Elementary Statistics
Chapter 8: Confidence Intervals
Confidence Intervals with small samples
When sample sizes are small, we must use the t-distribution instead of the normal
curve (z-distribution). This table (given in Appendix C page 477) relies on ‘degrees of
freedom’, which is always n – 1. The difference with this curve is that the shape changes
slightly depending on the sample size, which means the area under the curve (probability)
will also change (it is lower in the middle and higher on the tails). As the sample size gets
larger, the shape approaches the normal curve. The formula for the confidence interval is
exactly the same, except for the fact that instead of using z /2 you will use t/2.
 s 
X  t 2 

 n
to
 s 
X  t 2 

 n
Conditions for using the student t distribution:
1. The sample is small (n  30); and
2.  is unknown; and
3. The parent population has a distribution that is essentially normal.
1. Create a 95% confidence interval for the starting salaries of 20 college graduates who
have taken a statistics course if the mean salary is $43,704, and the standard deviation is
$9879.
2. With destructive testing, sample items are destroyed in the process of testing them.
Crash testing of cars is one very expensive example of destructive testing. If you were
responsible for such crash tests, there is no way you would want to tell your supervisor that
you must crash and destroy more than 30 cars so that you could use the normal distribution.
Let’s assume that you have crash tested 12 Dodge Viper sports cars (current list price: over
$60,000) under a variety of conditions that simulate typical collisions. Analysis of the 12
damaged cars results in repair costs having a distribution that appears to be bell shaped,
with a mean of $26,227 and a standard deviation of $15,873 (based on data from the
Highway Loss Data Institute). Find the 99% confidence interval estimate of , the mean
repair cost for all Dodge vipers involved in collisions.
3. Bowling league members bowled (what else?) at two different locations. At the location
with oiled lanes, 40 bowling scores had a mean of 205.5 and a standard deviation of 35.4.
At the location with dry lanes, 40 bowling scores had a mean of 185.0 and a standard
deviation of 28.4.
a. Construct a 95% confidence interval estimate of the mean for all scores at the
location with the oiled lanes.
b. Construct a 95% confidence interval estimate of the mean for all scores at the
location with the dry lanes.
c. Compare and interpret the results. Does it appear that the lane surface has an
effect on the bowling scores?
4. A random sample of 16 students was selected from the student body at a local college to
determine the average amount of money that students carry with them. If the sample
mean, X , was equal to $28, with a sample standard deviation of s = 6, then construct a 95%
and a 99% confidence interval for estimating the mean amount of money all students carry
with them.
5. Compute a 99% confidence interval for the salaries of statistics professors, with n = 60,
mean salary = $85,678, and s = $12,345 (yea, I wish!).
Homework
1. Which is relatively better: a score of 75 on a history test or a score of 27 on a
psychology test? (Explain). Scores on the history test have a mean of 80 and a standard
deviation of 12. Scores on the psychology test have a mean of 30 and a standard deviation
of 8.
2. The mean gas mileage of the sample is 23.5 mpg with a standard deviation of 7.82
mpg, using 130 data values. Find the 95% confidence interval for the mean.