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# hws52 - 6.13 a The population from which we are randomly...

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°±²³² ´µ ¶·¸¹º»¼½¾»¿À´ ±»Á·Â¸»ÃÄ 6.13. a. The population from which we are randomly sampling n = 35 measurements is not necessarily normally distributed. However, the sampling distribution of x does have an approximately normal distribution, with mean μ and standard deviation n / σ . The probability of interest is [ ] [ ] [ ] [ ] 1 ) ( 1 ) ( 1 ) ( 1 1 - < - - < - = < - < - = < - μ μ μ μ x P x P x P x P Standardizing, we have: [ ] [ ] [ ] [ ] ° ± ² ³ ´ µ - + - < - ° ± ² ³ ´ µ - + < = + - < - + < = - < - - < - 35 / 12 ) 1 ( 35 / 12 ) 1 ( 1 1 1 ) ( 1 ) ( μ μ μ μ μ μ μ μ z P z P x P x P x P x P [ ] [ ] 3758 . 0 ) 1879 . 0 ( 2 493 . 0 493 . 0 = = - < - < = z P z P where, since σ is unknown, we use the sample standard deviation s = 12 (we can do this, since the sample size is large enough). b. No. There are many possible values for x , the actual percent tax savings, as given by the probability distribution for x . 6.16. a. The sample mean x is normally distributed, since the original strength measurements are Normally distributed, with mean μ and standard deviation 10 / 2 / = n σ .

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