The contrary also holds in that a very small sample size will result
in a confidence interval that is quite large, because both the t-value for a given confidence level is larger for a smaller sample size and the small n
in the denominator of the calculation does not reduce the interval size
significantly. The larger interval for a desired level of confidence is
required to account for the additional uncertainty introduced by the
At this point, apply your knowledge and develop a confidence interval using a t-
distribution. Consider a consumer safety group that performs crash
tests on vehicles to determine safety ratings. Crashing cars is very
expensive since the damaged cars have little or no salvage value. This
consequence calls for a small sample size, since the budget for cars to
crash is limited.
Suppose a particular auto manufacturer has produced a new model
vehicle. The consumer safety group needs to determine a safety rating
and calculate how accurate their estimate is based on the small sample
size. After choosing a random sample of five of the new model cars from
the manufacturer, the test is performed and the estimate of the safety
rating is determined to be eight with a sample standard deviation of
.94. The consumer safety group wants to have a high degree of
confidence in its estimate and so it chooses to calculate a 99 percent
confidence interval. The t-score for 99 percent and n = 4 is found in the t-distribution table to be 4.604. Using the formula, the confidence interval is calculated as follows.
This result means that the consumer safety group can say that it is
99 percent confident the safety rating of the new model is between 6.06
and 9.94. To get a smaller confidence interval at the 99 percent
confidence level, more cars will have to be crash tested.
1. How does a t-distribution differ from a Z distribution?
2. When does a t-distribution begin to approximate a Z distribution?
3. What is a 95 percent confidence interval?
4. How is a 95 percent confidence interval for a population mean calculated?
5. What is the t-score for a 95 percent confidence interval for m in a sample of a random variable having a standard normal distribution of 16 data points?
6. A cleaning business operates in the city of New York and works
for the companies that lease office space in the city. The business
contracts to clean office space in increments of 100 square feet. The
business determines its margins by determining how long it takes a crew
to clean 100 square feet of office space, and bases its rates on this
Because the company is relatively new, it has to estimate the time
it takes to clean a 100 square feet of office space. The company
estimates that it should take 5.5 hours to clean 100 square feet (m
= 5.5). The company starts its business with this expectation and works
for a week straight, collecting data as it proceeds in order to be
certain that it is neither over- nor under-charging its clients. The
data collected by the company can be seen in the table below.
After collecting this data, the company wants to determine if the
time originally estimated to clean 100 square feet of office space was
reasonable. Assuming a 95 percent confidence level, was this estimation
7. If the cleaning company from the previous question had a sample
of seven rather than a sample of 12 upon which to base its conclusions,
what would be the boundaries of the 95 percent confidence interval for
the estimate of the number of hours? Assume that the sample mean and
standard deviation are equal to those calculated above.