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Hypothesis testing is used to make decisions concerning the value of a parameter.

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Null Hypothesis: H 0 a working hypothesis about the population parameter in question

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The value specified in the null hypothesis is often: a historical value a claim a production specification

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Alternate Hypothesis: H 1 any hypothesis that differs from the null hypothesis

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An alternate hypothesis is constructed in such a way that it is the one to be accepted when the null hypothesis must be rejected.

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A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long. Determine the null and alternate hypotheses.

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A manufacturer claims that their light bulbs burn for an average of 1000 hours.... The null hypothesis (the claim) is that the true average life is 1000 hours. H 0 : = 1000

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… A manufacturer claims that their light bulbs burn for an average of 1000 hours. We have reason to believe that the bulbs do not last that long.... If we reject the manufacturer’s claim, we must accept the alternate hypothesis that the light bulbs do not last as long as 1000 hours. H 1 : < 1000

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Type I Error rejecting a null hypothesis which is, in fact, true

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Type II Error not rejecting a null hypothesis which is, in fact, false

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Options in Hypothesis Testing H 0 is Our Choices:

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Errors in Hypothesis Testing H 0 is Our Choices: Type I error

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Errors in Hypothesis Testing H 0 is Our Choices: Type I error Type II error

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Errors in Hypothesis Testing H 0 is Our Choices: Correct decision Type I error Type II error Correct decision

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Level of Significance, Alpha ( ) the probability with which we are willing to risk a type I error

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Type II Error = beta =probability of a type II error (failing to reject a false hypothesis) small is normally is associated with a (relatively) large , and vice-versa. Choices should be made according to which error is more serious.

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Power of the Test = 1 – Beta The probability of rejecting H 0 when it is in fact false = 1 – . The power of the test increases as the level of significance ( ) increases. Using a larger value of alpha increases the power of the test but also increases the probability of rejecting a true hypothesis.

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Probabilities Associated with a Hypothesis Test

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Reject or... When the sample evidence is not strong enough to justify rejection of the null hypothesis, we fail to reject the null hypothesis. Use of the term “accept the null hypothesis” should be avoided. When the null hypothesis cannot be rejected, a confidence interval is frequently used to give a range of possible values for the parameter.

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Fail to Reject H 0 There is not enough evidence to reject H 0. The null hypothesis is retained but has not been proven.

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Reject H 0 There is enough evidence to reject H 0. Choose the alternate hypothesis with the understanding that it has not been proven.

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A fast food restaurant indicated that the average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. We wish to test the claim with a level of significance of = 0.01,

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… average age of its job applicants is fifteen years. We suspect that the true age is lower than 15. H 0 : = 15 H 1 : < 15 Describe Type I and Type II errors.

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H 0 : = 15 H 1 : < 15 = 0.01 A type I error would occur if we rejected the claim that the mean age was 15, when in fact the mean age was 15 (or higher). The probability of committing such an error is as much as 1%.

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H 0 : = 15 H 1 : < 15 = 0.01 A type II error would occur if we failed to reject the claim that the mean age was 15, when in fact the mean age was lower than 15. The probability of committing such an error is called beta.

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Types of Tests.When the alternate hypothesis contains the “not equal to” symbol ( ), perform a two-tailed test. When the alternate hypothesis contains the “greater than” symbol ( > ), perform a right-tailed test. When the alternate hypothesis contains the “less than” symbol ( < ), perform a left-tailed test.

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Two-Tailed Test H 0 : = k H 1 : k

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Two-Tailed Test – z 0 z If test statistic is at or near the claimed mean, we do not reject the Null Hypothesis either tail If test statistic is in either tail - the critical region - of the distribution, we reject the Null Hypothesis. H 0 : = k H 1 : k

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Right-Tailed Test H 0 : = k H 1 : > k

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Right-Tailed Test 0 z If test statistic is at, near, or below the claimed mean, we do not reject the Null Hypothesis the right tail If test statistic is in the right tail - the critical region - of the distribution, we reject the Null Hypothesis. H 0 : = k H 1 : > k

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Left-Tailed Test z 0 If test statistic is at, near, or above the claimed mean, we do not reject the Null Hypothesis the left tail If test statistic is in the left tail - the critical region - of the distribution, we reject the Null Hypothesis. H 0 : = k H 1 : < k

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