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ESTIMATION OF THE MEAN INTRO :: ESTIMATION Definition The assignment of plausible value(s) to a population parameter based on a value of a sample statistic is called estimation. 2 INTRO :: ESTIMATION The sample statistic used to estimate a population parameter is called estimator. Examples … sample mean estimator population mean … 3 INTRO :: ESTIMATION The estimation procedure involves … 1. 2. 3. 4. Select a sample. Collect the required information from the members of the sample. Calculate the value of the sample statistic. Assign plausible value(s) to the corresponding population parameter. 4 POINT ESTIMATES & INTERVAL ESTIMATES A Point Estimate An Interval Estimate 5 A Point Estimate Definition The value of a sample statistic that is used to estimate a population parameter is called a point estimate. 6 INTERVAL ESTIMATES Usually, whenever we use point estimation, we calculate the margin of error associated with that point estimation. Point Estimate w/out M.E. not very useful!!!! 7 Interval Estimates Definition In interval estimation, an interval is constructed around the point estimate, and it is stated that this interval is likely to contain the corresponding population parameter. Gives a range of plausible values for the parameter of interest. 8 Interval estimation. x $1130 x $1370 $1610 9 Interval Estimates Definition Each interval is constructed with regard to a given confidence level and is called a confidence interval. The confidence level associated with a confidence interval states how much confidence we have that this interval contains the true population parameter. The confidence level is denoted by (1 – α)100%. 10 INTERVAL ESTIMATION OF A POPULATION MEAN: The t Distribution Confidence Interval for μ Using the t Distribution 11 The t Distribution Conditions Under Which the t Distribution Is Used to Make a Confidence Interval About μ The t distribution is used to make a confidence interval about μ if The population from which the sample is drawn is (approximately) normally distributed. 2. The population standard deviation, σ, unknown. 1. 12 The t Distribution cont. The t distribution is a specific type of bellshaped distribution with a lower height and a wider spread than the standard normal distribution. As the sample size becomes larger, the t distribution approaches the standard normal distribution. The t distribution has only one parameter, called the degrees of freedom (df). The mean of the t distribution is equal to 0 and its standard deviation is df /( df 2) . 13 The t distribution for df = 9 and the standard normal distribution. The standard deviation of the t distribution is The standard deviation of the standard normal distribution is 1.0 9 /(9 2) 1.134 μ=0 14 Example Find the value of t for 16 degrees of freedom and .05 area in the right tail of a t distribution curve. 15 Determining t for 16 df and .05 Area in the Right Tail Area in the right tail Area in the Right Tail Under the t Distribution Curve df df .10 .05 .025 … .001 1 2 3 . 16 . 3.078 1.886 1.638 … 1.337 … 6.314 2.920 2.353 … 1.746 … 12.706 4.303 3.182 … 2.120 … … … … … … … 318.309 22.327 10.215 … 3.686 … The required value of t for 16 df and .05 area in the right tail 16 The value of t for 16 df and .05 area in the right tail. .05 df = 16 0 1.746 t This is the required value of t 17 The value of t for 16 df and .05 area in the left tail. df = 16 .05 -1.746 0 t 18 Confidence Interval for μ Using the t Distribution The (1 – α)100% confidence interval for μ is x tsx s where s x n The value of t is obtained from the t distribution table for n – 1 degrees of freedom and the given confidence level. 19 Examples … 20