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Chapter 8 2114

# Chapter 8 2114 - Chapter 8 Tests of Hypotheses Based on a...

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Chapter 8 Tests of Hypotheses Based on a Single Sample 1

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Problem to which hypotheses testing can be applied: If the average daily high temperature in Pennsylvania in January averaged 26.6˚F from 1998 to 2008 but was 27.8˚F in January of 2009, does this indicate a change in the average temperature in the state? To answer: We need to know the sample size on which this average was based as well as the standard deviation of the population of temperature data or of the sample data. We need to know how many standard deviation units the value of 27.8 is away from a population mean of 26.6. We’ll return to this later. 2
Hypothesis Testing To decide between two contradictory claims regarding a parameter: μ = 100 μ ≠ 100 The objective is to decide, based upon sample information, which of two hypotheses is correct

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Hypotheses 4 The null hypothesis , denoted H 0 , is the claim that is initially assumed to be true. The alternative hypothesis , denoted by H a , is the assertion that is contrary to H 0 . Possible conclusions from hypothesis-testing analysis are reject H 0 or fail to reject H 0 . The null hypothesis will be in the form: H 0 : θ = θ o (= null value) The alternative hypothesis will have the form: H a : θ > θ o One-tailed test (upper tailed test) or H a : θ < θ o One-tailed test (lower tailed test) or H a : θ ≠ θ o Two-tailed test
A Test of Hypotheses 5 A test of hypotheses is a method for using sample data to decide whether the null hypothesis should be rejected.

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Test Procedure is specified by: 6 A test procedure is a rule, based on sample data, for deciding whether to reject H 0 . (Reject H 0 if x ≤ 10) 1. A test statistic , a function of the sample data on which the decision is to be based. 2. A rejection region , the set of all test statistic values for which H 0 will be rejected (null hypothesis rejected iff the test statistic value falls in this region.)
Errors in Hypothesis Testing 7 A type I error consists of rejecting the null hypothesis H 0 when it was true. The probability of a type I error is denoted by α. A type II error involves not rejecting H 0 when H 0 is false. The probability of a type II error is denoted by β.

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Errors in Hypothesis Testing Defendant Innocent of Crime Defendant Guilty of Crime Convict Error (Type I) Correct Decision Acquit Correct Decision Error (Type II) Decision Reality
9 We are concerned about the average strength for pipe welds in a nuclear power plant. Is the mean greater than 100 psi? Type I error - If the mean equals 100psi we could get an unrepresentative sample of weak welds and reject H 0 . Type II error - If the mean is less than 100psi we could get an unrepresentative sample of strong welds and not reject. (Less likely as the true mean weld strength gets farther below 100 psi.) If we could afford a test that examined an entire population, we could eliminate the possibility of errors. Since this is not usually possible we must look for tests for which the probability of either type of error is very small.

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Example: We are concerned about the average strength for pipe welds in a nuclear power plant. Does the mean weld strength equal 100 psi?
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Chapter 8 2114 - Chapter 8 Tests of Hypotheses Based on a...

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