12. Consider a population (a random variable X) with mean = 100 and variance 2 =900. The sample size = 30, so the central limit theorem applies. a) What are the mean and variance of the sampling distribution for x , the sample mean? b) What is the probability that x bar > 109? c) What is the probability that x bar is 96 and 211? d) What is the probability that x bars <202?

13. The lifespans of light bulbs manufactured by a company have an average of 1,200 hours and a standard deviation of 400 hours. The probability distribution of this variable obeys the normal law. Suppose you buy 9 bulbs; you can consider as a random sample from that manufacturer's production. a) What is the mean of the sample mean? b) What is the variance of the sample mean? c) What is the standard deviation of the sample mean? d) What is the probability that the sample mean exceeds 93.0 units? 14. The average sale price for condominiums in the town of Green Valley was $215,000. (population average). The population standard deviation was $ 25,000. A researcher took a random sample of size 100 units. a) What is the probability that the sample mean is greater than $ 210,000? b) What is the probability that the sample mean is greater than $ 213,000 but less than $217,000? c) Without doing the calculations, determine which of the following intervals would contain the sample mean: $213.00 to $215,000; $214,000 to $216,000; $ 215,000 to $217,000; $216,000 to $218,000 d) Suppose a friend suggests to you that the calculations you have just made do not are invalid because this variable - The average selling price for condominiums in the town of Green Valley - does not obey normal law. What would your answer be?