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STA220- Section 4.9 ̅ and the Central Limit Theorem The Sampling Distribution of 𝒙 Ex. 1: Interpreting the Central Limit Theorem Phone bills for residents of Cincinnati have a mean of $64 and a standard deviation of $9, as shown in the following graph. Random samples of 36 phone bills are drawn from the population and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution. Ex. 2: Interpreting the Central Limit Theorem The heights of four fully grown white oak trees are normally distributed, with a mean of 90 feet and a standard deviation of 3.5 feet. Random samples are drawn from this population, and the mean of each sample is determined. Find the mean and standard error of the mean of the sampling distribution. Then sketch a graph of the sampling distribution. Probability and the Central Limit Theorem In sections 4.3, you learned how to find the probability that a random variable, x, will fall in a given interval of population values. In a similar manner, you can find the probability that a sample mean, 𝑥̅ will fall in a given interval of the 𝑥̅ sampling distribution. To transform 𝑥̅ to a z-score, you can use the following equation. Ex. 3: Finding Probabilities for Sampling Distributions The graph at the right lists the length of time adults spend reading newspapers. You randomly select 50 adults ages 18 to 24. What is the probability that the mean time they spend reading the newspaper is between 8.7 and 9.5 minutes? Assume that = 1.5 minutes Ex. 4: Finding Probabilities for Sampling Distributions The mean rent of an apartment in a professionally managed apartment building is $780. You randomly select nine professionally managed apartments. What is the probability that the mean rent is less than $825? Assume that the rents are normally distributed with a mean of $780 and a standard deviation of $150. ̅. Ex. 5: Finding probabilities for x and 𝒙 Credit card balances are normally distributed, with a mean of $2870 and a standard deviation of $900. a. What is the probability that a randomly selected credit card holder has a credit card balance less than $2500? b. You randomly select 25 credit card holders. What is the probability that the mean credit card balance is less than $2500? c. Compare the probabilities from (a) and (b).