UNIT SEVEN:We useas our estimate ofxxμ, andis an example of a point estimate. Axxconfidence intervalmeasuresthe uncertainty associated with a point estimatePoint estimate ± margin of error. The margin of error is based on theconfidence level and the standard deviation of the sampling distribution of the statistic. Assumptions we need to make forinterval estimation: the sample data is a simple random sample from the population of interest and the population isnormally distributed. If n ≥ 30, the Central Limit Theorem will have an approximately normal distribution. In general, a(1-α)100% confidence interval for μ is given by:± zxxα/2σxx.[zα/2] is called the confidence coefficient and is a value from thestandard normal distribution. The entire term zα/2σxxis the margin of error. Percentage table goes: 60,70,80,90,95,98,99.Percentages for confidence intervals goes:80%: 1.282, 85%: 1.440, 90%: 1.645, 95%: 1.96, 99%: 2.576. A simpleinterpretationwould be: We are 95% confident that the true value of μ lies within the calculated interval. A morecomplicated interpretationwould be: In repeated sampling, 95% of the 95% confidence intervals calculated using thismethod will contain μ. A large margin of error causes less precision with a wider CI while a small margin of error causesmore precision with a narrower CI. We can increase the precision of our estimate by decreasing the margin of error.Margin of error:zα/2σ/√n. T-distribution: -xxμ / s/√n with σxx≈ s/√n. It has a bell shaped distribution, symmetric, centeredaround 0, lower peak, more area in the tails, one parameter (v) and degrees of freedom. As v increases, the t distributionapproaches the standard normal distribution. A (1-α)100% confidence interval for μ is given by:± txxα/2SE( )xx± txxα/2s/√n
