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AP Statistics
9.1A Notes
A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to
assess. The claim is a statement about a parameter, like the population proportion p or the population mean µ. We express the
results of a significance test in terms of a probability that measures how well the data and the claim agree.
The Reasoning of Significance Tests
Statistical tests deal with claims about a population. Tests asked if sample data give good evidence against a claim. A test might say,
“If we took many random samples and the claim were true, we would rarely get a result like this.” To get a numerical measure of
how strong the sample evidence is, replace the vague term “rarely” by a probability.
Stating Hypotheses
A significance test starts with a careful statement of the claims we want to compare. The claim we seek evidence against is called
the null hypothesis, abbreviated H0. Usually, the null hypothesis is a statement of “no difference.” The claim we hope or suspect to
be true instead of the null hypothesis is called the alternative hypothesis. We abbreviate the alternative hypothesis as H a.
Definition: Null hypothesis HO, alternative hypothesis Ha
The claim tested by a statistical test is called the null hypothesis (H 0). The test is designed to assess the strength of the evidence
against the null hypothesis. Often the null hypothesis is a statement of “no difference.”
The claim about the population that we trying to find evidence for is the alternative hypothesis (H a).
Example
Does the job satisfaction of assembly-line workers differ when their work is machine-paced rather than self-paced? One study chose
18 subjects at random from a company with over 200 workers who assembled electronic devices. Half of the workers were assigned
at random to each of two groups. Both groups did similar assembly work, but one group was allowed to pace themselves while the
other group used an assembly line that moved at a fixed pace. After two weeks, all the workers took a test of job satisfaction. Then
they switched work setups and took the test again after two more weeks. (This experiment uses a matched pairs design.) The
response variable is the difference in satisfaction scores, self-paced minus machine-paced.
1. Describe the parameter of interest in this setting.
2. State appropriate hypotheses for performing a significance test.
Caution: The hypotheses should express the hopes or suspicions we have before we see the data. It is cheating to look at the data
firs and then frame hypotheses to fit what the data show. For example, the data for the job satisfaction study showed that the
workers were more satisfied with self-paced work. But you should not change the alternative hypothesis to
H 0 :   0 after
looking at the data. If you do not have a specific direction firmly in mind in advance, use a two-sided alternative.
Definition: One-sided alternative hypothesis and two-sided alternative hypothesis
The alternative hypothesis is one-sided if it states that a parameter is larger than the null hypothesis value or if it states that the
parameter is smaller than the null value. It is two-sided if it states that the parameter is different from the null hypothesis value (it
could be either larger or smaller).
In any significance test, the null hypothesis has a form H 0: parameter = value. The alternative hypothesis has one of the forms Ha:
parameter < value, Ha: parameter > value, or Ha: parameter ≠ value. To determine the correct form of H a, read the problem
carefully.
Hypotheses always refer to a population, not to a sample. Be sure to state H 0 and Ha in terms of population parameters. It is never
correct to write a hypothesis about a sample statistic, such as pˆ  0.64 or x  85.
Example
State the appropriate hypotheses for a significance test.
3. According to the web site sleepdeprivation.com, 85% of teens are getting less than eight hours of sleep a night. Jannie wonders
whether this result holds in her large high school. She asks an SRS of 100 students at the school how much sleep they get on a
typical night. In all, 75 of the responders said less than 8 hours.
4. As part of its 2010 census marketing campaign, the U.S. Census Bureau advertised “10 questions, 10 minutes – that’s all it
takes.” On the census form itself, we read, “The U.S. Census Bureau estimates that, for the average household, this form will
take about 10 minutes to complete, including the time for reviewing the instructions and answers.” We suspect that the actual
trim it takes to complete the form may be longer than advertised.