Question

Explain how z scores are incorporated into r calculations and CI.

Answer 1

Z-scores are a stand-in for the actual measurement, and they represent the distance of a value from the mean measured in standard deviations. So a z-score of 2.0 means the measurement is 2 standard deviations away from the mean.

It is easy to calculate the Z-scores in R.

First, specify a Normal distribution say, of weights of students of a class :  N(²) where are mean and standard deviation respectively.

Let us use the z-score to find the probability of finding someone who is 72 kg of weight

i.e for a value x=72 we want to find the z score. Input values for in R as :

We first generate 20 samples from the normal distribution and find the Z score for x=72

R-code :

Now we want to find the probability of a student having weight more than 72. The z-score will be used to determine the area [probability] underneath the distribution curve past the z-score value that we are interested in.

Both R and typical z-score tables will return the area under the curve from -infinity to the specified value on the graph (which in this case is x=72)

We generally use the z-score and look it up in a table and find the probability that way. But R has a function ‘pnorm’ which will give you a more precise answer than a table in a book.[‘pnorm’ stands for “probability normal distribution”.]

R code :

So our required probability of finding a student having weight more than 54 kg is (1-0.9772499)=0.022

Confidence Interval with Z score :

CI for the population mean of the normal distribution is (- _/2*, + _/2*)

input values in R and formulate the above to get the confidence interval for the mean. Say, alpha=0.1 we know _/2=1.645

The qnorm( ) function gives critical z-values corresponding to a given lower-tailed area We will use this function to verify the value of _/2=1.645 and hence find the CI.

Hence we get the confidence interval of population mean.