Chapter12_STAT1100_LC.ppt", filename="Chapter12_STAT1100_LC.ppt", filename="Chap

# Chapter12_STAT1100_LC.ppt", filename="Chapter12_STAT1100_LC.ppt", filename="Chap

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Inference About a Population Statistics for Management and Economics Chapter 12

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Inference About A Population Identify the parameter to be estimated or tested. Specify the parameter’s estimator and its sampling distribution. Derive the interval estimator and test statistic . Parameter Population Sample Statistic Inference
Inference About A Population We will develop techniques to estimate and test three population parameters: Population Mean μ Population Variance σ 2 Population Proportion p Parameter Population Sample Statistic Inference

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Inference With Variance Unknown Previously, we looked at estimating and testing the population mean when the population standard deviation ( σ ) was known or given: But how often do we know the actual population variance? Instead, we use the Student t-statistic , given by:
Inference With Variance Unknown When σ is unknown, we use its point estimator s and the z-statistic is replaced by the the t- statistic, where the number of “degrees of freedom” υ , is n–1.

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When the population standard deviation is unknown and the population is normal, the test statistic for testing hypotheses about is: which is Student t distributed with υ = n–1 degrees of freedom. The confidence interval estimator of μ is given by: Testing μ when σ is unknown
Example The manager of a frozen yogurt store claims that a medium-size serving contains an average of more than 4 ounces of yogurt. From a random sample of 14 servings, he obtains a mean of 4.31 ounces and a standard deviation of 0.52 ounce. Test, with α =0.05, the manager’s claim. Assume that the distribution of weight per serving is normal.

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Example Our objective is to determine if the population mean weight for a medium size serving is really more than 4 oz. State the null and alternative hypotheses: IDENTIFY H 0 : μ = 4.0 H 1 : μ >4.0
Example Our test statistic is: With n=14 data points, we have n–1=13 degrees of freedom. Our hypothesis under question is: H 1 : μ > 4.0 Our rejection region becomes: Thus we will reject the null hypothesis in favor of the alternative if our calculated test static falls in this region. COMPUTE 771 . 1 13 , 05 . 0 , = = t t t ν α

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Example From the data given, we know = 4.31, s   =0.52 and thus: Since t = 2.23 > t α , ν = 1.771 We reject H 0 in favor of H 1 , that is, there is sufficient evidence to conclude that the medium size serving weighs more than 4 ounces. COMPUTE 23 . 2 14 / 52 . 0 0 . 4 31 . 4 / = - = - = n s x t μ
Example Alternatively, we can use t-test:Mean from Tools > Data Analysis Plus in Excel… COMPUTE t statistic Critical t-value for testing hypotheses (both one- and two-tailed) Also gives p- value (for both one- and two- tailed tests)

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Example Can we estimate the weight for frozen yogurt served at this store? We are given a
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Chapter12_STAT1100_LC.ppt", filename="Chapter12_STAT1100_LC.ppt", filename="Chap

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