Introduction to Estimating Population Means Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. HAWKES LEARNING SYSTEMS math courseware specialists Section 8.1 Introduction to Estimating Population Means
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means Definitions: 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. (e.g. 0.95, 0.99, etc.) 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.
The best point estimate of a population mean is the sample mean. HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means Find the best point estimate: 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. 45 68 72 91 100 71 69 83 86 55 89 97 76 92 75 84 70 81 90 85 74 88 99 93 96 Solution: The best point estimate of a population mean is the sample mean.
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means Confidence Interval for Population Means:
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means Construct the confidence 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. Solution: Lower endpoint: 15.7 - 2.2 = 13.5 hours Upper endpoint: 15.7 + 2.2 = 17.9 hours 13.5 < < 17.9
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means How to choose z or t: σ is the population standard deviation σ is known, n=30 or more, and any distribution or σ is known, n=29 or less, and normally distributed choose z σ is unknown, normally distributed, and n=30 or more σ is unknown, normally distributed, and n=29 or less choose z choose t
σ is known, n=29 or less, and normally distributed
σ is known, n=30 or more
σ is known, n=29 or less, and not normally distributed
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means How to choose z or t: σ is the population standard deviation σ is known, n=30 or more, and any distribution or σ is known, n=29 or less, and normally distributed choose z σ is unknown, normally distributed, and n=30 or more σ is unknown, normally distributed, and n=29 or less choose z choose t
σ is unknown, not normally distributed
σ is unknown, normally distributed, and n=29 or less
HAWKES LEARNING SYSTEMS math courseware specialists Confidence Intervals 8.1 Introduction to Estimating Population Means How to choose z or t: σ is the population standard deviation σ is known, n=30 or more, and any distribution or σ is known, n=29 or less, and normally distributed choose z σ is unknown, normally distributed, and n=30 or more σ is unknown, normally distributed, and n=29 or less choose z choose t
σ is unknown, normally distributed, and n=29 or less
Lower = Mean – Margin of Error = 50 – 1.5 = 48.5 Upper = Mean + Margin of Error = 50 + 1.5 = 51.5