Questions are from The Practice of Statistics for Business and Economics, 3rd Ed

Question

Questions are from The Practice of Statistics for Business and Economics, 3rd Ed.

4.123 MANUFACTURING DEFECTS. Newly manufactured automobile radiators may have small leaks. Most have no leaks, but some have 1,2 or more. The number of leaks in radiators made by one supplier has mean 0.15 and standard deviation 0.4. The distribution of number of leaks cannot be Normally because only whole number counts are possible. The supplier ships 400 radiators per day to an auto assembly plant. Take x to be the mean number of leaks in these 400 radiators. Over several years of daily shipments, what range of values will contain the middle 95% of the many x’s?

Explanation / Answer

Apply the central limit theorem.

Although the population is not normally distributedm, the sample is large enough for the distribution of sample mean to be normally distributed.

So we can say that x is normal distributed with mean 0.15 (which is the same as that of population), and standard deviation of 0.02 (which is population's standard deviation 0.4 divided by the size of the sample, which is 400). For any normal distribution, the range of values that contain 95% of the distribution lies within the range of two standard deviations from mean. That is, for any normal distribution, the range of values that contain 95% of the distribution lies within the range of -2 and +2. For the distribution of sample mean, these values are 0.11 [=0.15-2(0.02)] and 0.19 [=0.15+2(0.02)].

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