Review of Z-Scores and the Normal Distribution

What 68% of you should know 99.7% of the time; what 99.7% of you should know 68% of the time; and what all of

you should know all of the time!

Section 1.3 – a critical part

Go to Normal Distribution Applet How is the z-score calculated? Why does it make sense that it depends on two

characteristics of the population? What is an N(0,1) distribution? What fraction of a population is found between z = -0.85

and z = 1.65? Sketch this. How would relate a 95% confidence level to a N-

distribution? How would it appear as a sketch on the N-distribution curve?

What is the 68-95-99.7 rule?

Examples…

Do all data sets fit a N-distribution? How about this:

Testing the data for normality… Histogram

Normal Quantile Plot

X zX

z

Applying the N-distribution

Look at the last mid-term: On March 18, 2007 CBC hosted the first national

IQ test. The test pitted seven studio groups against each other and also had an on-line participation of over 100 000 Canadians. We know from large population samples that IQ scores are normally distributed.

1. If the average IQ is 100 what fraction of the population will you expect to score within ± 1 s of the mean? (1 marks)

2. The highest scoring studio participant scored 137 which placed him in the top 2% of national IQ scores. Estimate the standard deviation of IQ scores from this. (3 marks)

Example

The King’s grading policy (like most universities) stipulates that a “typical” class should have a 13% failure rate, and 17% receiving A- or above. Suppose a large Psych class final exam (n = 100) had a roughly normal distiribution of test scores with = 8.9% and = 62.4%. Where should the pass-fail and A- and above grade cut-offs be set?

Group Work…

Do the following problems dealing with z-scores: 1.97, 1.98 and 1.99 (pg 88)

An interesting Minitab problem – 1.132