Review for confidence intervals
Tip: It is often helpful to highlight or jot down important information as you read the problem.
1. A recent poll found that “433 of the 1548 randomly-selected adults questioned felt that unemployment
compensation should be extended an additional six months while the country is in its current economic downturn.” We
want to use this information to construct a 95% confidence interval to estimate the proportion of the U.S. adults who
feel this way. Tip: Do you know what the problem is talking about? Figure out your context first.
a) Interpret the confidence level of 95% in the context of the problem.
b) Find the appropriate critical value (include a picture) for a 95% confidence interval and the standard error of
the sample proportion.
c) Use this information to estimate the proportion of all US adults who feel this way. Don’t forget your 4 step
process.
d) Suppose you wanted to estimate the proportion of US adults who feel that unemployment compensation
should be extended an additional six months while the country is in its economic downturn with 95%
confidence to within ± 3%. How large of a sample would you need?
e) If you wanted to have a margin of error of ± 3% with 90% confidence, would your sample have to be larger,
smaller, or the same size as the sample in part (d)? Explain.
2. You want to estimate the mean amount spent by customers at a local gas station with 98% confidence and a
margin of error of no more than $2. Preliminary data suggests that 𝞼 = $5.1 is a reasonable estimate for the standard
deviation for all customers. How large a sample do you need?
Tip: Think about what formulas you need.
3. Considerable research is being done in bioremediation—the use of living organisms to clean up pollution. In a
recently conducted experiment, researchers used fungi to degrade hydrocarbons, the by-products of incomplete
combustion of fossil fuels. These by-products can cause cancer, and their accumulation in water and soil poses an
environmental hazard. In order to be cost-effective, a minimum of 1.00 micromoles/minute/gram of soil of a particular
hydrocarbon must be degraded. The researchers added fungi and growth media (essentially sugar) to the soil in 9
containers with random samples of soil and obtained the following degradation rates in micromoles/min/gram:
0.83
0.91
1.00
1.08
1.08
1.09
(Tip: Do you know what the problem is talking about?
1.10
1.23
1.36
Figure out your context first.)
a) The sample mean is xbar = 1.076 and the sample standard deviation is s = 0.158 . Calculate and interpret the
standard error of the mean for these data.
b) Using this information, estimate the true mean degradation rate in micromoles/min/gram.
c) Can the researchers be confident that this process will be cost-effective?