Lecture 6 – Z scores
An Introduction to Making Inferences
descriptive statistics – summarize important characteristics of known population data
inferential statistics – we use sample data to make inferences or generalizations about a
population (review population and sample)
sampling statistic - any statistic that describes the distribution of values for a
variable, or relationships between variables in a sample
population parameter – the estimated characteristics of a population derived from
sampling statistics
 must have a well chosen sample!!! (sound sampling procedures)
z-scores (and the normal distribution) enable us to standardize values so they can
be compared (example: SAT); also remember 68-95-99.7 rule
looking at data: the smaller the standard deviation in relation to the range of responses,
the more homogeneous are the responses
standard score (or z-score) – the number of standard deviations that a given value x is
above or below the mean. In a z distribution, the mean = 0, and the standard deviation is
1. Thus, a z-score of 1.5 is 1 ½ sd above the mean, and a z-score of –2 is 2 sd below the
mean.
sample z: z 
xX
s
population z: z 
x

Examples:
Suppose you took 3 midterm examinations in Math, English, and Biology, and
you want to compare your scores:
your score
Mean
Standard deviation
z-score
Math
90
85
5
English
85
82
2
Biology
93
94
1
1. Find the following probabilities using Appendix A page 469-473:
z < 1.56
z < -.68
z > 2.34
z between 0 and 2.1
z between –1.23 and .90
2. What proportion of all young women are less than 68 inches tall? This proportion is
the area under the normal curve with a mean of 64.5 and standard deviation of 2.5”.
3. The level of cholesterol in the blood is important because high cholesterol levels may
increase the risk of heart disease. The distribution of blood cholesterol levels in a large
population of people of the same age and sex is roughly normal. For 14-year old boys,
the mean is 170 mg of cholesterol per deciliter of blood (mg/dl) and the standard
deviation is 30mg/dl. Levels above 240 mg/dl may require medical attention. What
percent of 14 year-old boys have more than 240 mg/dl of cholesterol?
What percent of 14-year-old boys have blood cholesterol between 140 and 200 mg/dl?
4. Scores on the SAT verbal test in recent years follow approximately the normal
distribution, with mean of 505 and sd 110. How high must a student score in order to
place in the top 10% of all students taking the SAT?
5. One classic use of the normal distribution is inspired by a letter to “Dear
Abby” in which a wife claimed to have given birth 308 days after a brief visit from her
husband, who was serving in the Navy. The lengths of pregnancies are normally
distributed with a mean of 268 days and a standard deviation of 15 days. Given this
information, find the probability of a pregnancy lasting 308 days or longer. What does
the result suggest?
6. Domino sugar packets are labeled as containing 3.5 g. Assume that those packets are
actually filled with amounts that are normally distributed with a mean of 3.586 g and a
standard deviation of 0.074 g. What percentage of packets have less than 3.5 g? Are
many consumers being cheated?
7. A study compared the facial behavior of non-paranoid schizophrenic persons with that
of a control group of normal persons. The control group was timed for eye contact
during a period of 5 minutes, or 300 seconds. The eye-contact times were normally
distributed with a mean of 184 sec and a standard deviation of 55 sec (based on data from
“Ethological Study of Facial Behavior in Non-paranoid and Paranoid Schizophrenic
Patients,” by Pitman, Kolb, Orr, and Singh, Psychiatry, 114:1). Because results showed
that non-paranoid schizophrenic patients had much lower eye-contact times than did the
control group, you have decided to further analyze people in the control group who are in
the bottom 5%. For the control group, find the eye-contact time separating the bottom
5% from the rest.