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

Section 8.1Introduction 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.

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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:

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The best point estimate of a population mean is the sample mean.

Solution:

Confidence Intervals

8.1 Introduction to Estimating

Population Means

45 68 72 91 100 7169 83 86 55 89 9776 68 92 75 84 7081 90 85 74 88 9976 91 93 85 96 100

is the best point estimate of

• We can’t find the exact by surveying the entire population.

• We did take a sample and found the .• That’s as good as we can do.

• But we want to express some uncertainity.• Being uncertain in a mathematically precise

way.

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Confidence Interval for Population Means:

Confidence Intervals

8.1 Introduction to Estimating

Population Means

E is theMargin 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:

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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:

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Confidence Intervals

8.1 Introduction to Estimating

Population Means

z

z t

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 thepopulation’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, continuedWhen pop. Normal or n ≥ 30)

Large sample:Use z, the normal distrib’n.

Small sample:Use the Student t-distribution.

z t