Download Understanding Confidence Intervals, Z-Scores, and t-Distributions and more Study notes Statistics in PDF only on Docsity! 9. Confidence Intervals and Z-Scores Weโre accumulating a lot of terminology to refer to locations or areas of distributions: standard error, standard deviation, percentile, quartile, etc. This lab will help clarify how all these relate to one another. We will also look at the difference between a normal population distribution and a sampling t-distribution. Much of what is covered here progresses straight into t-tests, which we will cover in the next lab. 9.1 t-distribution (a.k.a. The Student t-distribution) ๏ท While the normal distribution typically describes our raw data, the t-distribution is the distribution of our sample statistics (e.g. sample means). The t-distribution is very similar to the standard normal distribution in shape however the tails of the t-distribution are generally fatter and the variance is greater than 1. ๏ท Note that there is a different t-distribution for each sample size, in other words, it is a class of distributions. So when we speak of a specific t-distribution, we have to specify the degrees of freedom, where . As our sample size increases (and degrees of freedom increase), the t- distribution approaches the shape of the normal distribution. The normal distribution vs the t-distribution 9.2 Converting between raw data, t-values, and p-values Percentile - the value below which a given percentage of observations within a group fall Quartile - (1st, 2nd, 3rd, 4th) points that divide the data set into 4 equal groups, each group comprising a quarter of the data Alpha level - predetermined probability where we make some sort of decision P-value - (percentiles) the probability the observed value or larger is due to random chance Critical t-value - the t-value that corresponds to the ๐ผโ๐๐๐ฃ๐๐ Actual t-value - the t-value that corresponds to the raw data value being tested with the ๐ผโ๐๐๐ฃ๐๐ (signal- to-noise ratio) Signal - the difference between the test and mean values Noise โ a measure of the distribution of the data 9.3 Confidence Intervals ๏ท The following points discuss and provide examples of how we can calculate confidence intervals and convert between the above scales in R ๏ท The functions qnorm()and pnorm()convert from units of standard deviations (= standard error in this case) to percentiles (= probabilities) (and vice-versa) for a normal distribution. Remember, pnorm() expects a standard deviation / error and returns a percentile / probability and qnorm() does the opposite. pnorm(x,mean,sd) ๏ท The command above will return the area to the left of x under the normal distribution curve with a mean of mean and a standard deviation of sd. If you donโt specify mean or sd, R will assume a mean of zero and an SD of 1. You can use the qnorm()statement to work things backwards from probabilities also: qnorm(p,mean,sd) ๏ท The command above will return the number x for which the area to the left of x under the normal distribution curve with a mean of mean and an standard deviation of sd is equal to p. Again, if you donโt specify mean or sd, R will assume a mean of zero and an SD of 1. That in mind, we can calculate some basic confidence intervals with these commands. ๏ท We know that one standard error of the mean (SEx) for large sample sizes (or one standard deviation of a normally distributed population) is equivalent to the ~68% confidence interval of the mean because ~34% of values fall within 1 SE either side of the mean. We can confirm this with the pnorm() command. Remember that pnorm() gives you the total area under the curve to the LEFT of the number that you specify. pnorm(1) pnorm(-1) qnorm(0.16) qnorm(0.84)
