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Example A population has a mean of 200 and a standard deviation of 50. A random sample of size 100 will be taken and the sample mean mean. x̄ will be used to estimate the the population a-) What is the expected value of x̄ ? b-) what is the standard deviation of x̄ ? c-) Shoe the sampling distribution of x̄ ? d-) What does the sampling distribution of x̄ show? Example Assume the population standard deviation is σ = 25 . Compute the standard error of the mean, σx̄ for sample sizes of 50, 100, 150 and 200. What can you say about the size of the standard error of the mean as the sample size is increased ? Example A population has a mean of 200 and a standard deviation of 50. Suppose a random sample of size 100 is selected and the sample mean x̄ will be used to estimate the population mean. a-) What is the probability that the sample mean will be within ± 5 of the population mean ? b-) What is the probability that the sample mean will be within ± 10 of the population mean ? Example 3: If a random sample of size 30 is taken from binomial distribution with n=9 and p= 0.5 Q: Find the probability that the sample mean exceeds 5. Exercise - 1 A package-filling process at a Cement company fills bags of cement to an average weight of µ but µ changes from time to time. The standard deviation is σ = 3 pounds. A sample of 25 bags has been taken and their mean was found to be 150 pounds. Assume that the weights of the bags are normally distributed. Find the 90% confidence limits for µ. Exercise - 3 An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the upper end point in a 99% confidence interval for the average income? Exercise - 4 An economist is interested in studying the incomes of consumers in a particular region. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in an average income of $15,000. What is the width of the 90% confidence interval?