Statistics and Central Limit Theorem

Question

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 8.8 ppm and standard deviation 1.6 ppm. 37 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the probability that one randomly selected city's waterway will have more than 7.9 ppm pollutants? 
4. For the 37 cities, find the probability that the average amount of pollutants is more than 7.9 ppm. 
5. For part d), is the assumption that the distribution is normal necessary? YesNo
6. Find the IQR for the average of 37 cities.
7. Q1 =  ppm
8. Q3 =  ppm
9. IQR:  ppm

Answer 1

May I assume that XX and xx, wherever they appear mean X and x? If os,

1. You are given this information in the problem. Normally distributed with mean = __ and standard deviation = ___
2. The central limit theorem indicates that the sampling distribution of the mean, i.e., if you took an infinite number of samples from this population, has mean = population mean and standard deviation = the standard error of the mean which equals the population standard deviation divided by the square root of the sample size. In this case, sigma/sqrt(37)
3. This part of the question involves working with the population and any randomly drawn observation. Convert 7.9 ppm to a Z score using the given population standard deviation as the standard deviation. Z = (x - mu)/sigma. Use any Z-table or Excel's NORM.S.DIST() fundtion to find the probability of a value <= Z. 1 - this probability gives you the probability of a z > Z.
4. This part of the question involves working with the sampling distribution of the mean. Convert 7.9 ppm to a Z score using the standard error you calculated as the standard deviation. Complete work with this Z score as done in part 3.
5. Review your textbook's description of the central limit theorem and the steps you did for part 4.
6. - 9. More Z score work using the population mean and the standard error you calculated in part 2.