Download AP Statistics AP Review Inference – Confidence Intervals

AP Statistics
AP Review Inference – Confidence Intervals
Vocabulary
Point estimate
Margin of error
Critical value
Standard error
Degrees of freedom
Robust procedures
Notes
To estimate an unknown population parameter, start with a statistic that provides a reasonable guess. The
chosen statistic is a point estimator for the parameter. The specific value of the point estimator that we use
gives a point estimate for the parameter.
A C% confidence interval uses sample data to estimate an unknown population parameter with an indication
of how precise the estimate is and of how confident we are that the result is correct.
A confidence interval gives an interval of plausible values for the parameter. The interval is computed from
the data and has the form: point estimate ± margin of error
When calculating a confidence interval, it is common to use the form:
statistic ± (critical value)(standard deviation of statistic)
To interpret a C% confidence interval, say, “We are C% confident that the[parameter in context] is between
___ and ___ .” Be sure that your interpretation describes a parameter and not a statistic.
The confidence level C is the success rate of the method that produces the interval. If you use 95% confidence
intervals often, in the long run about 95% of your intervals will contain the true parameter value. You don’t
know whether a 95% confidence interval calculated from a particular set of data actually captures the true
parameter value.
Other things being equal, the margin of error of a confidence interval gets smaller as


The confidence level C decreases
The sample size n increases
Remember that the margin of error for a confidence interval includes only chance variation, not other sources
of error like nonresponse and undercoverage.
The conditions for constructing a confidence interval about a population proportion are



Random: The data were produced by a well-designed random sample or randomized experiment.
10%: When sampling without replacement, we check that the population is at least 10 times as large
as the sample.
Large counts: The sample is large enough that npˆ and n 1  pˆ  , the counts of successes and failures in the
sample, are both at least 10.
Confidence intervals for the population proportion p are based on the sampling distribution of the sample proportion pˆ.
when the conditions for inference are met, the sampling distribution of p̂ is approximately Normal with mean p and
standard deviation
p 1  p 
n
.
In practice, we use the sample proportion p̂ to estimate the unknown parameter p. we therefore replace the standard
deviation of p̂ with its standard error when constructing a confidence interval. The C% confidence interval for p is
pˆ  z *
pˆ 1  pˆ 
n
where z * is the standard Normal critical value with C% of its area between  z * and z * .
When constructing a confidence interval, remember to include:


Conditions – random, normal, independent
Calculations – either the name or formula as well as the answer

Conclusion – “We are C% confident that the [parameter in context] is between ___ and ___ .”
The sample size needed to obtain a confidence interval with approximate margin of error ME for a population
proportion involves solving ME  z *
pˆ 1  pˆ 
n
for n, where p̂ is a guessed value for the sample proportion, and
z * is the critical value for the confidence level you want. Use pˆ  0.5 if you don’t have a good idea about the value of
p̂ .
Problems
1. How much do the volumes of bottles of water vary? A random sample of 50 “20 oz.” water bottles is
collected and the contents are measured. A 90% confidence interval for the population mean is 19.10
to 20.74.
A. Interpret the confidence interval in context
B. Interpret the confidence level in context.
C. Based on this interval, what can you say about the contents of the bottles in the sample? What can
you say about the contents of bottles in the population?
2. A large company is interested in developing a new bake ware product for consumers. In an effort to
determine baking habits of adults, a researcher selects a random sample of 50 addresses in a large,
Midwestern, metropolitan area. She calls each selected home in the late-morning to collect information
on their baking habits. The proportion of adults who bake at least twice a week is calculated and a 90%
confidence interval is constructed.
Discuss whether or not each of the conditions for constructing a confidence interval has been met. If any
have not been met, discuss the implications on the interpretation of the interval.
3. According to a recent study, not everyone can roll their tongue. A researcher observed a random sample
of 300 adults and found 68 who could roll their tongue. Construct and interpret a 90% confidence interval
for the true proportion of adults who can roll their tongue.
4. A researcher would like to estimate the proportion of adults who can roll their tongues. However, unlike
the previous example, she’d like the estimate to be within 2% at a 95% confidence level. How large a
sample is needed?
5. The amount of sugar in soft drinks is increasingly becoming a concern. To test sugar content, a researcher
randomly sampled 8 soft drinks from a particular manufacturer and measured the sugar content in
grams/serving. The following data were produced:
26, 31, 23, 22, 11, 22, 14, 31
Use these data to construct and interpret a 95% confidence interval for the mean amount of sugar in this
manufacturer’s soft drinks.
6. A researcher would like to estimate the mean amount of time it takes to accomplish a particular task. A
previous study indicates the time required varies in the population with a standard deviation of 4 seconds.
He would like to estimate the true mean time within 0.5 seconds at 95% confidence. How large a sample
is needed?
7. An airline wishes to know what is the difference between the proportion of international passengers and
passengers flying within the U.S. who would be willing to pay an additional $300 to fly first class. To
answer this question they asked each person in a random sample of passengers on international flights
and in a random sample of passengers on flights within the U.S. if they would pay extra. The resulting data
are as follows:
Flight type
n
# who would pay extra
International
99
73
Within U.S.
91
56
Construct and interpret a 95% interval to answer the airline’s question.
8. Each year, most students take a test that measures their reading achievement. The score reports this in
terms of the grade level at which the child is reading. The following scores are for a random sample of six
children in a particular U.S. public school. Suppose it is reasonable to believe that the assumptions for
inference are met. How does the mean grade level at which the students are reading change with each
additional year of school?
Predictor
Constant
Grade in School
S = 0.222195
Coef
– 0.0676
1.01488
R-Sq = 99.8%
SE Coef
0.1714
0.02424
T
– 0.39
41.86
P
0.713
0.000
R-Sq (adj) = 99.7%
A. Use the following computer output to construct a 95% interval to answer the question.
B. Is there convincing evidence that the mean grade level score increases by more than 1 with each
additional year of school?
9. When a virus is placed on a tobacco leaf, small lesions appear on the leaf. To compare the mean number of
lesions produced by two different strains of virus, one strain is applied to half of each of eight tobacco
leaves, and the other strain is applied to the other half of each leaf. The strain that goes on the right half
of the tobacco leaf is decided by a coin flip. The lesions that appear on each half are then counted. The
data are given below:
Leaf
1
2
3
4
5
6
7
8
Strain 1
31
20
18
17
9
8
10
7
Strain 2
18
17
14
11
10
7
5
6
What is the mean difference in the number of lesions produced by the two different strains of virus?