View
219
Download
0
Category
Preview:
DESCRIPTION
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
Citation preview
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.
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
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.
HAWKES LEARNING SYSTEMS
math courseware specialists
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:
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
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
Recommended