Sample variance 1 1 1 in list mf26 1 1 1 1 n s n x x

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sample variance 1 1 1 ( ) (in List MF26) 1 1 ( ) 1 ( ) 1 n s n x x x x n n n x a x a n n Example 8.8 : A manufacturer claims that the average life span of his electric light bulbs is 2000 hours. A random sample of 64 bulbs is tested and the life span x in hours recorded. The results obtained are as follows: 2 127808, ( ) 9694.6 x x x   Is there sufficient evidence, at 2% level of significance, that the manufacturer is over-estimating the lifetime of his light bulbs? Assume that the distribution of the life span of light bulb is normal. Solution : Unbiased estimates for population mean and population variance, 127808 1997 64 x , 2 2 1 1 ( ) 153.8825 1 n i i s x x n Let denote the mean lifetime of an electric light bulb. If the manufacturer is over- estimating the lifetime, the population mean is lower than the claimed value. When the sample size is large ( 50 n ) and distribution of X is not normally distributed, the distribution of X is approximately normal by Central Limit Theorem. Hence (i) 2 ~ N , X n if population variance is known, or (ii) 2 ~ N , s X n approximately, if population variance is unknown. The test statistic is N(0,1 ~ ) / X Z n or N(0,1 ~ ) / X Z s n approximately
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