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4/9/2013 8‐1: Estimating µ When σ is Known Point Estimate • An estimate of a population parameter given by a single number. 1 4/9/2013 Margin of Error • Even if we take a very large sample size, differ from µ. will x Margin of Error x Confidence Levels • A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc. 2 4/9/2013 Critical Values Critical Value Example • Use Table 5 to find z0.99 such that 99% of the area under the standard normal curve lies between –z0.99 and z0.99. 3 4/9/2013 Critical Value Example Observe P(z < ‐ 2.58) ≈ 0.005 so use z0.99 = 2.58. Thus, P(‐2.58 < z < 2.58) ≈ 0.99. Common Confidence Levels 4 4/9/2013 Recall From Sampling Distributions • If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics: Mean of x x x Standard Deviation of x x x n 5 4/9/2013 A Probability Statement • In words, c is the probability that the sample mean will differ from the population mean by at most Maximal Margin of Error • Since µ is unknown, the margin of error | ‐ µ| is unknown. x • Using confidence level c, we can say that differs from µ by at most: x 6 4/9/2013 Confidence Intervals 7 4/9/2013 Confidence Interval Example • Salaries of community college math instructors are historically known to be normally distributed with a standard deviation of $3,500. Using the telephone, you take a random sample of 33 instructors and calculate the mean salary to be $53,128. Determine a 0.95 confidence interval for the mean salary of all community college math instructors. Confidence Interval Example n 33 x 53,128 3,500 c 0.95 zc 1.96 3,500 E zc 1.96 1,194.17 n 33 xE xE 53,128 1,194.17 53,128 1,194.17 51,933.83 54,322.17 8 4/9/2013 Confidence Interval Example • Thus, we conclude with 95% confidence that the interval from $51,933.83 to $54,322.17 contains the mean salary for all community college math instructors. Multiple Confidence Intervals • If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not! – The confidence level is the approximate proportion of intervals that will capture the population parameter. 9 4/9/2013 Multiple Confidence Intervals Choosing Sample Sizes • When designing statistical studies, it is good practice to decide in advance: – The confidence level – The maximal margin of error • Then, we can calculate the required minimum sample size to meet these goals. 10 4/9/2013 Sample Size for Estimating μ • If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s. Always round n up to the next integer!! Ex: (p. 324 #1 a, d) 11 4/9/2013 ASSIGNMENT: p. 324‐326 #2a,d; 3a,d; 4‐7 12