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Chapter 7

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Statistics

STATS 2B03

Aaron Childs

Fall

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Stats 2B03: Statistical Methods for Science
Chapter 7: Hypothesis Testing
7.1 Introduction
- Purpose of hypothesis testing is to aid the clinician, researcher, or
administrator in reaching a conclusion concerning a population by examining
a sample from that population
- Hypothesis may be defined simply as a statement about one or more
populations
- Research hypothesis is the conjecture or supposition that motivates the
research leading to statistical hypotheses
- Statistical hypotheses are hypotheses that are stated in such a way that
they may be evaluated by appropriate statistical techniques
- Hypothesis testing steps:
Data: the nature of the data that form the basis of the testing
procedures must be understood, since this determines the particular
test to be employed.
Assumptions: a general procedure is modified depending on the
assumptions. The same assumptions that are of importance in
estimation are important in hypothesis testing. Including assumptions
about the normality of the population distribution, equality of
variances, and independence of samples.
Hypotheses:
Null hypothesis: is the hypothesis to be tested, it is a statement
of agreement with conditions presumed to be true in the
population of interest
Alternative hypothesis: statement of what we will believe is
true if our sample data causes us to reject the null hypothesis
Test statistic: some statistic that may be computed from the data of
the sample
Distribution of test statistic: key to statistical inference is the sampling
distribution
Decision rule: all possible values that the test statistic can assume are
points on the horizontal axis of the graph of the distribution of the test
statistic and are divided into two groups; one group constitutes what
is known as the rejection region and the other group makes up the
nonrejection region. The values of the test statistic forming the
rejection region are those values that are less likely to occur if the null
hypothesis is true, while the values making up the acceptance region
are more likely to occur if the null hypothesis is true. The decision rule
tells us to reject the null hypothesis if the value of the test statistic
that we compute from our sample is one of the values in the rejection
region and to not reject the null hypothesis if the computed value of
the test statistic is one of the values in the nonrejection region. Calculation of test statistic: from data contained in the sample we
compute a value of the test statistic and compare it with the rejection
and nonrejection regions that have been specified
Statistical decision: consists of rejecting or not rejecting the null
hypotheses.
Conclusion: if H0is rejected, we conclude that H iA true. If H i0 not
rejected, we conclude that H m0y be true.
P values: number that tells us how unusual our sample results are,
given that the null hypothesis is true.
- Rules for stating statistical hypotheses:
For deciding what statement does in the null hypothesis and what
statement does in the alternative hypothesis:
What you hope or expect to be able to conclude as a result of
the test usually should be placed in the alternative hypothesis
The null hypothesis should contain a statement of equality,
either =, ≤, or ≥.
The null hypothesis is the hypothesis that is tested
The null and alternative hypotheses are complementary. That
is, the two together exhaust all possibilities regarding the value
that the hypothesized parameter can assume.
- A precaution: neither hypothesis testing nor statistical inference leads to the
proof of a hypothesis
- General formula for test statistic:
- Significance level: the level of significance α is a probability and, in fact, is
the probability of rejecting a true null hypothesis
- Types of errors:
Type I error: error committed when a true null hypothesis is rejected
Type II error: error committed when a false null hypothesis is not
rejected, designated by β (generally larger than α)
- Confusion matrix
Condit

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