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Example: Phone bills for residents of Cincinnati have a mean of $64 and a standard deviation of $9, as shown in the graph below. Random samples of 36 phone bills 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. What is the probability that a randomly selected sample of 36 phone bills will have a mean that falls between $61 and $67? (Use the sampling mean and standard error) 61 â 64 đ§1 = = â2 â 0.0228 đđ đĄđđđđ 1.50 67 â 64 đ§2 = = 2 â 0.9772 đđ đĄđđđđ 1.5 đ. đđđđ â đ. đđđđ = đ. đđđđ Probability and the Central Limit Theorem Distribution must be approximately normally distributed and this is to find the probability for the mean of a sample. z-score method: Calculator Method: -Convert desired mean value(s) to a z-score: -Find sampling distribution mean, īx and the standard error, īŗ īŊ īŗ zīŊ x ī īx īŗx īŊ xīī ( īŗ n ) -Look up z-score(s) on table for the probability x n -Use normalcdf function with sampling mean and standard error If you are asked to find the probability of a single value then you use the population mean and standard deviation. 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 balance less than $2500? đ= đđđđ â đđđđ = âđ. đđ â đ. đđđđ đđ đđđđđ = đđ. đđ% đđđ b) If you randomly select 25 credit card holders, what is the probability that their mean credit card balance is less than $2500? đđ = đđđ đđ = đđđ đđđđ â đđđđ đ= = âđ. đđ â đ. đđđđ đđ đđđđđ = đ. đđ% đđđ Which is more likely? The heights of American women aged 20 â 29 are normally distributed with a mean of 64 inches and a standard deviation of 2.4 inches. Are you more likely to select randomly one woman with a height more than 62 inches or are you more likely to select a sample of 20 women with an average height more than 62 inches? Single: đ§= 62 â64 2.4 Sample: đđĨ = 2.4 20 = â0.83 â 0.2033 đđ đĄđđđđ = 20.33% = 0.54 so z = 62 â64 0.54 = â3.70 â 0 đđ đĄđđđđ = 0%