Download Chap 9 Notes - duPont Manual High School

Chapter 9
Estimation
Using a Single Sample
9.1: Point Estimation
A point estimate of a population characteristic is a single number that is based on sample data
and represents a plausible value of the characteristic.
Could be mean, trimmed mean, median, max, min, whatever…
Example
A sample of 200 students at a large university is selected to estimate the proportion of students
that wear contact lens. In this sample 47 wore contact lens.
Let π = the true proportion of all students at this university who wear contact lens. Consider
“success” being a student who wears contact lens.
Example
Example
Bias
A statistic with mean value equal to the value of the population characteristic being estimated is
said to be an unbiased statistic. A statistic that is not unbiased is said to be biased.
Criteria
Given a choice between several unbiased statistics that could be used for estimating a population
characteristic, the best statistic to use is the one with the smallest standard deviation.
9.2: Large-sample Confidence Interval for a Population Proportion
A confidence interval for a population characteristic is an interval of plausible values for the
characteristic. It is constructed so that, with a chosen degree of confidence, the value of the
characteristic will be captured inside the interval.
Confidence Level
The confidence level associated with a confidence interval estimate is the success rate of the
method used to construct the interval.
Most say it is the confidence of a mean being within the range of the max-min of CI, but this is not
correct.
Recall
Some considerations
Some considerations
The 95% Confidence Interval
Example
For a project, a student randomly sampled 182 other students at a large university to determine if
the majority of students were in favor of a proposal to build a field house. He found that 75 were in
favor of the proposal.
Let π = the true proportion of students that favor the proposal.
Example - continued
The General Confidence Interval
Finding a z Critical Value
Finding a z critical value for a 98% CI.
Finding a z Critical Value
Finding a z critical value for a CI value other than those typically used (e.g. 75%).
1 – 0.75 = 0.25
Take ½ because 2 tails, => 0.125
1.0 - 0.125 = 0.8750
Look up the z score for the cumulative area for 0.8750 in the body of the table we find 0.8749 = z
of approximately 1.15
Some Common Critical Values
Terminology
Terminology
Sample Size
Sample Size Calculation Example
If a TV executive would like to find a 95% confidence interval estimate within 0.03 for the
proportion of all households that watch Lost regularly. How large a sample is needed if a prior
estimate for π was 0.15.
Sample Size Calculation Example revisited
Suppose a TV executive would like to find a 95% confidence interval estimate within 0.03 for the
proportion of all households that watch Lost regularly. How large a sample is needed if we have
no reasonable prior estimate for π.
Another Example
A college professor wants to estimate the proportion of students at a large university who favor
building a field house with a 99% confidence interval accurate to 0.02. If one of his students
performed a preliminary study and estimated π to be 0.412, how large a sample should he take.
9.3: CI for μ
One-Sample z Confidence Interval for μ
One-Sample z Confidence Interval for μ
Example
Example I (continued)
Unknown σ - Small Size Samples
[All Size Samples]
t Distributions
t Distributions
t Distributions
t Distributions
Notice: As df increase, t distributions approach the standard normal distribution.
One-Sample t Procedures
Confidence Interval Example
Ten randomly selected shut-ins were each asked to list how many hours of television they watched
per week. The results are
82
66
90
84
75
88
80
94
110 91
Find a 90% confidence interval estimate for the true mean number of hours of television watched
per week by shut-ins.
Confidence Interval Example
Sample Size
The sample size required to estimate a population mean μ to within an amount B with 95% confidence is:
n = (1.96σ/B)2
The value of σ may be estimated by prior information, or for a population that is not to skewed, by using
(range)/4.
Again, round the result up to the nearest integer.
Confidence Interval Example
Confidence Interval Example
9.4: Communicating & Interpreting the Results of Stats Analysis
We calculated a 90% CI related to students owning a computer & got (.36, .58)
Interpretation of interval: We can be 90% confident that between 36%-58% of the students at this university own a computer
Interpretation of confidence level: We used a method to produce this estimate that is successful in capturing the actual
population % 90% of the time
Ways you might see in literature: CI, estimate +- bound on error, estimate +- standard error
Estimate +- bound on error often called “Margin of error” in media
Communicating & Interpreting the Results of Stats Analysis cont.
Caution:
• Be leery of point estimates that don’t include a bound error or some other measure of accuracy
• A CI that is wide we don’t have precise info on the population. Be leery of precise point estimates if the interval is wide.
• I am quite confident that 99% of my students will have a class average between 20% & 100%. High accuracy, but the
range is so wide that it doesn’t tell me anything about the actual class average.
• Accuracy depends on sample size, not population size. It is sample size “n” that is used in the equations, not population
size. Also, n is in the denominator of the equations, so the bounds decrease (tighten up) as the sample size increases.
• All the calculations in this chapter are based on normal populations, so you must do the checks for normalcy to use them.