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Confidence Intervals Means Remember: In sampling distribution of means….. x x n Approximately normal if is large ( n 30) – CLT The population is normal n So how will I find a confidence intervals for a mean? How is it different from a proportion? Formula for a z-confidence interval: x z 2 n Suppose you work for a consumer advocate agency & want to find the mean repair cost of a washing machine. You randomly select 40 repair costs and find the mean to be $100. The standard deviation is $17.50. construct a 95% interval for the population mean. Find the minimum required sample size if you want to be 95% confident that the sample mean is within 2 units of the population mean if the population standard deviation is 4.8. But what if the sample size is small or the population standard deviation is unknown? We use the t-distribution if the population standard deviation is unknown. It’s bell shaped – centered at 0. Each t-distribution is more spread out than the z-distribution (normal). As the sample size increases, the spread decreases and actually approaches the normal distribution. Based on degrees of freedom (df = n-1) Degrees of freedom: (n-1) Suppose A + B + C + D = 18 Free to be anything Thus there are 3 degrees of freedom The t- distribution compared to the normal distribution http://www.nku.edu/~longa/stats/taryk/TDist .html In normal sampling but with t-distribution z x n x t s n Reading the t-chart The Raman Arches is an Italian restaurant. The manager wants to estimate the average amount a customer spends on lunch. A random sample of 115 customers’ lunch tabs gave a mean of $9.74 with a standard deviation of $2.93. Find and interpret a 90% confidence interval for the average amount spent on lunch by all customers. A sample of 20 students had a test average of 100 with a standard deviation of 4.2 points. Find and interpret a 90% confidence interval for the average test score for all students on this test. A random sample of large tents listed in Consumer Reports: Special Outdoor Issue gave the following prices. Find a 90% confidence interval for the mean price of all such tents. 115 110 140 135 80 110 150 210 250 120 230 130 Homework - Worksheet