Methods of Inferential Statistics Estimation oPoint Estimation Single number calculated based on sample data oInterval Estimation Pair of numbers within which the parameter is expected to lie. Hypothesis Testing Point Estimate vs Interval Estimate
Properties of Point Estimators Property 1: A good estimator has a sampling distribution that is centered at the parameter An estimator with this property is unbiased The sample mean is an unbiased estimator of the population mean The sample proportion is an unbiased estimator of the population proportion Property 2: A good estimator has a small standard error compared to other estimators This means it tends to fall closer than other estimates to the parameter Confidence Interval A confidence interval is an interval containing the most believable values for a parameter The probability that this method produces an interval that contains the parameter is called the confidence level This is a number chosen to be close to 1, most commonly 0.95 Margin of Error The margin of error measures how accurate the point estimate is likely to be in estimating a parameter The distance of 1.96 standard errors in the margin of error for a 95% confidence interval Finding the 95% Confidence Interval for a Population Proportion We symbolize a population proportion by p The point estimate of the population proportion is the sample proportion A 95% confidence interval uses a margin of error = 1.96(standard errors) [point estimate ± margin of error] = p ± 1.96(standard errors) Confidence Interval for a Population Proportion, p A confidence interval for a population proportion p is: n)pˆ-(1pˆzpˆ
Effects of Confidence Level and Sample Size on Margin of Error Example 19: The average zinc concentration recovered from a sample of zinc measurements in 36 different locations is found to be 2.6 grams per milliliter. Find the 95% and 99% confidence intervals for the mean zinc concentration in the river. Assume that the population standard deviation is 0.3. Example 20: A random sample of 100 automobile owners shows that, in the state of Virginia, an automobile is driven on the average 23,500 kilometers per year with a standard deviation of 3900 kilometers. Assume the distribution of measurements to be approximately normal. (a) Construct a 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia. (b) What can we assert with 99% confidence about the possible size of our error if we estimate the average number of kilometers driven by car owners in Virginia to be 23.500 kilometers per year?