Section 8.1 Introduction to Estimating Population Means HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.

Point Estimate – a single number estimate of a population parameter. The best point estimate of a population mean is the sample mean. Interval Estimate – a range of possible values for the population parameter. Level of Confidence, c – the degree of certainty that the interval estimate contains the population parameter. Confidence Interval – an interval estimate associated with a certain level of confidence. Margin of Error, E – the largest possible distance from the point estimate that a confidence interval will cover. HAWKES LEARNING SYSTEMS math courseware specialists Definitions: Confidence Intervals 8.1 Introduction to Estimating Population Means

Find the best point estimate for the population mean of test scores on a standardized biology final exam. The following is a simple random sample taken from these test scores. Find the best point estimate: HAWKES LEARNING SYSTEMS math courseware specialists The best point estimate of a population mean is the sample mean. Solution: Confidence Intervals 8.1 Introduction to Estimating Population Means

HAWKES LEARNING SYSTEMS math courseware specialists Confidence Interval for Population Means: Confidence Intervals 8.1 Introduction to Estimating Population Means E is the Margin of Error μ is between these values, μ is in this (low,high) interval

A college student researching study habits collects data from a random sample of 250 college students on campus and calculates that the sample mean is 15.7 hours per week. If the margin of error for the data using a 95% level of confidence is 2.2 hours, construct a 95% confidence interval for the data. Construct the confidence interval: HAWKES LEARNING SYSTEMS math courseware specialists Lower endpoint: 15.7  2.2  13.5 hours Upper endpoint: 15.7  2.2  17.9 hours Solution: 13.5 <  < 17.9 Confidence Intervals 8.1 Introduction to Estimating Population Means Here at the beginning: They make it easy and just tell us what E is. Later: We have a special formula to calculate E.

How to choose z or t: HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means z zt Sometimes E is calculated according to Normal Distribution (z) Sometimes E is calculated according to the Student t-Distrib.

Choosing the distribution, z or t First decision: Do we know the population’s standard deviation, σ ? (not sample’s, but population’s)

If we know what σ is And if the population is known to be normally distributed, then use z Or if sample size n ≥ 30, then use z Otherwise, “advanced techniques”

But if we don’t know what σ is… First, you need either Population is normally distributed Or large sample size, n ≥ 30 (If neither of those apply, then some advanced techniques are required.)

If we don’t know σ, continued When pop. Normal or n ≥ 30) Large sample: Use z, the normal distrib’n. Small sample: Use the Student t-distribution. z t