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Chapter 7

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Statistical Sciences

Statistical Sciences 2244A/B

Jennifer Waugh

Spring

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7.1 Overview
● This chapter describes the statistical procedure for testing hypotheses
● The two major activities of inferential statistics are the estimation of population
parameters and hypothesis testing
● In statistics a hypothesis is a claim or statement about a property of a population
● Ahypothesis test (or test of significance) is a standard procedure for testing a claim about
a property of a population
● Recall the rare event rule for inferential statistics:
○ if, under a given assumption, the probability of a particular observed event is
exceptionally small, we conclude that the assumption is probably not correct
● Following this rule, we test a claim by analyzing sample data in an attempt to distinguish
between results that can easily occur by chance and results that are highly unlikely to
occur by chance
7.2 Basics of Hypothesis Testing
● In this section we describe the formal components used in hypothesis testing: null
hypothesis, alternative hypothesis, test statistic, critical region, significance level, critical
value, P-value, type I error, type II error
Components of a Formal Hypothesis Test
● The null hypothesis (H ) isoa statement that the value of a population parameter such as
proportion, mean or standard deviation, is equal to some claimed value
○ we test the null hypothesis directly in the sense that we assume it is true and reach
a conclusion to either reject H oo fail to reject H o
● The alternative hypothesis (H or H1) is tae statement that the parameter has a value that
somehow differs from the null hypothesis; symbolic form of alternative hypothesis must
use < or > or ≠
● Identifying H andoH : 1
○ Identify the specific claim or hypothesis to be tested, and express it in symbolic
form
○ Give the symbolic form that must be true when the original claim is false
○ Of the two symbolic expressions obtained so far, let the alternative hypothesis H 1
be the one not containing equality so that H us1s the symbol < or > or ≠; let the
null hypothesis H bo the symbolic expression that the parameter equals the fixed
value being considered
● The test statistic is a value computed from the sample data and it is used in making the
decision about the rejection of the null hypothesis
● It is found by converting the sample statistic (such as sample mean or sample proportion)
to a score (like z or t) with the assumption that the null hypothesis is true
Critical Region, Significance Level, Critical Value, and P-Value
● The critical region (or rejection region) is the set of all values of the test statistic that
cause us to reject the null hypothesis
● The significance level (denoted by α) is the probability that the test statistic will fall in
the critical region when the null hypothesis is actually true; if the test statistic falls in the
critical region, we will reject the null hypothesis so α is the probability of making the
mistake of rejecting the null hypothesis when it is true ● Acritical value is any value that separates from the critical region (where we reject the
null hypothesis) from the values of the test statistic that do not lead to rejection of the null
hypothesis
○ value depends on the nature of the null hypothesis, the sampling distribution that
applies, and the significance level α
Two Tailed, Left Tailed, and Right Tailed
● The tails in a distribution are the extreme regions bounded by critical values; some
hypothesis are two-tailed, some are right-tailed, and some are left-tailed
○ Two tailed test: the critical region is in the two extreme regions under the curve
○ Left tailed test: the critical region is in the extreme left region under the curve
○ Right tailed test, the critical region is in the extreme right region under the curve
● In two tailed tests, the significance level α is divided equally between the two tails that
constitute the critical region; in tests that are one tailed, the area of the critical region in
that one tail is α
● The P-value (or p-value or probability value) is the probability of getting a value of the
test statistic that is at least as extreme as the one representing the sample data, assuming
that the null hypothesis is true
○ null hypothesis is rejected if the P-value is very small (<0.05)
● If a test is two tailed, look at where the test statistic is located in reference to the center
○ if the test statistic is to the right of the center, P-value = twice the area to the right
of the test statistic
○ if the test statistic to the left of the center, P-value = twice the area to the left of
the test statistic
Decisions and Conclusions
● Our initial conclusion will always be one of the following:
○ Reject the null hypothesis
○ Fail to reject the null hypothesis
● Decision criterion: the decision to reject or fail to reject the null hypothesis is usually
made using either the traditional method (or classical method) of testing hypothesis, the
P-value method, or the decision is sometimes based on confidence intervals
● Traditional method:
○ reject H if0the test statistic falls within the critical region
○ fail to reject H i0 the test statistic does not fall within the critical region
● P-value method:
○ reject H if P-value <= α (where α is the significance level such as 0.05)
0
○ fail to reject H i0 P-value > α
● Another option: instead of using a significance level such as α=0.05, simply identify the P
value and leave the decision to the reader
● Confidence intervals: because a confidence interval estimate of a population parameter
contains the likely values of that parameter, reject a claim that the population parameter
has a value that is not included in the confidence interval
Wording of the Final Conclusion
● Note, only one case leads to a statement that the sample data actually supports the
conclusion
● If you want to support some claim, state it in such a way that it becomes the alternative
hypothesis and then hope that the null hypothesis gets rejected ● Note about accept/fail to reject, we should recognize that we are not proving the null
hypothesis, rather we are merely saying that the sample evidence is not strong enough to
warrant rejection of the null hypothesis
Type I and Type II Errors
● When testing a null hypothesis, we arrive at a conclusion of rejecting or failing to reject
that null hypothesis; such conclusions are sometimes correct and sometimes wrong (even
when everything you did was correct)
● Type I error: the mistake of rejecting the null hypothesis when it is actually true; the
symbol α is used to repre

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