us to quantify precision of the sample mean as an estimate of population mean ( ?) In order to solve the problem of not having a population standard deviation for the confidence interval we have to estimate the population standard deviation from the sample standard deviation - This gives estimated standard deviation of the sample mean se(x)= s where s is the standard deviation of the sample The estimated standard deviation of a statistic is its standard error - Standard error of the sample mean (se(x)), is different for each sample Using the standard error to standardise gives a statistic
X-p SIvn where X and S are the sample mean and sample standard deviation Because the sample standard deviation is a random quantity (different for different samples) - Adds extra uncertainty to results - Means that the confidence interval may drop from 95% to 91% o 91% chance that the population mean is within 1.96 standard deviations of the sample mean we found o May need to go out more standard deviations to increase confidence interval as a results Student's t Distribution T= S/Vn X and S are the sample mean and sample standard deviation from a random sample T is a random variable with a distribution known as Student's t distribution There is a different t distribution for every sample size. The amount of info we have about the population variability is important - This quantity is called degrees of freedom E.g. you call the distribution of T from samples of size 5 the t distribution with 4 degrees of freedom. This is written t (4) - t values calculated from samples of size n will have the t (n-1) distribution 5.1.3 Confidence Interval for a Mean To calculate confidence interval for a population mean u , using the