26.1: Confidence Intervals for the Mean (σ is known) Estimation is a primary aspect of inferential statistics. It allows us to estimate the value of a parameter based on information collected from a sample.For example:“The average kindergarten student has seen more than 5000 hours of television”“Four out of five doctors recommend …”Since the populations for these values are large, the values are estimates of the parameters.
3Section 6.1Point estimate – a specific numerical value estimate for a population parameter.The best point estimate of the population mean is the sample mean.This is because the sample mean varies less than other statistics.Estimator – a statistic used to estimate a parameter.
4Three Properties of a Good Estimator Section 6.1Three Properties of a Good EstimatorThe estimator should be an unbiased estimator.The expected value (or the mean on the estimates) should be equal to the parameter being estimated.The estimator should be consistent.As sample size increases, the value of the estimator approaches the value of the parameter being estimated.The estimator should be a relatively efficient estimator.Of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.
5Section 6.1Example 1:An economics researcher is collecting data about grocery store employees in a county. The data listed below represents a random sample of the number of hours worked by 40 employees from several grocery stores in the county. Find a point estimate of the population mean.
6Section 6.1Interval estimate – an interval or range of values used to estimate the parameter.This may or may not contain the value of the parameter being estimated.A degree of confidence (usually a percent) can be assigned before the estimation is made. The larger the percent, the larger the interval.Level of Confidence – the probability that the interval estimate will contain the parameter.This assumes that a large number of samples are selected and that the estimation process on the same parameter is repeated.Confidence interval – estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.
7Section 6.1The three common confidence intervals used are: 90%, 95%, and 99%.In the confidence interval formula, α(alpha) represents the total area of both tails of the standard normal distribution curve.
8Section 6.1Margin of error E (a.k.a. maximum error of the estimate) – the greatest possible difference between the point estimate and the value of the parameter.Notation:Must fulfill the following conditions:The sample is randomThe population is normally distributed OR n ≥ 30When rounding a confidence interval, round to the same number of decimal places as the mean.
9Section 6.1Example 2:Using the information from Example 1, determine the margin of error (E) for a confidence of 95% (z = 1.96). Assume the standard deviation for the population is 7.9 hours.
10Formula for the Confidence Interval of the Mean for a Specificα Section 6.1Formula for the Confidence Interval of the Mean for a SpecificαFor a 90% confidence interval, use:For a 95% confidence interval, use:For a 99% confidence interval, use:
11Section 6.1Example 3:Use the data from Examples 1 and 2 to construct a 95% confidence interval for the mean number of hours worked by the grocery store employees.
12Homework: Section 6.1 Pg 305: #’s 1 – 32 Read and Take Notes on Section 6.1 (pgs )