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Testing Claims About Population Means with Known Standard Deviation

This section outlines the process of testing claims about a population mean when the standard deviation (σ) is known. Key assumptions include having a simple random sample and either normal distribution of the population or a sample size greater than 30. It describes the use of both traditional hypothesis testing methods and P-value methods, particularly utilizing the TI-83/84 calculator. The rationale for hypothesis testing is detailed, focusing on the comparison between sample results and the null hypothesis, and the implications of small or large discrepancies.

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Testing Claims About Population Means with Known Standard Deviation

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  1. Section 8-4 Testing a Claim About a Mean: σ Known

  2. ASSUMPTIONS FOR TESTING CLAIMS ABOUT POPULATION MEANS • The sample is a simple random sample. • The value of the population standard deviation σ is known. • Either or both of these conditions are satisfied: • The population is normally distributed. • The sample size is larger than 30; that is, n > 30.

  3. TEST STATISTIC FOR TESTING A CLAIM ABOUT A MEAN

  4. TRADITIONAL ANDP-VALUE METHODS We will use both the traditional method and P-­value method for hypothesis testing about a mean. The process is the same as that described in the last two sections.

  5. P-VALUE METHOD USING THE TI-83/84 CALCULATOR • Press STAT. • Arrow over to TESTS. • Select 1:Z-Test…. • Select the Inpt type (Data or Stats) • Enter as μ0 the value of μ from the null hypothesis. • Enter the values for σ(population standard deviation), x (sample mean), and n (sample size). • Select the type of test. Select either ≠μ0, < μ0 , or >μ0. This corresponds to the alternative hypothesis. • Highlight Calculate, and press ENTER.

  6. UNDERLYING RATIONALE FOR HYPOTHESIS TESTING • If, under a given observed assumption, the probability of getting the sample is exceptionally small, we conclude that the assumption is probably not correct. • When testing a claim, we make an assumption (null hypothesis) that contains equality. We then compare the assumption and the sample results and we form one of the following conclusions:

  7. UNDERLYING RATIONALE FOR HYPOTHESIS TESTING • If the sample results can easily occur when the assumption (null hypothesis) is true, we attribute the relatively small discrepancy between the assumption and the sample results to chance. • If the sample results cannot easily occur when that assumption (null hypothesis) is true, we explain the relatively large discrepancy between the assumption and the sample by concluding that the assumption is not true.

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